Differential graded bocses and $A_{\infty}$-modules

TL;DR

This paper develops the theory of twisted modules over triangular differential graded bocses, establishing their categorical properties and linking them to $A_{ abla}$-algebra modules, thus advancing the understanding of homotopical and algebraic structures.

Contribution

It introduces the category of twisted modules over triangular differential graded bocses and proves their categorical properties, connecting them to $A_{ abla}$-algebra modules.

Findings

Idempotents split in the category of twisted modules.

The category admits a Frobenius structure.

Homotopically trivial modules correspond to acyclic complexes.

Abstract

We introduce and study the category of twisted modules over a triangular differential graded bocs. We show that in this category idempotents split, that it admits a natural structure of a Frobenius category, that a twisted module is homotopically trivial if and only if its underlying complex is acyclic, and that any homotopy equivalence of differential graded bocses determines an equivalence of the corresponding homotopy categories of twisted modules. The category of modules over an $A_{\infty}$-algebra is equivalent to the category of twisted modules over a triangular differential graded bocs, so all preceding statements lift to the former category.

Equations

$$M=\underset{j\in \mathbb{Z}}{⨁}{M}_j$$

$${\mathrm{Hom}}_{\mathrm{GMod}-S}(M,N)=\underset{d\in \mathbb{Z}}{⨁}{\mathrm{Hom}}_{\mathrm{GMod}-S}^d(M,N).$$

$$(M{\otimes }_{S}N{)}_j=\underset{s+t=j}{⨁}{M}_s{\otimes }_S{N}_t.$$

$$[f\otimes g](x\otimes y):=(-1{)}^{|g||x|}f(x)\otimes g(y).$$

$$(h\otimes t)(f\otimes g)=(-1{)}^{|t||f|}hf\otimes tg.$$

$$\{m_n:A^{\otimes n}to20.0pt\text{rightarrowfill}A{\}}_{n\in \mathbb{N}},$$

$$S_n:\sum _{\begin{array}{c}r+s+t=n\\ s\ge 1;r,t\ge 0\end{array}}(-1{)}^{r+st}{m}_{r+1+t}(i{d}^{\otimes r}\otimes m_s\otimes i{d}^{\otimes t})=0.$$

$${\Sigma }_n=\sum _{\begin{array}{c}r+s+t=n\\ r,t\ge 0;s\ge 1\end{array}}(-1{)}^{r+st}{f}_{r+1+t}(i{d}^{\otimes r}\otimes m_s\otimes i{d}^{\otimes t}),$$

$${\Sigma }_n^{\prime }=\sum _{\begin{array}{c}1\le r\le n\\ i_1+\cdots +i_r=n\end{array}}(-1{)}^{\mathrm{sgn}(i_1,\dots ,i_r)}{m}_r^{\prime }(f_{i_1}\otimes \cdots \otimes f_{i_r}),$$

$$\mathrm{sgn}(i_1,\dots ,i_r)=(r-1)(i_1-1)+(r-2)(i_2-1)+\cdots +2(i_{r-2}-1)+(i_{r-1}-1).$$

$$(g\circ f{)}_n=\sum _{\begin{array}{c}1\le s\le n\\ i_1+\cdots +i_s=n\end{array}}(-1{)}^{\mathrm{sgn}(i_1,\dots ,i_s)}{g}_s(f_{i_1}\otimes \cdots \otimes f_{i_s}).$$

$$m_n^{\prime }:=(-1{)}^{\frac{n(n-1)}{2}}{m}_n.$$

$$S_n^{\prime }:\sum _{\begin{array}{c}r+s+t=n\\ s\ge 1;r,t\ge 0\end{array}}(-1{)}^{rs+t}{m}_{r+t+1}^{\prime }(i{d}^{\otimes r}\otimes m_s\otimes i{d}^{\otimes t})=0.$$

$$Z_n=\sum _{\begin{array}{c}r+s+t=n\\ s\ge 1;r,t\ge 0\end{array}}(-1{)}^{r+st}{m}_{r+t+1}(i{d}^{\otimes r}\otimes m_s\otimes i{d}^{\otimes t})$$

$$Z_n^{\prime }=\sum _{\begin{array}{c}r+s+t=n\\ s\ge 1;r,t\ge 0\end{array}}(-1{)}^{rs+t}{m}_{r+t+1}^{\prime }(i{d}^{\otimes r}\otimes m_s^{\prime }\otimes i{d}^{\otimes t}),$$

$$Z_n^{\prime }=(-1{)}^{\frac{n(n-1)}{2}}{Z}_n.$$

$$Z_n^{\prime }=\sum _{\begin{array}{c}r+s+t=n\\ s\ge 1;r,t\ge 0\end{array}}(-1{)}^{rs+t}(-1{)}^{\frac{(r+t+1)(r+t)}{2}}(-1{)}^{\frac{s(s-1)}{2}}{m}_{r+t+1}(i{d}^{\otimes r}\otimes m_s\otimes i{d}^{\otimes t}).$$

$$rs+t+\frac{(r+t+1)(r+t)}{2}+\frac{s(s-1)}{2}\equiv r+st+\frac{n(n-1)}{2}.$$

$$m_n^M:M\otimes A^{\otimes (n-1)}to20.0pt\text{rightarrowfill}M$$

$${\Sigma }_n^{+}=\sum _{\begin{array}{c}r+s+t=n\\ t\ge 0;r,s\ge 1\end{array}}(-1{)}^{r+st}{m}_{r+1+t}^M(i{d}^{\otimes r}\otimes m_s\otimes i{d}^{\otimes t})$$

$${\Sigma }_n^0=\sum _{\begin{array}{c}s+t=n\\ t\ge 0;s\ge 1\end{array}}(-1{)}^{st}{m}_{1+t}^M(m_s^M\otimes i{d}^{\otimes t}).$$

$$f_n:M{\otimes }_S{A}^{\otimes (n-1)}to20.0pt\text{rightarrowfill}N$$

$${\Sigma }_n^{f+}=\sum _{\begin{array}{c}r+s+t=n\\ t\ge 0;r,s\ge 1\end{array}}(-1{)}^{r+st}{f}_{r+1+t}(i{d}^{\otimes r}\otimes m_s\otimes i{d}^{\otimes t}),$$

$${\Sigma }_n^{f0}=\sum _{\begin{array}{c}s+t=n\\ t\ge 0;s\ge 1\end{array}}(-1{)}^{st}{f}_{1+t}(m_s^M\otimes i{d}^{\otimes t}),$$

$${\Sigma }_n^{f-}=-\sum _{\begin{array}{c}r+s=n\\ r\ge 1;s\ge 0\end{array}}(-1{)}^{(r+1)s}{m}_{1+s}^N(f_r\otimes i{d}^{\otimes s}).$$

$$g\circ f=\{(g\circ f{)}_n{\}}_{n\in \mathbb{N}}:Mto20.0pt\text{rightarrowfill}L$$

$$(g\circ f{)}_n=\sum _{\begin{array}{c}r+s=n\\ r\ge 1;s\ge 0\end{array}}(-1{)}^{(r+1)s}{g}_{1+s}(f_r\otimes i{d}^{\otimes s}).$$

$$h_n:M{\otimes }_S{A}^{\otimes (n-1)}to20.0pt\text{rightarrowfill}N$$

$$H_n^{(1)}=\sum _{\begin{array}{c}r+s=n\\ r\ge 1;s\ge 0\end{array}}(-1{)}^{rs}{m}_{1+s}^N(h_r\otimes i{d}^{\otimes s}),$$

$$H_n^{(2)}=\sum _{\begin{array}{c}s+t=n\\ s\ge 1;t\ge 0\end{array}}(-1{)}^{st}{h}_{1+t}(m_s^M\otimes i{d}^{\otimes t}),$$

Full text

††2010 Mathematics Subject Classification: Primary 06B15, 16T15, 18E30; Secondary 16G20.††Key words and phrases: differential graded bocses, differential graded coalgebras, $A_{\infty}$-algebras, $A_{\infty}$-modules, Frobenius category, twisted graded modules.

Dedicated to J.A. De la Peña on the occasion of his 60th birthday

1 Introduction

There is some analogy between the theory of modules over bocses and the theory of $A_{\infty}$-modules over an $A_{\infty}$-algebra. This can be noticed, for instance, in the fact that the morphisms $f:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ of modules over a given triangular bocs can be handled as pairs $(f^{0},f^{1})$ of morphisms, where the properties of the first component $f^{0}:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ reflect on the properties of the whole morphism $f$, similarly, the properties of a morphism $g:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ of $A_{\infty}$-modules over a given $A_{\infty}$-algebra depend on the properties of the first component $g_{1}:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$.

For instance, the natural exact structure on the category of modules of a given triangular bocs is determined by the first components of morphisms, while in the category of modules over a given $A_{\infty}$-algebra the natural exact structure is determined by the first components of morphisms.

Motivated by this analogy, we work here with the natural notions of differential graded bocses and their category of twisted graded modules, as defined below in (2.1) and (2.5). The study of these algebraic structures led us to simpler proofs of some well known facts on categories of $A_{\infty}$-modules.

In order to describe more precisely our constribution, let us fix some notation and recall some well known concepts. Throughout this article, we denote by $k$ a fixed ground field, which will act centrally on any $k$-algebra and on any $k$-$k$-bimodule we consider. We also fix a finite-dimensional semisimple $k$-algebra $S$. Moreover, we ask that $S\otimes_{k}S^{op}$ is also semisimple. This holds, for instance when $k$ is a perfect field or if $S$ is a finite product of copies of the ground field. We will make explicit any further requirement on $S$, when needed. We consider $S$ as a $\mathbb{Z}$-graded $k$-algebra concentrated at degree [math].

We will consider graded right $S$-modules (or graded left $S$-modules, or graded $S$-$S$-bimodules) $M$, so $M$ is equipped with a direct sum decomposition

[TABLE]

of right $S$-submodules $M_{j}$ (resp. left $S$-submodules, or $S$-$S$-subbimodules). The elements $x\in M_{j}$ are called homogeneous of degree $j$, and we indicate this fact by $|x|=j$. Given two graded right $S$-modules, a homogeneous morphism $f:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ of degree $|f|=d$ satisfies that $f(M_{j})\subseteq N_{j+d}$, for any $j\in\mathbb{Z}$. We denote by ${\rm Hom}_{{\rm GMod}\hbox{-}S}^{d}(M,N)$ the space of homogeneous morphisms of graded right $S$-modules $f:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ of degree $d$. Then, we have the graded category ${\rm GMod}\hbox{-}S$ of graded right $S$-modules with hom spaces

[TABLE]

Similarly, we have the graded category ${\rm GMod}\hbox{-}S\hbox{-}S$ of graded $S$-$S$-bimodules.

The tensor product $M\otimes_{S}N$ of two graded objects $M$ and $N$ is equipped with the standard grading given by the homogeneous components

[TABLE]

Assume that $f:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ is a homogeneous morphism of graded right $S$-modules and $g:M^{\prime}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N^{\prime}$ is a homogeneous morphism of graded left $S$-modules. Then, their tensor product $f\otimes g$ is the homogeneous linear map $M\otimes_{S}M^{\prime}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N\otimes_{S}N^{\prime}$ of degree $|f\otimes g|=|f|+|g|$ defined, for any homogeneous elements $x\in M$ and $y\in M^{\prime}$, by the following formula.

[TABLE]

So, if $f:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$, $h:N\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}L$ are morphisms of graded right $S$-modules, and $g:M^{\prime}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N^{\prime}$, $t:N^{\prime}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}L^{\prime}$ are morphisms of graded left $S$-modules, we have

[TABLE]

Now we recall some basic definitions of the theory of $A_{\infty}$-algebras.

Definition 1.1**.**

An $A_{\infty}$-algebra $A$ is a graded $S$-$S$-bimodule $A$, equipped with a sequence of homogeneous morphisms of $S$-$S$-bimodules

[TABLE]

where each $m_{n}$ has degree $|m_{n}|=2-n$, such that, for each $n\in\mathbb{N}$, the following Stasheff identity holds.

[TABLE]

Given two $A_{\infty}$-algebras $(A,\{m_{n}\})$ and $(B,\{m^{\prime}_{n}\})$, a morphism of $A_{\infty}$-algebras $f:(A,\{m_{n}\})\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(B,\{m^{\prime}_{n}\})$ is a family of homogeneous $S$-$S$-bimodule morphisms $\{f_{n}:A^{\otimes n}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}B\}_{n\in\mathbb{N}}$ such that each $f_{n}$ is homogeneous of degree $|f_{n}|=1-n$ and the equality $\Sigma_{n}=\Sigma^{\prime}_{n}$ holds for all $n\in\mathbb{N}$, where

[TABLE]

[TABLE]

where $i_{1},\ldots,i_{r}\geq 1$ and

[TABLE]

Given two morphisms of $A_{\infty}$-algebras $f:A\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}B$ and $g:B\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}C$, their composition $g\circ f:A\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}C$ is the family $(g\circ f)=\{(g\circ f)_{n}\}_{n\in\mathbb{N}}$ defined, for each $n\in\mathbb{N}$, by

[TABLE]

The preceding notions give rise to a $k$-category ${\rm Alg}_{\infty}$ of $A_{\infty}$-algebras with morphisms of $A_{\infty}$-algebras. There is a well known full and faithful functor $\Psi:{\rm Alg}_{\infty}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\rm DGCoalg}$, where ${\rm DGCoalg}$ denotes the category of differential graded $S$-coalgebras, which has been used successfully to study the category ${\rm Alg}_{\infty}$, see [6], [7] and their references. The functor $\Psi$ is given by the bar construction: it maps each $A_{\infty}$-algebra $(A,\{m_{n}\})$ onto its reduced tensor $S$-coalgebra $\overline{T}_{S}(A[1])$, equipped with a differential $\delta$ induced by the family $\{m_{n}\}$, see [6](3.6).

There is some notational difference with [6] and [7], which is a minor one as explained by the following.

Lemma 1.2**.**

Let $A$ be a graded $S$-$S$-bimodule equipped with a sequence of homogeneous morphisms of $S$-$S$-bimodules $\{m_{n}:A^{\otimes n}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}A\}_{n\in\mathbb{N}}$. For $n\in\mathbb{N}$, define

[TABLE]

Then, the equation $S_{n}$ appearing in the last definition holds iff the following equation holds

[TABLE]

Moreover, if we make

[TABLE]

and

[TABLE]

we have that

[TABLE]

Proof.

By definition,

[TABLE]

The non-negative integers $r,s,t$ satisfy $r+s+t=n$ and $s\geq 1$. Then, our statement follows from the next congruence modulo $2$

[TABLE]

∎

Thus, the choice for the signs in the formulas we made in the last definition, which is also adopted by other authors, is not essential. It will reflect on some signs in the formulas involving $A_{\infty}$-objects.

Definition 1.3**.**

Let $A=(A,\{m_{n}\})$ be an $A_{\infty}$-algebra. Then, a graded right $S$-module $M$ is called a right $A_{\infty}$-module over $A$ iff $M$ is equipped with a family $\{m_{n}^{M}\}_{n\in\mathbb{N}}$ of morphism of right $S$-modules such that, each map

[TABLE]

is homogeneous of degree $|m^{M}_{n}|=2-n$ and, for every $n\in\mathbb{N}$, the condition $\Sigma_{n}^{+}+\Sigma_{n}^{0}=0$ is satisfied, where

[TABLE]

and

[TABLE]

In the sum $\Sigma_{n}^{+}$, for simplicity, we abuse of the language writing $id^{\otimes r}$ instead of $id_{M}\otimes id_{A}^{\otimes(r-1)}$. In case $n=1$, the condition reduces to $\Sigma^{0}_{1}=0$ which is equivalent to say that $m^{M}_{1}$ satisfies $(m^{M}_{1})^{2}=0$.

Given $M=(M,\{m^{M}_{n}\})$ and $N=(N,\{m_{n}^{N}\})$ right $A_{\infty}$-modules over $A$. Then, a morphism of right $A_{\infty}$-modules $f:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ over $A$ is a family $f=\{f_{n}\}_{n\in\mathbb{N}}$, where each

[TABLE]

is a homogeneous morphism of graded right $S$-modules of degree $|f_{n}|=1-n$ such that, for each $n\in\mathbb{N}$, the equality $\Sigma_{n}^{f+}+\Sigma_{n}^{f0}+\Sigma_{n}^{f-}=0$, where

[TABLE]

[TABLE]

and

[TABLE]

The condition in case $n=1$ is equivalent to $m_{1}^{N}f_{1}=f_{1}m_{1}^{M}$, that is, to the requirement that the map $f_{1}:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ is a morphism of complexes of right $S$-modules. The morphism $f$ is called a quasi-isomorphism iff $f_{1}$ is so.

The class of right $A_{\infty}$-modules together with the morphisms between them is a category with the following composition. If $f:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ and $g:N\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}L$ are morphisms of right $A_{\infty}$-modules over $A$, their composition

[TABLE]

is defined, for each $n\in\mathbb{N}$, by

[TABLE]

Given a left $A_{\infty}$-module $M$, the identity ${\rm I\hskip-2.0ptI}_{M}=\{h_{n}\}:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}M$ is given by $h_{1}=id_{M}$ and $h_{n}=0$, for all $n\geq 2$. We shall denote this category by ${\rm Mod}_{\infty}\hbox{-}A$.

For the study of the category ${\rm Mod}_{\infty}\hbox{-}A$ one of the main tools is the graded category ${\rm DGComod}\hbox{-}{\cal B}_{A}$, the category of differential graded comodules over the differential tensor $S$-coalgebra ${\cal B}_{A}=(T_{S}(A[1]),\mu,\epsilon,\delta)$, with $\mu$ the comultiplication given by the bar construction, $\epsilon:T_{S}(A[1])\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}S$ is the canonical projection, and $\delta$ is the differential induced by the family of operations $\{m_{n}\}_{n\in\mathbb{N}}$ of the $A_{\infty}$-algebra $A$. This is so, because of there is a full and faithful functor $\Phi:{\rm Mod}_{\infty}\hbox{-}A\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\rm DGComod}^{0}\hbox{-}{\cal B}_{A}$. The superindex [math] indicates the subcategory of ${\rm DGComod}\hbox{-}{\cal B}_{A}$ with the same objects but with only degree zero homogeneous morphisms.

Finally, we recall the appropriate notions of homotopy for the preceding categories ${\rm Alg}_{\infty}$ and ${\rm Mod}_{\infty}\hbox{-}A$.

Definition 1.4**.**

Let $A$ be an $A_{\infty}$-algebra and $f,g:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ morphisms of right $A_{\infty}$-modules over $A$. Then, a homotopy from $f$ to $g$ is a family of maps $\{h_{n}\}_{n\in\mathbb{N}}$ where, for each $n\in\mathbb{N}$,

[TABLE]

is a homogeneous morphism of right $S$-modules of degree $|h_{n}|=-n$, such that, for all $n\in\mathbb{N}$, we have $f_{n}-g_{n}=H_{n}^{(1)}+H_{n}^{(2)}+H_{n}^{(3)}$, where

[TABLE]

[TABLE]

and

[TABLE]

A morphism of $A_{\infty}$-modules $f:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ is called null-homotopic iff there is a homotopy from $f$ to [math].

Definition 1.5**.**

Let $A$ and $B$ be $A_{\infty}$-algebras and let $f,g:A\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}B$ be morphisms of $A_{\infty}$-algebras. A homotopy $h$ from $f$ to $g$ is a family $h=\{h_{n}\}_{n\in\mathbb{N}}$, where each

[TABLE]

is a homogeneous morphism of $S\hbox{-}S$-bimodules with degree $|h_{n}|=-n$, such that, for each $n\in\mathbb{N}$, we have

[TABLE]

where

[TABLE]

and

[TABLE]

where $i_{1},\ldots,i_{r},j_{1},\ldots,j_{t}\geq 1$ and ${\rm sgn}={\rm sgn}(i_{1},\ldots,i_{r},s,j_{1},\ldots,j_{t})$ is given by the sum

[TABLE]

The preceding notions of homotopy are known to be equivalence relations on the corresponding categories and give rise to the homotopy categories $\underline{{\rm Mod}}_{\infty}\hbox{-}A$ and $\underline{{\rm Alg}}_{\infty}$, respectively.

The categories ${\rm DGCoalg}$ and ${\rm DGComod}^{0}\hbox{-}{\cal B}_{A}$ have their own classical homotopy relations. The functors

[TABLE]

preserve and reflect the preceding homotopy relations and, therefore, induce full and faithful functors $\underline{\Psi}:\underline{{\rm Alg}}_{\infty}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}\underline{{\rm DGCoalg}}$ and $\underline{\Phi}:\underline{{\rm Mod}}_{\infty}\hbox{-}A\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}\underline{{\rm DGComod}}^{0}\hbox{-}{\cal B}_{A}$.

Many important properties of $\underline{{\rm Alg}}_{\infty}$ and $\underline{{\rm Mod}}_{\infty}\hbox{-}A$ are derived, using the full and faithful functors $\Psi$ and $\Phi$, from the corresponding properties for $\underline{{\rm DGCoalg}}$ and $\underline{{\rm DGComod}}^{0}\hbox{-}{\cal B}_{A}$ which are better understood categories.

In this paper, we propose that the use of the categories of twisted modules over differential graded $S$-bocses can be a fresh and simpler tool for some studies of ${\rm Mod}_{\infty}\hbox{-}A$. We illustrate this with three applications.

Namely, we can describe with quite good precision the structure of ${\rm Mod}_{\infty}\hbox{-}A$ as a Frobenius category. In particular, we prove that idempotents split in ${\rm Mod}_{\infty}\hbox{-}A$, a fact which seems to have remained unnoticed. We obtain this as an application of the study of the category ${\rm TGMod}\hbox{-}{\cal B}$, for a general triangular differential graded $S$-bocs, see (7.16). By “general” we mean that it is not necessarily a differential graded tensor $S$-coalgebra. We compare our description with the one given in [6](Proposition 5.2 and 8.4).

The important fact that each quasi-isomorphism of $A_{\infty}$-modules is a homotopy equivalence, see [6](Theorem 4.2), is obtained here, for the case where $S$ is a finite product of copies of the ground field $k$, as a consequence of our theorem (8.3). The latter is proved by an induction argument using the triangularity of the given triangular differential graded $S$-bocs. The proof does not resort to any model theoretical considerations.

Finally, we show the fact that any homotopy equivalence $f:A\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}B$ of $A_{\infty}$-algebras determines a restriction functor $R_{f}:\underline{{\rm Mod}}_{\infty}\hbox{-}A\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}\underline{{\rm Mod}}_{\infty}\hbox{-}A$ which is an equivalence of categories, see [6](Proposition 6.2). We obtain this as a consequence of the corresponding result for triangular differential graded $S$-bocses, see (5.5), which essentially relies on the remarkable property that a morphism $f:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ in the category of modules over a triangular differential graded $S$-bocs is an isomorphism if and only if its first component is so, see (4.8).

The twisted modules we consider in (2.5) are constructed from the differential graded category ${\rm GMod}\hbox{-}{\cal B}$, of graded modules over a differential graded $S$-bocs ${\cal B}$. These twisted modules are a special kind of the twisted complexes over a general differential graded category considered in [3]. Here we focus on a naive and very concrete approach to the study of the category ${\rm TGMod}\hbox{-}{\cal B}$ for a triangular differential graded $S$-bocs ${\cal B}$ and extract from this the preceding applications to the study of $A_{\infty}$-modules.

2 Differential graded bocses, twisted modules

In this paper, we use the word bocs in the following specific sense.

Definition 2.1**.**

A graded $S$-bocs ${\cal B}$ is a triple ${\cal B}=(C,\mu,\epsilon)$, where $C$ is a graded $S$-$S$-bimodule, so we have a decomposition $C=\bigoplus_{j\in\mathbb{Z}}C_{j}$ as a direct sum of $S$-$S$-bimodules, and $\mu:C\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}C\otimes_{S}C$ and $\epsilon:C\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}S$ are homogeneous $S$-$S$-bimodule maps of degree 0 such that the following diagrams commute

[TABLE]

where $\lambda_{C}$ and $\rho_{C}$ are the left and right $S$-multiplications on $C$, respectively. The $S$-$S$-bimodule $S$ is considered as a graded $S$-bimodule concentrated at 0.

A coderivation $\delta$ on ${\cal B}$ is a homogeneous morphism of $S$-$S$-bimodules $\delta:C\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}C$, of degree 1, such that the following square commutes.

[TABLE]

A coderivation $\delta$ on ${\cal B}$ is called a differential iff furthermore $\delta^{2}=0$. In this case, ${\cal B}=(C,\mu,\epsilon,\delta)$ is called a differential graded $S$-bocs. A morphism $f:(C,\mu,\epsilon,\delta)\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(C^{\prime},\mu^{\prime},\epsilon^{\prime},\delta^{\prime})$ of differential graded $S$-bocses is a homogeneous morphism $f:C\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}C^{\prime}$ of graded $S$-$S$-bimodules of degree [math] such that $\mu^{\prime}f=(f\otimes f)\mu$, $\epsilon^{\prime}f=\epsilon$ and $\delta^{\prime}f=f\delta$.

So a (differential) graded $S$-bocs is exactly the same concept as a (differential) graded $S$-coalgebra. A morphism of (differential) graded $S$-bocses is the same as a morphism of (differential) graded $S$-coalgebras.

The following useful property is well known, but we include a proof for the sake of the reader.

Lemma 2.2**.**

For any differential graded $S$-bocs ${\cal B}=(C,\mu,\epsilon,\delta)$ we have $\epsilon\delta=0$.

Proof.

We know that $\rho(id_{C}\otimes\epsilon)\mu=id_{C}$ and $\lambda(\epsilon\otimes id_{C})\mu=id_{C}$. Thus, for $c\in C$, if $\mu(c)=\sum_{i}c_{i}\otimes c^{\prime}_{i}$, we get

[TABLE]

Since $\delta$ is a coderivation, we have $[(id_{C}\otimes\delta)+(\delta\otimes id_{C})]\mu=\mu\delta$. As a consequence, $\rho(id_{C}\otimes\epsilon)(id_{C}\otimes\delta)\mu+\rho(id_{C}\otimes\epsilon)(\delta\otimes id_{C})\mu=\rho(id_{C}\otimes\epsilon)\mu\delta=\delta$. Hence,

[TABLE]

Evaluate at the element $c$ to obtain $\sum_{i}(-1)^{|c_{i}|}c_{i}\epsilon\delta(c^{\prime}_{i})+\sum_{i}\delta(c_{i})\epsilon(c^{\prime}_{i})=\delta(c)$. Since we also have

[TABLE]

we know that $\sum_{i}(-1)^{|c_{i}|}c_{i}\epsilon\delta(c^{\prime}_{i})=0$. Applying the morphism $\epsilon$, we have $\sum_{i}(-1)^{|c_{i}|}\epsilon(c_{i})\epsilon\delta(c^{\prime}_{i})=0$. Since $\epsilon:C\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}S$ is homogeneous of degree [math], we have $\epsilon(c_{i})=0$ whenever $|c_{i}|\not=0$, so we finally get

[TABLE]

∎

Definition 2.3**.**

Given a graded $S$-bocs ${\cal B}=(C,\mu,\epsilon)$, we can consider the $k$-category of graded right ${\cal B}$-modules denoted by ${\rm GMod}\hbox{-}{\cal B}$. A graded right ${\cal B}$-module is by definition a graded right $S$-module. Given graded ${\cal B}$-modules $M$ and $N$, and $d\in\mathbb{Z}$, a morphism $f:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ of right ${\cal B}$-modules of degree $d$ is a homogeneous morphism of right $S$-modules $f:M\otimes_{S}C\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ of degree $d$. So, their set of morphisms in the category ${\rm GMod}\hbox{-}{\cal B}$ is by definition

[TABLE]

If we have a pair $f:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$, $g:N\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}L$ of composable morphisms in ${\rm GMod}\hbox{-}{\cal B}$, their composition in ${\rm GMod}\hbox{-}{\cal B}$ is defined as the following composition of morphisms of right $S$-modules

[TABLE]

It is not hard to show that ${\rm GMod}\hbox{-}{\cal B}$ is indeed a graded $k$-category (the unit morphism at a graded ${\cal B}$-module $M$, denoted by $\mathbb{I}_{M}$, is the morphism of right $S$-modules $\rho_{M}(id_{M}\otimes\epsilon)$, where $\rho_{M}:M\otimes_{S}S\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}M$ is the right multiplication on $M$. We will denote by ${\rm GMod}^{0}\hbox{-}{\cal B}$ the subcategory of ${\rm GMod}\hbox{-}{\cal B}$ with the same objects but only the degree zero morphisms of ${\rm GMod}\hbox{-}{\cal B}$.

Proposition 2.4**.**

Let ${\cal B}=(C,\mu,\epsilon,\delta)$ be a differential graded $S$-bocs. Given $M,N\in{\rm GMod}\hbox{-}{\cal B}$, define on the graded hom space

[TABLE]

the homogeneous linear map $\hat{\delta}:{\rm Hom}_{{\rm GMod}\hbox{-}{\cal B}}(M,N)\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\rm Hom}_{{\rm GMod}\hbox{-}{\cal B}}(M,N)$ of degree 1 by the following recipe, for any homogeneous morphism $f:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$,

[TABLE]

Then, the category ${\rm GMod}\hbox{-}B$ is a differential graded category. Namely, $\hat{\delta}^{2}=0$, $\hat{\delta}(\mathbb{I}_{M})=0$, for all $M\in{\rm GMod}\hbox{-}{\cal B}$, and the following Leibniz rule holds

[TABLE]

for any homogeneous morphisms $f:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ and $g:N\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}L$ in ${\rm GMod}\hbox{-}{\cal B}$.

Proof.

For this proof make $\hat{\underline{\delta}}(f)=f(id_{M}\otimes\delta)$. Then, we have

[TABLE]

Then,

[TABLE]

∎

Definition 2.5**.**

Let ${\cal B}=(C,\mu,\epsilon,\delta)$ be a differential graded $S$-bocs. Then, a twisted ${\cal B}$-module is a pair $(M,u)$, where $M\in{\rm GMod}\hbox{-}{\cal B}$ and $u$ is a homogeneous morphism $u\in{\rm Hom}_{{\rm GMod}\hbox{-}{\cal B}}^{1}(M,M)$ such that the following Maurer-Cartan equation for $u$ holds

[TABLE]

If $(M,u)$ and $(N,v)$ are twisted ${\cal B}$-modules, then a homogeneous morphism of twisted ${\cal B}$-modules $f:(M,u)\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(N,v)$ of degree $d$ is a homogeneous morphism $f:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ in ${\rm GMod}\hbox{-}{\cal B}$ of degree $d$ such that

[TABLE]

We will denote by ${\rm Hom}_{{\rm GMod}\hbox{-}{\cal B}}^{d}((M,u),(N,v))$ the space of homogeneous morphisms of twisted ${\cal B}$-modules of degree $d$. Moreover, we make

[TABLE]

Lemma 2.6**.**

With the preceding notations, we can form the graded category of twisted ${\cal B}$-modules ${\rm TGMod}\hbox{-}{\cal B}$ with objects the twisted ${\cal B}$-modules, morphisms of twisted ${\cal B}$-modules, and the same composition $*$ of the category ${\rm GMod}\hbox{-}{\cal B}$.

Proof.

Let $f:(M,u)\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(N,v)$ and $g:(N,v)\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(L,w)$ be homogeneous morphisms of twisted ${\cal B}$-modules. We have $\hat{\delta}(f)+v*f-(-1)^{|f|}f*u=0$ and $\hat{\delta}(g)+w*g-(-1)^{|g|}g*v=0$. Then, we have

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

∎

Proposition 2.7**.**

Let ${\cal B}_{1}=(C_{1},\mu_{1},\epsilon_{1},\delta_{1})$ and ${\cal B}_{2}=(C_{2},\mu_{2},\epsilon_{2},\delta_{2})$ be differential graded $S$-bocses and $\psi:{\cal B}_{1}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\cal B}_{2}$ a morphism of differential graded $S$-bocses. Then, there is a functor of differential graded categories (called the restriction functor associated to the morphism $\psi$)

[TABLE]

such that $R_{\psi}(M)=M$, for $M\in{\rm GMod}\hbox{-}{\cal B}_{2}$, and $R_{\psi}(f)=f(id_{M}\otimes\psi)$, for any morphism $f:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ of ${\rm GMod}\hbox{-}{\cal B}_{2}$.

Proof.

The identity morphism of $M\in{\rm GMod}\hbox{-}{\cal B}_{2}$ is $\rho(id_{M}\otimes\epsilon_{2}):M\otimes_{S}C_{2}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}M$, where $\rho:M\otimes_{S}S\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}M$ is the multiplication map. Then,

[TABLE]

So, $R_{\psi}$ preserves identities.

Given composable morphisms $f:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ and $g:N\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}L$ in ${\rm GMod}\hbox{-}{\cal B}_{2}$, we have

[TABLE]

Finally, given a homogeneous morphism $f:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ in ${\rm GMod}\hbox{-}{\cal B}_{2}$, we have

[TABLE]

∎

Proposition 2.8**.**

Let ${\cal B}_{1}=(C_{1},\mu_{1},\epsilon_{1},\delta_{1})$ and ${\cal B}_{2}=(C_{2},\mu_{2},\epsilon_{2},\delta_{2})$ be differential graded $S$-bocses and $\psi:{\cal B}_{1}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\cal B}_{2}$ a morphism of differential graded $S$-bocses. Then, the restriction functor $R_{\psi}:{\rm GMod}\hbox{-}{\cal B}_{2}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\rm GMod}\hbox{-}{\cal B}_{1}$ induces a $k$-functor (called again the restriction functor associated to the morphism $\psi$)

[TABLE]

such that $R_{\psi}(M,u)=(M,R_{\psi}(u))$, for $(M,u)\in{\rm TGMod}\hbox{-}{\cal B}_{2}$, and $R_{\psi}(f)=f(id_{M}\otimes\psi)$, for any morphism $f:(M,u)\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(N,v)$ of ${\rm TGMod}\hbox{-}{\cal B}_{2}$.

Proof.

Given $(M,u)\in{\rm TGMod}\hbox{-}{\cal B}_{2}$, we have

[TABLE]

so, $(M,R_{\psi}(u))\in{\rm TGMod}\hbox{-}{\cal B}_{1}$.

Given a homogeneous morphism $f:(M,u)\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(N,v)$ in ${\rm TGMod}\hbox{-}{\cal B}_{2}$, we have that

[TABLE]

coincides with

[TABLE]

So $R_{\psi}(f):(M,R_{\psi}(u))\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(N,R_{\psi}(v))$ is a homogeneous morphism of twisted ${\cal B}_{1}$-modules. ∎

3 Bocses, coalgebras and homotopy

Let us see how that the preceding notions relate to the category of comodules and differential comodules over a given differential graded $S$-bocs ${\cal B}$. We first recall some basic notions.

Definition 3.1**.**

Given a graded $S$-coalgebra ${\cal B}=(C,\mu,\epsilon)$, that is a graded $S$-bocs, we will denote by ${\rm GComod}\hbox{-}{\cal B}$ the category of the graded right ${\cal B}$-comodules. Recall that a graded right ${\cal B}$-comodule is a pair $(M,\mu_{M})$, where $M$ is a graded right $S$-module and $\mu_{M}:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}M\otimes_{S}C$ is a homogeneous morphism of right $S$-modules of degree 0 such that the following diagrams commute

[TABLE]

If $(M,\mu_{M})$ and $(N,\mu_{N})$ are graded right ${\cal B}$-comodules and $d\in\mathbb{Z}$, a morphism of graded right ${\cal B}$-comodules $h:(M,\mu_{M})\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(N,\mu_{N})$ of degree $d$ is a homogeneous morphism of right $S$-modules $h:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ with degree $d$, such that the following square commutes

[TABLE]

We denote by ${\rm Hom}^{d}_{{\rm GComod}\hbox{-}{\cal B}}(M,N)$ the space of morphisms of graded right ${\cal B}$-comodules of degree $d$ from $M$ to $N$ and we make

[TABLE]

We denote by ${\rm GComod}^{0}\hbox{-}{\cal B}$ the subcategory of ${\rm GComod}\hbox{-}{\cal B}$ with the same objects and only degree zero morphisms.

Definition 3.2**.**

Given a differential graded $S$-coalgebra ${\cal B}=(C,\mu,\epsilon,\delta)$, that is a differential graded $S$-bocs, we will denote by ${\rm DGComod}\hbox{-}{\cal B}$ the graded category of the differential graded right ${\cal B}$-comodules. Recall that a differential graded right ${\cal B}$-comodule is a triple $(M,\mu_{M},\delta_{M})$, where $(M,\mu_{M})$ is a graded right ${\cal B}$-comodule and $\delta_{M}:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}M$ is a differential on $(M,\mu_{M})$. Recall that a map $\delta_{M}:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}M$ is a coderivation on $(M,\mu_{M})$ iff it is a homogeneous morphism of graded right $S$-modules of degree $1$ such that the following diagram commutes

[TABLE]

Such a map $\delta_{M}$ is called a differential if, furthermore, $\delta_{M}^{2}=0$.

Whenever $(M,\mu_{M},\delta_{M})$ and $(N,\mu_{N},\delta_{N})$ are differential graded right ${\cal B}$-comodules and $d\in\mathbb{Z}$, we agree that a morphism of differential graded right ${\cal B}$-comodules $h:(M,\mu_{M},\delta_{M})\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(N,\mu_{N},\delta_{N})$ of degree $d$ is a homogeneous morphism of graded right ${\cal B}$-comodules $h:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ with degree $d$, such that $\delta_{N}h=(-1)^{d}h\delta_{M}$. We denote by ${\rm Hom}^{d}_{{\rm DGComod}\hbox{-}{\cal B}}(M,N)$ the space of homogeneous morphisms of differential graded right ${\cal B}$-comodules of degree $d$ from $M$ to $N$ and we make

[TABLE]

As before, with ${\rm DGComod}^{0}\hbox{-}{\cal B}$ we denote the subcategory of ${\rm DGComod}\hbox{-}{\cal B}$ with the same objects but only degree zero morphisms.

Remark 3.3**.**

Let ${\cal B}=(C,\mu,\epsilon,\delta)$ be a differential graded $S$-coalgebra, $M,N\in{\rm DGComod}\hbox{-}{\cal B}$, and $h:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ a homogeneous morphism of graded right ${\cal B}$-comodules. Then,

[TABLE]

Proposition 3.4**.**

Let ${\cal B}=(C,\mu,\epsilon,\delta)$ be a differential graded $S$-bocs.

There is a full and faithful functor of graded categories

[TABLE]

such that $\Phi(M)={\rm Ind}_{\cal B}(M):=(M\otimes_{S}C,id_{M}\otimes\mu)$, for $M\in{\rm GMod}\hbox{-}{\cal B}$, and, for any $f\in{\rm Hom}_{{\rm GMod}\hbox{-}{\cal B}}(M,N)$, we have $\Phi(f)=(f\otimes id_{C})(id_{M}\otimes\mu)$. 2. 2.

There is a full and faithful functor of graded categories

[TABLE]

such that $\Phi(M,u)=({\rm Ind}_{\cal B}(M),\delta_{\Phi(M,u)})$, where $\delta_{\Phi(M,u)}=id_{M}\otimes\delta+\Phi(u)$, for each $(M,u)\in{\rm TGMod}\hbox{-}{\cal B}$, and $\Phi(f)=(f\otimes id_{C})(id_{M}\otimes\mu)$, for any $f\in{\rm Hom}_{{\rm TGMod}\hbox{-}{\cal B}}((M,u),(N,v))$.

Proof.

(1): It is easy to see that the association $\Phi$ is well defined. In order to show that $\Phi$ preserves composition, take $f\in{\rm Hom}_{{\rm GMod}\hbox{-}{\cal B}}(M,N)$ and $g\in{\rm Hom}_{{\rm GMod}\hbox{-}{\cal B}}(N,L)$. Then,

[TABLE]

Given $M\in{\rm GMod}\hbox{-}{\cal B}$, we consider the identity morphism $\mathbb{I}_{M}=\rho_{M}(id_{M}\otimes\epsilon)$ in the category ${\rm GMod}\hbox{-}{\cal B}$. Then,

[TABLE]

Thus $\Phi$ is a degree preserving functor. It is full and faithful because, for $d\in\mathbb{Z}$,

[TABLE]

is an isomorphism, with inverse $\Phi^{\prime}_{d}$ given by $\Phi^{\prime}_{d}(\phi)=q_{N}\phi$, where $q_{N}:=\rho_{N}(id_{N}\otimes\epsilon).$

(2): Take $(M,u)\in{\rm TGMod}\hbox{-}{\cal B}$. If we denote by ${\rm Coder}({\rm Ind}_{\cal B}(M))$ the set of coderivations of ${\rm Ind}_{\cal B}(M)$, we have a bijection

[TABLE]

given by $\Theta(h)=id_{M}\otimes\delta+h$. Then, we know that $\delta_{\Phi(M,u)}=\Theta\Phi_{1}(u)$ is a coderivation on ${\rm Ind}_{\cal B}(M)$. Since $\delta_{\Phi(M,u)}^{2}$ is a homogeneous morphism of graded right ${\cal B}$-comodules of degree $2$, to show that $\delta_{\Phi(M,u)}^{2}=0$ is equivalent to show that $q_{M}\delta_{\Phi(M,u)}^{2}=0$.

From (2.2), we have

[TABLE]

Now, take a homogeneous morphism $f:(M,u)\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(N,v)$ in ${\rm TGMod}\hbox{-}{\cal B}$. We already know that $\Phi(f):{\rm Ind}_{\cal B}(M)\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\rm Ind}_{\cal B}(N)$ is a homogeneous morphism of graded ${\cal B}$-comodules. By definition, the map $\Phi(f):\Phi(M,u)\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}\Phi(N,v)$ is a morphism of differential graded ${\cal B}$-comodules if and only if the following difference $D$ is zero.

[TABLE]

By (3.3), we have that $D=0$ is equivalent to $q_{N}D=0$. Since $f$ is a morphism of twisted ${\cal B}$-modules, we have

[TABLE]

Hence $\Phi$ is a well defined faithful functor. In order to show that it is a full functor, we use again the fact that $\Phi_{d}$ is a bijection and reverse the preceding argument. ∎

Definition 3.5**.**

Let ${\cal B}=(C,\mu,\epsilon,\delta)$ be a differential graded $S$-bocs. Given morphisms $f,g\in{\rm Hom}^{0}_{{\rm TGMod}\hbox{-}{\cal B}}((M,u),(N,v))$, a homotopy $h$ from $f$ to $g$ is a morphism $h\in{\rm Hom}_{{\rm GMod}\hbox{-}{\cal B}}^{-1}(M,N)$ such that

[TABLE]

A morphism $f\in{\rm Hom}^{0}_{{\rm TGMod}\hbox{-}{\cal B}}((M,u),(N,v))$ is null-homotopic iff there is a homotopy from $f$ to [math].

We denote by ${\rm TGMod}^{0}\hbox{-}{\cal B}$ the subcategory ${\rm TGMod}\hbox{-}{\cal B}$ with the same objects and only zero degree homogeneous morphisms. Thus, the notion of homotopy is an equivalence relation in the category ${\rm TGMod}^{0}\hbox{-}{\cal B}$.

Proposition 3.6**.**

Let ${\cal B}=(C,\mu,\epsilon,\delta)$ be a differential graded $S$-bocs. Consider the full and faithful functor $\Phi:{\rm TGMod}\hbox{-}{\cal B}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\rm DGComod}\hbox{-}{\cal B}$ of (3.4). Then, a morphism $f\in{\rm Hom}^{0}_{{\rm TGMod}\hbox{-}{\cal B}}((M,u),(N,v))$ in ${\rm TGMod}^{0}\hbox{-}{\cal B}$ is is null-homotopic iff $\Phi(f):\Phi(M,u)\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}\Phi(N,v)$ is null-homotopic in ${\rm DGComod}^{0}\hbox{-}{\cal B}$. As a consequence, there is an induced full and faithful functor on the homotopy categories

[TABLE]

Proof.

Recall that, by definition, $\Phi(M,u)=({\rm Ind}_{\cal B}(M),id_{M}\otimes\delta+\Phi(u))$ and $\Phi(N,v)=({\rm Ind}_{\cal B}(N),id_{N}\otimes\delta+\Phi(v))$. The morphism $\Phi(f)$ is null-homotopic in ${\rm DGComod}^{0}\hbox{-}{\cal B}$ iff there is a morphism $\Phi(h)\in{\rm Hom}_{{\rm GComod}\hbox{-}{\cal B}}^{-1}({\rm Ind}_{\cal B}(M),{\rm Ind}_{\cal B}(N))$ with

[TABLE]

This is equivalent to

[TABLE]

and this is equivalent to

[TABLE]

that is, $f$ is null-homotopic in ${\rm TGMod}^{0}\hbox{-}{\cal B}$. ∎

Lemma 3.7**.**

Let ${\cal B}_{1}=(C_{1},\mu_{1},\epsilon_{1},\delta_{1})$ and ${\cal B}_{2}=(C_{2},\mu_{2},\epsilon_{2},\delta_{2})$ be differential graded $S$-bocses and $\psi:{\cal B}_{1}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\cal B}_{2}$ a morphism of differential graded $S$-bocses. Then, the restriction functor $R_{\psi}:{\rm TGMod}^{0}\hbox{-}{\cal B}_{2}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\rm TGMod}^{0}\hbox{-}{\cal B}_{1}$ maps null-homotopic morphisms onto null-homotopic morphisms. Hence it induces a $k$-functor on the corresponding homotopy categories

[TABLE]

Proof.

Given a null-homotopic morphism $f\in{\rm Hom}^{0}_{{\rm TGMod}\hbox{-}{\cal B}_{2}}((M,u),(N,v))$, there is $h\in{\rm Hom}_{{\rm GMod}\hbox{-}{\cal B}_{2}}^{-1}(M,N)$ with $f=\hat{\delta}_{2}(h)+v*h+h*u$. Then, applying $R_{\psi}$, we obtain

[TABLE]

Hence, $R_{\psi}(f)$ is null-homotopic. ∎

4 Triangular bocses

In this section we show some properties of the category ${\rm TGMod}\hbox{-}{\cal B}$, where ${\cal B}$ is a differential graded $S$-bocs of a special type, which we call triangular. We stress the fact that some proofs are inspired in the study of differential graded tensor algebras (or ditalgebras) and their module categories initiated by the Kiev school of representation theory, see [2].

Definition 4.1**.**

A normal graded $S$-bocs ${\cal B}$ is a differential unitary graded $S$-coalgebra $(C,\mu,\epsilon,\delta)$, with an $S$-$S$-bimodule decomposition $C=S\bigoplus\overline{C}$ such that $\epsilon:C\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}S$ is the projection map and $\mu(1)=1\otimes 1$.

Lemma 4.2**.**

If ${\cal B}=(C,\mu,\epsilon,\delta)$ is a normal graded $S$-bocs, then we have a well defined map

[TABLE]

and the differential $\delta$ restricts to a map $\overline{\delta}:\overline{C}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}\overline{C}$ of degree $1$. Moreover, the triple $\overline{\cal B}=(\overline{C},\overline{\mu},\overline{\delta})$ is a differential graded $S$-coalgebra (without counit). We call $\overline{\cal B}$ the reduced differential graded $S$-bocs of ${\cal B}$.

Proof.

See, for instance, [2](3.2). ∎

The following reinterpretation of the $k$-category ${\rm GMod}\hbox{-}{\cal B}$ is useful.

Definition 4.3**.**

Given a normal graded $S$-bocs ${\cal B}=(C,\mu,\epsilon,\delta)$, we can form the following graded category ${\rm G}\mathbb{M}{\rm od}\hbox{-}{\cal B}$. Its objects are the graded right $S$-modules and, given $d\in\mathbb{Z}$, a morphism $f:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ of degree $d$ in ${\rm G}\mathbb{M}{\rm od}\hbox{-}{\cal B}$ is a pair of maps $f=(f^{0},f^{1})$, where $f^{0}:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ and $f^{1}:M\otimes_{S}\overline{C}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ are homogeneous morphisms of right $S$-modules of degree $d$. Thus, its hom spaces are given by

[TABLE]

If $f=(f^{0},f^{1})\in{\rm Hom}_{{\rm G}\mathbb{M}{\rm od}\hbox{-}{\cal B}}(M,N)$ and $g=(g^{0},g^{1})\in{\rm Hom}_{{\rm G}\mathbb{M}{\rm od}\hbox{-}{\cal B}}(N,L)$, their composition $g*f=((g*f)^{0},(g*f)^{1})\in{\rm Hom}_{{\rm G}\mathbb{M}{\rm od}\hbox{-}{\cal B}}(M,L)$ is defined by $(g*f)^{0}=g^{0}f^{0}$ and $(g*f)^{1}$ is the following composition in ${\rm GMod}\hbox{-}S$

[TABLE]

It is not hard to show that ${\rm G}\mathbb{M}{\rm od}\hbox{-}{\cal B}$ is indeed a graded $k$-category, where the identity morphism $\mathbb{I}_{M}$ on each $M\in{\rm G}\mathbb{M}{\rm od}\hbox{-}{\cal B}$ is just $(id_{M},0)$. Again, with ${\rm G}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B}$, we denote the subcategory of ${\rm G}\mathbb{M}{\rm od}\hbox{-}{\cal B}$ with the same objects but with only zero degree morphisms.

Lemma 4.4**.**

For any normal graded $S$-bocs ${\cal B}$, there is an equivalence of graded categories

[TABLE]

such that $F(M)=M$, for each $M\in{\rm GMod}\hbox{-}{\cal B}$ and, given any morphism $f\in{\rm Hom}_{{\rm GMod}\hbox{-}{\cal B}}(M,N)={\rm Hom}_{{\rm GMod}\hbox{-}S}(M\otimes_{S}C,N)$, we have $F(f)=(f^{0},f^{1})$, where $f^{0}:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ is given by $f^{0}(m)=f(m\otimes 1)$, for $m\in M$, and the morphism $f^{1}:M\otimes_{S}\overline{C}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ is just the restriction of the map $f$.

Proof.

It is clear that $F$ defined as above determines a linear isomorphism

[TABLE]

So we only have to show that $F$ preserves compositions and identities. Assume that $f:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ and $g:N\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}L$ are morphisms in ${\rm GMod}\hbox{-}{\cal B}$. Then, for $m\in M$, we have

[TABLE]

For $x\in\overline{C}$, we have $\mu(x)=\overline{\mu}(x)+1\otimes x+x\otimes 1$ and $\overline{\mu}(x)=\sum_{j}y_{j}\otimes z_{j}$, so

[TABLE]

Thus, we obtain that $F(g*f)=F(g)*F(f)$. Given $M\in{\rm GMod}\hbox{-}{\cal B}$, recall that $\mathbb{I}_{M}=\rho_{M}(id_{M}\otimes\epsilon)$, thus $F(\mathbb{I}_{M})^{0}(m)=\rho(id_{M}\otimes\epsilon)(m\otimes 1)=m.$ For $x\in\overline{C}$, we have $F(\mathbb{I}_{M})^{1}(m\otimes x)=\rho(id_{M}\otimes\epsilon)(m\otimes x)=0$. Hence, $F(\mathbb{I}_{M})=(id_{M},0)=\mathbb{I}_{F(M)}$. ∎

Remark 4.5**.**

Given $f:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ in ${\rm GMod}\hbox{-}{\cal B}$, we often refer to the maps $f^{0}$ and $f^{1}$ in $F(f)=(f^{0},f^{1})$ as the components of $f$.

Notice that any morphism $(f^{0},f^{1})\in{\rm Hom}_{{\rm G}\mathbb{M}{\rm od}\hbox{-}{\cal B}}(M,N)$ decomposes as a sum of morphisms $(f^{0},f^{1})=(f^{0},0)+(0,f^{1})$ in ${\rm Hom}_{{\rm G}\mathbb{M}{\rm od}\hbox{-}{\cal B}}(M,N)$.

This notation as pairs of maps was relevant in the study of categories of modules over differential tensor algebras in [2]. We exploit it here in the context of graded $S$-bocses.

Definition 4.6**.**

A normal graded $S$-bocs ${\cal B}=(C,\mu,\epsilon,\delta)$ is called triangular iff there is a sequence

[TABLE]

of graded $S$-$S$-subbimodules of $\overline{C}$ such that $\overline{C}=\bigcup_{i\in\mathbb{N}}\overline{C}_{i}$,

[TABLE]

Proposition 4.7**.**

Let ${\cal B}=(C,\mu,\epsilon,\delta)$ be a triangular graded $S$-bocs. Consider any endomorphism in ${\rm G}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B}$ of the form $f=(0,f^{1}):M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}M$. Then, for each $x\in\overline{C}$, there is a natural number $n_{x}$ such that $(f^{n})^{1}[m\otimes x]=0$, for all $n\geq n_{x}$ and $m\in M$. In this case, we can consider the morphism in ${\rm G}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B}$

[TABLE]

Proof.

Adopt the notations of (4.6). We proceed to show, by induction on $i\in\mathbb{N}$, that there is a natural number $n_{i}$ such that for any $x\in\overline{C}_{i}$ and $m\in M$, we have, $(f^{n_{i}})^{1}[m\otimes x]=0$.

From the definition of the composition of morphisms is ${\rm G}\mathbb{M}{\rm od}\hbox{-}{\cal B}$, we have that $(f^{n+1})^{1}(m\otimes x)=f^{n}(f^{1}\otimes id_{\overline{C}})(id_{M}\otimes\overline{\mu})[m\otimes x]=0$, for $x\in\overline{C}_{1}$ and $n\geq 1$, and we can take $n_{1}=2$.

Assume that we have proved that $n_{i}$ exists for a fixed $i\in\mathbb{N}$, such that for any $x\in\overline{C}_{i}$ and $m\in M$, we have $(f^{n_{i}})^{1}[m\otimes x]=0$. Then, if $x\in\overline{C}_{i+1}$, we have that $\overline{\mu}(x)=\sum_{j=1}^{t}x_{j}\otimes y_{j}$, with $x_{j},y_{j}\in\overline{C}_{i}$. Thus, for $m\in M$, we have

[TABLE]

The statement of the Proposition follows from this. ∎

Corollary 4.8**.**

Let ${\cal B}$ be a triangular graded $S$-bocs. Take any morphism $f=(f^{0},f^{1}):M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ in ${\rm G}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B}$. Then, $f:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ is an isomorphism in ${\rm G}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B}$ iff $f^{0}:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ is an isomorphism of ${\rm GMod}\hbox{-}S$.

Proof.

We first show that an endomorphism of the form $f=(id_{M},f^{1}):M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}M$ is an automorphism. Indeed, we can write $f=\mathbb{I}_{M}-h$ where $h=(0,-f^{1})$. Then

[TABLE]

Similarly, we have $(\sum_{i=0}^{\infty}h^{i})*(\mathbb{I}_{M}-h)=\mathbb{I}_{M}$. Thus $f$ is an automorphism.

Now, consider any morphism $f=(f^{0},f^{1}):M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ with $f^{0}:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ isomorphism and consider the inverse $g^{0}:N\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}M$ of $f^{0}$. Then, $f*(g^{0},0)=(id_{N},u)$ and $(g^{0},0)*f=(id_{M},v)$ are isomorphisms. So, $f$ is a section and a retraction in ${\rm G}\mathbb{M}{\rm od}\hbox{-}{\cal B}$. It follows that $f$ is an isomorphism in ${\rm G}\mathbb{M}{\rm od}\hbox{-}{\cal B}$. ∎

Corollary 4.9**.**

Let ${\cal B}$ be a triangular graded $S$-bocs. Take any morphism $f:(M,u)\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(N,v)$ in ${\rm TGMod}^{0}\hbox{-}{\cal B}$. Then, $f:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ is an isomorphism in ${\rm G}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B}$ iff its first component $f^{0}:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ is an isomorphism of ${\rm GMod}\hbox{-}S$.

Proof.

From (4.8), it will be enough to show the following.

*Claim: Fix any homogeneous morphism $f:(M,u)\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(N,v)$ in ${\rm TGMod}^{0}\hbox{-}{\cal B}$. Then, $f:(M,u)\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(N,v)$ is an isomorphism in ${\rm TGMod}^{0}\hbox{-}{\cal B}$ iff $f:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ is an isomorphism in ${\rm GMod}\hbox{-}{\cal B}$. *

We now prove the claim. Since $f:(M,u)\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(N,v)$ is a morphism in ${\rm TGMod}^{0}\hbox{-}{\cal B}$, we have that $\hat{\delta}(f)+v*f-f*u=0$. If $f$ is an isomorphism in ${\rm GMod}^{0}\hbox{-}{\cal B}$, it has a homogeneous two-sided inverse $g:N\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}M$ with $|g|=0$. Since $g*f=\mathbb{I}_{M}$, we have $0=\hat{\delta}(\mathbb{I}_{M})=\hat{\delta}(g*f)=\hat{\delta}(g)*f+g*\hat{\delta}(f)$. So,

[TABLE]

So, $g:(N,v)\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(M,u)$ is a morphism in ${\rm TGMod}^{0}\hbox{-}{\cal B}$, an inverse for $f$. ∎

Proposition 4.10**.**

Suppose that ${\cal B}$ is a triangular differential graded $S$-bocs. Consider a composable pair of morphisms in ${\rm G}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B}$

[TABLE]

such that $g*f=0$ and the sequence $0\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits^{f^{0}}}E\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits^{g^{0}}}N\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}0$ is an exact sequence in ${\rm GMod}\hbox{-}S$. Then, there is a commutative diagram in ${\rm G}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B}$

[TABLE]

where $h$ is an automorphism.

Proof.

Step 1: There is an automorphism $h$ of $E$ such that $(h*f)^{1}=0$.

We have $f=\underline{f}^{0}+\underline{f}^{1}$, where $\underline{f}^{0}=(f^{0},0)$ and $\underline{f}^{1}=(0,f^{1})$ are morphisms in ${\rm G}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B}$. Since $S$ is semisimple, the exact sequence in the statement of our proposition splits and we have a morphism $p:E\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}M$ in ${\rm GMod}\hbox{-}S$ such that $pf^{0}=id_{M}$. Consider the morphism $\underline{p}=(p,0):E\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}M$ in ${\rm G}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B}$. Then, we have $\underline{p}*\underline{f}^{0}=\mathbb{I}_{M}$ in ${\rm G}\mathbb{M}{\rm od}\hbox{-}{\cal B}$. Make $u:=\underline{f}^{1}*\underline{p}$ and notice that $u^{0}=0$. Then, we have the automorphism $h$ of $E$ defined by

[TABLE]

We have that $h*f=\underline{f}^{0}+\underline{f}^{1}+v*\underline{f}^{0}+v*\underline{f}^{1}$, where

[TABLE]

Therefore, we obtain $(h*f)^{1}=0$.

Step 2: Now, we may assume that $f=(f^{0},0)$ and finish the proof.

As before, we have $g=\underline{g}^{0}+\underline{g}^{1}$, where $\underline{g}^{0}=(g^{0},0)$ and $\underline{g}^{1}=(0,g^{1})$ are morphisms in ${\rm G}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B}$. Since $S$ is semisimple, there is a morphism $s:N\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}E$ in ${\rm GMod}\hbox{-}S$ such that $g^{0}s=id_{N}$. If we take $\underline{s}=(s,0)$ we have $\underline{g}^{0}*\underline{s}=\mathbb{I}_{N}$ in ${\rm G}\mathbb{M}{\rm od}\hbox{-}{\cal B}$. Now, make $u=\underline{s}*\underline{g}^{1}$ and notice that $u^{0}=0$. Now, consider the automorphism $h$ of $E$ defined by

[TABLE]

We have that $g*h=\underline{g}^{0}+\underline{g}^{1}+\underline{g}^{0}*w+\underline{g}^{1}*w$, where

[TABLE]

Therefore, we obtain $(g*h)^{1}=0$. Notice that

[TABLE]

Hence, $h^{-1}=\mathbb{I}_{E}+u$. Since $g*f=0$, we have that $\underline{g}^{1}*f=0$ and, therefore, $h^{-1}*f=f+u*f=f$. So, $(h^{-1}*f)^{1}=0$ and we are done. ∎

Lemma 4.11**.**

Let ${\cal B}$ be a differential graded $S$-bocs, assume that $(M,u)\in{\rm TGMod}\hbox{-}{\cal B}$, and $N\in{\rm GMod}\hbox{-}{\cal B}$. Then, for any homogeneous isomorphism $h:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ in ${\rm GMod}\hbox{-}{\cal B}$, there is a unique morphism $v\in{\rm Hom}_{{\rm GMod}\hbox{-}{\cal B}}^{1}(N,N)$ such that $h:(M,u)\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(N,v)$ is an isomorphism in ${\rm TGMod}\hbox{-}{\cal B}$.

Proof.

By assumption, $\hat{\delta}(u)+u*u=0$ and we are looking for a morphism $v\in{\rm Hom}_{{\rm GMod}\hbox{-}{\cal B}}^{1}(N,N)$ such that $\hat{\delta}(h)+v*h-(-1)^{|h|}h*u=0$. Then, the only possible choice is $v=(-1)^{|h|}h*u*h^{-1}-\hat{\delta}(h)*h^{-1}$. It remains to show that $\hat{\delta}(v)+v*v=0$. We shall compute each term separately.

From Leibniz rule, we have that $0=\hat{\delta}(\mathbb{I}_{N})=\hat{\delta}(h*h^{-1})=\hat{\delta}(h)*h^{-1}+(-1)^{|h|}h*\hat{\delta}(h^{-1})$. Then, we have $\hat{\delta}(h^{-1})=-(-1)^{|h|}h^{-1}*\hat{\delta}(h)*h^{-1}$, and

[TABLE]

Moreover, we have

[TABLE]

Since $\hat{\delta}(u)+u*u=0$, we obtain $\hat{\delta}(v)=-v*v$, as we wanted. ∎

Proposition 4.12**.**

Suppose that ${\cal B}$ is a triangular graded $S$-bocs. Consider a composable pair of homogeneous morphisms in ${\rm TGMod}^{0}\hbox{-}{\cal B}$

[TABLE]

such that $g*f=0$ and the sequence of first components

[TABLE]

is exact in ${\rm GMod}\hbox{-}S$. Then, there is a commutative diagram in ${\rm TGMod}^{0}\hbox{-}{\cal B}$

[TABLE]

where $h$ is an isomorphism and the second components of $r$ and $s$ are zero.

Proof.

Consider the underlying diagrams in ${\rm GMod}^{0}\hbox{-}{\cal B}$ of our hypothesis. Then, there is an isomorphism $h:E\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}E$ in ${\rm GMod}^{0}\hbox{-}{\cal B}$ as in (4.10). We have $(E,u_{E})\in{\rm TGMod}\hbox{-}{\cal B}$ and, so, we can apply (4.11) to $h$ and obtain a morphism $u^{\prime}_{E}\in{\rm Hom}_{{\rm GMod}\hbox{-}{\cal B}}^{1}(E,E)$ such that $h:(E,u_{E})\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(E,u^{\prime}_{E})$ is an isomorphism in ${\rm TGMod}^{0}\hbox{-}{\cal B}$. ∎

5 Scalar restriction functors and homotopy

Let us recall the classical definition of homotopy of differential graded $S$-coalgebras (or $S$-bocses).

Definition 5.1**.**

Let $\overline{\cal B}_{A}=(\overline{C}_{A},\overline{\mu}_{A},\overline{\delta}_{A})$ and $\overline{\cal B}_{B}=(\overline{C}_{B},\overline{\mu}_{B},\overline{\delta}_{B})$ be differential graded $S$-bocses (without counit). Let $\overline{\phi},\overline{\psi}:\overline{C}_{A}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}\overline{C}_{B}$ be morphisms of differential graded $S$-bocses. A homotopy $\overline{h}$ from $\overline{\phi}$ to $\overline{\psi}$ is a morphism $\overline{h}\in{\rm Hom}^{-1}_{{\rm GMod}\hbox{-}S\hbox{-}S}(\overline{C}_{A},\overline{C}_{B})$ such that

[TABLE]

The morphisms $\overline{\phi}$ and $\overline{\psi}$ are called homotopic iff there is such a homotopy $\overline{h}$. A morphism $\overline{h}\in{\rm Hom}^{-1}_{{\rm GMod}\hbox{-}S\hbox{-}S}(\overline{C}_{A},\overline{C}_{B})$ is called a $\overline{\phi}\hbox{-}\overline{\psi}$-coderivation when the preceding left equality is satisfied. We denote by ${\rm Coder}^{-1}_{\overline{\phi}\hbox{-}\overline{\psi}}(\overline{C}_{A},\overline{C}_{B})$ the space of $\overline{\phi}\hbox{-}\overline{\psi}$-coderivations.

The definition of homotopy of morphisms of differential counitary graded $S$-bocses is similar, we just remove the overlines and add $\epsilon_{A}$ and $\epsilon_{B}$ to the given triples, respectively.

The proof of the following statement is straightforward.

Lemma 5.2**.**

Let ${\cal B}_{A}=({C}_{A},{\mu}_{A},\epsilon_{A},{\delta}_{A})$ and ${\cal B}_{B}=({C}_{B},{\mu}_{B},\epsilon_{B},{\delta}_{B})$ be differential normal graded $S$-bocses. Let $\overline{\phi},\overline{\psi}:\overline{\cal B}_{A}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}\overline{\cal B}_{B}$ be morphisms of the corresponding reduced differential graded $S$-bocses and assume that $\overline{h}$ is a homotopy from $\overline{\phi}$ to $\overline{\psi}$. Extend $\overline{\phi}$ and $\overline{\psi}$ to morphisms of differential unitary graded $S$-bocses $\phi,\psi:{\cal B}_{A}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\cal B}_{B}$ defining $\phi(s+x)=s+\overline{\phi}(x)$ and $\psi(s+x)=s+\overline{\psi}(x)$, for $s\in S$ and $x\in\overline{C}_{A}$; extend $\overline{h}$ to $h:C_{A}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}C_{B}$ defining $h(s+x)=\overline{h}(x)$. Then, we have that $h$ is a homotopy from $\phi$ to $\psi$. That is $h\in{\rm Hom}_{{\rm GMod}\hbox{-}S\hbox{-}S}^{-1}(C_{A},C_{B})$ satisfies

[TABLE]

Lemma 5.3**.**

Under the assumptions of the last lemma, for $d\in\mathbb{Z}$, define

[TABLE]

such that $R_{h}(u)=(-1)^{|u|}u(id_{M}\otimes h)$, for $u\in{\rm Hom}_{{\rm GMod}\hbox{-}{\cal B}_{B}}^{d}(M,N)$. Then, the following holds.

Whenever $u:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ and $v:N\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}L$ are homogeneous morphisms in ${\rm GMod}\hbox{-}{\cal B}_{B}$, we have

[TABLE] 2. 2.

For any homogeneous morphism $u:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ in ${\rm GMod}\hbox{-}{\cal B}_{B}$, we have

[TABLE]

Proof.

(1): Having in mind (2.7), we get

[TABLE]

(2): Since $\hat{\delta}_{B}(u)=(-1)^{|u|+1}u(id_{M}\otimes\delta_{B})$, we have

[TABLE]

∎

Proposition 5.4**.**

Adopt the assumptions of (5.2) and assume, furthermore, that ${\cal B}_{A}$ is a triangular $S$-bocs. Then, there is an isomorphism of functors

[TABLE]

where $\underline{R}_{\phi},\underline{R}_{\psi}:\underline{{\rm TGMod}}^{0}\hbox{-}{\cal B}_{B}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}\underline{{\rm TGMod}}^{0}\hbox{-}{\cal B}_{A}$ are the functors induced on the homotopy categories by $R_{\phi},R_{\psi}:{\rm TGMod}^{0}\hbox{-}{\cal B}_{B}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\rm TGMod}^{0}\hbox{-}{\cal B}_{A}$, respectively, see (3.7). For each $(M,u)\in{\rm TGMod}\hbox{-}{\cal B}_{B}$, the map $\underline{\eta}_{(M,u)}:R_{\phi}(M,u)\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}R_{\psi}(M,u)$ is the class modulo homotopy of the morphism

[TABLE]

Proof.

We first show that $\mathbb{I}_{M}+R_{h}(u)\in{\rm Hom}^{0}_{{\rm GMod}\hbox{-}{\cal B}_{A}}(M,M)$ is an isomorphism in the category ${\rm GMod}\hbox{-}{\cal B}_{A}$. In order to see this, consider the equivalence $F:{\rm GMod}\hbox{-}{\cal B}_{A}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\rm G}\mathbb{M}{\rm od}\hbox{-}{\cal B}_{A}$ of (4.4). Consider the morphism $F(R_{h}(u))=(h^{0},h^{1}):M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}M$ in ${\rm G}\mathbb{M}{\rm od}\hbox{-}{\cal B}$ and notice that, for $m\in M$, we have

[TABLE]

because $h(1)=0$. Therefore, $h^{0}=0$ and, by (4.8), we know that $\mathbb{I}_{M}+R_{h}(u)$ is an isomorphism in ${\rm GMod}^{0}\hbox{-}{\cal B}_{A}$.

Now we show that $\mathbb{I}_{M}+R_{h}(u)\in{\rm Hom}^{0}_{{\rm TGMod}\hbox{-}{\cal B}_{A}}(R_{\phi}(M,u),R_{\psi}(M,u))$. By definition, this is equivalent to show the identity

[TABLE]

We know that $u:M\otimes_{S}C_{B}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}M$ is a homogeneous morphism of degree $1$ such that $\hat{\delta}_{B}(u)+u*u=0$. Then, applying $R_{h}$ and using (5.3), we get

[TABLE]

That is $\hat{\delta}_{A}R_{h}(u)+R_{\psi}(u)*R_{h}(u)-R_{h}(u)*R_{\phi}(u)+R_{\psi}(u)-R_{\phi}(u)=0$, which is equivalent to the identity $(*)$.

Finally, we show that the family of classes $\underline{\eta}_{(M,u)}:R_{\phi}(M,u)\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}R_{\psi}(M,u)$ modulo homotopy of the morphisms $\eta_{(M,u)}=\mathbb{I}_{M}+R_{h}(u):R_{\phi}(M,u)\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}R_{\psi}(M,u)$ is a natural transformation. Take any homogeneous morphism $f:(M,u)\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(N,v)$ of degree [math] in ${\rm TGMod}\hbox{-}{\cal B}_{B}$. Hence, we have $\hat{\delta}_{B}(f)+v*f-f*u=0$. Applying $R_{h}$ to the preceding equality, we obtain

[TABLE]

Hence,

[TABLE]

Let us compute the difference $D=\eta_{(N,v)}*R_{\phi}(f)-R_{\psi}(f)*\eta_{(M,u)}$.

[TABLE]

Hence, $D$ is null-homotopic in ${\rm TGMod}^{0}\hbox{-}{\cal B}_{A}$. This finishes the proof. ∎

Corollary 5.5**.**

Let $\phi:{\cal B}_{A}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\cal B}_{B}$ be a homotopy equivalence of triangular $S$-bocses. Then, the corresponding restriction functor determines an equivalence of categories

[TABLE]

Proof.

If $\phi$ is a homotopy equivalence, its class modulo homotopy $\underline{\phi}$ has an inverse $\underline{\psi}$, for some morphism of triangular $S$-bocses $\psi:{\cal B}_{B}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\cal B}_{A}$. Then we have $\underline{\phi\psi}=\underline{\phi}\,\underline{\psi}\sim\underline{id}_{{\cal B}_{B}}$ and $\underline{\psi\phi}=\underline{\psi}\,\underline{\phi}\sim\underline{id}_{{\cal B}_{A}}$. Therefore, we get isomorphisms of functors $\underline{R}_{\phi}\underline{R}_{\psi}=\underline{R}_{\psi\phi}\cong id_{\underline{{\rm TGMod}}^{0}\hbox{-}{\cal B}_{A}}$ and $\underline{R}_{\psi}\underline{R}_{\phi}=\underline{R}_{\phi\psi}\cong id_{\underline{{\rm TGMod}}^{0}\hbox{-}{\cal B}_{B}}$. ∎

6 Splitting idempotents for twisted modules

In this section we show that idempotents split in the category ${\rm TGMod}^{0}\hbox{-}{\cal B}$, where ${\cal B}$ is any triangular differential graded $S$-bocs. We start with some simple considerations on the additivity of the category ${\rm TGMod}\hbox{-}{\cal B}$.

Lemma 6.1**.**

Given a graded $S$-bocs ${\cal B}=(C,\mu,\epsilon)$, there is a functor of graded $k$-categories

[TABLE]

which acts as the identity on objects and for any homogeneous morphism $f:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ of right graded $S$-modules, by definition

[TABLE]

As a consequence, ${\rm GMod}\hbox{-}{\cal B}$ is an additive category. When $\epsilon$ is surjective, the functor $L$ is faithful and we call it the canonical embedding.

Remark 6.2**.**

If $\sigma_{j}:M_{j}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}\bigoplus_{i=1}^{n}M_{i}$ and $\pi_{j}:\bigoplus_{i=1}^{n}M_{i}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}M_{j}$ denote, respectively, the injection and projection maps corresponding to the direct sum $M=\bigoplus_{i=1}^{n}M_{i}$ in ${\rm GMod}\hbox{-}S$, we have that $\underline{\sigma}_{j}:=L(\sigma_{j}):M_{j}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}\bigoplus_{i=1}^{n}M_{i}$ and $\underline{\pi}_{j}:=L(\pi_{j}):\bigoplus_{i=1}^{n}M_{i}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}M_{j}$ are, respectively, the injection and projection morphisms corresponding to the direct sum $\bigoplus_{i=1}^{n}M_{i}$ in ${\rm GMod}\hbox{-}{\cal B}$. We will work with the usual matrix notation $(g_{j,i})$ for a morphism $g:\bigoplus_{i=1}^{m}M_{i}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}\bigoplus_{j=1}^{n}N_{j}$ in an additive category, the context will always permit the reader to avoid confusion with this notation: in ${\rm GMod}\hbox{-}{\cal B}$, this means that $g_{j,i}=\underline{\pi}_{j}*g*\underline{\sigma_{i}}$ and, therefore, $g=\sum_{i,j}\underline{\sigma}_{j}*g_{j,i}*\underline{\pi}_{i}$.

Notice also that, for a differential graded $S$-bocs ${\cal B}$ with differential $\delta$, from the definition of the differential $\widehat{\delta}$ on the category ${\rm GMod}\hbox{-}{\cal B}$ and (2.2), we immediately obtain that $\widehat{\delta}(L(h))=0$ for any morphism $h:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ in ${\rm GMod}\hbox{-}S$.

Moreover, if ${\cal B}$ is a differential graded $S$-bocs, the category ${\rm TGMod}\hbox{-}{\cal B}$ is additive too: given $(M_{1},u_{1}),(M_{2},u_{2})\in{\rm TGMod}\hbox{-}{\cal B}$, their direct sum is

[TABLE]

The injections are the morphisms $\underline{\sigma}_{i}:(M_{i},u_{i})\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(M_{1}\oplus M_{2},u)$, while the projections are the morphisms $\underline{\pi}_{i}:(M_{1}\oplus M_{2},u)\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(M_{i},u_{i})$. Indeed, since $u=\underline{\sigma}_{1}*u_{1}*\underline{\pi}_{1}+\underline{\sigma}_{2}*u_{2}*\underline{\pi}_{2}$, we obtain $u*\underline{\sigma}_{i}=\underline{\sigma}_{i}*u_{i}*\underline{\pi}_{i}*\underline{\sigma}_{i}=\underline{\sigma}_{i}*u_{i}$. Therefore, $\widehat{\delta}(\underline{\sigma}_{i})+u*\underline{\sigma}_{i}-\underline{\sigma}_{i}*u_{i}=0$. Similarly, we have $\widehat{\delta}(\underline{\pi}_{i})+u_{i}*\underline{\pi}_{i}-\underline{\pi}_{i}*u=0$.

Given a morphism $g:\bigoplus_{i=1}^{m}M_{i}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}\bigoplus_{j=1}^{n}N_{j}$ in ${\rm GMod}\hbox{-}{\cal B}$, with associated matrix $(g_{j,i})$, from Leibniz rule, we have $\widehat{\delta}(g)=\sum_{i,j}\underline{\sigma}_{j}*\widehat{\delta}(g_{j,i})*\underline{\pi}_{i}$. Then, we have $\widehat{\delta}(g)_{t,s}=\underline{\pi}_{t}*\widehat{\delta}(g)*\underline{\sigma}_{s}=\underline{\pi}_{t}*\underline{\sigma}_{t}*\widehat{\delta}(g_{t,s})*\underline{\pi}_{s}*\underline{\sigma}_{s}=\widehat{\delta}(g_{t,s})$, which means that the image under $\widehat{\delta}$ of the matrix $(g_{j,i})$ is the matrix of images $(\widehat{\delta}(g_{j,i}))$.

Lemma 6.3**.**

Let ${\cal B}=(C,\mu,\epsilon,\delta)$ be a differential graded $S$-bocs. Assume that $e:(M,u)\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(M,u)$ is an idempotent morphism in the category ${\rm TGMod}^{0}\hbox{-}{\cal B}$ such that the idempotent $e:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}M$ splits in ${\rm GMod}^{0}\hbox{-}{\cal B}$, then $e$ splits in ${\rm TGMod}^{0}\hbox{-}{\cal B}$.

Proof.

By assumption, there is an isomorphism $h:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}M_{1}\oplus M_{2}$ in ${\rm GMod}^{0}\hbox{-}{\cal B}$ such that the following diagram commutes in ${\rm GMod}^{0}\hbox{-}{\cal B}$

[TABLE]

where $M_{1}\oplus M_{2}$ is the direct sum of graded right ${\cal B}$-modules.

According to (4.11), there is some $v\in{\rm Hom}^{1}_{{\rm GMod}\hbox{-}{\cal B}}(M_{1}\oplus M_{2},M_{1}\oplus M_{2})$ such that $h:(M,u)\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(M_{1}\oplus M_{2},v)$ is an isomorphism in ${\rm TGMod}^{0}\hbox{-}{\cal B}$. Then, $h*e*h^{-1}:(M_{1}\oplus M_{2},v)\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(M_{1}\oplus M_{2},v)$ is a morphism in ${\rm TGMod}^{0}\hbox{-}{\cal B}$. Thus, $\widehat{\delta}(h*g*h^{-1})+v*(h*e*h^{-1})-(h*e*h^{-1})*v=0$. But $\widehat{\delta}(h*e*h^{-1})=\widehat{\delta}\begin{pmatrix}\mathbb{I}_{M_{1}}&0\\ 0&0\\ \end{pmatrix}=0$, because $\widehat{\delta}(\mathbb{I}_{M_{1}})=0$. Hence, $(h*e*h^{-1})*v=v*(h*e*h^{-1})$. It follows that

[TABLE]

Then, from the equality $\widehat{\delta}(v)+v*v=0$, we obtain the equalities

[TABLE]

Then, $(M_{1},v_{1})$ and $(M_{2},v_{2})$ belong to ${\rm TGMod}\hbox{-}{\cal B}$, and

[TABLE]

is an isomorphism in ${\rm TGMod}^{0}\hbox{-}{\cal B}$ with $h*e*h^{-1}=\begin{pmatrix}\mathbb{I}_{M_{1}}&0\\ 0&0\end{pmatrix}$, thus $e$ splits in ${\rm TGMod}^{0}\hbox{-}{\cal B}$, as claimed. ∎

For the sake of computational simplicity, it is convenient to rewrite the morphisms of the category ${\rm G}\mathbb{M}{\rm od}\hbox{-}{\cal B}$ in a slightly different way.

Definition 6.4**.**

Given a normal graded $S$-bocs ${\cal B}=(C,\mu,\epsilon)$, we can consider the following category ${\rm G}\mathbb{M}{\rm od}\hskip-1.5pt\hbox{\rm I}\hbox{-}{\cal B}$. Its objects are the graded right $S$-modules and, for $d\in\mathbb{Z}$, a morphism $f:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ of degree $d$ is a pair $f=(f^{0},f^{1})$, where $f^{0}:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ and $f^{1}:\overline{C}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\rm Hom}_{{\rm GMod}\hbox{-}k}(M,N)$ are homogeneous morphisms, $f^{0}$ of right $S$-modules and $f^{1}$ of $S$-$S$-bimodules, of degree $d$. Thus, the hom spaces are given by

[TABLE]

The composition in this category is transfered from the composition in the category ${\rm G}\mathbb{M}{\rm od}\hbox{-}{\cal B}$ with the help of the natural isomorphism

[TABLE]

So, the composition of two morphisms $f:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ and $g:N\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}L$ in ${\rm G}\mathbb{M}{\rm od}\hskip-1.5pt\hbox{\rm I}\hbox{-}{\cal B}$ is given by $g*f=((g*f)^{0},(g*f)^{1})$, where $(g*f)^{0}=g^{0}f^{0}$ and the morphism of graded $S$-$S$-bimodules $(f*g)^{1}:\overline{C}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\rm Hom}^{*}_{{\rm GMod}\hbox{-}k}(M,N)$ is given by

[TABLE]

It is easy to show that the bijection $(f^{0},f^{1})\mapsto(f^{0},\eta(f^{1}))$ determines an isomorphism of categories ${\rm G}\mathbb{M}{\rm od}\hbox{-}{\cal B}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\rm G}\mathbb{M}{\rm od}\hskip-1.5pt\hbox{\rm I}\hbox{-}{\cal B}$. The formula for the composition of morphisms in ${\rm G}\mathbb{M}{\rm od}\hskip-1.5pt\hbox{\rm I}\hbox{-}{\cal B}$ is the translation of the composition in ${\rm G}\mathbb{M}{\rm od}\hbox{-}{\cal B}$ using the precedent bijection.

Lemma 6.5**.**

For any graded triangular $S$-bocs ${\cal B}$, idempotents split in ${\rm G}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B}$.

Proof.

In this proof, for the sake of notational simplicity, we write $gf$ instead of $g*f$ to indicate the composite morphism in ${\rm G}\mathbb{M}{\rm od}\hskip-1.5pt\hbox{\rm I}^{0}\hbox{-}{\cal B}$.

It will be enough to show that any idempotent morphism $e=(e^{0},e^{1}):M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}M$ splits in ${\rm G}\mathbb{M}{\rm od}\hskip-1.5pt\hbox{\rm I}^{0}\hbox{-}{\cal B}$. Since idempotents clearly split in ${\rm GMod}\hbox{-}S$, it will be enough to show that there is an isomorphism $h:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}M$ such that $(heh^{-1})^{1}=0$. Adopt the notation of (4.6), make $\overline{C}_{-1}:=0$, and let us first show the following.

Claim 1: There is a sequence of isomorphisms $M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits^{h_{1}}}M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits^{h_{2}}}M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits^{h_{3}}}\cdots$ in ${\rm G}\mathbb{M}{\rm od}\hskip-1.5pt\hbox{\rm I}^{0}\hbox{-}{\cal B}$ such that, for each $i\geq 1$, we have

Each isomorphism has the form $h_{i}=(id_{M},h^{1}_{i})$; 2. 2.

$h^{1}_{i}(\overline{C}_{i-2})=0$; and 3. 3.

For $e_{i}:=h_{i}\cdots h_{2}h_{1}eh_{1}^{-1}h_{2}^{-1}\cdots h_{i}^{-1}$, we have $e_{i}^{1}(\overline{C}_{i-1})=0$.

Proof of the Claim 1: The inductive argument is essentially the same as the one given in [2](5.12), but we recall it for the sake of the reader. At the base of the induction, we have the isomorphism $h_{1}=(id_{M},0):M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}M$ such that $h_{1}^{1}(\overline{C}_{-1})=0$ and the idempotent $e_{1}:=e$ such that $e_{1}^{1}(\overline{C}_{0})=0$.

Assume that we have constructed the isomorphisms $h_{1},\ldots,h_{i}$ satisfying conditions 1–3. Notice that $e_{i}$ is idempotent, and so is $e_{i}^{0}$. From $e_{i}^{1}(\overline{C}_{i-1})=0$ and the triangularity we obtain, for $c\in\overline{C}_{i}$ that $e_{i}^{1}(c)=(e_{i}^{2})^{1}(c)=e_{i}^{0}e_{i}^{1}(c)+e_{i}^{1}(c)e_{i}^{0}$. Hence, $e_{i}^{0}e_{i}^{1}(c)e_{i}^{0}=0$. Now, for $c\in\overline{C}$ define $h_{i+1}^{1}(c):=e_{i}^{1}(c)f_{i}^{0}$, where $f_{i}^{0}\in{\rm Hom}_{{\rm GMod}\hbox{-}S}(M,M)$ will be specified in a moment. Then, clearly $h_{i+1}^{1}(\overline{C}_{i-1})=0$ and $h_{i+1}^{1}\in{\rm Hom}_{{\rm GMod}\hbox{-}S\hbox{-}S}(\overline{C},{\rm Hom}_{{\rm GMod}\hbox{-}k}(M,M))$. Then, the pair $h_{i+1}=(1_{M},h_{i+1}^{1}):M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}M$ is a morphism in ${\rm G}\mathbb{M}{\rm od}\hskip-1.5pt\hbox{\rm I}^{0}\hbox{-}{\cal B}$. From (4.8), we know that $h_{i+1}$ is an isomorphism in ${\rm G}\mathbb{M}{\rm od}\hskip-1.5pt\hbox{\rm I}^{0}\hbox{-}{\cal B}$.

Let $g_{i+1}:=h_{i+1}^{-1}:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}M$. Then, $g_{i+1}^{0}=1_{M}$ and, since $(g_{i+1}h_{i+1})^{1}=0$, we obtain for $c\in\overline{C}_{i}$ that $g_{i+1}^{1}(c)=-h_{i+1}^{1}(c)$. Then, by triangularity, we have for $c\in\overline{C}_{i}$,

[TABLE]

Since $e_{i}^{0}e_{i}^{1}(c)e_{i}^{0}=0$, choosing $f_{i}^{0}:=1_{M}-2e_{i}^{0}$ we obtain the equality $e^{1}_{i+1}(c)=(h_{i+1}e_{i}g_{i+1})^{1}(c)=0$, as we wanted. $\square$

From (1) and (2), we obtain, for $i\geq 1$ and $c\in\overline{C}_{i-1}$, the equality

[TABLE]

Indeed, this is clear for $i=1$. For $i\geq 2$, we have $(h_{i}\cdots h_{2}h_{1})^{0}=id_{M}$ and $h^{0}_{i+1}=id_{M}$. By assumption, $\overline{\mu}(c)=\sum_{t}c^{1}_{t}\otimes c^{2}_{t}$, with $c^{1}_{t},c^{2}_{t}\in\overline{C}_{i-2}$. Then, we obtain

[TABLE]

Then, we can consider the map $h^{1}:\overline{C}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\rm Hom}_{{\rm GMod}\hbox{-}k}(M,M)$ defined by

[TABLE]

Since $h^{1}$ is a morphism of graded $S$-$S$-bimodules, we can consider the isomorphism $h=(id_{M},h^{1}):M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}M$ in ${\rm G}\mathbb{M}{\rm od}\hskip-1.5pt\hbox{\rm I}^{0}\hbox{-}{\cal B}$.

The statement (4.7) gives us in this context that any morphism of the form $g=(0,g^{1}):M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}M$ satisfies that for any $i\geq 1$ there is some $n_{i}\in\mathbb{N}$ such that for $c\in\overline{C}_{i}$ and $n\geq n_{i}$ we have $(g^{n})^{1}(c)=0$. So, we have a well defined morphism $\sum_{n=0}^{\infty}g^{n}:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}M$. So, consider $g:=(0,-h^{1})$ and notice that $h=\mathbb{I}_{M}-g$, so $h^{-1}=\sum_{n=0}^{\infty}g^{n}=\mathbb{I}_{M}+\sum_{n=1}^{\infty}g^{n}$. In particular, $(h^{-1})^{1}=\sum_{n=1}^{\infty}[(0,-h^{1})^{n}]^{1}$.

Consider the iterated comultiplication $\overline{\mu}^{n}:\overline{C}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}\overline{C}^{\otimes(n+1)}$, defined recursively, for $n\geq 1$, by $\overline{\mu}^{0}=id_{\overline{C}}$ and $\overline{\mu}^{n}=(id_{\overline{C}}\otimes\overline{\mu}^{(n-1)})\overline{\mu}$.

Claim 2: Given any morphism of the form $g=(0,g^{1}):M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}M$ in ${\rm G}\mathbb{M}{\rm od}\hskip-1.5pt\hbox{\rm I}^{0}\hbox{-}{\cal B}$, the following holds

For $n\geq 1$ and $c\in\overline{C}$, we have

[TABLE] 2. 2.

Moreover, for $n\geq 2$, $i\geq 1$, and $c_{i}\in\overline{C}_{i}$, we have $c_{t}^{1},\ldots,c_{t}^{n}\in\overline{C}_{i-1}$.

Proof of the Claim 2: We proceed by induction. Item (1) holds trivially for $n=1$. So assume item (1) holds for $n$. Then, for $c\in\overline{C}$, we have

[TABLE]

where $\overline{\mu}(c)=\sum_{s}c_{s}^{1}\otimes c_{s}^{2}$ and, for each $s$, $\overline{\mu}^{(n-1)}(c_{s}^{2})=\sum_{t_{s}}c_{t_{s}}^{2}\otimes\cdots\otimes c_{t_{s}}^{n+1}$. Thus,

[TABLE]

Now, if $n\geq 2$, $i\geq 1$, and $c\in\overline{C}_{i}$, we have from the triangularity condition that $c_{s}^{1}\in\overline{C}_{i-1}$. We also have that $c_{t_{s}}^{2},\ldots,c_{t_{s}}^{n+1}\in\overline{C}_{i-1}$ by the induction hypothesis for item (2). The Claim 2 is proved. $\square$

Assume that $n\geq 1$, $c\in\overline{C}_{i-1}$, and $\overline{\mu}^{(n-1)}(c)=\sum_{t}c_{t}^{1}\otimes\cdots\otimes c_{t}^{n}$, then

[TABLE]

and

[TABLE]

Claim 3: Let $p,q:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}M$ be any morphisms in ${\rm G}\mathbb{M}{\rm od}\hskip-1.5pt\hbox{\rm I}^{0}\hbox{-}{\cal B}$. Then,

Given a family of morphisms $\{p_{i}:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}M\}_{i\geq 1}$ in ${\rm G}\mathbb{M}{\rm od}\hskip-1.5pt\hbox{\rm I}^{0}\hbox{-}{\cal B}$ such that $p_{i}^{0}=p^{0}$ and $p^{1}_{i}(c)=p^{1}(c)$, for $i\geq 1$ and $c\in\overline{C}_{i-1}$, we have

[TABLE] 2. 2.

Given a family of morphisms $\{q_{i}:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}M\}_{i\geq 1}$ in ${\rm G}\mathbb{M}{\rm od}\hskip-1.5pt\hbox{\rm I}^{0}\hbox{-}{\cal B}$ such that $q_{i}^{0}=q^{0}$ and $q^{1}_{i}(c)=q^{1}(c)$, for $i\geq 1$ and $c\in\overline{C}_{i-1}$, we have

[TABLE]

Proof of Claim 3: (1): For $c\in\overline{C}_{i-1}$, we have $\overline{\mu}(c)=\sum_{t}c_{t}^{1}\otimes c_{t}^{2}$, with $c_{t}^{1},c_{t}^{2}\in\overline{C}_{i-1}$, so

[TABLE]

The proof of (2) is similar. $\square$

Then, for $c\in\overline{C}$, say with $c\in\overline{C}_{i-1}$, we have

[TABLE]

Therefore, $(heh^{-1})^{1}=0$, as we wanted to show. ∎

Definition 6.6**.**

Let ${\cal B}=(C,\mu,\epsilon)$ be a graded $S$-bocs. A graded right ${\cal B}$-comodule $(M,\mu_{M})$ is called induced iff it is isomorphic to a right ${\cal B}$-comodule of the form ${\rm Ind}_{\cal B}(N)=(N\otimes C,id_{N}\otimes\mu)$ for some $N\in{\rm GMod}\hbox{-}S$. We denote by ${\rm GCoind}\hbox{-}{\cal B}$ the full subcategory of ${\rm GComod}\hbox{-}{\cal B}$ formed by the induced ${\cal B}$-comodules. So ${\rm GCoind}\hbox{-}{\cal B}$ is an additive subcategory of ${\rm GComod}\hbox{-}{\cal B}$.

If ${\cal B}=(C,\mu,\epsilon,\delta)$ is a differential graded $S$-bocs, a differential graded right ${\cal B}$-comodule $(M,\mu_{M},\delta_{M})$ is called induced iff $(M,\mu_{M})\in{\rm GCoind}\hbox{-}{\cal B}$. The full subcategory of ${\rm DGComod}\hbox{-}{\cal B}$ formed by the differential graded induced ${\cal B}$-comodules will be denoted by ${\rm DGCoind}\hbox{-}{\cal B}$. So, ${\rm DGCoind}\hbox{-}{\cal B}$ is an additive subcategory of ${\rm DGComod}\hbox{-}{\cal B}$.

Here again the superindex [math] in ${\rm GCoind}^{0}\hbox{-}{\cal B}$ and ${\rm DGCoind}^{0}\hbox{-}{\cal B}$ indicates the subcategory with all objects and only degree zero morphisms.

Proposition 6.7**.**

Let ${\cal B}=(C,\mu,\epsilon,\delta)$ be a triangular differential graded $S$-bocs. Then, in the categories ${\rm GMod}^{0}\hbox{-}{\cal B}$, ${\rm GCoind}^{0}\hbox{-}{\cal B}$, ${\rm TGMod}^{0}\hbox{-}{\cal B}$, and ${\rm DGCoind}^{0}\hbox{-}{\cal B}$ idempotents split.

Proof.

From (6.5), idempotents split in ${\rm G}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B}$. But ${\rm GMod}^{0}\hbox{-}{\cal B}$ is equivalent to this category, so idempotents split in ${\rm GMod}^{0}\hbox{-}{\cal B}$. By (6.3), idempotents split in ${\rm TGMod}^{0}\hbox{-}{\cal B}$,

By (3.4), we know that the categories ${\rm GMod}^{0}\hbox{-}{\cal B}$ and ${\rm GCoind}^{0}\hbox{-}{\cal B}$ are equivalent categories, and that ${\rm TGMod}^{0}\hbox{-}{\cal B}$ and ${\rm DGCoind}^{0}\hbox{-}{\cal B}$ are equivalent categories, so idempotents split in ${\rm GCoind}^{0}\hbox{-}{\cal B}$ and in ${\rm DGCoind}^{0}\hbox{-}{\cal B}$. ∎

Remark 6.8**.**

Given a triangular graded $S$-bocs ${\cal B}$, we have the equivalent categories ${\rm GMod}\hbox{-}{\cal B}\simeq{\rm G}\mathbb{M}{\rm od}\hbox{-}{\cal B}\simeq{\rm G}\mathbb{M}{\rm od}\hskip-1.5pt\hbox{\rm I}\hbox{-}{\cal B}$. Then, we have canonical embeddings ${\rm GMod}\hbox{-}S\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\rm G}\mathbb{M}{\rm od}\hbox{-}{\cal B}$ and ${\rm GMod}\hbox{-}S\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\rm G}\mathbb{M}{\rm od}\hskip-1.5pt\hbox{\rm I}\hbox{-}{\cal B}$ which we denote with the same symbol $L$, as in (6.1).

Notice that given $h:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ in ${\rm GMod}\hbox{-}S$, we have $L(h)=(h,0)$ in ${\rm G}\mathbb{M}{\rm od}\hbox{-}{\cal B}$ and in ${\rm G}\mathbb{M}{\rm od}\hskip-1.5pt\hbox{\rm I}\hbox{-}{\cal B}$.

The following statement can be proved as in [2](6.2).

Lemma 6.9**.**

Let ${\cal B}$ be a triangular graded $S$-bocs. Then, any exact sequence

[TABLE]

in ${\rm GMod}\hbox{-}S$ determines an exact pair $M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits^{\ (f^{0},0)\ }}E\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits^{\ (g^{0},0)\ }}N$ in ${\rm G}\mathbb{M}{\rm od}\hskip-1.5pt\hbox{\rm I}\hbox{-}{\cal B}$.

Lemma 6.10**.**

Let ${\cal B}$ be a triangular graded $S$-bocs. Then, for any morphism $f=(f^{0},f^{1}):M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ in ${\rm G}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B}$ the following holds.

If $f^{0}$ is surjective, there is an automorphism $h$ of $M$ such that $(f*h)^{1}=0$. 2. 2.

If $f^{0}$ is injective, there is an automorphism $h$ of $N$ such that $(h*f)^{1}=0$.

Proof.

We will prove only (1). Assume that $f^{0}:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ is surjective in ${\rm GMod}\hbox{-}S$. Since $S$ is semisimple, $f^{0}$ is a retraction. Choose a right inverse $t^{0}:N\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}M$ for $f^{0}$. Adopt the notation of (4.6), skip the star in the notation for the composition in ${\rm G}\mathbb{M}{\rm od}\hskip-1.5pt\hbox{\rm I}^{0}\hbox{-}{\cal B}$, and make $\overline{C}_{-1}=0$. We first show the following.

Claim: There is a sequence of isomorphisms $\cdots\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits^{h_{3}}}M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits^{h_{2}}}M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits^{h_{1}}}M$ such that, for each $i\geq 1$, we have

Each isomorphism has the form $h_{i}=(id_{M},h^{1}_{i})$; 2. 2.

$h^{1}_{i}(\overline{C}_{i-2})=0$; and 3. 3.

For $f_{i}:=fh_{1}h_{2}\cdots h_{i}$, we have $f_{i}^{1}(\overline{C}_{i-1})=0$.

Proof of the Claim. The inductive argument is essentially the same as the one given in [2](5.7), but we rephrase it here for the sake of the reader. At the base of the induction, we have the isomorphism $h_{1}=(id_{M},0):M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}M$ such that $h_{1}^{1}(\overline{C}_{-1})=0$ and the morphism $f_{1}=f$ is such that $f_{1}^{1}(\overline{C}_{0})=0$.

Assume that we have constructed the isomorphisms $h_{1},\ldots,h_{i}$ satisfying conditions 1–3 and define $h_{i+1}^{1}(c):=-t^{0}f_{i}^{1}(c)$, for $c\in\overline{C}$.

Clearly $h_{i+1}^{1}(\overline{C}_{i-1})=0$ and $h_{i+1}^{1}\in{\rm Hom}^{0}_{{\rm GMod}\hbox{-}S\hbox{-}S}(\overline{C},{\rm Hom}_{{\rm GMod}\hbox{-}k}(M,M))$. Then, the pair $h_{i+1}=(id_{M},h_{i+1}^{1}):M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}M$ is a morphism in ${\rm G}\mathbb{M}{\rm od}\hskip-1.5pt\hbox{\rm I}^{0}\hbox{-}{\cal B}$. From (4.8), we know that $h_{i+1}$ is an isomorphism in ${\rm G}\mathbb{M}{\rm od}\hskip-1.5pt\hbox{\rm I}^{0}\hbox{-}{\cal B}$. Moreover, for $c\in\overline{C}_{i}$, we have $\overline{\mu}(c)=\sum_{t}c_{t}^{1}\otimes c_{t}^{2}$, with $c_{t}^{1},c_{t}^{2}\in\overline{C}_{i-1}$. Notice also that $f_{i}^{0}=(fh_{1}\cdots h_{i})^{0}=f^{0}$ and $h_{i+1}^{0}=id_{M}$. Then, we have

[TABLE]

as we wanted. $\square$

From (1) and (2), we obtain, for $i\geq 1$ and $c\in\overline{C}_{i-1}$, the equality

[TABLE]

Indeed, this is clear for $i=1$. For $i\geq 2$, we have $(h_{1}h_{2}\cdots h_{i})^{0}=id_{M}$ and $h^{0}_{i+1}=id_{M}$. By assumption, $\overline{\mu}(c)=\sum_{t}c^{1}_{t}\otimes c^{2}_{t}$, with $c^{1}_{t},c^{2}_{t}\in\overline{C}_{i-2}$. Then, we obtain

[TABLE]

Then, we can consider the map $h^{1}:\overline{C}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\rm Hom}_{{\rm GMod}\hbox{-}k}(M,M)$ defined by

[TABLE]

Since $h^{1}$ is a morphism of graded $S$-$S$-bimodules, we can consider the isomorphism $h=(id_{M},h^{1}):M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}M$ in ${\rm G}\mathbb{M}{\rm od}\hskip-1.5pt\hbox{\rm I}^{0}\hbox{-}{\cal B}$. For $c\in\overline{C}$, say with $c\in\overline{C}_{i-1}$, we have

[TABLE]

∎

Proposition 6.11**.**

Let ${\cal B}$ be a triangular differential $S$-bocs. Denote by ${\cal E}_{0}$ the class of composable pairs $M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits^{f}}E\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits^{g}}N$ in ${\rm G}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B}$ such that $g*f=0$ and

[TABLE]

is an exact sequence in ${\rm GMod}\hbox{-}S$. Then we have:

The pair $({\rm G}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B},{\cal E}_{0})$ is an exact category. The class ${\cal E}_{0}$ consists of the split exact pairs in ${\rm G}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B}$. 2. 2.

A morphism $f:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}E$ in ${\rm G}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B}$ is an ${\cal E}_{0}$-inflation iff $f^{0}:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}E$ is injective. 3. 3.

A morphism $g:E\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ in ${\rm G}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B}$ is an ${\cal E}_{0}$-deflation iff $g^{0}:E\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ is surjective.

Proof.

By (6.5), we already know in the category ${\rm G}\mathbb{M}{\rm od}\hskip-1.5pt\hbox{\rm I}^{0}\hbox{-}{\cal B}$ idempotents split. Follow the argument of the proof of [2](6.6 and 6.7), using (6.9) and (6.10), to show that $({\rm G}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B},{\cal E}_{0})$ is an exact category. From (4.10), (4.8), and (6.9) we get that any composable pair in ${\cal E}_{0}$ is a split exact pair in ${\rm G}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B}$. ∎

7 The Frobenius category of twisted modules

Given a triangular differential graded $S$-bocs ${\cal B}=(C,\mu,\epsilon,\delta)$ we will describe a natural structure of a Frobenius category on ${\rm TG}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B}$, in the following sense.

Definition 7.1**.**

Let ${\cal A}$ be an additive $k$-category where idempotents split, and let ${\cal E}$ be an exact structure on ${\cal A}$. Then $({\cal A},{\cal E})$ is called a Frobenius category iff it has enough ${\cal E}$-projectives and enough ${\cal E}$-injectives and, moreover, the class of ${\cal E}$-projectives coincides with the class of ${\cal E}$-injectives.

Definition 7.2**.**

Consider the class ${\cal E}$ of composable morphisms in ${\rm TG}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B}$

[TABLE]

such that $g*f=0$ and the short sequence of first components

[TABLE]

is an exact sequence in ${\rm GMod}\hbox{-}S$. Equivalently, the composable pair

[TABLE]

is a split exact pair in ${\rm G}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B}$.

Lemma 7.3**.**

Any composable pair of morphisms in ${\rm TG}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B}$ of the form

[TABLE]

belongs to ${\cal E}$ and is an exact pair in ${\rm TG}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B}$.

Proof.

Clearly $M_{1}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits^{s}}M_{1}\oplus M_{2}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits^{p}}M_{2}$ is a split exact pair in ${\rm G}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B}$.

In order to show that $s$ is the kernel of $p$ in ${\rm TG}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B}$, take any morphism $t:(N,w)\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(M_{1}\oplus M_{2},v)$ in ${\rm TG}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B}$ such that $p*t=0$. Then, there is a morphism $t^{\prime}:N\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}M_{1}$ in ${\rm G}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B}$ such that $t=s*t^{\prime}$.

We have $\widehat{\delta}(t)=\widehat{\delta}(s*t^{\prime})=s*\widehat{\delta}(t^{\prime})$. Since $s$ is a morphism in ${\rm TG}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B}$, we have $\widehat{\delta}(s)+v*s-s*u_{1}=v*s-s*u_{1}$, so $v*s=s*u_{1}$. We also have that $\widehat{\delta}(t)+v*t-t*w=0$. Then,

[TABLE]

Since $s$ is a monomorphism in ${\rm G}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B}$, we obtain $\widehat{\delta}(t^{\prime})+u_{1}*t^{\prime}-t^{\prime}*w=0$. So $t^{\prime}:(N,w)\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(M_{1},u_{1})$ is a morphism in ${\rm TG}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B}$. Hence, $s$ is the kernel of $p$ in ${\rm TG}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B}$. The proof of the fact that the cokernel of $s$ is $p$ in ${\rm TG}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B}$ is dual. ∎

Lemma 7.4**.**

Let ${\cal B}$ be a triangular differential $S$-bocs. Then, we have:

A morphism $g:(M,u)\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(M_{2},u_{2})$ in ${\rm TG}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B}$ is an $\cal E$-deflation iff $g:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}M_{2}$ is a retraction in ${\rm G}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B}$. 2. 2.

A morphism $f:(M_{1},u_{1})\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(M,u)$ in ${\rm TG}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B}$ is an $\cal E$-inflation iff $f:M_{1}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}M$ is a section in ${\rm G}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B}$. 3. 3.

Every composable pair in ${\cal E}$ is an exact pair in ${\rm TG}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B}$.

Proof.

(1): Assume that $g:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}M_{2}$ is a retraction in ${\rm G}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B}$. Since in ${\rm G}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B}$ idempotents split, the retraction $g$ has a kernel and, moreover, there is a commutative diagram in ${\rm G}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B}$

[TABLE]

where $h:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}M_{1}\oplus M_{2}$ is an isomorphism. From (4.11), we know there is $v\in{\rm Hom}^{1}_{{\rm GMod}\hbox{-}{\cal B}}(M_{1}\oplus M_{2},M_{1}\oplus M_{2})$ such that $h:(M,u)\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(M_{1}\oplus M_{2},v)$ is an isomorphism in ${\rm TG}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B}$. Therefore, $g*h^{-1}:(M_{1}\oplus M_{2},v)\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(M_{2},u_{2})$ is a morphism in ${\rm TG}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B}$, so $\widehat{\delta}(g*h^{-1})+u_{2}*(g*h^{-1})-(g*h^{-1})*v=0$. Since $\widehat{\delta}(g*h^{-1})=\widehat{\delta}(p)=0$, we obtain $u_{2}*(g*h^{-1})=(g*h^{-1})*v$. This implies that $v=\begin{pmatrix}u_{1}&x\\ 0&u_{2}\end{pmatrix}$, where $u_{1}\in{\rm Hom}^{1}_{{\rm G}\mathbb{M}{\rm od}\hbox{-}{\cal B}}(M_{2},M_{2})$ and $x\in{\rm Hom}^{1}_{{\rm G}\mathbb{M}{\rm od}\hbox{-}{\cal B}}(M_{2},M_{1})$. Moreover, the equation $\widehat{\delta}(v)+v*v=0$ implies that $(M_{1},u_{1})\in{\rm TG}\mathbb{M}{\rm od}\hbox{-}{\cal B}$. Notice also that $s:(M_{1},u_{1})\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(M_{1}\oplus M_{2},v)$ is a morphism in ${\rm TG}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B}$, because

[TABLE]

Then, the composable pair $(M_{1},u_{1})\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits^{\ f\ }}(M,u)\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits^{\ g\ }}(M_{2},u_{2})$ in the category ${\rm TG}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B}$ belongs to ${\cal E}$ and $g:(M,u)\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(M_{2},u_{2})$ is an ${\cal E}$-deflation.

The proof of (2) is similar.

(3): Consider any composable pair $(L,w)\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits^{f}}(M,u)\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits^{g}}(M_{2},u_{2})$ in ${\cal E}$. Then, $g:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}M_{2}$ is a retraction in ${\rm G}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B}$, so we can apply the preceding argument to construct the following commutative diagram in ${\rm TG}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B}$

[TABLE]

where $h$ is an isomorphism. From (7.3), we know that the second row is an exact pair in ${\rm TG}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B}$. Then, since $p*h*f=0$, there is a morphism $t:(L,w)\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(M_{1},u_{1})$ in ${\rm TG}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B}$ such that $s*t=h*f$. If we consider the first components of the underlying diagram in ${\rm GMod}\hbox{-}S$, we have the following commutative diagram where the vertical arrows are isomorphisms

[TABLE]

Hence $t:(L,w)\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(M_{1},u_{1})$ is an isomorphism in ${\rm TG}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B}$. It follows that the composable pair $(L,w)\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits^{f}}(M,u)\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits^{g}}(M_{2},u_{2})$ is exact. ∎

Proposition 7.5**.**

Let ${\cal B}$ be a triangular $S$-bocs, then we have the following.

The pair $({\rm TG}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B},{\cal E})$ is an exact category. 2. 2.

A morphism $f:(M,u_{M})\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(E,u_{E})$ in ${\rm TG}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B}$ is an ${\cal E}$-inflation iff $f^{0}:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}E$ is injective. 3. 3.

A morphism $g:(E,u_{E})\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(N,u_{N})$ in ${\rm TG}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B}$ is an ${\cal E}$-deflation iff $g^{0}:E\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ is surjective.

Proof.

From the last section, we know that idempotents split in ${\rm TG}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B}$. The description of the ${\cal E}$-inflations and the ${\cal E}$-deflations follows from (7.4) and (6.11). The fact that ${\cal E}$ is a class of exact pairs closed under isomorphisms, follows from (7.4).

Consider a morphism $f:\underline{Z}^{\prime}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}\underline{Z}$ and a deflation $d:\underline{Y}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}\underline{Z}$ in ${\rm TG}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B}$. Consider the morphism $(f,d):\underline{Z}^{\prime}\bigoplus\underline{Y}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}\underline{Z}$ in ${\rm TG}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B}$. Since $d:\underline{Y}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}\underline{Z}$ is a deflation, the morphism $d:Y\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}Z$ is a retraction in ${\rm G}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B}$ and, therefore, so is $(f,d):Z^{\prime}\oplus Y\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}Z$. Then, $(f,d):\underline{Z}^{\prime}\bigoplus\underline{Y}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}\underline{Z}$ is a deflation in ${\rm TG}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B}$ and it appears in an exact pair of ${\cal E}$

[TABLE]

Therefore, we have a pull-back diagram in the category ${\rm TG}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B}$

[TABLE]

and an exact sequence in ${\rm GMod}\hbox{-}S$ given by the first components

[TABLE]

So we have the pullback-diagram in the category ${\rm GMod}\hbox{-}S$

[TABLE]

Since $d^{0}$ is surjective, then $\widehat{d}^{0}$ is surjective, thus $\widehat{d}:Y^{\prime}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}Z^{\prime}$ is a retraction, and $\hat{d}:\underline{Y}^{\prime}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}\underline{Z}^{\prime}$ is an ${\cal E}$-deflation.

The other requirements in the definition of an exact structure are easy to verify for ${\cal E}$ using (7.4). ∎

We shall see that ${\rm TGMod}^{0}\hbox{-}{\cal B}$ is a Frobenius category with the following additional structure.

Definition 7.6**.**

A Frobenius category $({\cal A},{\cal E})$ is called special iff there are an exact automorphism $T:{\cal A}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\cal A}$ and an endofunctor $J:{\cal A}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\cal A}$ such that

$J(M)$ is ${\cal E}$-projective, for any $M\in{\cal A}$. 2. 2.

There are natural transformations $\alpha:id_{\cal A}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}J$ and $\beta:J\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}T$ such that, for each $M\in{\cal A}$ the pair

[TABLE]

is an exact pair in ${\cal E}$.

Remark 7.7**.**

In the following lemma, recall that we have already defined, for each graded $S$-module $M$ its shifting $M[1]$. Thus $M[1]=\bigoplus_{i\in\mathbb{Z}}M[1]_{i}$, with $M[1]_{i}=M_{i+1}$, for $i\in\mathbb{Z}$. We use also the canonical morphism $\sigma_{M}:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}M[1]$ of degree $-1$ determined by the identity map on $M$. The isomorphism $\underline{\sigma}_{M}:=L(\sigma_{M})\in{\rm Hom}^{-1}_{{\rm GMod}\hbox{-}{\cal B}}(M,M[1])$ plays an important role in the following.

Lemma 7.8**.**

Let ${\cal B}$ be a triangular differential $S$-bocs. Then, we have:

For each $M\in{\rm GMod}\hbox{-}{\cal B}$, make $T(M)=M[1]$ and, for any morphism $f:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ in ${\rm GMod}\hbox{-}{\cal B}$, make

[TABLE]

Then, we have a degree preserving autofunctor $T:{\rm GMod}\hbox{-}{\cal B}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\rm GMod}\hbox{-}{\cal B}$ with inverse $T^{-1}$ given by $T^{-1}(M)=M[-1]$ and

[TABLE]

It is called the shifting autofunctor of ${\rm GMod}\hbox{-}{\cal B}$. 2. 2.

For each $(M,u)\in{\rm TGMod}\hbox{-}{\cal B}$, make $T(M,u)=(M[1],-u[1])$ and, for any morphism $f:(M,u)\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(N,v)$ in ${\rm TGMod}\hbox{-}{\cal B}$, make

[TABLE]

Then, we have a degree preserving autofunctor $T:{\rm TGMod}\hbox{-}{\cal B}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\rm TGMod}\hbox{-}{\cal B}$ with inverse $T^{-1}$ given by $T^{-1}(M,u)=(M[-1],-u[-1])$ and

[TABLE]

The functor $T$ is called the shifting autofunctor of ${\rm TGMod}\hbox{-}{\cal B}$. 3. 3.

Furthermore, the functor $T:({\rm TGMod}^{0}\hbox{-}{\cal B},{\cal E})\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}({\rm TGMod}^{0}\hbox{-}{\cal B},{\cal E})$ is an automorphism of exact categories, that is

[TABLE]

As a consequence, the functors $T$ and $T^{-1}$ preserve the clases of ${\cal E}$-injectives and ${\cal E}$-projectives.

Proof.

The proof of (1) is easy. In order to prove (2), notice that given $h:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ in ${\rm GMod}\hbox{-}{\cal B}$, using that $\widehat{\delta}(\underline{\sigma}_{M})=0$ and Leibniz rule, we have

[TABLE]

Then, given $(M,u)\in{\rm TGMod}\hbox{-}{\cal B}$ we have $\widehat{\delta}(u)+u*u=0$. Thus,

[TABLE]

and $(M[1],-u[1])\in{\rm TGMod}\hbox{-}{\cal B}$. Moreover, given a morphism $f:(M,u)\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(N,v)$ in ${\rm TGMod}\hbox{-}{\cal B}$ we have

[TABLE]

so $f[1]:(M[1],-u[1])\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(N[1],-v[1])$ is a morphism in ${\rm TGMod}\hbox{-}{\cal B}$. Now, it is clear that $T$ is an endofunctor of ${\rm TGMod}\hbox{-}{\cal B}$. It is easy to see that $T$ is an autofunctor with inverse functor $T^{-1}$.

(3): Consider a composable pair $(M,u)\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits^{f}}(E,v)\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits^{g}}(N,w)$ in the category ${\rm TGMod}^{0}\hbox{-}{\cal B}$ and its image

[TABLE]

under the functor $T$. Clearly $g*f=0$ iff $g[1]*f[1]=0$. Notice that for any morphism $h:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ in ${\rm GMod}\hbox{-}S$, the first component $L(h)^{0}$ of $L(h)$ is precisely $h$. So, $\underline{\sigma}_{M}^{0}=\sigma_{M}$, for any $M\in{\rm GMod}\hbox{-}S$. Hence, we have $(f[1])^{0}=(\underline{\sigma}_{E}*f*\underline{\sigma}_{M}^{-1})^{0}=\underline{\sigma}^{0}_{E}f^{0}(\underline{\sigma}^{0}_{M})^{-1}=\sigma_{E}f^{0}\sigma_{M}^{-1}$ and similarly for $g$. So we have the commutative diagram in ${\rm GMod}\hbox{-}S$

[TABLE]

where the vertical morphisms are isomorphisms. Hence, the first row is exact iff the second one is so. ∎

Proposition 7.9**.**

For $(M,u_{M})\in{\rm TGMod}\hbox{-}{\cal B}$, we write $u_{M[1]}:=-u_{M}[1]$, so we have $T(M,u_{M})=(M[1],u_{M[1]})$. There is a degree preserving endofunctor

[TABLE]

such that, for any $(M,u_{M})\in{\rm TGMod}\hbox{-}{\cal B}$ we have

[TABLE]

and, given a morphism $f:(M,u_{M})\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(N,u_{N})$ in ${\rm TGMod}\hbox{-}{\cal B}$, the morphism $J(f)$ is given by the matrix

[TABLE]

Proof.

Given $(M,u)\in{\rm TGMod}\hbox{-}{\cal B}$, we have

[TABLE]

[TABLE]

So, indeed, we have $J(M,u)\in{\rm TGMod}\hbox{-}{\cal B}$.

Given a morphism $f:(M,u_{M})\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(N,u_{N})$ in ${\rm TGMod}\hbox{-}{\cal B}$, we have

[TABLE]

[TABLE]

So, indeed, we have that $J(f):J(M,u_{M})\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}J(N,u_{N})$ is a morphism in ${\rm TGMod}\hbox{-}{\cal B}$. Since $T:{\rm GMod}\hbox{-}{\cal B}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\rm GMod}\hbox{-}{\cal B}$ is a $k$-functor, so is $J$. ∎

Proposition 7.10**.**

For $(M,u_{M}),(N,u_{N})\in{\rm TGMod}\hbox{-}{\cal B}$, we have natural isomorphisms

[TABLE]

and

[TABLE]

Any morphism of twisted graded ${\cal B}$-modules $f=(f_{1},f_{2})^{t}:(M,u_{M})\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}J(N,u_{N})$, with underlying codomain $N\oplus N[1]$ in ${\rm GMod}\hbox{-}{\cal B}$ is mapped on $\eta_{1}(f)=f_{1}$, and any morphism $g=(g_{1},g_{2}):J(M,u_{M})\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(N,u_{N})$ of twisted graded ${\cal B}$-modules, with undelying domain $M\oplus M[1]$ in ${\rm GMod}\hbox{-}{\cal B}$ is mapped on $\eta_{2}(g)=g_{2}$.

Proof.

(1): Let $f:(M,u_{M})\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}J(N,u_{N})$ be a homogeneous morphism of twisted graded ${\cal B}$-modules with degree [math]. Thus, $f=(f_{1},f_{2})^{t}:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N\oplus N[1]$ is a morphism in ${\rm GMod}\hbox{-}{\cal B}$ which satisfies the equation

[TABLE]

Equivalently, it satisfies the equations:

[TABLE]

From the equality $(a_{1})$, we obtain the following expression of $f_{2}$ in terms of $f_{1}$

[TABLE]

Thus, the linear map $\eta_{1}$ is injective.

Let us see that $\eta_{1}$ is surjective. For this, take $f_{1}\in{\rm Hom}^{0}_{{\rm GMod}\hbox{-}{\cal B}}(M,N)$ and define $f_{2}$ by the equality $(a_{1})$, so $f:=(f_{1},f_{2})^{t}\in{\rm Hom}^{0}_{{\rm GMod}\hbox{-}{\cal B}}(M,N\oplus N[1])$. Then,

[TABLE]

We have the following

[TABLE]

[TABLE]

and

[TABLE]

Then, we have that equation $(b_{1})$ holds and $\eta_{1}(f)=f_{1}$. So $\eta_{1}$ is an isomorphism, which is clearly a natural transformation.

(2) Let $g:J(M,u_{M})\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(N,u_{N})$ be a homogeneous morphism of twisted graded ${\cal B}$-modules with degree [math]. Thus, $g=(g_{1},g_{2}):M\oplus M[1]\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ is a morphism in ${\rm GMod}\hbox{-}{\cal B}$ which satisfies the equation

[TABLE]

Equivalently, it satisfies the equations:

[TABLE]

From the equality $(b_{2})$, we obtain the following expression of $g_{1}$ in terms of $g_{2}$

[TABLE]

Thus, the linear map $\eta_{2}$ is injective.

Let us see that $\eta_{2}$ is surjective. For this, take $g_{2}\in{\rm Hom}^{0}_{{\rm GMod}\hbox{-}{\cal B}}(M[1],N)$ and define $g_{1}$ by the equality $(b_{2})$, so $g:=(g_{1},g_{2})\in{\rm Hom}^{0}_{{\rm GMod}\hbox{-}{\cal B}}(M\oplus M[1],N)$. Then,

[TABLE]

We have the equalities

[TABLE]

[TABLE]

and

[TABLE]

Then, we have that equation $(a_{2})$ holds and $\eta_{2}(g)=g_{2}$. So $\eta_{2}$ is an isomorphism, which is clearly a natural transformation. ∎

Corollary 7.11**.**

For each $(M,u)\in{\rm TGMod}\hbox{-}{\cal B}$, the twisted graded ${\cal B}$-module $J(M,u)$ is ${\cal E}$-projective and ${\cal E}$-injective in the exact category $({\rm TGMod}^{0}\hbox{-}{\cal B},{\cal E})$.

Proof.

Let us write $\underline{M}=(M,u_{M})$ for the objects of ${\rm TGMod}\hbox{-}{\cal B}$. For each exact pair $\underline{M}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits^{f}}\underline{E}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits^{g}}\underline{N}$ in ${\cal E}$, we have a commutative diagram

[TABLE]

where the second row is exact with $g_{*}$ surjective and $f_{*}$ injective. Since $\eta_{2}$ is a natural isomorphism, the first row is exact with $g_{*}$ surjective and $f_{*}$ injective. So $J(\underline{L})$ is ${\cal E}$-projective.

The argument showing that $J(\underline{L})$ is ${\cal E}$-injective is similar, now using $\eta_{1}$. ∎

For the rest of this section, we consider the restriction functors

[TABLE]

denoted with the same symbols used for the functors $T$ and $J$ considered before.

Proposition 7.12**.**

There are natural transformations $\alpha:id_{{\rm TGMod}^{0}\hbox{-}{\cal B}}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}J$ and $\beta:J\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}T$ such that for each $\underline{M}\in{\rm TGMod}\hbox{-}{\cal B}$ we have an ${\cal E}$-conflation

[TABLE]

For $\underline{M}\in{\rm G}\mathbb{M}{\rm od}\hbox{-}{\cal B}$, these morphisms are given by

[TABLE]

Proof.

Consider the [math]-degree homogeneous morphisms of twisted ${\cal B}$-modules

[TABLE]

which are described explicitely by the formulas for the inverses of $\eta_{1}$ and $\eta_{2}$ given in the proof of (7.10). ∎

From (7.11) and (7.12), we immediately obtain the following.

Corollary 7.13**.**

The ${\cal E}$-projective and the ${\cal E}$-injective objects in the exact category $({\rm TG}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B},{\cal E})$ are the direct summands of the objects $J(\underline{M})$, for $\underline{M}\in{\rm TG}\mathbb{M}{\rm od}\hbox{-}{\cal B}$.

Proposition 7.14**.**

Let $f:\underline{M}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}\underline{N}$ be a morphism in ${\rm TG}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B}$. Then, $f$ is homotopically trivial iff it factors through an ${\cal E}$-projective twisted ${\cal B}$-module.

Proof.

Since the ${\cal E}$-projective twisted ${\cal B}$-modules coincide with the ${\cal E}$-injective twisted ${\cal B}$-modules, a morphism $f:\underline{M}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}\underline{N}$ factors through an ${\cal E}$-projective iff it factors through $\alpha_{\underline{M}}$. So we have to show that $f$ factors through $\alpha_{\underline{M}}$ iff it is homotopically trivial.

Suppose first that $f$ factors through $\alpha_{\underline{M}}$ Then there is $h=(h_{1},h_{2})\in{\rm Hom}^{0}_{{\rm TG}\mathbb{M}{\rm od}\hbox{-}{\cal B}}(J(\underline{M}),\underline{N})$ such that $f=h*\alpha_{\underline{M}}$. Then, $f=h_{1}$ and the first component of $h$ is given by

[TABLE]

Hence,

[TABLE]

with $h_{2}*\underline{\sigma}_{M}\in{\rm Hom}^{-1}_{{\rm G}\mathbb{M}{\rm od}\hbox{-}{\cal B}}(M,N)$. So, $f$ is homotopically trivial.

Assume now that $f:\underline{M}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}\underline{N}$ is homotopically trivial and take a morphism $g\in{\rm Hom}^{-1}_{{\rm G}\mathbb{M}{\rm od}\hbox{-}{\cal B}}(M,N)$ such that $f=\widehat{\delta}(g)+u_{N}*g+g*u_{M}$. Now, consider the morphism $h_{2}:=g*\underline{\sigma}_{M}^{-1}\in{\rm Hom}^{0}_{{\rm G}\mathbb{M}{\rm od}\hbox{-}{\cal B}}(M[1],N)$. Then, we have

[TABLE]

Then, $h=(f,h_{2})\in{\rm Hom}^{0}_{{\rm TG}\mathbb{M}{\rm od}\hbox{-}{\cal B}}(J(\underline{M}),\underline{N})$ satisfies $f=h*\alpha_{\underline{M}}$. ∎

Definition 7.15**.**

Let ${\cal B}$ be a triangular differential graded $S$-bocs. Then, a sequence of morphisms

[TABLE]

in the homotopic category $\underline{{\rm TGMod}}^{0}\hbox{-}{\cal B}$ such that

[TABLE]

is an exact pair in ${\cal E}$ and we have a commutative diagram of the form

[TABLE]

in ${\rm TGMod}^{0}\hbox{-}{\cal B}$ is called a canonical triangle in $\underline{{\rm TGMod}}^{0}\hbox{-}{\cal B}$. Notice that, in this case, we have $\xi_{M}h=\xi$. Now, consider the class ${\cal T}$ of sequences of morphisms

[TABLE]

in $\underline{{\rm TGMod}}^{0}\hbox{-}{\cal B}$ which are isomorphic to canonical triangles. That is, there is a canonical triangle $M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits^{\underline{f}}}E\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits^{\underline{g}}}N\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits^{\underline{h}}}TM$ and isomorphisms $a,b,c$ such that the following diagram commutes

[TABLE]

The elements of ${\cal T}$ are called the triangles of $\underline{{\rm TGMod}}^{0}\hbox{-}{\cal B}$.

The following result follows from the general theory for special Frobenius categories, see [5] and [1].

Theorem 7.16**.**

Let ${\cal B}$ be a triangular differential graded $S$-bocs. Then the category ${\rm TGMod}^{0}\hbox{-}{\cal B}$ is a special Frobenius category with the automorphism $T$ of (7.8) and the endofunctor $J$ described in (7.9), both restricted to ${\rm TGMod}^{0}\hbox{-}{\cal B}$.

The stable category $\underline{{\rm TGMod}}^{0}\hbox{-}{\cal B}$ with the automorphism $T$ and the class of triangles ${\cal T}$ defined above is a triangulated category.

8 Quasi-isomorphisms of twisted modules

Before, the proof of our theorem (8.3), we fix some useful notation.

Remark 8.1**.**

Given a graded differential $S$-bocs ${\cal B}=(C,\mu,\epsilon,\delta)$, the equivalence of categories $F:{\rm GMod}\hbox{-}{\cal B}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\rm G}\mathbb{M}{\rm od}\hbox{-}{\cal B}$ described in (4.4) permits to transfer the differential $\widehat{\delta}$ of the category ${\rm GMod}\hbox{-}{\cal B}$ to a differential $\widehat{\delta}$ on the graded category ${\rm G}\mathbb{M}{\rm od}\hskip-1.5pt\hbox{\rm I}\hbox{-}{\cal B}$.

Thus, given a homogeneous morphism $f=(f^{0},f^{1}):M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ in ${\rm G}\mathbb{M}{\rm od}\hskip-1.5pt\hbox{\rm I}\hbox{-}{\cal B}$, we have $\widehat{\delta}(f)=(0,\widehat{\delta}(f)^{1})$, where for any homogeneous elements $m\in M$ and $c\in\overline{C}$, we have

[TABLE]

If $g:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ is a morphism in ${\rm GMod}\hbox{-}{\cal B}$, then $\widehat{\delta}(F(g))=F(\widehat{\delta}(g))$. So $F$ determines an equivalence of the differential graded categories ${\rm GMod}\hbox{-}{\cal B}$ and ${\rm G}\mathbb{M}{\rm od}\hskip-1.5pt\hbox{\rm I}\hbox{-}{\cal B}$.

Then, the category of twisted ${\cal B}$-modules ${\rm TG}\mathbb{M}{\rm od}\hskip-1.5pt\hbox{\rm I}\hbox{-}{\cal B}$ is formed by the pairs $\underline{M}=(M,u_{M})$, with $u_{M}\in{\rm Hom}^{1}_{{\rm G}\mathbb{M}{\rm od}\hskip-1.5pt\hbox{\scriptsize\rm I}\hbox{-}{\cal B}}(M,N)$ such that $\widehat{\delta}(u_{M})+u_{M}*u_{M}=0$.

A homogeneous morphism $f:\underline{M}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}\underline{N}$ of degree $d$ in ${\rm TG}\mathbb{M}{\rm od}\hskip-1.5pt\hbox{\rm I}\hbox{-}{\cal B}$ is a homogeneous morphism $f:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ of degree $d$ in ${\rm G}\mathbb{M}{\rm od}\hskip-1.5pt\hbox{\rm I}\hbox{-}{\cal B}$ such that the equality $\widehat{\delta}(f)+u_{N}*f-(-1)^{d}f*u_{M}=0$ holds.

A homogeneous morphism $f:\underline{M}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}\underline{N}$ in ${\rm TG}\mathbb{M}{\rm od}\hskip-1.5pt\hbox{\rm I}^{0}\hbox{-}{\cal B}$ is called homotopically trivial iff there is a homogeneous morphism $h:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ in ${\rm G}\mathbb{M}{\rm od}\hskip-1.5pt\hbox{\rm I}\hbox{-}{\cal B}$ of degree $-1$ such that $f=\widehat{\delta}(h)+u_{N}*h+h*u_{M}$.

For the sake of notational simplicity, from now on, given $f=(f^{0},f^{1}):M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ and $g=(g^{0},g^{1}):N\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}L$ in ${\rm G}\mathbb{M}{\rm od}\hskip-1.5pt\hbox{\rm I}\hbox{-}{\cal B}$, we write

[TABLE]

[TABLE]

and $g^{1}\cdot f^{1}:=(g*f)^{1}-g^{0}f^{1}-g^{1}f^{0}:\overline{C}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\rm Hom}_{{\rm GMod}\hbox{-}k}(M,L)$. Thus, the maps $g^{0}f^{1}$, $g^{1}f^{0}$ and $g^{1}\cdot f^{1}$ are morphisms of graded $S$-$S$-bimodules. Then, as mentioned in (6.4), there is a formula to compute $g^{1}\cdot f^{1}$. Namely, given $c\in\overline{C}$ with $\overline{\mu}(c)=\sum_{s}c^{1}_{s}\otimes c_{s}^{2}$, with $c^{1}_{s},c^{2}_{s}\in\overline{C}$, we have

[TABLE]

Therefore, $(g*f)^{0}=g^{0}f^{0}$ and $(g*f)^{1}=g^{0}f^{1}+g^{1}f^{0}+g^{1}\cdot f^{1}$.

Then, the equality $\widehat{\delta}(u_{M})+u_{M}*u_{M}=0$ is equivalent to

[TABLE]

A homogeneous morphism $f:\underline{M}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}\underline{N}$ of degree [math] in ${\rm TG}\mathbb{M}{\rm od}\hskip-1.5pt\hbox{\rm I}\hbox{-}{\cal B}$ is a morphism $f=(f^{0},f^{1}):M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ of degree [math] in ${\rm G}\mathbb{M}{\rm od}\hskip-1.5pt\hbox{\rm I}\hbox{-}{\cal B}$ such that

[TABLE]

In particular, $f^{0}:(M,u_{M}^{0})\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(N,u^{0}_{N})$ is a morphism of complexes of right $S$-modules.

For the proof of the following result, it is convenient to extend the preceding notations as follows.

Remark 8.2**.**

Let ${\cal B}=(C,\mu,\epsilon,\delta)$ be a triangular differential graded $S$-bocs and consider its triangular filtration $0=\overline{C}_{0}\subseteq\overline{C}_{1}\subseteq\cdots\subseteq\overline{C}_{i}\subseteq\overline{C}_{i+1}\subseteq\cdots$

For any $M,N,L\in{\rm GMod}\hbox{-}S$, $f_{i}^{1}\in{\rm Hom}_{{\rm GMod}\hbox{-}S}(M\otimes_{S}\overline{C}_{i},N)$, and $g^{1}_{i}\in{\rm Hom}_{{\rm GMod}\hbox{-}S}(N\otimes_{S}\overline{C}_{i},L)$ we can consider the morphism of graded right $S$-modules $g_{i}^{1}*f_{i}^{1}\in{\rm Hom}_{{\rm GMod}\hbox{-}S}(M\otimes_{S}\overline{C}_{i+1},L)$ defined as the composition

[TABLE]

For each $i\geq 0$, we have the canonical isomorphism

[TABLE]

We can define the product morphism: $\eta_{i}(g^{1}_{i})\cdot\eta_{i}(f^{1}_{i}):=\eta_{i+1}(g^{1}_{i}*f^{1}_{i})$.

Then, given any two morphisms $f_{i}^{1}\in{\rm Hom}_{{\rm GMod}\hbox{-}S\hbox{-}S}(\overline{C}_{i},{\rm Hom}_{{\rm GMod}\hbox{-}k}(M,N))$ and $g_{i}^{1}\in{\rm Hom}_{{\rm GMod}\hbox{-}S\hbox{-}S}(\overline{C}_{i},{\rm Hom}_{{\rm GMod}\hbox{-}k}(N,L))$, we have their product morphism $g_{i}^{1}\cdot f_{i}^{1}\in{\rm Hom}_{{\rm GMod}\hbox{-}S\hbox{-}S}(\overline{C}_{i+1},{\rm Hom}_{{\rm GMod}\hbox{-}k}(M,L))$ which is computed on each element $c\in\overline{C}_{i+1}$ as

[TABLE]

Given $g^{0}\in{\rm Hom}_{{\rm GMod}\hbox{-}S}(N,L)$ and $f_{i}^{1}\in{\rm Hom}_{{\rm GMod}\hbox{-}S\hbox{-}S}(\overline{C}_{i},{\rm Hom}_{{\rm GMod}\hbox{-}k}(M,N))$, the morphism $g^{0}f_{i}^{1}\in{\rm Hom}_{{\rm GMod}\hbox{-}S\hbox{-}S}(\overline{C}_{i},{\rm Hom}_{k}(M,L))$ is defined by $(g^{0}f^{1}_{i})(c)=g^{0}f^{1}_{i}(c)$. Similarly, given morphisms $g_{i}^{1}\in{\rm Hom}_{{\rm GMod}\hbox{-}S\hbox{-}S}(\overline{C}_{i},{\rm Hom}_{{\rm GMod}\hbox{-}k}(N,L))$ and $f^{0}\in{\rm Hom}_{{\rm GMod}\hbox{-}S}(M,N)$, $g_{i}^{1}f^{0}\in{\rm Hom}_{{\rm GMod}\hbox{-}S\hbox{-}S}(\overline{C}_{i},{\rm Hom}_{{\rm GMod}\hbox{-}k}(M,L))$ is defined by $(g^{1}_{i}f^{0})(c)=g_{i}^{1}(c)f^{0}$.

The following formulas hold:

[TABLE]

If the morphism $h^{1}_{i+1}$ restricts to $h^{1}_{i}$ and the morphism $f^{1}_{i+1}$ restricts to $f^{1}_{i}$, from the coassociativity of the comultiplication $\overline{\mu}$, we have the associativity formula

[TABLE]

Since the bocs ${\cal B}$ is triangular, we have $\delta(\overline{C}_{i+1})\subseteq Z(\overline{C}_{i+1})$. Then, for a homogeneous element $\widehat{f}^{1}_{i}\in{\rm Hom}_{{\rm GMod}\hbox{-}S\hbox{-}S}(\overline{C}_{i}+Z(\overline{C}_{i+1}),{\rm Hom}_{{\rm GMod}\hbox{-}k}(M,N))$, we can define the morphism $\widehat{d}(\widehat{f}^{1}_{i})\in{\rm Hom}_{{\rm GMod}\hbox{-}S\hbox{-}S}(\overline{C}_{i+1},{\rm Hom}_{{\rm GMod}\hbox{-}k}(M,N))$ by

[TABLE]

for any $c\in\overline{C}_{i+1}$ and any homogeneous element $m\in M$.

Given a homogeneous $\widehat{g}^{1}_{i}\in{\rm Hom}_{{\rm GMod}\hbox{-}S\hbox{-}S}(\overline{C}_{i}+Z(\overline{C}_{i+1}),{\rm Hom}_{{\rm GMod}\hbox{-}k}(N,L))$, the following Leibniz formula for morphisms $\overline{C}_{i+1}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\rm Hom}_{{\rm GMod}\hbox{-}k}(M,L)$ holds

[TABLE]

where $f^{1}_{i}$ and $g^{1}_{i}$ denote the restrictions of $\widehat{f}^{1}_{i}$ and $\widehat{g}^{1}_{i}$ from their common domain $\overline{C}_{i}+Z(\overline{C}_{i+1})$ to $\overline{C}_{i}$; thus the morphism $g_{i}^{1}\cdot f_{i}^{1}$ is defined on $\overline{C}_{i+1}$, hence it is defined on $\overline{C}_{i}+Z(\overline{C}_{i+1})$; the morphisms $\widehat{d}(\widehat{g}^{1}_{i})$ and $\widehat{d}(\widehat{f}^{1}_{i})$ are defined on $\overline{C}_{i+1}$, hence on $\overline{C}_{i}$.

Theorem 8.3**.**

Assume that $S$ is a finite product of copies of the field $k$. Let $(M,u)\in{\rm TG}\mathbb{M}{\rm od}\hbox{-}{\cal B}$. Then, the twisted ${\cal B}$-module $(M,u)$ is homotopically trivial in ${\rm TG}\mathbb{M}{\rm od}\hskip-1.5pt\hbox{\rm I}^{0}\hbox{-}{\cal B}$ iff the complex of right $S$-modules $(M,u^{0})$ is acyclic, that is $H^{i}(M,u^{0})=0$, for all $i\in\mathbb{Z}$.

Proof.

If $(M,u)$ is homotopically trivial, then $\mathbb{I}_{M}=(id_{M},0)$ is homotopic to the zero map in ${\rm TG}\mathbb{M}{\rm od}\hskip-1.5pt\hbox{\rm I}^{0}\hbox{-}{\cal B}$. Then, there is a morphism $h\in{\rm Hom}^{-1}_{{\rm G}\mathbb{M}{\rm od}\hskip-1.5pt\hbox{\scriptsize\rm I}\hbox{-}{\cal B}}(M,M)$ such that $\mathbb{I}_{M}=\widehat{\delta}(h)+u*h+h*u$. Looking at the first components, we obtain $id_{M}=u^{0}h^{0}+h^{0}u^{0}$. So, $(M,u^{0})$ is a homotopically trivial complex of right $S$-modules, which implies that $H^{i}(M,u^{0})=0$, for all $i\in\mathbb{Z}$.

Now, assume that $(M,u^{0})$ is an acyclic complex of right $S$-modules. Then, for instance from [4](0.3), we know that $(M,u^{0})$ is homotopically trivial. So, there is a homogeneous morphism $h^{0}:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}M$ of right $S$-modules of degree $-1$ such that the following equality holds

[TABLE]

We want to show that $(M,u)$ is homotopically trivial, so we are looking for a homogeneous morphism of $S$-$S$-bimodules $h^{1}:\overline{C}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\rm Hom}_{{\rm GMod}\hbox{-}k}(M,M)$ of degree $-1$ such that $h=(h^{0},h^{1})$ satisfies $\widehat{\delta}(h)+u*h+h*u=\mathbb{I}_{M}$. That is such that the following equality holds

[TABLE]

Write $E_{M}:={\rm Hom}_{{\rm GMod}\hbox{-}k}(M,M)$ and make $h^{1}_{0}=0\in{\rm Hom}_{{\rm GMod}\hbox{-}S\hbox{-}S}^{-1}(\overline{C}_{0},E_{M})$. We will construct, for each $i\geq 0$, a morphism $h^{1}_{i+1}\in{\rm Hom}_{{\rm GMod}\hbox{-}S\hbox{-}S}^{-1}(\overline{C}_{i+1},E_{M})$ such that

[TABLE]

and $h^{1}_{i+1}$ restricts to $h^{1}_{i}$, for all $i$. Here, $u^{1}_{i}$ denotes the restriction of $u^{1}$ to $\overline{C}_{i}$. Once we evaluate $u_{i}^{1}$, we can skip the subindex: $u^{1}_{i}(c)=u^{1}(c)$, for $c\in\overline{C}_{i}$.

Once we have done this, we can define $h^{1}\in{\rm Hom}_{{\rm GMod}\hbox{-}S\hbox{-}S}^{-1}(\overline{C},E_{M})$, by $h^{1}(c)=h^{1}_{i}(c)$, whenever $c\in\overline{C}_{i}$. So $\Delta^{1}(c)=\Delta^{1}_{i}(c)=0$, for $c\in\overline{C}_{i+1}$. Hence $\Delta^{1}=0$ and we are done.

We require for this construction a special vector space basis $\mathbb{B}_{i+1}$ of $\overline{C}_{i+1}$ consisting of homogeneous elements. In order to describe this basis, we will consider, for each $i\geq 0$, decompositions of graded $S$-$S$-bimodules of the form

[TABLE]

where $\overline{C}_{i}+Z(\overline{C}_{i+1})=\overline{C}_{i}\oplus V_{i+1}$ and $V_{i+1}\subseteq Z(\overline{C}_{i+1})$. By assumption, we have a decomposition $1=\sum_{s=1}^{n}e_{s}$ of the unit element of the algebra $S$ as a sum of central primitive orthogonal idempotents. The special basis we are interested in has the form $\mathbb{B}_{i+1}=\bigcup_{s,t\in[1,n]}\mathbb{B}_{i+1}(s,t)$, where each subset $\mathbb{B}_{i+1}(s,t)$ is the basis of $e_{t}\overline{C}_{i+1}e_{s}$ defined recursively as follows. At the base $i=0$, we have $\mathbb{B}_{1}(s,t):=\mathbb{B}^{v}_{1}(s,t)\cup\mathbb{B}^{w}_{1}(s,t)$, where $\mathbb{B}^{v}_{1}(s,t)$ and $\mathbb{B}^{w}_{1}(s,t)$ are basis formed by homogeneous elements of $e_{t}V_{1}e_{s}$ and $e_{t}W_{1}e_{s}$, respectively. Once we have defined a basis $\mathbb{B}_{i}$ of $\overline{C}_{i}$, for $i\geq 1$, we define

[TABLE]

where $\mathbb{B}^{v}_{i+1}(s,t)$ and $\mathbb{B}^{w}_{i+1}(s,t)$ are basis consisting of homogeneous elements of $e_{t}V_{i+1}e_{s}$ and $e_{t}W_{i+1}e_{s}$, respectively

Step 1: The construction of $h_{1}^{1}:\overline{C}_{1}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}E_{M}$.

In order to define the homogeneous morphism of $S$-$S$-bimodules $h_{1}^{1}$ we want, it will be enough to give, for each homogeneous basic element $c\in\mathbb{B}_{1}(s,t)$, a homogeneous element $h_{1}^{1}(c)\in{\rm Hom}_{{\rm GMod}\hbox{-}k}(Me_{t},Me_{s})$ of degree $|c|-1$ satisfying the equation $\Delta^{1}_{0}(c)=0$.

Start with a basic element $c\in\mathbb{B}_{1}^{v}(s,t)$, so $c\in Z(\overline{C}_{1})$. Consider the homogeneous morphism $f_{0}(c):Me_{t}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}Me_{s}$ of degree $|c|$ given by $f_{0}(c):=u_{1}^{1}(c)h^{0}+h^{0}u_{1}^{1}(c)$. Since $id_{M}=u^{0}h^{0}+h^{0}u^{0}$, we have

[TABLE]

Since $\overline{\mu}(c)=0$ and $\overline{\delta}(c)=0$, from (8.1)(A), we get $u^{0}u^{1}(c)+u^{1}(c)u^{0}=0$. Therefore,

[TABLE]

In the following, given $N\in{\rm GMod}\hbox{-}k$, we denote by $\sigma^{[i]}:N\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N[i]$ the homogeneous morphism of degree $-i$ which acts as the identity on the underlying non-graded spaces.

Consider the morphism $\tau:=\sigma^{[|c|]}:Me_{s}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}Me_{s}[|c|]$. Then, the homogeneous morphism $\tau f_{0}(c):Me_{t}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}Me_{s}[|c|]$ of degree [math] satisfies

[TABLE]

Thus $\tau f_{0}(c)$ is a morphism of complexes $(Me_{t},u^{0}_{|Me_{t}})\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(Me_{s}[|c|],\tau(u^{0}_{|Me_{s}})\tau^{-1})$.

The complex $(Me_{t},u^{0}_{|Me_{t}})$ is acyclic, so it is homotopically trivial. Then, so is the morphism $-\tau f_{0}(c)$. Hence, there is a homogeneous morphism $\underline{h}_{0}^{1}(c)\in{\rm Hom}^{-1}_{{\rm GMod}\hbox{-}k}(Me_{t},Me_{s}[|c|])$ such that

[TABLE]

Then, we have the homogeneous morphism $\widehat{h}^{1}_{0}(c):=\tau^{-1}\underline{h}_{0}^{1}(c):Me_{t}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}Me_{s}$, with degree $|c|-1$, such that $-f_{0}(c)=u^{0}\widehat{h}_{0}^{1}(c)+\widehat{h}^{1}_{0}(c)u^{0}$. Hence,

[TABLE]

We have defined $\widehat{h}^{1}_{0}(c)$ for any $c\in\mathbb{B}_{1}^{v}$ such that the preceding equality holds. Then, we can consider the homogeneous morphism

[TABLE]

of degree $-1$ determined by the given values $\widehat{h}^{1}_{0}(c)$ on the basic elements $c\in\mathbb{B}^{v}_{1}$. Thus $\widehat{h}^{1}_{0}\in{\rm Hom}_{{\rm GMod}\hbox{-}S\hbox{-}S}^{-1}(Z(\overline{C}_{1}),E_{M})$ extends $h^{1}_{0}:\overline{C}_{0}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}E_{M}$ and satisfies $\widehat{\Delta}^{1}_{0}(c)=0$, for all $c\in Z(\overline{C}_{1})$.

Now, take an element $c\in\mathbb{B}_{1}^{w}(s,t)$, so $\overline{\mu}(c)=0$ and $\delta(c)\in Z(\overline{C}_{1})$. Consider the homogeneous morphism $g_{0}(c):Me_{t}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}Me_{s}$ of degree $|c|$ given by

[TABLE]

Write $\lambda(c):=u_{1}^{1}(c)h^{0}+h^{0}u_{1}^{1}(c)$ and $\gamma(c):=\widehat{d}(\widehat{h}_{0}^{1})(c)$. Since $id_{M}=u^{0}h^{0}+h^{0}u^{0}$, we have

[TABLE]

Since $\delta(c)\in Z(\overline{C}_{1})$, we have $\widehat{\Delta}_{0}^{1}(\delta(c))=0$. Then, for any homogeneous element $m\in Me_{t}$, we have

[TABLE]

Thus, $u^{0}\gamma(c)-\gamma(c)u^{0}=[\widehat{d}(u_{1}^{1})h^{0}-h^{0}\widehat{d}(u_{1}^{1})](c)$. Then, from (8.1)(A), we have

[TABLE]

Thus $\tau g_{0}(c)$ is a morphism of complexes $(Me_{t},u^{0}_{|Me_{t}})\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(Me_{s}[|c|]),\tau(u^{0}_{|Me_{s}})\tau^{-1}).$ Proceeding as before, we get a homogeneous morphism $\widetilde{h}_{0}^{1}(c):Me_{t}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}Me_{s}$ with degree $|c|-1$ such that

[TABLE]

Then, we have the equality

[TABLE]

We have the homogeneous morphism $\widetilde{h}^{1}_{0}:W_{1}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\rm Hom}_{{\rm GMod}\hbox{-}k}(M,M)$ of degree $-1$ determined by the given values $\widetilde{h}_{0}^{1}(c)$ on the basic elements $c\in\mathbb{B}^{w}_{1}$. Then, the morphisms $\widehat{h}_{0}^{1}$ and $\widetilde{h}_{0}^{1}$ determine a homogeneous morphism $h^{1}_{1}:\overline{C}_{1}=Z(\overline{C}_{1})\oplus W_{1}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}E_{M}$ of degree $-1$ such that the equation $\Delta^{1}_{0}(c)=0$ is satisfied for all $c\in\overline{C}_{1}$, because either $\widehat{\Delta}^{1}_{0}(c)=0$ or $\widetilde{\Delta}_{0}^{1}(c)$ hold on basic elements $c\in\mathbb{B}_{1}$. Clearly, $h^{1}_{1}$ extends $h^{1}_{0}$.

Step 2: The construction of $h_{i+1}^{1}:\overline{C}_{i+1}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}E_{M}$, from $h_{i}^{1}:\overline{C}_{i}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}E_{M}$.

Assume we have already defined the homogeneous morphism $h_{i}^{1}:\overline{C}_{i}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}E_{M}$ of degree $-1$ such that $\Delta^{1}_{i-1}(c)=0$ holds for all $c\in\overline{C}_{i}$.

Given a basic element $c\in\mathbb{B}^{v}_{i+1}(s,t)$, we consider the homogeneous morphism $f_{i}(c):Me_{t}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}Me_{s}$ of degree $|c|$ given by

[TABLE]

Write $\lambda(c):=u^{1}_{i+1}(c)h^{0}+h^{0}u_{i+1}^{1}(c)$ and $\rho(c)=(u^{1}_{i}\cdot h^{1}_{i})(c)+(h^{1}_{i}\cdot u_{i}^{1})(c)$. Then, from (8.1)(A), we have

[TABLE]

and, since $c\in Z(\overline{C}_{i+1})$, we obtain

[TABLE]

Moreover, we have

[TABLE]

Now, we have $\Delta^{1}_{i-1}=0$. That is the following equality of morphisms from $\overline{C}_{i}$ to $E_{M}$ holds

[TABLE]

Mutiplying the equation $\Delta^{1}_{i-1}=0$ on the right by $u_{i}^{1}$, we have the following equality of morphisms from $\overline{C}_{i+1}$ to $E_{M}$

[TABLE]

Evaluating at our fixed element $c$, we obtain

[TABLE]

Similarly, multiplying the equation $\Delta_{i-1}^{1}=0$ on the left by $u^{1}_{i}$, we have

[TABLE]

Evaluating at the element $c$, we get

[TABLE]

Then, since $u^{1}_{i}h^{0}\cdot u^{1}_{i}=u^{1}_{i}\cdot h^{0}u^{1}_{i}$, we have

[TABLE]

Moreover, from the associativity of (8.2), we have

[TABLE]

Since $h_{i}^{1},u_{i}^{1}$ are defined on $\overline{C}_{i}$, they are defined on $\overline{C}_{i-1}+Z(\overline{C}_{i})$, then the morphisms $\widehat{d}(h^{1}_{i})$ and $\widehat{d}(u^{1}_{i})$ are defined on $\overline{C}_{i}$. Moreover, the morphisms $h^{1}_{i}$ and $u^{1}_{i}$ are defined on $\overline{C}_{i}$, hence their product $h_{i}^{1}\cdot u^{1}_{i}$ is defined on $\overline{C}_{i+1}$, hence it is defined on $\overline{C}_{i}+Z(\overline{C}_{i+1})$. Therefore, by the Leibniz formula of (8.2), we have

[TABLE]

where the left equality is due to the fact that $c\in\mathbb{B}_{i+1}^{v}(s,t)\subseteq V_{i+1}\subseteq Z(\overline{C}_{i+1})$. Thus, $(\widehat{d}(h^{1}_{i})\cdot u^{1}_{i})(c)=(h^{1}_{i}\cdot\widehat{d}(u^{1}_{i}))(c)$ and, similarly, we have $(\widehat{d}(u^{1}_{i})\cdot h_{i}^{1})(c)=(u^{1}_{i}\cdot\widehat{d}(h^{1}_{i}))(c)$. Then, we have

[TABLE]

From (8.1)(A), we know that $\widehat{d}(u^{1}_{i})+u^{0}u^{1}_{i}+u^{1}_{i}u^{0}+u^{1}_{i-1}\cdot u^{1}_{i-1}=0$ on $\overline{C}_{i}$. It follows that $u^{0}f_{i}(c)-f_{i}(c)u^{0}=u^{0}\lambda(c)-\lambda(c)u^{0}+u^{0}\rho(c)-\rho(c)u^{0}=0$. As before, we have a morphism of complexes $-\tau f_{i}(c):(Me_{t},u^{0}_{|Me_{t}})\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(Me_{s}[|c|],\tau u^{0}_{|Me_{s}}\tau^{-1})$ which is homotopically trivial. As a consequence, there is a homogeneous morphism $\underline{h}_{i}^{1}(c)\in{\rm Hom}_{{\rm GMod}\hbox{-}k}^{-1}(Me_{t},Me_{s}[|c|])$ such that

[TABLE]

The morphism $\widehat{h}^{1}_{i}(c):=\tau^{-1}\underline{h}_{i}^{1}(c):Me_{t}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}Me_{s}$ is homogeneous of degree $|c|-1$ such that $-f_{i}(c)=u^{0}\widehat{h}_{i}^{1}(c)+\widehat{h}_{i}^{1}(c)u^{0}$. Hence,

[TABLE]

We have defined $\widehat{h}^{1}_{i}(c)$ for any $c\in\mathbb{B}_{i+1}^{v}$ such that the preceding equality holds. Then, we can consider the homogeneous morphism

[TABLE]

of degree $-1$ determined by the given values $\widehat{h}^{1}_{i}(c)$ on the basic elements $c\in\mathbb{B}^{v}_{i+1}$ and by $\widehat{h}^{1}_{i}(c):=h^{1}_{i}(c)$ on the basic elements $c\in\mathbb{B}_{i}$. Therefore, the morphism $\widehat{h}^{1}_{i}\in{\rm Hom}_{{\rm GMod}\hbox{-}S\hbox{-}S}^{-1}(\overline{C}_{i}+Z(\overline{C}_{i+1}),E_{M})$ extends $h^{1}_{i}:\overline{C}_{i}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}E_{M}$ and satisfies $\widehat{\Delta}^{1}_{i}(c)=0$, for all $c\in Z(\overline{C}_{i+1})$.

Now, take an element $c\in\mathbb{B}_{i+1}^{w}(s,t)$. Consider the homogeneous morphism $g_{i}(c):Me_{t}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}Me_{s}$ of degree $|c|$ given by

[TABLE]

Write $\lambda(c):=u^{1}_{i+1}(c)h^{0}+h^{0}u_{i+1}^{1}(c)$, $\rho(c)=(u^{1}_{i}\cdot h^{1}_{i})(c)+(h^{1}_{i}\cdot u_{i}^{1})(c)$ and $\gamma(c)=\widehat{d}(\widehat{h}^{1}_{i})(c)$ . Then, the same calculations used at the beginining of the preceding case show that the following two equalities hold

[TABLE]

and

[TABLE]

Applying again Leibniz formula of (8.2), we have

[TABLE]

and

[TABLE]

Thus,

[TABLE]

From (8.1)(A), we obtain

[TABLE]

Since $u^{0}\gamma(c)-\gamma(c)u^{0}=u^{0}\widehat{d}(\widehat{h}^{1}_{i})(c)-\widehat{d}(\widehat{h}^{1}_{i})(c)u^{0}$, we have

[TABLE]

Moreover, $\delta(c)\in Z(\overline{C}_{i+1})$, so $\widehat{\Delta}_{i}^{1}(\delta(c))=0$. Hence,

[TABLE]

Thus,

[TABLE]

Therefore, $u^{0}g_{i}(c)-g_{i}(c)u^{0}=0$ and $\tau g_{i}(c)$ is a morphism of complexes $(Me_{t},u^{0}_{|Me_{t}})\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(Me_{s}[|c|]),\tau(u^{0}_{|Me_{s}})\tau^{-1}).$ Proceeding as before, we get a homogeneous morphism $\widetilde{h}_{i}^{1}(c):Me_{t}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}Me_{s}$ with degree $|c|-1$ such that

[TABLE]

Then, we have the equality

[TABLE]

Consider the homogeneous morphism $\widetilde{h}^{1}_{i}:W_{i+1}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\rm Hom}_{{\rm GMod}\hbox{-}k}(M,M)$ of degree $-1$ determined by the given values $\widetilde{h}_{i}^{1}(c)$ on the basic elements $c\in\mathbb{B}^{w}_{i+1}$. Then, the morphisms $h^{1}_{i}$, $\widehat{h}_{i}^{1}$ and $\widetilde{h}_{i}^{1}$ determine a homogeneous morphism

[TABLE]

of degree $-1$ such that the equation $\Delta^{1}_{i}(c)=0$ is satisfied for all $c\in\overline{C}_{i+1}$, because either $\Delta^{1}_{i-1}(c)=0$, $\widehat{\Delta}^{1}_{i}(c)=0$ or $\widetilde{\Delta}_{i}^{1}(c)=0$ hold on basic elements $c\in\mathbb{B}_{i+1}$. Clearly, $h^{1}_{i+1}$ extends $h^{1}_{i}$. ∎

Theorem 8.4**.**

Assume that $S$ is a finite product of copies of the field $k$. Let ${\cal B}$ be a triangular differential graded $S$-bocs and $f=(f^{0},f^{1}):(M,u_{M})\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(N,u_{N})$ a morphism in ${\rm TG}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B}$. Then $f$ is a homotopy equivalence iff the morphism of complexes of right $S$-modules $f^{0}:(M,u^{0}_{M})\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(N,u^{0}_{N})$ is a quasi-isomorphism.

Proof.

Consider the homotopy category $\underline{{\rm TG}\mathbb{M}{\rm od}}^{0}\hbox{-}{\cal B}$ with its triangulated structure as in (7.16). Then, there is a triangle of the form

[TABLE]

of $\underline{{\rm TG}\mathbb{M}{\rm od}}^{0}\hbox{-}{\cal B}$. By definition of this triangular structure, the preceding triangle is isomorphic to a triangle of the form

[TABLE]

where

[TABLE]

is an exact pair in the exact structure ${\cal E}$ of ${\rm TG}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B}$. Consider an isomorphism of triangles

[TABLE]

Then, the morphisms $s=(s^{0},s^{1}):(M,u_{M})\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(K,u_{K})$ and $t=(t^{0},t^{1}):(N,u_{N})\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(E,u_{E})$ are homotopy equivalences such that $\phi s$ is homotopic to $tf$. Therefore, the morphisms of complexes $\phi^{0}s^{0}$ and $t^{0}f^{0}$ from $(M,u^{0}_{M})$ to $(E,u^{0}_{E})$ are homotopic. As a consequence

[TABLE]

Since $H^{i}(s^{0})$ and $H^{i}(t^{0})$ are isomorphisms for all $i\in\mathbb{Z}$, then $H^{i}(f^{0})$ is an isomorphism if and only if $H^{i}(\phi^{0})$ is an isomorphism. Thus $f^{0}$ is a quasi-isomorphism iff $\phi^{0}$ is a quasi-isomorphism. Clearly, $f$ is a homotopy equivalence iff $\phi$ is so.

Since the exact pair $\xi$ belongs to ${\cal E}$, we have the exact sequence of graded right $S$-modules

[TABLE]

Then, we have the exact sequence of complexes of right $S$-modules

[TABLE]

In the homology long exact sequence associated to the preceding exact sequence of complexes, we see that $H^{i}(\phi^{0})$ is an isomorphism for all $i\in\mathbb{Z}$ iff $H^{i}(Q,u^{0}_{Q})=0$ for all $i\in\mathbb{Z}$. So $\phi^{0}$ is a quasi-isomorphism iff $(Q,u^{0}_{Q})$ is acyclic. Then, by (8.3), we obtain that $\phi^{0}$ is a quasi-isomorphism iff $(Q,u_{Q})$ is homotopically trivial in ${\rm TG}\mathbb{M}{\rm od}^{0}\hbox{-}{\cal B}$, which is equivalent to the fact that $\underline{\phi}:(K,u_{K})\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(E,u_{E})$ is an isomorphism in $\underline{{\rm TG}\mathbb{M}{\rm od}}^{0}\hbox{-}{\cal B}$. ∎

9 The Frobenius category of $A_{\infty}$-modules

Given a fixed $A_{\infty}$-algebra $A$, denote by ${\cal B}_{A}$ the differential tensor $S$-coalgebra (or differential tensor $S$-bocs) ${\cal B}_{A}=(T_{S}(A[1]),\mu,\epsilon,\delta)$ given by the bar construction. In order to describe precisely the connection of the category ${\rm Mod}_{\infty}\hbox{-}A$ of right $A_{\infty}$-modules over $A$ with the category ${\rm TG}\mathbb{M}{\rm od}\hbox{-}{\cal B}_{A}$ of twisted ${\cal B}_{A}$-modules, it is convenient to introduce the following categories.

Definition 9.1**.**

We will denote with ${\rm GMod}\hbox{-}A$ the following $k$-category. Its class of objects coincides with the class of objects of ${\rm GMod}\hbox{-}S$. Given two graded right $S$-modules $M$ and $N$, and $d\in\mathbb{Z}$, a homogeneous morphism $f:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ of degree $d$ in ${\rm GMod}\hbox{-}A$ is a collection of morphisms $f=\{f_{n}\}_{n\in\mathbb{N}}$, where each

[TABLE]

is a homogeneous morphism of graded right $S$-modules of degree $|f_{n}|=d+1-n$. We denote by ${\rm Hom}^{d}_{{\rm GMod}\hbox{-}A}(M,N)$ the space of homogeneous morphisms from $M$ to $N$ in ${\rm GMod}\hbox{-}A$, and we make

[TABLE]

If $f\in{\rm Hom}_{{\rm GMod}\hbox{-}A}(M,N)$ and $g\in{\rm Hom}_{{\rm GMod}\hbox{-}A}(N,L)$ are homogeneous morphisms, their composition

[TABLE]

is defined, for each $n\in\mathbb{N}$, by

[TABLE]

Given $M\in{\rm GMod}\hbox{-}A$, the identity morphism ${\rm I\hskip-2.0ptI}_{M}=\{h_{n}\}:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}M$ is given by $h_{1}=id_{M}$ and $h_{n}=0$, for all $n\geq 2$.

The category ${\rm GMod}\hbox{-}A$ is a graded category with differential $\delta_{\infty}$ defined for any homogeneous $f\in{\rm Hom}_{{\rm GMod}\hbox{-}A}(M,N)$ by

[TABLE]

for $n\geq 1$, thus $\delta_{\infty}(f)_{1}=0$.

A morphism $f=\{f_{n}\}_{n\in\mathbb{N}}:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ in ${\rm GMod}\hbox{-}A$ is called strict iff $f_{n}=0$, for all $n\geq 2$.

The fact that the preceding notions give rise indeed to a differential graded category is a consequence of the following.

Proposition 9.2**.**

Let ${\cal B}_{A}=(T_{S}(A[1]),\mu,\epsilon,\delta)$ denote the differential tensor $S$-bocs associated to the $A_{\infty}$-algebra $A$. Then there is an equivalence of differential graded $k$-categories

[TABLE]

Given $M\in{\rm GMod}\hbox{-}A$, by definition $\mathbb{G}(M)=M[1]$, so it acts as the usual translation on objects. Given two graded right $S$-modules $M$, $N$ and a homogeneous morphism $f=\{f_{n}\}_{n\in\mathbb{N}}:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ in ${\rm GMod}\hbox{-}A$ with degree $|f|=d$, we have the family of morphisms of right $S$-modules

[TABLE]

determined by the commutativity of the following diagrams

[TABLE]

where $n$ runs in $\mathbb{N}$ and $\zeta:M[1]\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}M[1]\otimes_{S}S$ is the canonical isomorphism. Each morphism $\hat{f}_{n}$ is homogeneous with degree $|\hat{f}_{n}|=d$. The family of maps $\{\hat{f}_{n}\}_{n\geq 0}$ extends to a homogeneous morphism of right $S$-modules of degree $d$

[TABLE]

By definition, we have $\mathbb{G}(f)=\hat{f}\in{\rm Hom}^{d}_{{\rm GMod}\hbox{-}{\cal B}_{A}}(M[1],N[1])$.

Proof.

Step 1: We have indeed a graded $k$-category ${\rm GMod}\hbox{-}A$ and $\mathbb{G}$ is an equivalence of graded $k$-categories.

Given composable morphisms $f:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ and $g:N\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}L$ in ${\rm GMod}\hbox{-}A$, we know that the composition $\mathbb{G}(g)*\mathbb{G}(f)=\hat{g}*\hat{f}:M[1]\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}L[1]$ in ${\rm GMod}\hbox{-}{\cal B}_{A}$ is given by the composition of maps

[TABLE]

where $C=T_{S}(A[1])$. Consider the restrictions $(\hat{g}*\hat{f})_{n}:M[1]\otimes A[1]^{\otimes n}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}L[1]$, for $n\geq 0$, and let us verify that the following diagrams commute

[TABLE]

where $n$ runs in $\mathbb{N}$. For this, it is convenient to write the restriction $\mu_{n}:A[1]^{\otimes n}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}C\otimes C$ of the bar comultiplication $\mu$ on $T_{S}(A[1])$ as follows

[TABLE]

Then, for $n\geq 1$, we have

[TABLE]

Write $(\hat{g}*\hat{f})_{n}=X_{n}^{(1)}+X_{n}^{(2)}+X_{n}^{(3)}$ with $X_{n}^{(1)}:=\sum_{\scriptsize\begin{matrix}r+s=n\\ r,s\geq 1;\end{matrix}}\hat{g}_{s}(\hat{f}_{r}\otimes id^{\otimes s})$, $X_{n}^{(2)}=\hat{g}_{0}(\zeta\hat{f}_{n})$ and $X_{n}^{(3)}=\hat{g}_{n}(\hat{f}_{0}\zeta\otimes id^{\otimes n})$. We want to show that, for $n\geq 1$, we have

[TABLE]

By definition of $g\circ f$, for $n\geq 1$, we have $(g\circ f)_{n}=Y_{n}^{(1)}+Y_{n}^{(2)}+Y_{n}^{(3)}$, where

[TABLE]

$Y_{n}^{(2)}=g_{1}(f_{n})$, $Y_{1}^{(3)}=0$ and, for $n\geq 2$, $Y_{n}^{(3)}=(-1)^{|f|(n-1)}g_{n}(f_{1}\otimes id^{\otimes(n-1)})$. We have

[TABLE]

We also have

[TABLE]

and

[TABLE]

Moreover, we have

[TABLE]

From the preceding calculations, we obtain that $\mathbb{G}(g\circ f)=\mathbb{G}(g)*\mathbb{G}(f)$. Take any $M\in{\rm GMod}\hbox{-}A$ and consider the morphism $h=\{h_{n}\}_{n\in\mathbb{N}}:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}M$ in ${\rm GMod}\hbox{-}A$ given by $h_{1}=id_{M}$ and $h_{n}=0$, for $n\geq 2$. Let us show that $\mathbb{G}(h)=\mathbb{I}_{M[1]}$, the identity morphism on the object $\mathbb{G}(M)\in{\rm GMod}\hbox{-}{\cal B}_{A}$.

Clearly, we have $\hat{h}_{0}=\sigma_{M}h_{1}(\zeta\sigma_{M})^{-1}=\sigma_{M}\sigma_{M}^{-1}\zeta^{-1}=\zeta^{-1}$ and, for $n\geq 1$, we have that $\hat{h}_{n}=\sigma_{M}h_{n+1}(\sigma_{M}\otimes\sigma^{\otimes n})^{-1}=0$. This means that the components $(\hat{h}^{0},\hat{h}^{1})$ of the morphism $\mathbb{G}(h)=\hat{h}$ are $(id_{M[1]},0)$, so $\mathbb{G}(h)=\mathbb{I}_{M[1]}$.

It is clear that the association $f\longmapsto\hat{f}$ is an isomorphism of graded vector spaces ${\rm Hom}_{{\rm GMod}\hbox{-}A}(M,N)\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\rm Hom}_{{\rm GMod}\hbox{-}{\cal B}_{A}}(M[1],N[1])$. It follows that ${\rm GMod}\hbox{-}A$ is a graded $k$-category and that $\mathbb{G}:{\rm GMod}\hbox{-}A\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\rm GMod}\hbox{-}{\cal B}_{A}$ is an equivalence of categories.

Step 2: We have indeed a differential $\delta_{\infty}$ on the graded category ${\rm GMod}\hbox{-}A$ and $\mathbb{G}$ is an equivalence of differential graded $k$-categories.

It will be enough to show that for any homogeneous morphism $f:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ in ${\rm GMod}\hbox{-}A$, we have

[TABLE]

Consider the restrictions $\widehat{\delta}(\hat{f})_{n}:M[1]\otimes A[1]^{\otimes n}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N[1]$, for $n\geq 0$. Let us verify that the following diagrams commute

[TABLE]

where $n$ runs in $\mathbb{N}$.

By definition, the differential $\widehat{\delta}$ of the category ${\rm GMod}\hbox{-}{\cal B}_{A}$ applied to the morphism $\mathbb{G}(f)=\hat{f}$ is

[TABLE]

Recall that the differential $\delta$ on the tensor $S$-coalgebra $T_{S}(A[1])$ satisfies that $\delta(S)=0$ and is determined by the morphisms of right $S$-modules

[TABLE]

given, for $n\geq 1$, by the following formula $\delta_{n}=\sum_{\scriptsize\begin{matrix}r+s+t=n\\ r,t\geq 0;s\geq 1\end{matrix}}id^{\otimes r}\otimes\hat{m}_{s}\otimes id^{\otimes t},$ where $\hat{m}_{s}$ is defined, for $s\geq 1$, by the following commutative square

[TABLE]

Here, if $r=0$ (or $t=0$) then $id^{\otimes r}$ (resp. $id^{\otimes t}$) is omitted.

Then, for $n\geq 1$, making $\Delta_{n}:=\widehat{\delta}(\hat{f})_{n}(\sigma_{M}\otimes\sigma^{\otimes n})$ we have

[TABLE]

Finally, we have

[TABLE]

This finishes the proof. ∎

Definition 9.3**.**

Given an $A_{\infty}$-algebra $A=(A,\{m_{n}\})$, we will denote by ${\rm TGMod}\hbox{-}A$ the following graded $k$-category. Its objects $(M,m^{M})$ are graded right $S$-modules $M$ equipped with a morphism $m^{M}:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}M$ in ${\rm GMod}\hbox{-}A$ with degree $|m^{M}|=1$ such that the following Maurer Cartan equation holds

[TABLE]

A homogeneous morphism $f:(M,m^{M})\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(N,m^{N})$ in ${\rm TGMod}\hbox{-}A$ with degree $|f|=d$ is a homogeneous morphism $f:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ is ${\rm GMod}\hbox{-}A$ with degree $d$ such that the following equation holds

[TABLE]

The composition in ${\rm TGMod}\hbox{-}A$ is the same composition of ${\rm GMod}\hbox{-}A$.

Again, the fact that the preceding notions give rise indeed to a graded $k$-category follows from the following statement, which is quite clear.

Proposition 9.4**.**

Let ${\cal B}_{A}$ denote the differential tensor $S$-bocs associated to the $A_{\infty}$-algebra $A$, then there is an equivalence of graded $k$-categories

[TABLE]

Given $(M,m^{M})\in{\rm GMod}\hbox{-}A$, by definition $\mathbb{G}(M,m^{M})=(M[1],\mathbb{G}(m^{M}))$.

Given a homogeneous morphism $f:(M,m^{M})\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(N,m^{N})$ in ${\rm TGMod}\hbox{-}A$ with degree $d$, we look at the underlying morphism $f:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ in ${\rm GMod}\hbox{-}A$ and apply $\mathbb{G}$ to obtain a morphism $\mathbb{G}(f):\mathbb{G}(M)\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}\mathbb{G}(N)$ in ${\rm GMod}\hbox{-}{\cal B}_{A}$, such that $\mathbb{G}(f):\mathbb{G}(M,m^{M})\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}\mathbb{G}(N,m^{N})$ is a morphism in ${\rm TGMod}\hbox{-}{\cal B}_{A}$.

Proposition 9.5**.**

Let ${\cal B}_{A}$ denote the differential tensor $S$-bocs associated to an $A_{\infty}$-algebra $A$. Then, we have:

The preceding requirements for a pair $(M,m^{M})\in{\rm TGMod}\hbox{-}A$ mean that $(M,m^{M})$ is a right $A_{\infty}$-module over $A$ in the usual sense of (1.3). 2. 2.

The preceding requirements for a family of morphisms $f=\{f_{n}\}_{n\in\mathbb{N}}$ to be a homogeneous morphism of degree $d$ from $(M,m^{M})$ to $(N,m^{N})$ in ${\rm TGMod}\hbox{-}A$ translate into the following. They mean that $f$ is a family $f=\{f_{n}\}_{n\in\mathbb{N}}$, where each

[TABLE]

is a homogeneous morphism of graded right $S$-modules of degree $|f_{n}|=d+1-n$ such that, for each $n\in\mathbb{N}$, the equality $\Sigma_{n}^{f+}+\Sigma_{n}^{f0}+\Sigma_{n}^{f-}=0$ holds, where

[TABLE]

[TABLE]

and

[TABLE]

The condition in case $n=1$ is equivalent to $m_{1}^{N}f_{1}=f_{1}m_{1}^{M}$, that is, to the requirement that the map $f_{1}:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ is a morphism of complexes of right $S$-modules.

Then, if we restrict to the case where $f$ has degree $|f|=0$, we have that $f:(M,m^{M})\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(N,m^{N})$ is a morphism of right $A_{\infty}$-modules in the usual sense of (1.3).

Therefore, the usual category ${\rm Mod}_{\infty}\hbox{-}A$ of right $A_{\infty}$-modules over the $A_{\infty}$-algebra $A$ is the subcategory ${\rm TGMod}^{0}\hbox{-}A$ of ${\rm TGMod}\hbox{-}A$ with the same objects and the homogeneous morphisms of degree [math] between them. Then, we can restrict the functor $\mathbb{G}$ to an equivalence functor $\mathbb{G}:{\rm Mod}_{\infty}\hbox{-}A\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\rm TGMod}^{0}\hbox{-}{\cal B}_{A}$.

Proof.

Item (1) follows from the observation that, for $n\geq 1$, we have

[TABLE]

Item (2) follows from the observation that, for $n\geq 1$, we have

[TABLE]

∎

Proposition 9.6**.**

Let $A$ be an $A_{\infty}$-algebra. Then, the category ${\rm GMod}^{0}\hbox{-}A$ is additive with split idempotents. Denote by ${\cal E}_{0}$ the class of composable morphisms $M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits^{f}}E\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits^{g}}N$ in ${\rm GMod}^{0}\hbox{-}A$ such that $g\circ f=0$ and

[TABLE]

is an exact sequence in ${\rm GMod}\hbox{-}S$. Then we have:

The pair $({\rm GMod}^{0}\hbox{-}A,{\cal E}_{0})$ is an exact category. The class ${\cal E}_{0}$ consists of the split exact pairs in ${\rm GMod}^{0}\hbox{-}A$. 2. 2.

A morphism $f:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}E$ in ${\rm GMod}^{0}\hbox{-}A$ is an ${\cal E}_{0}$-inflation iff $f_{1}:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}E$ is injective. 3. 3.

A morphism $g:E\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ in ${\rm GMod}^{0}\hbox{-}A$ is an ${\cal E}_{0}$-deflation iff $g_{1}:E\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ is surjective.

Proof.

Recall that the coalgebra ${\cal B}_{A}=(T_{S}(A[1]),\mu,\epsilon,\delta)$ associated by the bar construction to the $A_{\infty}$-algebra $A$ is a triangular differential graded $S$-bocs. Then, we have (6.7) and (6.11), and we can translate them to ${\rm GMod}^{0}\hbox{-}A$ with the equivalence $\mathbb{G}:{\rm GMod}^{0}\hbox{-}A\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\rm GMod}^{0}\hbox{-}{\cal B}_{A}$. ∎

Proposition 9.7**.**

Let $A$ be an $A_{\infty}$-algebra. Then, the category ${\rm Mod}_{\infty}\hbox{-}A$ is additive with split idempotents. Denote by ${\cal E}_{\infty}$ the class of composable morphisms in ${\rm Mod}_{\infty}\hbox{-}A$

[TABLE]

such that $g\circ f=0$ and the short sequence of first components

[TABLE]

is an exact sequence in ${\rm GMod}\hbox{-}S$. Then we have the following.

The pair $({\rm Mod}_{\infty}\hbox{-}A,{\cal E}_{\infty})$ is an exact category. The class ${\cal E}_{\infty}$ coincides with the class of composable morphisms such that when we forget the second components of the objects, we have a split exact pair in ${\rm GMod}^{0}\hbox{-}A$. 2. 2.

A morphism $f:(M,m^{M})\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(E,m^{E})$ in ${\rm Mod}_{\infty}\hbox{-}A$ is an ${\cal E}_{\infty}$-inflation iff $f_{1}:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}E$ is injective. 3. 3.

A morphism $g:(E,m^{E})\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(N,m^{N})$ in ${\rm Mod}_{\infty}\hbox{-}A$ is an ${\cal E}_{\infty}$-deflation iff $g_{1}:E\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ is surjective. 4. 4.

The functor $\mathbb{G}:({\rm Mod}_{\infty}\hbox{-}A,{\cal E}_{\infty})\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}({\rm TGMod}^{0}\hbox{-}{\cal B}_{A},{\cal E})$ is an equivalence of exact categories.

Proof.

As before, the $S$-coalgebra ${\cal B}_{A}$ associated by the bar construction to the $A_{\infty}$-algebra $A$ is a triangular differential graded $S$-bocs. Then, we can apply (6.7) to ${\rm TGMod}^{0}\hbox{-}{\cal B}_{A}$. Moreover, we can translate (7.5) to ${\rm Mod}_{\infty}\hbox{-}A={\rm TGMod}^{0}\hbox{-}A$ with the equivalence functor $\mathbb{G}:{\rm Mod}_{\infty}\hbox{-}A\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\rm TGMod}^{0}\hbox{-}{\cal B}_{A}$. ∎

Remark 9.8**.**

Given an $A_{\infty}$-algebra $A$, as we did in (7.7), we can associate to any given $M\in{\rm GMod}\hbox{-}A$ the homogeneous isomorphism $\underline{\sigma}_{M}:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}M[1]$ in ${\rm GMod}\hbox{-}A$ of degree $-1$ such that $\underline{\sigma}_{M}=\{\sigma_{n}\}_{n\in\mathbb{N}}$, where $\sigma_{1}:=\sigma_{M}:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}M[1]$ and, for $n\geq 2$, $\sigma_{n}:=0:M\otimes A^{\otimes(n-1)}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}M[1]$.

Lemma 9.9**.**

Let $A$ be an $A_{\infty}$-algebra. Then, we have:

For each $M\in{\rm GMod}\hbox{-}A$, make $T(M)=M[1]$ and, for any morphism $f:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ in ${\rm GMod}\hbox{-}A$, make

[TABLE]

Then, we have an autofunctor $T:{\rm GMod}\hbox{-}A\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\rm GMod}\hbox{-}A$ with inverse $T^{-1}$ given by $T^{-1}(M)=M[-1]$ and

[TABLE]

It is called the shifting autofunctor of ${\rm GMod}\hbox{-}A$. 2. 2.

For each $(M,m^{M})\in{\rm TGMod}\hbox{-}A$, make $T(M,m^{M})=(M[1],-m^{M}[1])$ and, for any morphism $f:(M,m^{M})\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(N,m^{N})$ in ${\rm GMod}\hbox{-}A$, make

[TABLE]

Then, we have an autofunctor $T:{\rm TGMod}\hbox{-}A\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\rm TGMod}\hbox{-}A$ with inverse $T^{-1}$ given by $T^{-1}(M,m^{M})=(M[-1],-m^{M}[-1])$ and

[TABLE]

It is called the shifting autofunctor of ${\rm TGMod}\hbox{-}A$. 3. 3.

The functor $\mathbb{G}:{\rm GMod}\hbox{-}A\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\rm GMod}\hbox{-}{\cal B}_{A}$ commutes with the corresponding shifting autofunctors, and the functor $\mathbb{G}:{\rm TGMod}\hbox{-}A\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\rm TGMod}\hbox{-}{\cal B}_{A}$ commutes with the corresponding shifting autofunctors. 4. 4.

Furthermore, the shifting functor $T:({\rm Mod}_{\infty}\hbox{-}A,{\cal E}_{\infty})\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}({\rm Mod}_{\infty}\hbox{-}A,{\cal E}_{\infty})$ is an automorphism of exact categories. As a consequence, $T$ and $T^{-1}$ preserve the classes of ${\cal E}_{\infty}$-injectives and ${\cal E}_{\infty}$-projectives.

Proof.

It follows from (7.8). ∎

Proposition 9.10**.**

For $(M,m^{M})\in{\rm TGMod}\hbox{-}A$, we write $m^{M[1]}:=-m^{M}[1]$, so we have $T(M,m^{M})=(M[1],m^{M[1]})$. There is an endofunctor

[TABLE]

such that, for any $(M,m^{M})\in{\rm TGMod}\hbox{-}A$ we have

[TABLE]

and, given a morphism $f:(M,m^{M})\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(N,m^{N})$ in ${\rm TGMod}\hbox{-}A$, the morphism $J(f)$ is given by the matrix

[TABLE]

The equivalence $\mathbb{G}:{\rm TGMod}\hbox{-}A\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\rm TGMod}\hbox{-}{\cal B}_{A}$ satisfies $\mathbb{G}J=J\mathbb{G}$.

Moreover, there are natural transformations $\alpha:id_{{\rm Mod}_{\infty}\hbox{-}A}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}J$ and $\beta:J\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}T$ of endofunctors of ${\rm Mod}_{\infty}\hbox{-}A$ such that, for each $\underline{M}=(M,m^{M})\in{\rm Mod}_{\infty}\hbox{-}A$, we have an ${\cal E}_{\infty}$-conflation

[TABLE]

For $\underline{M}\in{\rm Mod}_{\infty}\hbox{-}A$, the morphisms $\alpha_{\underline{M}}$ and $\beta_{\underline{M}}$ are strict morphisms with

[TABLE]

and

[TABLE]

Then, the category ${\rm Mod}_{\infty}\hbox{-}A$ has enough ${\cal E}_{\infty}$-projectives and enough ${\cal E}_{\infty}$-injectives and, moreover, the class of ${\cal E}_{\infty}$-projectives coincides with the class of ${\cal E}_{\infty}$-injectives. So, we obtain that ${\rm Mod}_{\infty}\hbox{-}A$ is a special Frobenius category.

Proof.

It follows from (7.9), and (7.12), which are translated into ${\rm Mod}_{\infty}\hbox{-}A$ with the help of the functor $\mathbb{G}$. Then, we use (9.7) and (9.9). ∎

Lemma 9.11**.**

Let $A$ be an $A_{\infty}$-algebra. Then, we have the following explicit descriptions for the shifting functor $T$ and the functor $J$.

Given $(M,m^{M})\in{\rm TGMod}\hbox{-}A$, its translate $(M,m^{M})[1]=(M[1],m^{M[1]})$ is given for each $n\in\mathbb{N}$, by

[TABLE] 2. 2.

Given a morphism $f:(M,m^{M})\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(N,m^{N})$ in ${\rm TGMod}\hbox{-}A$, its translate $f[1]:(M[1],m^{M[1]})\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(N[1],m^{N[1]})$ is given, for each $n\in\mathbb{N}$, by

[TABLE] 3. 3.

Given $(M,m^{M})\in{\rm TGMod}\hbox{-}A$, we have $J(M,m^{M})=(M\oplus M[1],m^{J(M)})$, where

[TABLE]

with

[TABLE] 4. 4.

Given a morphism $f:(M,m^{M})\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(N,m^{N})$ in ${\rm TGMod}\hbox{-}A$, its translate $J(f):(M\oplus M[1],m^{J(M)})\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(N\oplus N[1],m^{J(N)})$ is given, for each $n\in\mathbb{N}$, by

[TABLE]

with matrix

[TABLE]

Proof.

(1) and (2): It will be enough to show that for any morphism $f:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ in ${\rm GMod}\hbox{-}A$, its translate $f[1]:M[1]\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N[1]$ in ${\rm GMod}\hbox{-}A$ is given, for each $n\in\mathbb{N}$, by

[TABLE]

Indeed, from the definition of $f[1]$ and of the composition in ${\rm GMod}\hbox{-}A$, we have

[TABLE]

Items (3) and (4) clearly follow from (1) and (2). ∎

Remark 9.12**.**

Let $A$ be an $A_{\infty}$-algebra and consider a pair of morphisms $f,g\in{\rm Hom}_{{\rm Mod}_{\infty}\hbox{-}A}((M,m^{M}),(N,m^{N}))$. Then, there is a homotopy $h=\{h_{n}\}_{n\in\mathbb{N}}$ from $f$ to $g$, as in definition (1.4), iff there is a morphism $h\in{\rm Hom}^{-1}_{{\rm GMod}\hbox{-}A}(M,N)$ satisfying

[TABLE]

Indeed, the preceding formula translates into the usual formulas given in (1.4): For $n\geq 1$, we have $\delta_{\infty}(h)_{n}=H_{n}^{(3)}$, $(m^{N}\circ h)_{n}=H^{(1)}_{n}$, and $(h\circ m^{M})_{n}=H_{n}^{(2)}$.

Proposition 9.13**.**

Let $A$ be an $A_{\infty}$-algebra. Then, the functor

[TABLE]

preserves and reflects homotopies. Therefore, it induces an equivalence of these categories modulo homotopy

[TABLE]

Definition 9.14**.**

Let $A$ be an $A_{\infty}$-algebra. Then, a sequence of morphisms

[TABLE]

in the homotopy category $\underline{{\rm Mod}}_{\infty}\hbox{-}A$ such that

[TABLE]

is an exact pair in ${\cal E}_{\infty}$ and we have a commutative diagram of the form

[TABLE]

in ${\rm Mod}_{\infty}\hbox{-}A$ is called a canonical triangle in $\underline{{\rm Mod}}_{\infty}\hbox{-}A$. Now, consider the class ${\cal T}_{\infty}$ of sequences of morphisms

[TABLE]

in $\underline{{\rm Mod}}_{\infty}\hbox{-}A$ which are isomorphic to canonical triangles. That is, there is a canonical triangle $M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits^{\underline{f}}}E\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits^{\underline{g}}}N\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits^{\underline{h}}}TM$ and isomorphisms $a,b,c$ such that the following diagram commutes

[TABLE]

The elements of ${\cal T}_{\infty}$ are called the triangles of $\underline{{\rm Mod}}_{\infty}\hbox{-}A$.

As in (7.16), the next statement follows from the general result for special Frobenius categories.

Theorem 9.15**.**

Let $A$ be an $A_{\infty}$-algebra and consider its associated differential tensor $S$-bocs ${\cal B}_{A}$. The stable category $\underline{{\rm Mod}}_{\infty}\hbox{-}A$ with the automorphism $T$ and the class of triangles ${\cal T}_{\infty}$ defined above is a triangulated category. The functor $\mathbb{G}:{\rm Mod}_{\infty}\hbox{-}A\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\rm TGMod}^{0}\hbox{-}{\cal B}_{A}$ induces an equivalence of triangulated categories

[TABLE]

From the preceding theorem and (8.4), we immediately obtain the following important statement, see [6](5.2).

Theorem 9.16**.**

Assume that $S$ is a finite product of copies of the field $k$ and let $A$ be an $A_{\infty}$-algebra. Then, every quasi-isomorphism of $A_{\infty}$-modules is a homotopy equivalence, hence an invertible morphism in $\underline{{\rm Mod}}_{\infty}\hbox{-}A$.

10 Restriction functors for $A_{\infty}$-modules

In this section we show how the result (5.5) on restriction functors for triangular differential graded $S$-bocses translates into the corresponding result for restriction functors for $A_{\infty}$-algebras. We start by recalling the following link between homotopies in ${\rm Alg}_{\infty}$ and ${\rm DGCoalg}$, as defined in (1.5) and (5.1).

Lemma 10.1**.**

The functor $\Psi:{\rm Alg}_{\infty}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\rm DGCoalg}$ preserves and reflects null-homotopic morphisms.

For the sake of completeness, we notice that this follows, for instance, from the following statement.

Lemma 10.2**.**

Given morphisms of $A_{\infty}$-algebras $f,g:A\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}B$ and $d\in\mathbb{Z}$, we consider the set ${\cal H}^{d}_{\infty}(A,B)$ formed by the families $h=\{h_{n}\}_{n\in\mathbb{N}}$, such that, for each $n\in\mathbb{N}$, the map $h_{n}:A^{\otimes n}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}B$ is a homogeneous morphism of $S$-$S$-bimodules of degree $|h_{n}|=d+1-n$. Make $\phi:=\Psi(f)$ and $\psi=\Psi(g)$. Then,

We have a linear isomorphism

[TABLE]

constructed as follows. Given $h\in{\cal H}_{\infty}^{d}(A,B)$, we consider the morphism of $S$-$S$-bimodules $\widehat{h}:\overline{T}_{S}(A[1])\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}B[1]$ of degree $|\widehat{h}|=d$ determined by the commutativity of the squares

[TABLE]

Then, consider the $\overline{\Psi}(f)\hbox{-}\overline{\Psi}(g)$-coderivation $\overline{\Delta}(\widehat{h}):\overline{T}_{S}(A[1])\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}\overline{T}_{S}(B[1])$ given by $\widehat{h}$ and the universal property of the reduced tensor $S$-coalgebra $\overline{T}_{S}(A[1])$. Finally, extend $\overline{\Delta}(\widehat{h})$ to a $\Psi(f)\hbox{-}\Psi(g)$-coderivation of degree $d$

[TABLE] 2. 2.

If $h\in{\cal H}^{-1}_{\infty}(A,B)$, with the notation of (1.5), we have

[TABLE] 3. 3.

We have $\Psi(f)-\Psi(g)\in{\rm Coder}^{0}_{\phi,\psi}({\cal B}_{A},{\cal B}_{B})$. 4. 4.

For any $\phi\hbox{-}\psi$-coderivation $\xi\in{\rm Coder}^{-1}_{\phi,\psi}({\cal B}_{A},{\cal B}_{B})$, we have

[TABLE] 5. 5.

For any $h\in{\cal H}^{-1}_{\infty}(A,B)$, we have $\Delta(h^{\odot})=\Delta(h)^{\odot}$. 6. 6.

We have $\Delta(f-g)=\phi-\psi$. 7. 7.

The map $\Delta$ induces a bijection between the homotopy sets ${\cal H}(f,g)$ and ${\cal H}(\phi,\psi)$.

As a consequence, the morphisms $f$ and $g$ are homotopic in ${\rm Alg}_{\infty}$ if and only if the morphisms $\phi$ and $\psi$ are homotopic in ${\rm DGCoalg}$.

Proof.

The map ${\cal H}^{d}_{\infty}(A,B)\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\rm Hom}^{d}_{{\rm GMod}\hbox{-}S\hbox{-}S}(\overline{T}_{S}(A[1]),B[1])$ is clearly an isomorphism, and so is the map

[TABLE]

given by the universal property of the reduced tensor $S$-coalgebra. Recall that the inverse of $\overline{\Delta}$ maps each coderivation $\xi$ onto $p_{B}\xi$, where $p_{B}:\overline{T}_{S}(B[1])\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}B[1]$ is the projection. Finally, the extension $\overline{\Delta}(\widehat{h})\mapsto\Delta(h)$ is also bijective.

(2)–(4) are easy to show. In order to show (5), take $h\in{\cal H}^{-1}_{\infty}(A,B)$. Make $\overline{\Delta}(\widehat{h})^{\odot}:=\overline{\delta}_{B}\overline{\Delta}(\hat{h})+\overline{\Delta}(\hat{h})\overline{\delta}_{A}$. Notice that it will be enough to show the equality $\overline{\Delta}(\widehat{h^{\odot}})=\overline{\Delta}(\widehat{h})^{\odot}$. Since both terms are $\phi\hbox{-}\psi$-coderivations, it will be enough to show that $p_{B}\overline{\Delta}(\widehat{h^{\odot}})_{n}=p_{B}\overline{\Delta}(\widehat{h})^{\odot}_{n}$, for all $n\geq 1$. This means that

[TABLE]

We have $p_{B}(\overline{\delta}_{B}\overline{\Delta}(\hat{h})+\overline{\Delta}(\hat{h})\overline{\delta}_{A})_{n}=D_{n}+R_{n}$ where

[TABLE]

From explicit description of $\overline{\Delta}(\widehat{h})$, we have

[TABLE]

and

[TABLE]

We have

[TABLE]

Moreover, we have

[TABLE]

where

[TABLE]

Fix a vector $v=(i_{1},\ldots,i_{r},s,j_{1},\ldots,j_{t})$ and let us examine $Z_{n}(v)$. We have

[TABLE]

equals

[TABLE]

and

[TABLE]

where ${\rm sgn}^{\prime}=\sum_{u=1}^{r}i_{u}$. The last term equals

[TABLE]

where the sign was defined in (1.5). Hence, $\sigma_{B}^{-1}D_{n}\sigma_{A}^{\otimes n}=H_{f,g}(h)_{n}$

Therefore, for $n\geq 1$, we get $\sigma_{B}^{-1}(D_{n}+R_{n})\sigma_{A}^{\otimes n}=(h^{\odot})_{n},$ or, equivalently,

[TABLE]

(6): Since $\overline{\Psi}(f)-\overline{\Psi}(g)$ is a $\overline{\Psi}(f)\hbox{-}\overline{\Psi}(g)$-coderivation, from (1), in order to prove that $\overline{\Psi}(f)-\overline{\Psi}(g)=\overline{\Delta}(\widehat{f-g})$, we have to show that

[TABLE]

This is clear, because, for $n\in\mathbb{N}$, both terms coincide with $\widehat{f}_{n}-\widehat{g}_{n}$.

(7): If $h$ is a homotopy from $f$ to $g$, then $f-g=h^{\odot}$. Then, we have

[TABLE]

where $\xi=\overline{\Delta}(\widehat{h})$ is a homotopy from $\overline{\Psi}(f)$ to $\overline{\Psi}(g)$.

Conversely, if $\xi:\overline{C}_{A}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}\overline{C}_{B}$ is a homotopy from $\overline{\Psi}(f)$ to $\overline{\Psi}(g)$, we have $\xi=\overline{\Delta}(\widehat{h})$, for some $h\in{\cal H}_{\infty}^{-1}(A,B)$ and

[TABLE]

It follows that $f-h=h^{\odot}$, so $h$ is a homotopy from $f$ to $g$. ∎

Proposition 10.3**.**

Let $A$ and $B$ be $A_{\infty}$-algebras and consider a morphism $\phi:A\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}B$ of $A_{\infty}$-algebras. Then,

The morphism $\phi$ induces a restriction functor of differential graded categories

[TABLE]

defined as follows. Given $M\in{\rm GMod}\hbox{-}B$, the $A$-module $R_{\phi}(M)=M$. Given a morphism $f:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ in ${\rm GMod}\hbox{-}B$, the morphism $R_{\phi}(f)=\{R_{\phi}(f)_{n}\}_{n\in\mathbb{N}}:R_{\phi}(M)\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}R_{\phi}(N)$ in ${\rm GMod}\hbox{-}A$ is given, for $n\geq 2$, by

[TABLE]

and $R_{\phi}(f)_{1}=f_{1}$, where ${\rm sgn}(i_{1},\ldots,i_{r})$ is as in (1.1). 2. 2.

The morphism $\phi$ induces a restriction functor of graded categories

[TABLE]

If $(M,\{m_{n}^{M}\})\in{\rm TGMod}\hbox{-}B$, by definition, $R_{\phi}(M,m^{M})=(M,R_{\phi}(m^{M}))$. Given a morphism $f:(M,m^{M})\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(N,m^{N})$ in ${\rm TGMod}\hbox{-}B$, the morphism $R_{\phi}(f)=\{R_{\phi}(f)_{n}\}_{n\in\mathbb{N}}:R_{\phi}(M)\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}R_{\phi}(N)$ in ${\rm GMod}\hbox{-}A$ is in fact a morphism $R_{\phi}(M,m^{M})\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}R_{\phi}(N,m^{N})$ in ${\rm TGMod}\hbox{-}A$. 3. 3.

Let $\overline{{\cal B}}_{A}$ and $\overline{{\cal B}}_{B}$ denote the reduced tensor $S$-coalgebras associated to $A$ and $B$ respectively, and denote by ${\cal B}_{A}$ and ${\cal B}_{B}$ the corresponding tensor $S$-coalgebras. Then the morphism of $A_{\infty}$-algebras $\phi:A\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}B$ determines a morphism of differential $S$-coalgebras $\overline{\Psi}(\phi):\overline{{\cal B}}_{A}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}\overline{{\cal B}}_{B}$, which extends to a morphism $\Psi(\phi):{\cal B}_{A}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\cal B}_{B}$, as in (5.2). Then, we have commutative squares of functors

[TABLE]

where $\mathbb{G}_{A}$ and $\mathbb{G}_{B}$ are the functors defined in (9.2) and (9.4). 4. 4.

Moreover, $R_{\phi}$ maps null-homotopic morphisms of ${\rm Mod}_{\infty}\hbox{-}B$ onto null-homotopic morphisms of ${\rm Mod}_{\infty}\hbox{-}A$. So, it induces a functor $\underline{R}_{\phi}$ and a commutative square of functors

[TABLE]

Proof.

In order to prove (1), it is enough to show the commutativity of the first diagram in (3), because we already know that $R_{\Psi(\phi)}$ is a functor and $\mathbb{G}_{A}$ and $\mathbb{G}_{B}$ are equivalences, and they all preserve the differentials of the categories. Clearly, $\mathbb{G}_{A}R_{\phi}$ and $R_{\Psi(\phi)}\mathbb{G}_{B}$ coincide on objects. Take a morphism $f:M\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}N$ in ${\rm GMod}\hbox{-}B$ and let us show that $\mathbb{G}_{A}R_{\phi}(f)=R_{\Psi(\phi)}\mathbb{G}_{B}(f)$. For this we have to show that, for each $n\geq 0$, we have $\mathbb{G}_{A}(R_{\phi}(f))_{n}=R_{\Psi(\phi)}\mathbb{G}_{B}(f)_{n}$. Since $R_{\Psi(\phi)}\mathbb{G}_{B}(f)_{n}=R_{\Psi(\phi)}(\widehat{f})_{n}=\widehat{f}(id_{M[1]}\otimes\Psi(\phi))_{n}$, the former is equivalent to show that $\widehat{R_{\phi}(f)}_{n}=\widehat{f}(id_{M[1]}\otimes\Psi(\phi))_{n}$. For this we will show that the following diagrams commute

[TABLE]

where $n$ runs in $\mathbb{N}$ and $\zeta:M[1]\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}M[1]\otimes_{S}S$ is the canonical isomorphism.

Recall that, for $n\geq 1$, we have

[TABLE]

where $\widehat{\phi}_{i}\sigma^{\otimes i}=\sigma\phi_{i}$, for all $i\in\mathbb{N}$. Then, if $Q:=\widehat{f}(id_{M[1]}\otimes\Psi(\phi))_{n}(\sigma_{M}\otimes\sigma^{\otimes n})$, we have

[TABLE]

For $n=0$, we have

[TABLE]

(2) and the commutativity of the second square in (3) follow from the commutativity of the first square and the fact that ${\rm TGMod}\hbox{-}A$ is constructed from ${\rm GMod}\hbox{-}A$ in a similar way that ${\rm TGMod}\hbox{-}{\cal B}_{A}$ is constructed from ${\rm GMod}\hbox{-}{\cal B}_{A}$, using the corresponding differentials, and the functor $R_{\phi}:{\rm TGMod}\hbox{-}B\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\rm TGMod}\hbox{-}A$ is constructed from $R_{\phi}:{\rm GMod}\hbox{-}B\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\rm GMod}\hbox{-}A$ in a similar way that the functor $R_{\Psi(\phi)}:{\rm TGMod}\hbox{-}{\cal B}_{B}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\rm TGMod}\hbox{-}{\cal B}_{A}$ is constructed from the functor $R_{\Psi(\phi)}:{\rm GMod}\hbox{-}{\cal B}_{B}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\rm GMod}\hbox{-}{\cal B}_{A}$.

(4) Given morphisms $f,g:(M,m^{M})\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}(N,m^{N})$ in ${\rm Mod}_{\infty}\hbox{-}A$, hence in ${\rm TGMod}^{0}\hbox{-}A$, any homotopy $h$ from $f$ to $g$ is a homogeneous morphism in ${\rm GMod}\hbox{-}A$ with degree $|h|=-1$ and is mapped by the restriction functor $R_{\phi}:{\rm GMod}\hbox{-}B\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\rm GMod}\hbox{-}A$ onto the homotopy $R_{\psi}(h)$ from $R_{\psi}(f)$ to $R_{\psi}(g)$, see (9.12). Hence, there is an induced functor $\underline{R}_{\phi}$ which clearly makes the diagram of (4) to commute. ∎

Corollary 10.4**.**

Let $f,g:A\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}B$ be homotopic morphisms of $A_{\infty}$-algebras. Then, there is an isomorphism of functors

[TABLE]

where $\underline{R}_{f},\underline{R}_{g}:\underline{{\rm Mod}}_{\infty}\hbox{-}B\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}\underline{{\rm Mod}}_{\infty}\hbox{-}A$ are the functors induced on the homotopy categories by the restriction functors $R_{f},R_{g}:{\rm Mod}_{\infty}\hbox{-}B\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\rm Mod}_{\infty}\hbox{-}A$, respectively. Any homotopy equivalence $f:A\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}B$ of $A_{\infty}$-algebras determines an equivalence of categories $\underline{R}_{f}:\underline{{\rm Mod}}_{\infty}\hbox{-}B\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}\underline{{\rm Mod}}_{\infty}\hbox{-}A$.

Proof.

Let ${\cal B}_{A}$ and ${\cal B}_{B}$ be the differential tensor $S$-coalgebras (or differential tensor $S$-bocses) associated to $A$ and $B$, respectively. Then, $\phi=\Psi(f)$ and $\psi=\Psi(g)$ are homotopic morphisms of graded $S$-coalgebras ${\cal B}_{A}\smash{\mathop{\hbox to20.0pt{\rightarrowfill}}\limits}{\cal B}_{B}$. We have the following commutative diagram

[TABLE]

and similarly for the morphisms $g$ and $\psi$. From (5.4), we know there is an isomorphism of functors $\underline{R}_{\phi}\cong\underline{R}_{\psi}$. Denote by $\mathbb{G}^{\prime}_{A}$ a quasi inverse for the equivalence $\mathbb{G}_{A}$. From the isomorphism of functors $\underline{R}_{\phi}\underline{\mathbb{G}}_{B}\cong\underline{R}_{\psi}\underline{\mathbb{G}}_{B}$, we obtain

[TABLE]

∎

Acknowledgements. The second author acknowledges the hospitality of Centro de Ciencias Matemáticas, UNAM, during his sabbatical year and the support of CONACyT sabbatical grant 2018-000008-01NACV.