Asymptotic Grothendieck groups and c.b.l.f. positive dg-algebras

TL;DR

This paper introduces asymptotic Grothendieck groups for certain categories, explores their properties, and computes these groups for specific positive dg-algebras with bounded, locally finite dimensions.

Contribution

It defines asymptotic Grothendieck groups for abelian and triangulated categories and relates them to topological Grothendieck groups, providing new insights and computations for positive dg-algebras.

Findings

Asymptotic Grothendieck groups are isomorphic under certain conditions.

Connection established between asymptotic and topological Grothendieck groups.

Computed asymptotic Grothendieck groups for specific positive dg-algebras.

Abstract

We introduce the notion of asymptotic Grothendieck groups for abelian and triangulated categories that are both AB4 and AB4*. We study when the asymptotic Grothendieck group of the heart of a triangulated category with a t-structure is isomorphic to the asymptotic Grothendieck group of the triangulated category itself. We also explain a connexion with the notion of topological Grothendieck group from Achar-Stroppel. Finally, we compute the asymptotic Grothendieck group of some positive (dg-)algebras having a cone bounded, locally finite dimension.

Equations

$$\frac{1}{1+q^2+\cdots +q^{2n-2}},$$

$$[R]=1+q^2+\cdots +q^{2n-2}[\mathbb{k}].$$

$$[\mathbb{k}]=\frac{1}{1+q^2+\cdots +q^{2n-2}}[R],$$

$$\frac{1-{\lambda }^2}{1-q^2},$$

$$\chi (R)=(1-{\lambda }^2)(1+q^2+q^4+\cdots )=\frac{1-{\lambda }^2}{1-q^2}.$$

$$\cdots \hookrightarrow x^{2}R\hookrightarrow (x^{2}R+x\omega R)\hookrightarrow xR\hookrightarrow (xR+\omega R)\hookrightarrow R,$$

$$\frac{R}{(xR+\omega R)}$$

$$\cdots \to q^2{\lambda }^{4}R[2]\oplus {\lambda }^{6}R[3]\stackrel{\left(\begin{array}{cc}\omega & x\\ 0& \omega \end{array}\right)}{\to }{q}^2{\lambda }^{2}R[1]\oplus {\lambda }^{4}R[2]\stackrel{\left(\begin{array}{cc}\omega & -x\\ 0& \omega \end{array}\right)}{\to }{q}^{2}R\oplus {\lambda }^{2}R[1]\stackrel{\left(\begin{array}{c}x\\ \omega \end{array}\right)}{\to }R\to L,$$

$$[R]$$

$$X=F_{m+1}\stackrel{f_m}{\to }\cdots \stackrel{f_2}{\to }{F}_2\stackrel{f_1}{\to }{F}_1\stackrel{f_0}{\to }{F}_0=Y,$$

$$[Y]-[X]=\sum _{i=0}^m[\mathrm{cok}{f}_i]-[\ker f_i].$$

$$\displaystyle\begin{split}X&=\lim\bigl{(}\cdots\xrightarrow{f_{2}}F_{2}\xrightarrow{f_{1}}F_{1}\xrightarrow{f_{0}}F_{0}=Y\bigr{)},\\ \text{(resp. }\quad Y&=\operatorname{colim}\bigl{(}X=F_{0}\xrightarrow{f_{0}}F_{1}\xrightarrow{f_{1}}F_{2}\xrightarrow{f_{2}}\cdots\bigr{)}\text{ )},\end{split}$$

$$\text{“ }[X]$$

$$0\to A_{\bullet }\to B_{\bullet }\to C_{\bullet }\to 0,$$

$$[B]=[A]+[C],$$

$$0\to A\to B\to C\to \tilde{A}\to \tilde{B}\to \tilde{C}\to 0.$$

$$[X]-[\tilde{X}]$$

$$\text{(resp. }[Y]-[\tilde{Y}]$$

$$[B]-[\tilde{B}]=[A]-[\tilde{A}]+[C]-[\tilde{C}],$$

$$K_0^{\oplus }(\mathcal{C})\cong \prod _{i\in I}\mathbb{Z}\cdot [X_i],$$

$${\mathbf{G}}_0(\mathcal{C})\cong \prod _{i\in I}\mathbb{Z}\cdot [S_i],$$

Full text

1. Introduction

Categorification is a trendy research area in mathematics nowadays. In short, the philosophy behind categorification is to lift classical objects to the level of categories, turning set elements into objects, functions into functors, equalities into isomorphisms, and revealing a whole new layer of structure achieving these isomorphisms. It is particularly relevant when one categorifies representations, turning a vector space into a category and the action (of a group or an algebra) into endofunctors. One usually refer to this procedure as higher representation theory (see for example [chuangrouquier, FKS, lauda, soergel]). Categorification also has applications in low-dimensional topology (see for example [khovanov, webster]) and deep connections with algebraic geometry (see [catgeom] for a nice survey). While most results live in an integral setting, it appears to be not enough to handle all situations, such as in the categorification of the Jones-Wenzl projectors [frenkelstroppelsussan], or in the categorification of Verma modules [naissevaz1, naissevaz2, naissevaz3], which is related by [naissevaztriply] to Khovanov–Rozansky triply–graded link homology [KRhomology].

In the first case, it is necessary to categorify fractions of the form

[TABLE]

where $q$ is a formal variable. The polynomial $1+q^{2}+\cdots+q^{2n-2}$ is the graded dimension of $R:=\Bbbk[x]/(x^{n})$, $\deg(x)=2$, the cohomology ring of $\mathbb{C}\mathbb{P}^{n-1}$, as a graded $\Bbbk$-module. Thus, in the Grothendieck group $K_{0}(\Bbbk\operatorname{\mathrm{-}vect})$ of $\mathbb{Z}$-graded $\Bbbk$-module, we have

[TABLE]

Moreover, $\Bbbk$ admits a free resolution over $R$ with (graded) ranks matching Eq. 1 expanded as a Laurent series in $\Bbbk\llbracket q\rrbracket[q^{-1}]$. Therefore, the authors of [frenkelstroppelsussan] suggested to categorify the fraction Eq. 1 by the Ext-algebra $\operatorname{Ext}^{*}_{R}(\Bbbk,\Bbbk)$. However, we usually cannot say that in $K_{0}(\Bbbk\operatorname{\mathrm{-}vect})$ we have

[TABLE]

since it would require to make sense of a convergent series, as the projective resolution is infinite. In [acharstroppel], Achar and Stroppel developed a framework to handle such situations. They start with a nice graded, finite length abelian category with enough projectives, and explain how to construct a derived category where (minimal) projective resolutions of the objects exist. Then, they develop a notion of topological Grothendieck group, by modding out relations coming from these (potentially infinite) projective resolutions. We can sum up their framework as objects are finite, but can admit infinite resolutions. This is the notion used in [frenkelstroppelsussan].

In the case of the categorification of Verma modules [naissevaz1, naissevaz2, naissevaz3], one needs to categorify the rational fraction

[TABLE]

which appears as a structure constant. The solution used in [naissevaz1, naissevaz2] was to interpret $\lambda$ as a second grading, the minus sign on the numerator as a parity degree (thus working in a “super” setting), and the denominator as the formal Laurent series $(1+q^{2}+q^{4}+\cdots)$. From that, the author developed with Vaz in [naissevaz1, Appendix] notions of Krull–Schmidt (resp. Jordan–Hölder) categories for infinite direct sums (resp. infinite composition series), controlled in a certain way. These categories also come with some ‘topological’ versions of the Grothendieck groups. Here, the object are not finite anymore, but only locally finite with respects to the grading. However, the framework in [naissevaz1, Appendix] does not handle infinite projective resolutions, or more generally derived categories.

In this paper, we propose a framework bringing both the topological Grothendieck groups of [acharstroppel] and of [naissevaz1] under the same roof, and generalizing it even further. We call this generalization an asymptotic Grothendieck group. Such a construction is necessary to handle properly the (de)categorification of (parabolic) Verma modules for all quantum Kac–Moody algebras, as done in [naissevaz3]. It also yields a more natural categorification of $\mathfrak{sl}_{2}$-Verma modules from [naissevaz1, naissevaz2] in the world of homological algebra rather than in the super-setting, with the minus sign on the numerator of Eq. 2 induced by a difference in homological degree. Consequently, asymptotic Grothendieck groups are also used in other constructions, such as [naissevaztriply], [twoblob] and [tensorvermas], which all connect categorification of Verma modules to low-dimensional topology. The results in [frenkelstroppelsussan] can also be rewritten in our setting.

1.1. Main example and goal of the paper

As a main example for the paper, we consider the following algebra, whose definition is motivated by [KRhomology] and [naissevaz1]. Consider the $\Bbbk$-algebra $R:=\Bbbk[x,\omega]/(\omega^{2})$. We endow it with a $\mathbb{Z}\times\mathbb{Z}^{2}$ grading, declaring that $\deg(x):=(0,2,0)$ and $\deg(\omega):=(1,0,2)$. We interpret the first $\mathbb{Z}$-grading as homological, while we call each of the components of the $\mathbb{Z}^{2}$-grading the $q$- and the $\lambda$-grading respectively. We equip $R$ with a trivial differential, turning it into a $\mathbb{Z}^{2}$-graded dg-algebra $(R,0)$, with the differential grading given by the homological one. Note that the graded Euler characteristic of $R$ is

[TABLE]

Furthermore, $R$ admits a unique (up to isomorphism and grading shift) projective module, given by $R$ itself, and a unique (up to isomorphism and grading shift) simple module $L:=\Bbbk$. They both lift to $\mathbb{Z}^{2}$-graded dg-modules when equipped with a trivial differential. Moreover, we obtain a filtration

[TABLE]

with quotients

[TABLE]

and a resolution

[TABLE]

where $q$ and $\lambda$ are grading shifts in the $\mathbb{Z}^{2}$-grading and $[1]$ in the homological grading. Our goal is to construct a suitable derived category of $(R,0)$ such that its asymptotic Grothendieck group is a free $\mathbb{Z}(\!(q,\lambda)\!)$-module, and generated either by the class of $[R]$ or $[L]$, with formulas

[TABLE]

obtained from the categorical relations in Eq. 3 and in Eq. 4. Throughout the paper, we will study different categories of modules over $R$ or its dg-version $(R,0)$, and their respective (asymptotic) Grothendieck group.

1.2. Sketch of the construction

Let us give an insight of what we mean by asymptotic Grothendieck group. Let $\mathcal{A}$ be an abelian category. Given a sequence or arrows

[TABLE]

one can deduce that in the (abelian) Grothendieck group $G_{0}(\mathcal{A})$ there is an equality

[TABLE]

The idea of the asymptotic Grothendieck group is to take this phenomenon to the limit. One can extend such a composition of arrows to the left (resp. to the right)

[TABLE]

giving a filtered limit (resp. filtered colimit). Naively, we could be tempted to ask that in this case we have

[TABLE]

However, filtered limits (resp. filtered colimits) are in general not exact. This is problematic since for example given a short exact sequence of filtered limits

[TABLE]

with $A:=\lim(\cdots\rightarrow A_{1}\rightarrow A_{0})$ and similarly for $B$ and $C$, then Eq. 7 would imply that

[TABLE]

while $B$ could be very far from being an extension of $C$ by $A$. Hopefully, when $\mathcal{A}$ is a full subcategory of an abelian category having countable products (resp. coproducts), it is possible to compute by how much a filtered limit $X$ (resp. colimit $Y$) is not exact, by considering its derived limit $\tilde{X}$ (resp. derived colimit $\tilde{Y}$). In particular, for a short exact sequence of limits as Eq. 8, there is an exact sequence

[TABLE]

Thus, we suggest to define the asymptotic Grothendieck group $\boldsymbol{G}_{0}(\mathcal{A})$ by taking $G_{0}(\mathcal{A})$ modded out by the relations

[TABLE]

whenever we are in the case of Eq. 6, see LABEL:def:topG0 for a precise definition of $\boldsymbol{G}_{0}(\mathcal{A})$. In particular, this definition implies that given Eq. 8, then

[TABLE]

which agrees with Eq. 9. We use a similar definition in the case of a triangulated category, using Milnor limits and Milnor colimits instead of filtered ones, and replacing $[\operatorname{cok}f_{r}]-[\ker f_{r}]$ by the class of the mapping cone $[\operatorname{Cone}(f_{r})]$, see LABEL:def:toptriangulatedK0.

1.3. Outline of the paper

LABEL:sec:classK0 is a reminder of the basics about Grothendieck groups. In LABEL:sec:triangcat we recall the construction of the derived category of a dg-algebra, which we will use as main example for later, and we fix the conventions (in particular all our differentials are of degree $-1$). In LABEL:sec:finitemultigradings we explain what we mean by a multigraded category and how it influences the Grothendieck group. In LABEL:sec:formalLaurent we recall the construction of the ring of formal Laurent series $\Bbbk(\!(x_{1},\dots,x_{n})\!)$.

Our contributions start in LABEL:sec:cbladditive where we extend the notion of locally Krull–Schmidt category of [naissevaz1, Appendix] to the non-graded case (LABEL:def:locKS). One of the main results is:

Corollary LABEL:cor:lockrullschmidt.

Let $\mathcal{C}$ be a locally Krull–Schmidt category with a full collection of pairwise non-isomorphic indecomposable objects $\{X_{i}\}_{i\in I}$. If $\mathcal{C}$ admits locally finite direct sums of indecomposable objects, then its split Grothendieck group is a free $\mathbb{Z}$-module

[TABLE]

*with basis $\{[X_{i}]\}_{i\in I}$. * We also recall the notions of cone bounded, locally finite (c.b.l.f.) direct sums and of c.b.l. additive category from [naissevaz1, Appendix] (appearing under the name ‘locally finite’ in the reference). We also recall the notion of c.b.l.f. dimensional $\Bbbk$-vector space, which basically means the graded dimension of the vector space is an element of $\mathbb{Z}(\!(x_{1},\dots,x_{n})\!)$.

In LABEL:sec:topG0 we introduce the notion of asymptotic Grothendieck group for abelian categories. We refine the notion of ‘local Jordan–Hölder category’ from [naissevaz1, Appendix]. The main difference being that we require in this paper filtered limits and filtered colimits to be exact (thus derived limits and derived colimits to be zero) in a locally Jordan–Hölder category (LABEL:def:locJH). Then, based on the results in [naissevaz1, Appendix], we show the following:

Theorem LABEL:thm:locallyJHG0.

Let $\mathcal{C}$ be a locally Jordan–Hölder category with a full collection of pairwise non-isomorphic simple objects $\{S_{i}\}_{i\in I}$. If $\mathcal{C}$ admits locally finite direct sums of $\{S_{i}\}_{i\in I}$, then its asymptotic Grothendieck group is a free $\mathbb{Z}$-module

[TABLE]

with basis $\{[S_{i}]\}_{i\in I}$.

In particular, the asymptotic Grothendieck group of a locally Jordan–Hölder category is equivalent to the topological one of [naissevaz1, Appendix]. This allows us to compute the asymptotic Grothendieck group for certain categories of graded modules. Let $R$ be a c.b.l.f. dimensional, $\mathbb{Z}^{n}$-graded $\Bbbk$-algebra. Consider the category $R\operatorname{\mathrm{-}mod}$ of $\mathbb{Z}^{n}$-graded $R$-modules with degree zero maps, and let $R\operatorname{\mathrm{-}mod_{cblf}}$ be the full subcategory of $R\operatorname{\mathrm{-}mod}$ consisting of c.b.l.f. dimensional modules.