Gorenstein projective bimodules via monomorphism categories and filtration categories
Introduction
Auslander [2] initiated Gorenstein homological algebra by introducing modules of G-dimension zero over a Noetherian commutative local ring, which coincides the maximal Cohen–Macaulay modules over Gorenstein commutative local ring. Later, Auslander and Bridger [3] generalized these modules to two-sided Noetherian rings, which are now called Gorenstein projective modules. The notion of Gorenstein projective modules over an arbitrary ring was introduced by Enochs and Jenda in [15]. These modules identify with maximal Cohen–Macaulay modules over Gorenstein Noetherian rings in the work of Buchweitz in [5].
Not only the place of Gorenstein projective modules in Gorenstein homological algebra is the same as that of projective modules in homological algebra, but the Gorenstein projective modules (namely, maximal Cohen–Macaulay modules) also play an important role in singularity theory. Buchweitz in [5] proved that the stable category of the category of Gorenstein projective modules over a Gorenstein Noetherian ring is triangle equivalent to the stable derived category which is just the singularity category defined by Orlov [28]. Note that Happel obtained the same result for Gorenstein algebras independently [18]. In particular, CM-finiteness (A ring is CM-finite if it has only finitely many isoclasses of indecomposable Gorenstein projective modules) is closely related to simple singularities, see [21], [6], [13]. For CM-finite Artin algebras, there is an Auslander-type result for Gorenstein projective modules [8].
Thus, one of the most important tasks is to describe Gorenstein projective modules, especially in non-commutative case. Up to now, there are some partial results, mainly concentrated on Artin algebras, such as -extension of an Artin algebra [25], upper triangular matrix Artin algebra [41], [46], [16], Artin algebras with radical square zero [10], [34], Nakayama algebras [30], monomial algebras [12], and some cases of tensor product of two algebras—we will show it explicitly below. In this paper, we study Gorenstein projective modules over the tensor product of two algebras. This has been done only in some special cases. Li and Zhang's result of -extension can be viewed as describing the Gorenstein projective modules over , where Q is the Dynkin quiver of type . Ringel and Zhang [35] completely described the Gorenstein projective modules of , where Q is a finite acyclic quiver and . In [26], Luo and Zhang studied the Gorenstein projective modules over , where Q is a finite acyclic quiver. Later, Luo and Zhang [27] generalize it to Q with monomial relations. The approaches of [25], [26], [27] mainly use monomorphism categories. In this paper, besides via monomorphism categories for which we give a general homological definition, we give another approach to the Gorenstein projective modules over tensor product of algebras, via filtration categories.
-via monomorphism category. The category of monomorphisms in a module category can be viewed as the first example of monomorphism category. This goes back to G. Birkhoff's problem [4] of classifying all subgroups of abelian p-groups by matrices. This question were related to representations of partial order sets, such as [1], [37], [38]. Ringel and Schmidmeier [31], [32], [33] renewed this subject by studying the representation type and the Auslander–Reiten theory of the submodule category for an Artin algebra, and in particular for . Kussin, Lenzing and Meltzer [22], [23] related submodule categories to weighted projective lines and singularity theory. Chen [9], [11] studied these categories from the viewpoint of triangulated categories and Gorenstein homological algebra.
For an algebra A, the category of monomorphisms in the module category of A can be viewed as a full subcategory of the module category of the tensor product , where is the quiver . Zhang [45] introduced monomorphism categories of type , say the category of successive monomorphisms, which can be viewed as a full subcategory of the module category of , where is the quiver . The Auslander–Reiten theory of this category was studied in [43]. In [26], Luo and Zhang generalized the above notion and introduced monomorphism categories over finite acyclic quivers, and then over finite acyclic quivers with monomial relations [27]. Recently, the Ringel–Schmidmeier–Simson (RSS for short) equivalence on monomorphism categories is introduced by Zhang et al. [42], [44]. Based on the combinatorial information of the quiver (and the monomial relations), the monomorphism category was applied to describe Gorenstein projective modules over the tensor product of an algebra and a path algebra (modulo the ideal generated by the monomial relations) over a field.
In this paper, we define monomorphism categories over an arbitrary finite dimensional algebra, using homological conditions (see Definition 3.1). Our definition is equivalent to the previous ones when restricting to finite acyclic quivers (with monomial relations). The homological definition seems to simplify many arguments. For two finite dimensional k-algebras A and B, we denote the monomorphism category of B over A by . For a full additive subcategory of A-mod , we can also define the full subcategory of . We show that is a Frobenius exact category and is a full subcategory of where (see Proposition 3.10, Proposition 4.3). Furthermore, we have the following main result.
Theorem A Theorem 4.5, Theorem 4.6 Let A and B be two finite dimensional k-algebras. Assume either A or B is Gorenstein. Then if and only if B is CM-free.
Thus once the modules in are known, it is relatively easy to describe the modules in . This gives a satisfactory answer in this case.
-via filtration category. In the definition of monomorphism categories , the role of B and A are not symmetric (see Example 3.3). This motivates us to consider the filtration category , that is, the class of -modules which are direct summands of iterated extensions of tensor products of Gorenstein projective A-modules and Gorenstein projective B-modules. The modules in the filtration category are always Gorenstein projective. The converse does not hold in general (see Example 5.6 below). However, in case that both algebras are Gorenstein, or one of the algebras is a triangular algebra, we obtain the following main results:
Theorem B Theorem 5.2 Let A and B be Gorenstein algebras. Assume that k is a splitting field for A or B. Then
Theorem C Theorem 5.5 Let A and B be two finite dimensional k-algebras. Assume that B is triangular and k is a splitting field for B. Then the three categories , and are the same, where is the smallest full subcategory of closed under extensions.
We can even show that a similar result holds for infinitely generated Gorenstein projective modules. Its proof essentially uses Quillen's powerful small object argument.
This paper is organized as follows. Section 2 contains some preliminary materials, including basic facts about Gorenstein projective modules. In Section 3, we introduce our (homological version of) monomorphism categories and study their basic properties. The next two sections are the heart of this paper. We describe Gorenstein projective bimodules using monomorphism categories and using filtration categories. Finally, we deal with infinitely generated Gorenstein projective bimodules by Quillen's small object argument in Subsection 5.2.
We should remark that Dawei Shen [36] also obtained some results of this paper independently, in particular, Proposition 2.6 (2) and Theorem 4.5.
Section snippets
Preliminaries
In this section, we recall some basic definitions and facts that needed in our later proofs.
Monomorphism categories
In this section, starting from Luo and Zhang's definition [26] of monic representations of an acyclic quiver over algebra, we introduce monic representations of an arbitrary finite dimensional algebra over an algebra (Definition 3.1), and study the category of all such monic representations and its full subcategories.
Gorenstein projective bimodules via monomorphism categories
Throughout this section, we fix two finite dimensional k-algebras A and B, and set to be their tensor product. It is natural to ask whether one can describe Gorenstein projective Λ-modules in terms of Gorenstein projective modules over A and B. In this section, we shall give an approach to Gorenstein projective Λ-modules via monomorphism categories.
At fist, we use the monomorphism categories to describe the category of projective Λ-modules. We get the following result.
Lemma 4.1 Let A and B be two
Finitely generated Gorenstein projective modules
To study the Gorenstein projective modules over tensor products, another strategy is to describe in terms of and . As before, we write Λ for throughout this subsection. At first, for arbitrary and , we have by Proposition 2.6. However, in general, Gorenstein projective Λ-modules may not be of this form. For instance, if both A and B are selfinjective, then so is Λ. In this case . In general, there are Λ-modules which are
Acknowledgements
The author W. Hu is grateful to NSFC (No. 11471038, No. 11331006) and the Fundamental Research Funds for the Central Universities for partial support. X.-H. Luo is supported by NSFC (No. 11401323, No. 11771272) and Jiangsu Government Scholarship for Oversea Studies (No. JS-2016-059). B.-L. Xiong is supported by NSFC (No. 11301019, No. 11471038). G. Zhou is supported by NSFC (No. 11671139) and by STCSM (No. 13dz2260400). The main part of this work was completed during the author B.-L. Xiong
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