Cohen-Macaulay differential graded modules and negative Calabi-Yau configurations

Abstract

In this paper, we introduce the class of Cohen-Macaulay (=CM) dg (=differential graded) modules over Gorenstein dg algebras and study their basic properties. We show that the category of CM dg modules forms a Frobenius extriangulated category, in the sense of Nakaoka and Palu, and it admits almost split extensions. We also study representation-finite d-self-injective dg algebras A in detail for some positive integer d. In particular, we classify the Auslander-Reiten (=AR) quivers of $\mathsf{CM}A$ for a large class of d-self-injective dg algebras A in terms of $(-d)$-Calabi-Yau (=CY) configurations, which are Riedtmann's configurations for the case $d=1$. For any given $(-d)$-CY configuration C, we show there exists a d-self-injective dg algebra A, such that the AR quiver of CMA is given by C. For type $A_n$, by using a bijection between $(-d)$-CY configurations and certain purely combinatorial objects which we call maximal d-Brauer relations given by Coelho Simões, we construct such A through a Brauer tree dg algebra.

Introduction

The notion of Cohen-Macaulay (CM) modules is classical in commutative algebra [9], [42], and has natural generalizations for non-commutative algebras [10], [22], [24], often called Gorenstein projective modules [3], [11], [17]. The category of CM modules has been studied by many researchers in representation theory (see, for example, [16], [40], [55], [62]). On the other hand, the derived categories of differential graded (dg) categories introduced by Bondal-Kapranov [7] and Keller [33], [35] is an active subject appearing in various areas of mathematics [43], [57], [60]. Among others, we refer to [2], [29], [30], [32], [53] for the representation theory of dg categories.

In this paper, we introduce Cohen-Macaulay dg modules over dg algebras and develop their representation theory to build a connection between these two subjects. One of the main properties of the category of Cohen-Macaulay dg modules is that it has a structure of extriangulated category and the stable category is equivalent to the singularity category, which is an analogue of Buchweitz's equivalence. Moreover, it admits almost split extensions and we can study it by Auslander-Reiten theory. In fact, there are many nice dg algebras (including those given in this paper), whose categories of Cohen-Macaulay dg modules can be well understood, while the derived categories of dg algebras are usually wild and it is hopeless to classify all the indecomposable objects.

To make everything works well, we need to add some restrictions on dg algebras. More precisely, we work on dg algebras A over a field k satisfying the following assumptions.

Assumption 0.1

• A is non-positive, i.e. ${\mathrm{H}}^i(A)=0$ for $i>0$ (without loss of generality, we may assume $A^i=0$ for $i>0$, see Section 1.3);

• A is proper, i.e. ${\dim }_k{⨁}_{i\in \mathbb{Z}}{\mathrm{H}}^i(A)<\infty$;

• A is Gorenstein, i.e. the thick subcategory $\mathsf{per}A$ of the derived category $\mathsf{D}(A)$ generated by A coincides with the thick subcategory generated by DA, where $D={\mathrm{Hom}}_k(?,k)$ is the k-dual.

In this case, we define Cohen-Macaulay dg A-modules as follows, where we denote by ${\mathsf{D}}^{\mathrm{b}}(A)$ the full subcategory of $\mathsf{D}(A)$ consisting of the dg A-modules whose total cohomology is finite-dimensional.

Definition 0.2 Definition 2.1

• A dg A-module M in ${\mathsf{D}}^{\mathrm{b}}(A)$ is called a Cohen-Macaulay dg A-module if ${\mathrm{H}}^i(M)=0$ and ${\mathrm{Hom}}_{\mathsf{D}A}(M,A[i])=0$ for $i>0$;

• We denote by $\mathsf{CM}A$ the subcategory of ${\mathsf{D}}^{\mathrm{b}}(A)$ consisting of Cohen-Macaulay dg A-modules.

Definition 0.2 is motived by the fact that if A is concentrated in degree zero, then the condition (1) above gives an alternative description of classical Cohen-Macaulay modules [25, Theorem 3.10]. Moreover, in this case the category $\mathsf{CM}A$ forms a Frobenius category in the sense of [21] and the stable category $\underset{\_}{\mathsf{CM}}A$ is a triangulated category which is triangle equivalent to the singularity category ${\mathsf{D}}_{\mathrm{sg}}(A)={\mathsf{D}}^{\mathrm{b}}(\mathsf{mod}A)/{\mathsf{K}}^{\mathrm{b}}(\mathsf{proj}A)$ introduced by Buchweitz [10] and Orlov [45]. However, $\mathsf{CM}A$ does not necessarily have a natural structure of exact category in our setting. Instead, the following result shows it has a natural structure of extriangulated category introduced by Nakaoka and Palu [44].

Theorem 0.3 Theorems 2.4, 3.1 and 3.7

Let A be a non-positive proper Gorenstein dg algebra. Then

• $\mathsf{CM}A$ is functorially finite in ${\mathsf{D}}^{\mathrm{b}}(A)$;

• $\mathsf{CM}A$ is a Frobenius extriangulated category with $\mathsf{Proj}(\mathsf{CM}A)=\mathsf{add}A$;

• The stable category $\underset{\_}{\mathsf{CM}}A:=(\mathsf{CM}A)/[\mathsf{add}A]$ is a triangulated category;

• The composition $\mathsf{CM}A\hookrightarrow {\mathsf{D}}^{\mathrm{b}}(A)\to {\mathsf{D}}^{\mathrm{b}}(A)/\mathsf{per}A$ induces a triangle equivalence

  $$\underset{\_}{\mathsf{CM}}A=(\mathsf{CM}A)/[\mathsf{add}A]\simeq {\mathsf{D}}^{\mathrm{b}}(A)/\mathsf{per}A={\mathsf{D}}_{\mathrm{sg}}(A);$$

• $\underset{\_}{\mathsf{CM}}A$ admits a Serre functor and $\mathsf{CM}A$ admits almost split extensions.

The main examples considered in this paper are trivial extension dg algebras and truncated polynomial dg algebras. We determine all indecomposable Cohen-Macaulay dg modules over truncated polynomial dg algebras concretely and give their AR quivers (see Theorem 4.2 for the details). We also show that, in this case, the stable categories are cluster categories by using a criterion given by Keller and Reiten [38] (see Theorem 4.9).

One of the traditional subjects is the classification of Gorenstein rings which are representation-finite in the sense that they have only finitely many indecomposable Cohen-Macaulay modules. Riedtmann [50], [51] and Wiedemann [58] considered the classification of representation-finite self-injective algebras and Gorenstein orders respectively. In both classifications, configurations play an important role. We may regard Wiedemann's configurations as “0-Calabi-Yau” since they are preserved by Serre functor $\mathbb{S}$ and regard Riedtmann's configurations as “$(-1)$-Calabi-Yau” since they are preserved by $\mathbb{S}\circ [1]$. Inspired by this, we introduce the negative Calabi-Yau configurations to study the AR quivers of $\mathsf{CM}A$.

Definition 0.4 Definition 5.1, Definition 5.2

Let $\mathcal{T}$ be a k-linear Hom-finite Krull-Schmidt triangulated category and let C be a set of indecomposable objects of $\mathcal{T}$. We call C a $(-d)$-Calabi-Yau configuration (or $(-d)$-CY configuration for short) for $d\ge 1$ if the following conditions hold.

• ${\dim }_k{\mathrm{Hom}}_{\mathcal{T}}(X,Y)={\delta }_{X,Y}$ for $X,Y\in C$;

• ${\mathrm{Hom}}_{\mathcal{T}}(X,Y[-j])=0$ for any two objects $X,Y$ in C and $0<j\le d-1$;

• For any indecomposable object M in $\mathcal{T}$, there exists $X\in C$ and $0\le j\le d-1$, such that ${\mathrm{Hom}}_{\mathcal{T}}(X,M[-j])\ne 0$.

We call C a d-simple-minded system (or d-SMS), if it satisfies (1), (2) and

(3′) $\mathcal{T}=\mathsf{add}\mathsf{Filt}\{C,C[1],\cdots ,C[d-1]\}$.

It is precisely Riedtmann's configuration if $d=1$ and $\mathcal{T}$ is the mesh category of $\mathbb{Z}\mathrm{\Delta }$ for a Dynkin diagram Δ (see [50, Definition 2.3] for the details). It is easy to see d-SMS implies $(-d)$-CY configuration and the converse is also true if $\mathsf{Filt}(C)$ is functorially finite in $\mathcal{T}$ due to [15, Proposition 2.13]. “$(-d)$-Calabi-Yau configuration” are also introduced as “left d-Riedtmann configuration” in [13], and further studied in [14], [15]. When the AR quiver of $\mathcal{T}$ is $\mathbb{Z}\mathrm{\Delta }/G$ for some Dynkin diagram Δ and some group G, Calabi-Yau configuration can be characterized combinatorially (see Section 5.3). Our name “$(-d)$-Calabi-Yau configuration” is motivated by the following theorem, which is new even for $d=1$ (see Remark 5.4).

Theorem 0.5 Theorem 5.3

Let $\mathcal{T}$ be a k-linear Hom-finite Krull-Schmidt triangulated category with a Serre functor $\mathbb{S}$. Let C be a $(-d)$-CY configuration in $\mathcal{T}$, then $\mathbb{S}[d]C=C$.

We say a dg k-algebra A in Assumption 0.1 is d-self-injective (resp. d-symmetric) if $\mathsf{add}A=\mathsf{add}DA[d-1]$ in $\mathsf{D}(A)$ (resp. $\mathsf{D}(A^{\mathrm{e}})$). The following result, characterizing simple dg A-modules as a $(-d)$-CY configuration, generalizes [50, Proposition 2.4].

Theorem 0.6 Theorem 5.6

Let A be a d-self-injective dg algebra. Then the set of simple dg A-modules is a d-SMS, and hence a $(-d)$-CY configuration in $\underset{\_}{\mathsf{CM}}A$.

Let Δ be a Dynkin digram. For a subset C of vertices of $\mathbb{Z}\mathrm{\Delta }$, we define a translation quiver ${(\mathbb{Z}\mathrm{\Delta })}_C$ by adding to $\mathbb{Z}\mathrm{\Delta }$ a vertex $p_c$ and two arrows $c\to p_c\to {\tau }^{-1}(c)$ for each $c\in C$ (see Definition 1.14). Our main result in this paper states that the converse of Theorem 0.6 also holds in the following sense.

Theorem 0.7 Theorem 7.1

Let Δ be a Dynkin digram. Let C be a subset of vertices of $\mathbb{Z}\mathit{\Delta }/\mathbb{S}[d]$. The following are equivalent.

• C is a $(-d)$-CY configuration;

• There exists a d-symmetric dg k-algebra A with AR quiver of $\mathsf{CM}A$ being ${(\mathbb{Z}\mathit{\Delta })}_C/\mathbb{S}[d]$.

To study the classification of configurations, Riedtmann [50] gave a geometrical description of configurations by Brauer relations, and Luo [41] gave a description of Wiedemann's configuration by 2-Brauer relations. Similarly, we introduce maximal d-Brauer relations (see Definition 8.2). It gives a nice description of $(-d)$-CY configurations of type $A_n$. This geometric model has been studied by Coelho Simões [13, Theorem 6.5]. By using this model, we show the number of $(-d)$-CY configurations in $\mathbb{Z}{A}_n/\mathbb{S}[d]$ is $\frac{1}{n+1}\left(\begin{array}{c}(d+1)n+d-1\\ n\end{array}\right)$ (Corollary 8.20). We develop several technical concepts and results on maximal d-Brauer relations and by using them we give another proof of Theorem 0.7 for the case $\mathrm{\Delta }=A_n$ (Theorem 8.32). In this case, for any given $(-d)$-CY configuration C, the corresponding d-symmetric dg k-algebra is given explicitly by Brauer tree dg algebra (see Section 8.3 for the details). The following table explains the comparison among different configurations.

The paper is organized as follows. Section 1 provides the necessary material on dg algebras, extriangulated categories and translation quivers. In Section 2, we introduce Cohen-Macaulay dg modules and show some basic properties of them. Section 3 deals with the Auslander-Reiten theory in $\mathsf{CM}A$. We compute the AR quiver of a truncated polynomial dg algebra in Section 4. From its own point of view, this example is also interesting. We introduce (negative) CY-configurations and combinatorial configurations in Section 5 and then show they coincide with each other in our context. In Section 6, we construct a class of self-injective dg algebras by taking trivial extension. In Section 7, we prove our main theorem that any CY configuration is given by simples of symmetric dg algebras. In section 8, we introduce the maximal d-Brauer relations and give a formula of the number of $(-d)$-CY configurations in $\mathbb{Z}{A}_n/\mathbb{S}[d]$. We prove Theorem 0.7 for the case $\mathrm{\Delta }=A_n$ by constructing a Brauer tree dg algebra from given maximal d-Brauer relation. In Appendix A, we give a new proof of the bijection between $(-d)$-CY configurations and maximal d-Brauer relations.

Acknowledgments The author would like to thank his supervisor Osamu Iyama for many useful discussions and for his consistent support. He also thanks Raquel Coelho Simões and David Pauksztello for pointing out their results on d-Riedtmann configurations in [13], [14], [15].