Generating degrees for graded projective resolutions

TL;DR

This paper develops a framework linking theories on the linearity of graded modules, focusing on tensor products of graded bimodules and providing a tool to evaluate degrees in graded projective resolutions.

Contribution

It introduces a new framework connecting existing theories and offers a method to determine degrees in projective resolutions based on initial generating degrees.

Findings

Established a connection between theories of linearity in graded modules.

Provided a tool to evaluate degrees in projective resolutions.

Enhanced understanding of tensor products of graded bimodules.

Abstract

We provide a framework connecting several well known theories related to the linearity of graded modules over graded algebras. In the first part, we pay a particular attention to the tensor products of graded bimodules over graded algebras. Finally, we provide a tool to evaluate the possible degrees of a module appearing in a graded projective resolution once the generating degrees for the first term of some particular projective resolution are known.

Equations

$$aTb:=\big{(}{\rm Ext}^{n}_{C}(L_{b},Z)\circ{\rm Ext}^{n}_{C}(Y,L_{a})\big{)}(T),\hbox{ for }T\in{\rm Ext}^{n}_{C}(Y,Z),a\in A,b\in B.$$

$$0\to N\stackrel{f_{n-1}}{\to }{T}_{n-1}\stackrel{f_{n-2}}{\to }\cdots \stackrel{f_0}{\to }{T}_0\stackrel{f_{-1}}{\to }M\to 0.$$

$$E^{i,j}_{2}={\rm Ext}^{i}_{Gr(A^{op}\otimes B)}\big{(}X,{\rm ext}^{j}_{C}(Y,Z)\big{)}\mbox{ and }\tilde{E}^{i,j}_{2}={\rm Ext}^{i}_{Gr(A^{op}\otimes C)}\big{(}{\rm Tor}_{j}^{B}(X,Y),Z\big{)}$$

$$\cdots \stackrel{d_n(X)}{\to }{P}_n(X)\stackrel{d_{n-1}(X)}{\to }\cdots \stackrel{d_0(X)}{\to }{P}_0(X)(\stackrel{{\mu }_X}{\to }X)$$

$$(Z\stackrel{{\iota }_Z}{\to })I^0(Z)\stackrel{d^0(Z)}{\to }\cdots \stackrel{d^{n-1}(Z)}{\to }{I}^n(Z)\stackrel{d^n(Z)}{\to }\dots$$

$${\rm Hom}_{Gr(A^{op}\otimes B)}\Big{(}P_{i}(X),{\rm hom}_{C}\big{(}Y,I^{j}(Z)\big{)}\Big{)}$$

$${\rm Hom}_{Gr(A^{op}\otimes C)}\big{(}P_{j}(X)\otimes_{B}Y,I^{i}(Z)\big{)}.$$

$$F_{\bullet}={\rm Hom}_{Gr(A^{op}\otimes C)}\big{(}P_{\bullet}(X)\otimes_{B}Y,I^{\bullet}(Z)\big{)}\cong{\rm Hom}_{Gr(A^{op}\otimes B)}\Big{(}P_{\bullet}(X),{\rm hom}_{C}\big{(}Y,I^{\bullet}(Z)\big{)}\Big{)},$$

$$E^{i,j}_{1}={\rm Hom}_{Gr(A^{op}\otimes B)}\big{(}P_{i}(X),{\rm ext}^{j}_{C}(Y,Z)\big{)}\mbox{ and }E^{i,j}_{2}={\rm Ext}^{i}_{Gr(A^{op}\otimes B)}\big{(}X,{\rm ext}^{j}_{C}(Y,Z)\big{)},$$

$$\tilde{E}^{i,j}_{1}={\rm Hom}_{Gr(A^{op}\otimes C)}\big{(}{\rm Tor}_{j}^{B}(X,Y),I^{i}(Z)\big{)}\mbox{ and }\tilde{E}^{i,j}_{2}={\rm Ext}^{i}_{Gr(A^{op}\otimes B)}\big{(}{\rm Tor}_{j}^{B}(X,Y),Z\big{)}.$$

$$\chi_{s}(i)=\left\{\begin{array}[]{ll}\frac{is}{2}&\text{ if $i$ is even,}\\ \frac{(i-1)s}{2}+1&\text{ if $i$ is odd.}\\ \end{array}\right.$$

$$({\mathcal{S}}\otimes{\mathcal{R}})_{i}=\bigcup\limits_{\scriptsize\begin{array}[]{c}j+k=i\\ j,k\geqslant 0\end{array}}\{n+m\mid n\in{\mathcal{S}}_{j},m\in{\mathcal{R}}_{k}\}.$$

$$E^{2}_{i,j}={\rm Ext}^{i}_{Gr(A^{op}\otimes B)}\big{(}X,{\rm ext}^{j}_{C}(Y,Z)\big{)}\mbox{ and }\tilde{E}^{2}_{i,j}={\rm Ext}^{i}_{Gr(A^{op}\otimes C)}\big{(}{\rm Tor}_{j}^{B}(X,Y),Z\big{)}$$

Full text

2010 MSC: 16S37, 18G10.

Keywords: Koszul, linear modules, Gröbner bases.

1 Introduction

Koszul algebras were introduced by S. Priddy in [pr]. We will apply the notion of a Koszul algebra for algebras presented by quivers with relations. It can be stated as follows. Suppose that ${\bf k}$ is a field, $Q$ is a finite quiver, $I$ is a homogeneous ideal of the path algebra ${\bf k}Q$ and $A={\bf k}Q/I$. The algebra $A$ is called Koszul if the maximal semisimple graded quotient $A_{0}$ of $A$ has a graded $A$-projective resolution $(P_{\bullet},d_{\bullet})$ such that for all $i\geqslant 0$, the $A$-module $P_{i}$ is generated in degree $i$. Such a resolution is called a linear resolution and it is minimal whenever it exists, in the sense that $d_{i}(P_{i+1})\subset P_{i}A_{>0}$ for all $i\geqslant 0$.

E. Green and R. Martínez Villa proved in [gm] that the quadratic algebra $A$ is Koszul if and only if its Yoneda algebra, $E(A)=\oplus_{i\geqslant 0}{\rm Ext}^{i}_{A}(A_{0},A_{0})$ is generated in degrees 0 and 1, which in turn is equivalent to the Yoneda algebra being isomorphic to the quadratic dual $A^{!}$ of $A$.

Koszulness has been generalized to various settings. Next we describe some of these generalizations.

R. Berger introduced in [be] the notion of “nonquadratic Koszul algebra” for algebras of the form $A=T_{{\bf k}}V/I$, where $V$ is a finite dimensional ${\bf k}$-vector space and $I$ is a two-sided ideal generated in degree $s$, for some $s\geq 2$. He required the trivial $A$-module ${\bf k}$ to have a minimal graded projective resolution $(P_{\bullet},d_{\bullet})$ such that each $P_{i}$ is generated in degree $\frac{is}{2}$ for $i$ even and $\frac{(i-1)s}{2}+1$ for $i$ odd.

The authors of [gmmz] considered, under the name of $s$-Koszul algebras, non necessarily quadratic Koszul algebras of the form $A={\bf k}Q/I$, with $Q$ a finite quiver and $I$ an ideal generated by homogeneous elements of degree $s$, connecting this notion with the Yoneda algebra: the algebra $A$ is $s$-Koszul if and only if $E(A)$ is generated in degrees 0, 1 and 2. Observe that $2$-Koszul algebras are just Koszul algebras.

Later on, E. Green and E. Marcos generalized this notion defining $\delta$-Koszul and $\delta$-determined algebras. See [gm1] for details.

Moreover, E. Green and E. Marcos also introduced in [gm2] a family of algebras that they called $2$-$s$-Koszul. They proved that these algebras also have the property that their Yoneda algebras are generated in degrees 0, 1, and 2.

The main objective of the current work is to place all these definitions in a unique framework. We next sketch how we will do this.

Let $A=\oplus_{i\geqslant 0}A_{i}$ be a graded ${\bf k}$-algebra generated in degrees 0 and 1, such that $A_{0}$ is a finite direct product of fields and $A_{1}$ is finite dimensional. Given a graded $A$-module $X$, we consider a minimal graded projective resolution $(P_{\bullet}(X),d_{\bullet}(X))$ and we take into account in which degrees $P_{i}(X)$ is generated for each $i\geqslant 0$. We are specially interested in what we call ${\mathcal{S}}$-determined case.

In Section 2 In Section 2 we prove that, given graded ${\bf k}$-algebras $A,B$ and $C$, a graded $A-B$ bimodule $X$ and a graded $B-C$ bimodule $Y$, if $X$ has a linear minimal $A-B$ -projective graded resolution, $Y$ has a linear minimal $B-C$ -projective graded resolution, and ${\rm Tor}_{i}^{B}(X,Y)$ vanishes for $i\geqslant 1$, then $X\otimes_{B}Y$ has a linear minimal $A-C$ -projective graded resolution. This is a particular case of Theorem 2 below. Note that this theorem shows that any graded bimodule over a Koszul algebra which is linear as a right module and flat as a left module is also linear as a bimodule and the tensor product with such a module gives a functor from the category of linear graded modules to the category of linear graded modules. Moreover, it recovers and generalizes the fact that the tensor product of two Koszul algebras is Koszul. The same holds for the ${\mathcal{S}}$-determined (see Definition 1).

Section LABEL:grobner is devoted to Gröbner bases. Loosely speaking, we show how one can use them to obtain generating degrees for the $n$-th term of the minimal graded projective resolution of a module if one knows the generating degrees for terms of its projective presentation of a special form.

We fix a field ${\bf k}$. All algebras will be ${\bf k}$-algebras and all modules will be right $A$-modules unless otherwise stated. We will simply write $\otimes$ for $\otimes_{{\bf k}}$ and $\mathbb{N}_{0}$ for the set of non negative integer numbers.

We thank the referee for the suggestions and for a careful reading of a previous version of this paper.

2 Tensor products of ${\mathcal{S}}$-determined modules

In this section we will prove Lemma 1, which is a graded version of the spectral sequences (2) and (3) from [CE, page 345].

Let $A,B$ and $C$ be ${\bf k}$-algebras. Let $X$ be an $A-B$-bimodule, $Y$ a $B-C$-bimodule and $Z$ an $A-C$-bimodule. Given $a\in A$, we will denote left multiplication by $a$ on $X$ by $L_{a}\in{\rm End}_{B}(X)$. We recall that for each $n\in\mathbb{N}$, ${\rm Ext}^{n}_{C}(Y,Z)$ is an $A-B$ bimodule with the structure given by

[TABLE]

Suppose now that $A$ is a ${\mathbb{Z}}$-graded algebra and $M$ is a graded $A$-module. Given $i\in{\mathbb{Z}}$, $M[i]$ will denote the $i$-shifted graded $A$-module with underlying $A$-module structure as before, whose grading is such that $M[i]_{r}=M_{i+r}$. For any graded $A$-module $N$ and any $n\in\mathbb{N}$, we will denote by ${\rm Hom}_{GrA}(M,N)$ the set of degree preserving $A$-module maps from $M$ to $N$ and by ${\rm Ext}^{n}_{GrA}(M,N)$ the set of equivalence classes of exact sequences of graded $A$-modules with degree zero morphisms

[TABLE]

Let us consider as usual ${\rm ext}_{A}^{n}(M,N):=\oplus_{i\in\mathbb{Z}}{\rm Ext}^{n}_{GrA}(M,N[i])$, which is a subset of ${\rm Ext}_{A}^{n}(M,N)$ in a natural way. Moreover, if $M$ has an $A$-projective resolution with finitely generated modules, then both sets coincide.

Suppose now that $A$, $B$ and $C$ are ${\mathbb{Z}}$-graded algebras, and that the bimodules $X,Y$ and $Z$ are graded. For each $n\geqslant 0$, the $A-B$-bimodule structure on ${\rm Ext}^{n}_{C}(Y,Z)$ induces a graded $A-B$-bimodule structure on ${\rm ext}_{C}^{n}(Y,Z)$ whose $i$-th component is ${\rm Ext}^{n}_{GrC}(Y,Z[i])$. Note also that ${\rm ext}_{C}^{n}(Y,Z[i])\cong{\rm ext}_{C}^{n}(Y,Z)[i]$ as graded $A-B$-bimodule, moreover for any $n\geqslant 0$, ${\rm Tor}_{n}^{B}(X,Y)$ is a graded $A-C$-bimodule in a natural way.

The main tool of this section is the following lemma.

Lemma 1**.**

Let $A$, $B$ and $C$ be ${\mathbb{Z}}$-graded algebras, $X$ a graded $A-B$-bimodule, $Y$ a graded $B-C$-bimodule, and $Z$ a graded $A-C$-bimodule. There are two first quadrant cohomological spectral sequences with second pages

[TABLE]

that converge to the same graded space.

• Proof.

Let

[TABLE]

be a graded $A-B$-projective resolution of $X$ and

[TABLE]

be a graded $A-C$-injective resolution of $Z$. Consider two bicomplexes whose $(i,j)$-components are respectively

[TABLE]

and

[TABLE]

Since there is an isomorphism of complexes

[TABLE]

the respective total complexes are isomorphic. Here, as usually, for two complexes of graded modules \big{(}U_{\bullet},d_{U,\bullet}\big{)} and \big{(}V^{\bullet},d^{V,\bullet}\big{)} over the algebra $D$ we denote by ${\rm Hom}_{GrD}(U_{\bullet},V^{\bullet})$ the complex with \big{(}{\rm Hom}_{GrD}(U_{\bullet},V^{\bullet})\big{)}_{n}=\oplus_{i\in{\mathbb{Z}}}{\rm Hom}_{GrD}(U_{i-n},V^{-i}) and differential $d_{\bullet}$ defined by the equality $d_{n}(f)=d^{V,-i}f+(-1)^{n}fd_{U,i-n}$ for $f\in{\rm Hom}_{GrD}(U_{i-n},V^{-i})$.

The first two pages of the spectral sequence $E$ corresponding to the first bicomplex are

[TABLE]

while the first two pages of the spectral sequence $\tilde{E}$ corresponding to the second bicomplex are

[TABLE]

Since both spectral sequences converge to the homology of $F_{\bullet}$, the lemma is proved.

$\Box$

From now on any ${\mathbb{Z}}$-graded algebra $A$ is assumed to be non negatively graded, that is $A=\oplus_{i\geqslant 0}A_{i}$, where $A_{0}$ is isomorphic to a finite product of copies of ${\bf k}$ as an algebra, $\dim_{{\bf k}}A_{1}<\infty$, and $A$ is generated as an algebra by $A_{0}\oplus A_{1}$. This is equivalent to say that $A\cong({\bf k}Q)/I$ where $Q$ is a finite quiver and $I$ is an ideal generated by homogeneous elements of degree bigger or equal 2.

Definition 1**.**

Let ${\mathcal{S}}=({\mathcal{S}}_{i})_{i\geqslant 0}$ be a collection of subsets ${\mathcal{S}}_{i}\subset{\mathbb{Z}}$. A graded $A$-module $X$ is called ${\mathcal{S}}$-determined if it has a graded projective resolution $P_{\bullet}(X)$ such that $P_{i}(X)$ is generated as $A$-module by elements of degrees belonging to ${\mathcal{S}}_{i}$, i.e. $P_{i}(X)=\langle\oplus_{j\in{\mathcal{S}}_{i}}P_{i}(X)_{j}\rangle_{A}$ for all $i\geqslant 0$. We say that $X$ is ${\mathcal{S}}$-determined up to degree $r$ if the condition on $P_{i}(X)$ holds for $0\leqslant i\leqslant r$.

If the set ${\mathcal{S}}_{i}=\{i\}$, i.e. each $P_{i}(X)$ is generated in degree $i$, then we say that the resolution is linear.

Equivalently, a graded $A$-module $X$ is ${\mathcal{S}}$-determined if and only if for any $i\geqslant 0$ and any graded $A$-module $Y$ with support not intersecting ${\mathcal{S}}_{i}$ – that is, $\oplus_{j\in{\mathcal{S}}_{i}}Y_{j}=0$ – the space ${\rm Ext}^{i}_{GrA}(X,Y)$ is zero. Analogously, the graded $A$-module $X$ is ${\mathcal{S}}$-determined up to degree $r$ if and only if the last mentioned condition holds for $0\leqslant i\leqslant r$.

The notion of an ${\mathcal{S}}$-determined module provides a general framework for some well-known situations. We will now exhibit some well known examples of ${\mathcal{S}}$-determined modules.

• •

Consider a function $\delta:{\mathbb{Z}}_{\geqslant 0}\rightarrow{\mathbb{Z}}$, and define ${\mathcal{S}}_{i}=\{\delta(i)\}$ for all $i\geqslant 0$, the ${\mathcal{S}}$-determined modules are called $\delta$-determined modules. If $A_{0}$ is a $\delta$-determined module over $A$, then the graded algebra $A$ is called $\delta$-determined.

• •

With the same notations, if moreover the ${\rm Ext}$ algebra, $E(A)$, of $A$ is finitely generated, then $A$ is called $\delta$-Koszul, see [gm1]. In particular, if $\delta$ is the identity, then $\delta$-determined modules are called linear modules and $\delta$-Koszul algebras are exactly Koszul algebras [pr].

• •

Also, given $s\in\mathbb{N}$, let us define $\chi_{s}:\mathbb{N}_{0}\rightarrow{\mathbb{Z}}$ by

[TABLE]

The $\chi_{s}$-linear modules are called $s$-linear modules and $\chi_{s}$-Koszul algebras are $s$-Koszul algebras, see [be]. Denoting by ${\mathcal{S}}_{i}$ the set $\{j\mid j\leqslant\chi_{s}(i)\}$, ${\mathcal{S}}$-linear modules correspond to $2$-$s$-linear modules. If moreover $A_{0}$ is a $2$-$s$-determined module over $A$, then the graded algebra $A$ is called $2$-$s$-determined, see [gm2].

Using minimal graded projective resolutions, it is not difficult to see that the $A$-module $A_{0}$ is ${\mathcal{S}}$-determined if and only if $A$ is an ${\mathcal{S}}$-determined module over $A^{op}\otimes A$. This fact follows, for example, from [Skol, Theorem 2].

Given two collections ${\mathcal{S}}=({\mathcal{S}}_{i})_{i\geqslant 0}$ and ${\mathcal{R}}=({\mathcal{R}}_{i})_{i\geqslant 0}$ of subsets of ${\mathbb{Z}}$ we define the collection {\mathcal{S}}\otimes{\mathcal{R}}=\big{(}({\mathcal{S}}\otimes{\mathcal{R}})_{i}\big{)}_{i\geqslant 0} by

[TABLE]

Lemma 1 allows us to prove the following theorem, which generalizes some well known results about Koszul algebras and Koszul modules concerning tensor products.

Theorem 2**.**

Let ${\mathcal{S}}=({\mathcal{S}}_{i})_{i\geqslant 0}$ and ${\mathcal{R}}=({\mathcal{R}}_{i})_{i\geqslant 0}$ be two collections of subsets of ${\mathbb{Z}}$. Let $A$, $B$ and $C$ be ${\mathbb{Z}}$-graded algebras, and finally let $X$ be a graded $A-B$-bimodule which is ${\mathcal{S}}$-determined as bimodule and $Y$ be a graded $B-C$-bimodule which is ${\mathcal{R}}$-determined as $C$-module. If ${\rm Tor}_{i}^{B}(X,Y)=0$ for $1\leqslant i\leqslant r-1$, then $X\otimes_{B}Y$ is an ${\mathcal{S}}\otimes{\mathcal{R}}$-determined until $r$-th degree $A-C$-bimodule. In particular, if ${\rm Tor}_{i}^{B}(X,Y)=0$ for all $i\geqslant 1$, then $X\otimes_{B}Y$ is an ${\mathcal{S}}\otimes{\mathcal{R}}$-determined $A-C$-bimodule.

• Proof.

Let us fix $n$ and $r$ such that $0\leqslant n\leqslant r$. For any graded $A-C$-bimodule $Z$ such that $\oplus_{m\in({\mathcal{S}}\otimes{\mathcal{R}})_{n}}Z_{m}=0$, we will prove that ${\rm Ext}^{n}_{Gr(A^{op}\otimes C)}(X\otimes_{B}Y,Z)=0$. By Lemma 1 there are spectral sequences

[TABLE]

that converge to the same graded space $T_{\bullet}$. It follows easily from the condition on ${\rm Tor}_{*}^{B}(X,Y)$ that $T_{n}={\rm Ext}^{n}_{Gr(A^{op}\otimes C)}(X\otimes_{B}Y,Z)$ if $n<r$ and that $T_{r}={\rm Ext}^{r}_{Gr(A^{op}\otimes C)}(X\otimes_{B}Y,Z)\oplus V$ for some V\subset{\rm Hom}_{Gr(A^{op}\otimes C)}\big{(}{\rm Tor}_{r}^{B}(X,Y),Z\big{)}.

Thus, it is enough to prove that $E^{2}_{i,j}=0$ for all integers $i,j\geqslant 0$ such that $i+j=n$. Let us fix such $i$ and $j$. If $k\in{\mathcal{S}}_{i}$, then for any $l\in{\mathcal{R}}_{j}$ it is clear that $k+l\in({\mathcal{S}}\otimes{\mathcal{R}})_{n}$ and so $Z[k]_{l}=Z_{k+l}=0$. Since $Y$ is an ${\mathcal{R}}$-linear $C$-module, we know that ${\rm ext}^{j}_{C}(Y,Z)_{k}={\rm Ext}^{j}_{GrC}(Y,Z[k])=0$ for any $k\in{\mathcal{S}}_{i}$; from this, since $X$ is an ${\mathcal{S}}$-linear $A^{op}\otimes B$-module, E^{2}_{i,j}={\rm Ext}^{i}_{Gr(A^{op}\otimes B)}\big{(}X,{\rm ext}^{j}_{C}(Y,Z)\big{)}=0. We have proven that for any $0\leqslant n\leqslant r$ and any graded $A-C$-bimodule $Z$ such that $\oplus_{m\in({\mathcal{S}}\otimes{\mathcal{R}})_{n}}Z_{m}=0$ one has ${\rm Ext}^{n}_{Gr(A^{op}\otimes C)}(X\otimes_{B}Y,Z)=0$. Consequently, $X\otimes_{B}Y$ is an ${\mathcal{S}}\otimes{\mathcal{R}}$-determined until $r$-th degree $A-C$-bimodule.