Gorenstein-Projective Modules over a Class of Morita Rings

Asefa, Dadi

doi:https://doi.org/10.1155/2022/3732360

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Research Article | Open Access

Volume 2022 | Article ID 3732360 | https://doi.org/10.1155/2022/3732360

Gorenstein-Projective Modules over a Class of Morita Rings

Dadi Asefa¹

Academic Editor: Li Guo

Received07 Jun 2022

Accepted09 Aug 2022

Published10 Oct 2022

Abstract

Let be a Morita ring such that the bimodule homomorphisms are zero. In this paper, we give sufficient conditions for a -module to be Gorenstein-projective. As an application, we give sufficient conditions when the algebras and inherit the strongly CM-freeness of .

1. Introduction

Gorenstein algebra and Gorenstein-projective modules are important topics of research in Gorenstein homological algebra. A fundamental problem in Gorenstein homological algebra is determining all the Gorenstein-projective -modules for a given algebra . The class of Gorenstein-projective modules is a key component of relative homological algebra and has received a great deal of attention in the study of representation theory (e.g., [1–6, 8–13, 16–18, 20, 23–27]).

For algebras and , bimodules and , and a -bimodule map , and an -bimodule map satisfying some special conditions. Bass [7] introduced Morita algebra , where the special conditions for and are to guarantee that the multiplication of has the associativity. Morita algebras give a very large class of algebras, and many important algebras can be realized as Morita algebras. For example, the matrix algebra over , the algebra , the upper triangular matrix algebra , the algebras defined by finite quivers and relations. Thus, researching Morita rings is pivotal.

Asefa [1] obtained sufficient conditions for Gorenstein-projective module over , implying that is a Gorenstein-projective -module and is a Gorenstein-projective B-module. Gao and Psaroudakis [13] constructed Gorenstein-projective modules over a Morita ring . They stated that [13], [Theorem 3.10] does not give sufficient conditions for a module Gorenstein-projective ([13], Remark 3.13). As a result, it is natural to ask, “When is a module Gorenstein-projective?”. This paper is motivated to answer this question. In the following main result, we give sufficient conditions for to be a Gorenstein-projective module over a Morita ring .

Theorem 1. Let be a Morita ring. Assume that(i) and have finite flat dimensions.(ii) and have finite projective dimensions.Then, if each of the following conditions holds, a -module is Gorenstein-projective.(1) is Gorenstein-projective -module;(2) is Gorenstein-projective -module; and(3) and .Lastly, we give sufficient conditions when the algebras and inherit the strongly -freeness of .

2. Preliminaries

This section discusses some basic definitions and facts that will be used throughout the paper.

Throughout, rings mean a ring with unity and an -module mean a left -module. Let be a ring. Let be an -module, then the projective(injective and flat) dimension of will be denoted by ( and ). The class of modules isomorphic to direct summands of direct sums of copies of is denoted by .

An -module is Gorenstein-projective if there exists an exact sequence of projective -modules

such that is exact for an arbitrary projective -module and that . The class of Gorenstein-projective -modules will be denoted by .

Let and be rings, and bimodules, and and bimodules homomorphism. This paper focuses on the case of . Then,is a Morita ring, where the addition is that of a matrix, and multiplication of this Morita ring is given as follows:

The case is a subclass of the general Morita rings(e.g., [7, 13–15, 21].)

2.1. Modules over

A left module over Δ(0.0) is given as , where is an -module, is a -module, andwhere is an -module map and is a -map.

A -module morphism is given by , where is a homomorphism in and is a homomorphism in such that the following diagrams are commutative.

Lemma 1 (see [13]). Letbe a Morita ring.(1)A sequenceis exact in-Mod if and only if the sequenceis exact inand the sequenceis.(2)Leta morphism in-module and consider the mapsand. Then,is given bywhere the mapsandare induced from the commutative diagrams given below.

Similarly, the Cokernel of can be described.

2.2. We Now Recall Functors Given in [16]

(1)The functor -Mod -Mod is given by for any object in -Mod.(2)The functor -Mod -Mod is given by for any object in -Mod.(3)The functor -Mod -Mod is given by for any object in -Mod.(4)The functor -Mod -Mod is given by for any object in -Mod.(5)Let be any object in Mod, then we denote by the map -module given by involution. The functor -Mod -Mod is given by for any object in -Mod.(6)Let be any object in -Mod, the we denote by the map -module given by involution. The functor -Mod -Mod is given by for any object in -Mod.(7)The functor -Mod -Mod is defied by for any object in -Mod. The functor -Mod -Mod can be similarly defined.

More information about the functors given above can be found in the following result.

Proposition 1. ([16], Proposition 2.4]), Letbe Morita ring. Then,(1)The functors,,, and, are fully faithful.(2)The pairs,, andare adjoint pairs.(3)The functorsandare exact.

Lemma 2. Let be Morita ring.(1)[19], [Theorem 7.3] A left-moduleis projective if and only iffor some projective left-moduleand projective left-module.(2)[22], [Corollary 2.2] A left-moduleis injective if and only iffor some injective left-moduleand injective left-module.

3. Gorenstein-Projective Modules over

This section aims to construct Gorenstein-projective modules over .

The following lemmas are required in order to prove the main theorems of this paper.

Lemma 3. Let be a ring and a--bimodule with finite flat dimension. If a complex of flat-modules is exact, then, the sequence is also exact.

Proof. Assume that is an exact complex of flat -modules. Because has a finite flat dimension, we have the following flat resolution of .We obtain the following exact sequence of complexes because all terms in the complex are flat.Since the complexes are exact for all , so is . □

Lemma 4. Let be a ring. If a -module has finite injective dimension and the complex of projective -modules,is exact, then so is .

Lemma 5. Let be a Morita ring with zero bimodule homomorphisms. Then(1)[13], [Lemma 3.8] For each-Mod and each-Mod we have the following exact sequences in-Mod. and(2)[13], [Lemma 3.9] For all-Mod and-Mod, we have the following isomorphisms:and

The following result provides sufficient conditions for the functor and the functor to preserve Gorenstein-projective modules.

Proposition 2. (1)Assume that has a finite flat dimension and that has a finite projective dimension. is a Gorenstein-projective -module if X is a Gorenstein-projective -module.(2)Assume that has a finite flat dimension and that has a finite projective dimension. is a Gorenstein-projective -module if is a Gorenstein-projective -module.

Proof. We show (1) and (2) can be proved in a similar manner. Since an -module is a Gorenstein-projective, there is an exact sequence of projective -modules,such that , and exact for any projective -module . Lemma 3 states that the assumption that has finite flat dimension implies the sequence is exact. Hence we get the exact sequence of projective -modules,such that . Now, it is left to show that is exact for any projective -module . By Lemma 2, this can be proved by showing the exactness of and for any projective -module , and any projective -module . By Proposition 1 the functor is fully faithful. Thus, . Hence because is exact. Since are adjoint pairs, we have the following equation:A module has finite projective dimension because it is isomorphic to a direct summand of direct sums of copies of . Since is a complete -projective resolution, the complex is exact(see [18], [Proposition 2]). Thus, is exact. Hence is exact for any projective -module . Therefore, is a Gorenstein-projective -module.

In the following result, we give sufficient conditions for a -module to be Gorenstein-projective.

Theorem 2. Let be a Morita ring. Assume that(i)and have finite flat dimensions.(ii)and have finite projective dimensions.Then, if each of the following conditions holds, a -module is Gorenstein-projective.(1)is a Gorenstein-projective -module;(2)is a Gorenstein-projective -module; and(3)and .

Proof. Suppose that conditions (1)–(3) are true. Since is a Gorenstein-projective -module, there exists an exact complex of projective -modules,such that and is exact for each projective -module . Thus, we get the following exact sequence,because has a finite flat dimension. Since is a Gorenstein-projective -module, there exists a complete projective resolutions,of -modules such that .
Let and . Consider the following commutative diagram of A-modules.Since ψ = 0, the above equation implies that there exists an -map that is unique and . Thus, from it follows that is an injective -map. Thus, we get the exact sequence as follows:Similarly, the sequenceis exact.
Since each has finite projective dimension, and since each is a Gorenstein-projective -module, we have that . Applying generalized Horseshoe Lemma([26], Lemma 1.6 ) to the exact sequences (18) and (21), we obtain an exact sequence as follows:with , , , such that the following diagramis commutative. The dual argument obtains the commutative diagram with exact rows shown below.When we combine (24) and (25), we get the exact sequence shown below.with .
We now construct an exact sequence similar to (26) for a left -module . Since each has finite projective dimension as -module by assumption on , and is a Gorenstein-projective -module, it follows that . Thus, by ([26], Lemma 1.6 ) again, we obtain the exact sequence as follows:with , and , , such that the diagramis commutative. The dual argument gives the commutative diagram with the exact rowsAs a result, combining (28) and (29) yields the following exact sequence, which is similar to the following equation:with .
Glue together the exact sequences (26) and (30) to obtain the following sequence:with .
The morphism , is a -map becauseandare commutative diagrams.
Since the complexes (26) and (30) are exact, it follows from Lemma 1 (1) that the sequence is exact. The object arises as the kernel of the morphism , and we see from Lemma 1 (2) that and . However, based on the commutative diagram of -modules shown below,We know that is uniquely determined by . Similarly, is uniquely determined by .
We are now left with showing that is exact for each projective -module . We can deduce from Lemma 2 that it is enough to show that and are exact for each projective -module and for each projective -module . By Lemma 5 (1) the sequence is exact. Since each term in the complex is a projective -module, the sequenceis exact. By Lemma 5 (2) we have the following equations,The complex is exact because is a complete projective resolution. Thus, the complex is exact. Also, by Lemma 5 (2), we haveTo show the exactness of , we know that a -module has finite projective dimension, since is isomorphic to direct summand of a direct sum of copies of . Thus, is exact by [18], [Proposition 2.3], which implies is exact. Hence from the exact sequence of complexes in (35) it follows that the complex is exact. Similarly, the complexis exact. Thus, is exact for each projective -module . Therefore, a -module is a Gorenstein-projective.

If the converse of Theorem 2 holds, then Gorenstein-projective modules over will be fully determined. However, whether the converse is true or not is an open problem.

Corollary 1. Let be Morita ring, and -module. If , are Gorenstein-projective -modules such that and , then is a Gorenstein-projective -module.

4. Application

In this section, we study when the class of all Gorenstein-projective -modules and -modules coincides with the class of projective -modules and -modules, respectively.

If each finitely generated projective left -module is projective, then a ring is said to be left -free. And is said to be strongly left -free if each Gorenstein-projective left module is projective(see [12]).

The results that follow provide sufficient conditions for the algebras and to inherit the strongly-freeness of .

Proposition 3. Let be Morita ring.(1)Assume that has finite flat dimension, is projective, and is a strongly CM-free, thenis a strongly CM-free.(2)Assume that has finite flat dimension, is projective, and is a strongly CM-free, thenis a strongly CM-free.

Proof. (1)Assume is projective and is a strongly CM-free. Let be a Gorenstein-projective -module. Because has a finite projective dimension, Proposition 2 (1) asserts that is a Gorenstein-projective -module. The assumption that is a strongly CM-free implies that is a projective -module. By Lemma 2 (1), for some projective -module or for some projective -module . Hence , or . An -module is projective because it is isomorphic to a direct summand of a direct sum of copies of and is projective. Thus, is a projective -module. Therefore, is a strongly CM-free.(2)Assume is projective and is a strongly CM-free. Let be a Gorenstein-projective -module. By similar argument as in(1), is a projective -module. Therefore, is a strongly CM-free.

As a consequence we have the following corollary.

Corollary 2. Let be Morita ring. If is a strongly CM-free, thenis a strongly CM-free.

Data Availability

No datasets were generated or analyzed during the study.

Conflicts of Interest

The author declares no conflicts of interest.

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Copyright © 2022 Dadi Asefa. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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