# Gorenstein-Projective Modules over Upper Triangular Matrix Artin Algebras

Authors: Dadi Asefa
Journal: Journal of Mathematics
 (2021)

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## Abstract

Gorenstein-projective module is an important research topic in relative homological algebra, representation theory of algebras, triangulated categories, and algebraic geometry (especially in singularity theory). For a given algebraA, how to construct all the Gorenstein-projective A-modules is a fundamental problem in Gorenstein homological algebra. In this paper, we describe all complete projective resolutions over an upper triangular Artin algebra Λ=AMBA0B. We also give a necessary and sufficient condition for all finitely generated Gorenstein-projective modules over Λ=AMBA0B.


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## Body

## 1. Introduction

Auslander and Bridger [1] generalized finitely generated projective modules to modules of Gorenstein dimension zero over two-sided Noetherian rings. After two decades, Enochs and Jenda [2] generalized it to an arbitrary ring and called it Gorenstein-projective modules. After that, this class of modules got special attention and has been studied by many authors (see, e.g., [3–15]). Given an algebra A, it is a fundamental problem to determine all the indecomposable finitely generated Gorenstein-projective A-modules, which is usually not easy.Zhang [14] introduced the notion of compatible bimodules and then determined all the Gorenstein-projective modules over upper triangular matrix Artin algebra Λ=AMBA0B, in terms of Gorenstein-projective A and B-modules. This gives us a strong motivation to study Gorenstein-projective modules over upper triangular matrix Artin algebras.The main aim of this paper is to describe all the complete projective resolutions and all finitely generated Gorenstein-projective modules over an upper triangular matrix Artin algebraΛ=AMBA0B, by giving the corresponding sufficient and necessary conditions. Although these conditions ((GP1)–(GP5)) are long, they may be helpful, especially in some special cases.Throughout the paper,A−Mod denotes the category of left A-modules and A−mod denotes the category of finitely generated left A-modules for a ring A.


## 2. Preliminaries

In this section, we recall some basic definitions and facts that will be used throughout the paper.Following [16], an A-module M is said to be Gorenstein-projective, in A−Mod (resp., in A−Mod), if there is an exact sequence of projective modules in A−Mod (resp., in A−Mod)(1)P•≔⋯⟶P−1⟶P0⟶d0P1⟶⋯,with HomAP•,Q being also exact for any projective module Q in A−Mod (resp., in A−mod), such that M≅Kerd0. Such P• is called a complete projective resolution in A−Mod (resp., A−mod).


### 2.1. Modules over Upper Triangular Matrix Algebra

LetMBA be an A−B bimodule and(2)Λ=AMBA0B=am0ba∈A,b∈B,m∈Mbe an Artin algebra. Note that addition of Λ is the addition of matrix, and the multiplication is given by(3)am0b⋅a′m′0b′=aa′am′+mb′0bb′.Left modules overΛ are described with a triple XYϕ, where X is an A-module and Y is a B-module, and ϕ:M⊗BY⟶X is an A-map. Λ-map is described with a pair fg:XYϕ⟶X′Y′ϕ′, where f:X⟶X is an A-map and g:Y⟶Y′ is a B-map, such that the diagramis commutative.A sequence(4)0⟶X1Y1ϕ1⟶f1g1X2Y2ϕ2⟶f2g2X3Y3ϕ3⟶0in Λ−mod is exact if and only if a sequence(5)0⟶X1⟶f1X2⟶f2X3⟶0is exact in A−mod, and a sequence(6)0⟶Y1⟶g1Y2⟶g2Y3⟶0is exact in B−mod.


### 2.2. Projective Modules

The indecomposable projective (left)Λ-modules are exactly P0 and M⊗BQQId, where P runs over indecomposable projective A-modules and Q runs over indecomposable projective B-modules.


## 2.1. Modules over Upper Triangular Matrix Algebra

LetMBA be an A−B bimodule and(2)Λ=AMBA0B=am0ba∈A,b∈B,m∈Mbe an Artin algebra. Note that addition of Λ is the addition of matrix, and the multiplication is given by(3)am0b⋅a′m′0b′=aa′am′+mb′0bb′.Left modules overΛ are described with a triple XYϕ, where X is an A-module and Y is a B-module, and ϕ:M⊗BY⟶X is an A-map. Λ-map is described with a pair fg:XYϕ⟶X′Y′ϕ′, where f:X⟶X is an A-map and g:Y⟶Y′ is a B-map, such that the diagramis commutative.A sequence(4)0⟶X1Y1ϕ1⟶f1g1X2Y2ϕ2⟶f2g2X3Y3ϕ3⟶0in Λ−mod is exact if and only if a sequence(5)0⟶X1⟶f1X2⟶f2X3⟶0is exact in A−mod, and a sequence(6)0⟶Y1⟶g1Y2⟶g2Y3⟶0is exact in B−mod.


## 2.2. Projective Modules

The indecomposable projective (left)Λ-modules are exactly P0 and M⊗BQQId, where P runs over indecomposable projective A-modules and Q runs over indecomposable projective B-modules.


## 3. Complete Projective Resolutions

The aim of this section is to describe all complete projective resolutions over an upper triangular matrix Artin algebraΛ=AMBA0B.


### 3.1. Morphisms of ProjectiveΛ-Modules

Note that any projectiveΛ-module is of the form(7)X=P⊕M⊗BQQ,where P and Q are projective A-module and projective B-module, respectively. For each i∈ℤ, we write(8)Xi=Pi⊕M⊗BQiQi.For such a projectiveΛ-module Xi as above and for any datum(9)Pi,Qi,αi,βi,gi,where αi:Pi⟶Pi+1 and βi:Pi⟶M⊗BQi+1 are A-maps and gi:Qi⟶Qi+1 is B-map ∀i∈ℤ, we put(10)fi=αi0βiIdM⊗gigi:Pi⊕M⊗BQiQi⟶Pi+1⊕M⊗BQi+1Qi+1.Lemma 1.
LetXi and Pi,Qi,αi,βi,gi be given as in (8) and (9). Then, fi:Xi⟶Xi+1 is a Λ-map if and only if fi is given in (10).Proof.
First, assume thatfi:Xi⟶Xi+1 is a Λ-map, where fi is given in (10). Since the diagram
is commutative, we see thatfi:Xi⟶Xi+1 is a Λ-map, ∀i∈ℤ.
Conversely, letfi:Xi⟶Xi+1 be an arbitrary Λ-map where Xi=Pi⊕M⊗BQiQi. Then, fi is of the form(11)αiωiβiδigi:Xi⟶Xi+1,where αi:Pi⟶Pi+1, βi:Pi⟶M⊗BQi+1, ωi:M⊗BQi⟶Pi+1, and δi:M⊗BQi⟶M⊗BQi+1 are A-maps and gi:Qi⟶Qi+1 is B-map, ∀i∈ℤ. By the definition of morphisms in Λ-mod, the diagram
is commutative. This implies that(12)ωi=0,δi=IdM⊗gi∀i∈ℤ.
Thus,fi is given as in (10). This completes the proof.


### 3.2. The Exactness of Morphisms of Projective Modules

Consider the following conditions:GP1P•=⋯⟶α−2P−1⟶α−1P0⟶α0P1⟶α1⋯ is a complex.GP2Q•=⋯⟶g−2Q−1⟶g−1Q0⟶g0Q1⟶g1⋯ is an exact sequence.GP3βi+1αi+IdM⊗gi+1βi=0,∀i∈ℤ.GP4 For any pi+1,xi+1∈Pi+1⊕M⊗BQi+1 with αi+1pi+1=0, βi+1pi+1+IdN⊗gi+1xi+1=0, there is p′i,x′i∈Pi⊕M⊗BQi such that pi+1=αip′i and xi+1=βip′i+IdM⊗gix′i,∀i∈ℤ.Lemma 2.
LetXi be a projective Λ-module given as in (8) and fi:Xi⟶Xi+1 be a Λ-map given as in (10). Then, Kerfi+1=Imfi if and only if conditions (GP1)–(GP4) are satisfied.Proof.
By Lemma1, fi is of the form fi=αi0βiIdM⊗gigi:Pi⊕M⊗BQiQi⟶Pi+1⊕M⊗BQi+1Qi+1,
Whereαi:Pi⟶Pi+1 and βi:Pi⟶M⊗BQi+1 and gi:Qi⟶Qi+1,∀i∈ℤ. We see that Kerfi+1=Imfi,∀i∈ℤ, if and only if(13)Kerαi+10βi+1IdM⊗gi+1=Imαi0βiIdM⊗gi,Kergi+1=Imgi,where(14)αi0βiIdM⊗gi:Pi⊕M⊗BQi⟶Pi+1⊕M⊗BQi+1,∀i∈ℤ.
Note thatKergi+1=Imgi,∀i∈ℤ, if and only if condition (GP2) is satisfied. Thus, it suffices to show that(15)Kerαi+10βi+1IdM⊗gi+1=Imαi0βiIdM⊗gi.
Now,Imαi0βiIdN⊗gi⊆Kerαi+10βi+1IdN⊗gi+1 if and only if αi+10βi+1IdN⊗gi+1αi0βiIdN⊗gi=0, i.e.,(16)αi+1αi0βi+1αi+IdN⊗gi+1βiIdN⊗gi+1gi=0,if and only if conditions (GP1)–(GP3) are satisfied.
We also see thatKerαi+10βi+1IdN⊗gi+1⊆Imαi0βiIdN⊗gi,∀i∈ℤ, if and only if condition GP4 is satisfied.
Thus,Kerfi+1=Imfi,∀i∈ℤ, if and only if conditions (GP1)–(GP4) are satisfied.
Consider the following condition (GP5)∀i∈ℤ:GP5 For any ci+1∈HomAPi+1,A,di+1∈HomAPi+1,M,ti+1∈HomBQi+1,B, such that(17)ci+1αi=0,di+1αi+IdM⊗ti+1βi=0,ti+1gi=0,there arec′i+2∈HomAPi+2,A,d′i+2∈HomAPi+2,M,t′i+2∈HomBQi+2,B, such that(18)ci+1=c′i+2αi+1,di+1=d′i+2αi+1+IdM⊗t′i+2βi+1,ti+1=t′i+2gi+1.Lemma 3.
LetXi be a projective Λ-module given as in (8) and f:Xi⟶Xi+1 be a Λ-map given as in (10). Assume that Kerfi+1=Imfi, i.e., conditions GP1–GP4 are satisfied. Consider the homomorphism of abelian groups fi∗=HomΛfi,Λ:HomΛXi+1,Λ⟶HomΛXi,Λ. Then, Kerfi∗=Imfi+1∗,∀i∈ℤ, if and only if condition (GP5) is satisfied.Proof.
Assume thatKerfi+1=Imfi. By Lemma 1, we know that fi:Xi⟶Xi+1 is given as(19)fi=αi0βiIdM⊗gigi:Pi⊕M⊗BQiQi⟶Pi+1⊕M⊗BQi+1Qi+1.
Any element in the abelian groupHomΛXi,Λ=HomΛPi⊕M⊗BQiQi,A⊕MB,∀i∈ℤ, will be written as(20)hi=cieidisiti:Pi⊕M⊗BQiQi⟶A⊕MB.
By definition of morphisms inΛ-mod, the diagram
is commutative. This implies that(21)ei=0,si=IdM⊗ti∀i∈ℤ.
Thus,(22)hi=ci0diIdN⊗titi:Pi⊕M⊗BQiQi⟶A⊕MB.
Hence,fi∗=HomΛfi,Λ:HomΛXi+1,Λ⟶HomΛXi,Λ,∀i∈ℤ, is written as(23)fi∗=HomΛαi0βiIdM⊗gigi,Λ:HomΛPi+1⊕M⊗BQi+1Qi+1,A⊕MB⟶HomΛPi⊕M⊗BQiQi,A⊕MB.
Thus,fi∗ maps(24)hi+1=ci+10di+1IdM⊗ti+1ti+1to(25)fi∗hi+1=hi+1fi=ci+10di+1IdM⊗ti+1αi0βiIdM⊗giti+1gi=ci+1αi0di+1αi+IdM⊗ti+1βiIdM⊗ti+1giti+1gi.
This means thathi+1=ci+10di+1IdM⊗ti+1ti+1∈Kerfi∗ if and only if(26)ci+1αi=0,di+1αi+IdM⊗ti+1βi=0,ti+1gi=0.
We see thathi+1=ci+10di+1IdM⊗ti+1ti+1∈Imfi+1∗ if and only if there is h′i+2=c′i+20d′i+2IdM⊗t′i+2t′i+2 such that(27)c′i+20d′i+2IdM⊗t′i+2αi+10βi+1IdM⊗gi+1t′i+2gi=ci+10di+1IdM⊗ti+1ti+1,which means that(28)ci+1=c′i+2αi+1,di+1=d′i+2αi+1+IdM⊗t′i+2βi+1,ti+1=t′i+2gi+1.
By conditions (G1)–(G4), we have(29)fi+1fi=αi+10βi+1IdM⊗gi+1αi0βiIdM⊗gigi+1gi=αi+1αi0βi+1αi+IdN⊗gi+1βiIdM⊗gi+1gigi+1gi=0.
Thus,fi∗fi+1∗hi+2=hi+2fi+1fi=0, i.e., Imfi+1∗⊆Kerfi∗. Therefore, Imfi+1∗=Kerfi∗ if and only if Imfi+1∗⊇Kerfi∗, if and only if for any hi=ci+10di+1IdN⊗ti+1ti+1:Pi+1⊕M⊗BQi+1Qi+1⟶A⊕MB with(30)ci+1αi=0,di+1αi+IdM⊗ti+1βi=0,ti+1gi=0,there is(31)h′i+2=c′i+20d′i+2IdM⊗t′i+2t′i+2:Pi+2⊕M⊗BQi+2Qi+2⟶A⊕MB,such that(32)ci+1=c′i+2αi+1,di+1=d′i+2αi+1+IdM⊗t′i+2βi+1,ti+1=t′i+2gi+1.
So, by the above arguments, we see thatImfi+1∗=Kerfi∗,∀i∈ℤ, if and only if condition (GP5) is satisfied. This completes the proof.


### 3.3. Main Result

Now, by Lemmas2 and 3, we get the following sufficient and necessary conditions of any complete projective resolutions over Λ=AMBA0B.Theorem 1.
LetXi be a projective Λ-module, given as in (8), and fi:Xi⟶Xi+1 be a Λ-map, given as in (10). Then,(33)X•=⋯⟶f−2X−1⟶f−1X0⟶f0X1⟶f1⋯is a complete Λ-projective resolution if and only if conditions (GP1)–(GP5) are satisfied.


### 3.4. Special Complete Projective Resolutions

We consider the special case off:Xi⟶Xi+1 where(34)fi=αi0βiIdM⊗gigi:Pi⊕M⊗BQiQi⟶Pi+1⊕M⊗BQi+1Qi+1,and we take βi=0∀i∈ℤ, i.e.,(35)fi=αi00IdM⊗gigi:Pi⊕M⊗BQiQi⟶Pi+1⊕M⊗BQi+1Qi+1.Then the condition (GP3) holds for eachi∈Z. We then get the following special kind of complete projective resolutions over Λ=AMBA0B.Proposition 1.
LetXi be a projective Λ-module, given as in (8), and fi:Xi⟶Xi+1 be a Λ-map, given as in (35). Then,(36)X•=⋯⟶f−2X−1⟶f−1X0⟶f0X1⟶f1⋯is a complete Λ-projective resolution if and only if(37)P•=⋯⟶α−2P−1⟶α−1P0⟶α0P1⟶α1⋯is a complete A-projective resolution,(38)Q•=⋯⟶g−2Q−1⟶g−1Q0⟶g0Q1⟶g1⋯is a complete B-projective resolution, and M⊗BQ• and HomAP•,M are exact sequences.Proof.
Assume that (36) is a complete Λ-projective resolution. Then, by Theorem 1, we know that conditions (GP1), (GP2), (GP4), and (GP5) are satisfied.
Letpi+1∈Kerαi+1. Since βi+1=0, it follows from (GP4) (by taking xi+1=0) that pi+1∈Imαi, i.e., Kerαi+1⊆Imαi. Together with (GP1), we see that the sequence P• in (37) is exact.
Sinceβi=0, (GP4) means that M⊗BQ• is an exact sequence.
Sinceβi=0, (GP5) means that HomAP•,A, HomAP•,M, and HomBQ•,B are exact sequences.
SinceP• and HomAP•,A are exact sequences, it follows that P• is a complete A-projective resolution. Similarly, Q• is a complete B-projective resolution.
Conversely, assume thatP• is a complete A-projective resolution, Q• is a complete B-projective resolution, and M⊗BQ• and HomAP•,M are exact sequences. Then, conditions (GP1), (GP2), (GP4), and (GP5) are satisfied, where βi=0,∀i∈ℤ. It follows from Theorem 1 that (36) is a complete Λ-projective resolution.


## 3.1. Morphisms of ProjectiveΛ-Modules

Note that any projectiveΛ-module is of the form(7)X=P⊕M⊗BQQ,where P and Q are projective A-module and projective B-module, respectively. For each i∈ℤ, we write(8)Xi=Pi⊕M⊗BQiQi.For such a projectiveΛ-module Xi as above and for any datum(9)Pi,Qi,αi,βi,gi,where αi:Pi⟶Pi+1 and βi:Pi⟶M⊗BQi+1 are A-maps and gi:Qi⟶Qi+1 is B-map ∀i∈ℤ, we put(10)fi=αi0βiIdM⊗gigi:Pi⊕M⊗BQiQi⟶Pi+1⊕M⊗BQi+1Qi+1.Lemma 1.
LetXi and Pi,Qi,αi,βi,gi be given as in (8) and (9). Then, fi:Xi⟶Xi+1 is a Λ-map if and only if fi is given in (10).Proof.
First, assume thatfi:Xi⟶Xi+1 is a Λ-map, where fi is given in (10). Since the diagram
is commutative, we see thatfi:Xi⟶Xi+1 is a Λ-map, ∀i∈ℤ.
Conversely, letfi:Xi⟶Xi+1 be an arbitrary Λ-map where Xi=Pi⊕M⊗BQiQi. Then, fi is of the form(11)αiωiβiδigi:Xi⟶Xi+1,where αi:Pi⟶Pi+1, βi:Pi⟶M⊗BQi+1, ωi:M⊗BQi⟶Pi+1, and δi:M⊗BQi⟶M⊗BQi+1 are A-maps and gi:Qi⟶Qi+1 is B-map, ∀i∈ℤ. By the definition of morphisms in Λ-mod, the diagram
is commutative. This implies that(12)ωi=0,δi=IdM⊗gi∀i∈ℤ.
Thus,fi is given as in (10). This completes the proof.


## 3.2. The Exactness of Morphisms of Projective Modules

Consider the following conditions:GP1P•=⋯⟶α−2P−1⟶α−1P0⟶α0P1⟶α1⋯ is a complex.GP2Q•=⋯⟶g−2Q−1⟶g−1Q0⟶g0Q1⟶g1⋯ is an exact sequence.GP3βi+1αi+IdM⊗gi+1βi=0,∀i∈ℤ.GP4 For any pi+1,xi+1∈Pi+1⊕M⊗BQi+1 with αi+1pi+1=0, βi+1pi+1+IdN⊗gi+1xi+1=0, there is p′i,x′i∈Pi⊕M⊗BQi such that pi+1=αip′i and xi+1=βip′i+IdM⊗gix′i,∀i∈ℤ.Lemma 2.
LetXi be a projective Λ-module given as in (8) and fi:Xi⟶Xi+1 be a Λ-map given as in (10). Then, Kerfi+1=Imfi if and only if conditions (GP1)–(GP4) are satisfied.Proof.
By Lemma1, fi is of the form fi=αi0βiIdM⊗gigi:Pi⊕M⊗BQiQi⟶Pi+1⊕M⊗BQi+1Qi+1,
Whereαi:Pi⟶Pi+1 and βi:Pi⟶M⊗BQi+1 and gi:Qi⟶Qi+1,∀i∈ℤ. We see that Kerfi+1=Imfi,∀i∈ℤ, if and only if(13)Kerαi+10βi+1IdM⊗gi+1=Imαi0βiIdM⊗gi,Kergi+1=Imgi,where(14)αi0βiIdM⊗gi:Pi⊕M⊗BQi⟶Pi+1⊕M⊗BQi+1,∀i∈ℤ.
Note thatKergi+1=Imgi,∀i∈ℤ, if and only if condition (GP2) is satisfied. Thus, it suffices to show that(15)Kerαi+10βi+1IdM⊗gi+1=Imαi0βiIdM⊗gi.
Now,Imαi0βiIdN⊗gi⊆Kerαi+10βi+1IdN⊗gi+1 if and only if αi+10βi+1IdN⊗gi+1αi0βiIdN⊗gi=0, i.e.,(16)αi+1αi0βi+1αi+IdN⊗gi+1βiIdN⊗gi+1gi=0,if and only if conditions (GP1)–(GP3) are satisfied.
We also see thatKerαi+10βi+1IdN⊗gi+1⊆Imαi0βiIdN⊗gi,∀i∈ℤ, if and only if condition GP4 is satisfied.
Thus,Kerfi+1=Imfi,∀i∈ℤ, if and only if conditions (GP1)–(GP4) are satisfied.
Consider the following condition (GP5)∀i∈ℤ:GP5 For any ci+1∈HomAPi+1,A,di+1∈HomAPi+1,M,ti+1∈HomBQi+1,B, such that(17)ci+1αi=0,di+1αi+IdM⊗ti+1βi=0,ti+1gi=0,there arec′i+2∈HomAPi+2,A,d′i+2∈HomAPi+2,M,t′i+2∈HomBQi+2,B, such that(18)ci+1=c′i+2αi+1,di+1=d′i+2αi+1+IdM⊗t′i+2βi+1,ti+1=t′i+2gi+1.Lemma 3.
LetXi be a projective Λ-module given as in (8) and f:Xi⟶Xi+1 be a Λ-map given as in (10). Assume that Kerfi+1=Imfi, i.e., conditions GP1–GP4 are satisfied. Consider the homomorphism of abelian groups fi∗=HomΛfi,Λ:HomΛXi+1,Λ⟶HomΛXi,Λ. Then, Kerfi∗=Imfi+1∗,∀i∈ℤ, if and only if condition (GP5) is satisfied.Proof.
Assume thatKerfi+1=Imfi. By Lemma 1, we know that fi:Xi⟶Xi+1 is given as(19)fi=αi0βiIdM⊗gigi:Pi⊕M⊗BQiQi⟶Pi+1⊕M⊗BQi+1Qi+1.
Any element in the abelian groupHomΛXi,Λ=HomΛPi⊕M⊗BQiQi,A⊕MB,∀i∈ℤ, will be written as(20)hi=cieidisiti:Pi⊕M⊗BQiQi⟶A⊕MB.
By definition of morphisms inΛ-mod, the diagram
is commutative. This implies that(21)ei=0,si=IdM⊗ti∀i∈ℤ.
Thus,(22)hi=ci0diIdN⊗titi:Pi⊕M⊗BQiQi⟶A⊕MB.
Hence,fi∗=HomΛfi,Λ:HomΛXi+1,Λ⟶HomΛXi,Λ,∀i∈ℤ, is written as(23)fi∗=HomΛαi0βiIdM⊗gigi,Λ:HomΛPi+1⊕M⊗BQi+1Qi+1,A⊕MB⟶HomΛPi⊕M⊗BQiQi,A⊕MB.
Thus,fi∗ maps(24)hi+1=ci+10di+1IdM⊗ti+1ti+1to(25)fi∗hi+1=hi+1fi=ci+10di+1IdM⊗ti+1αi0βiIdM⊗giti+1gi=ci+1αi0di+1αi+IdM⊗ti+1βiIdM⊗ti+1giti+1gi.
This means thathi+1=ci+10di+1IdM⊗ti+1ti+1∈Kerfi∗ if and only if(26)ci+1αi=0,di+1αi+IdM⊗ti+1βi=0,ti+1gi=0.
We see thathi+1=ci+10di+1IdM⊗ti+1ti+1∈Imfi+1∗ if and only if there is h′i+2=c′i+20d′i+2IdM⊗t′i+2t′i+2 such that(27)c′i+20d′i+2IdM⊗t′i+2αi+10βi+1IdM⊗gi+1t′i+2gi=ci+10di+1IdM⊗ti+1ti+1,which means that(28)ci+1=c′i+2αi+1,di+1=d′i+2αi+1+IdM⊗t′i+2βi+1,ti+1=t′i+2gi+1.
By conditions (G1)–(G4), we have(29)fi+1fi=αi+10βi+1IdM⊗gi+1αi0βiIdM⊗gigi+1gi=αi+1αi0βi+1αi+IdN⊗gi+1βiIdM⊗gi+1gigi+1gi=0.
Thus,fi∗fi+1∗hi+2=hi+2fi+1fi=0, i.e., Imfi+1∗⊆Kerfi∗. Therefore, Imfi+1∗=Kerfi∗ if and only if Imfi+1∗⊇Kerfi∗, if and only if for any hi=ci+10di+1IdN⊗ti+1ti+1:Pi+1⊕M⊗BQi+1Qi+1⟶A⊕MB with(30)ci+1αi=0,di+1αi+IdM⊗ti+1βi=0,ti+1gi=0,there is(31)h′i+2=c′i+20d′i+2IdM⊗t′i+2t′i+2:Pi+2⊕M⊗BQi+2Qi+2⟶A⊕MB,such that(32)ci+1=c′i+2αi+1,di+1=d′i+2αi+1+IdM⊗t′i+2βi+1,ti+1=t′i+2gi+1.
So, by the above arguments, we see thatImfi+1∗=Kerfi∗,∀i∈ℤ, if and only if condition (GP5) is satisfied. This completes the proof.


## 3.3. Main Result

Now, by Lemmas2 and 3, we get the following sufficient and necessary conditions of any complete projective resolutions over Λ=AMBA0B.Theorem 1.
LetXi be a projective Λ-module, given as in (8), and fi:Xi⟶Xi+1 be a Λ-map, given as in (10). Then,(33)X•=⋯⟶f−2X−1⟶f−1X0⟶f0X1⟶f1⋯is a complete Λ-projective resolution if and only if conditions (GP1)–(GP5) are satisfied.


## 3.4. Special Complete Projective Resolutions

We consider the special case off:Xi⟶Xi+1 where(34)fi=αi0βiIdM⊗gigi:Pi⊕M⊗BQiQi⟶Pi+1⊕M⊗BQi+1Qi+1,and we take βi=0∀i∈ℤ, i.e.,(35)fi=αi00IdM⊗gigi:Pi⊕M⊗BQiQi⟶Pi+1⊕M⊗BQi+1Qi+1.Then the condition (GP3) holds for eachi∈Z. We then get the following special kind of complete projective resolutions over Λ=AMBA0B.Proposition 1.
LetXi be a projective Λ-module, given as in (8), and fi:Xi⟶Xi+1 be a Λ-map, given as in (35). Then,(36)X•=⋯⟶f−2X−1⟶f−1X0⟶f0X1⟶f1⋯is a complete Λ-projective resolution if and only if(37)P•=⋯⟶α−2P−1⟶α−1P0⟶α0P1⟶α1⋯is a complete A-projective resolution,(38)Q•=⋯⟶g−2Q−1⟶g−1Q0⟶g0Q1⟶g1⋯is a complete B-projective resolution, and M⊗BQ• and HomAP•,M are exact sequences.Proof.
Assume that (36) is a complete Λ-projective resolution. Then, by Theorem 1, we know that conditions (GP1), (GP2), (GP4), and (GP5) are satisfied.
Letpi+1∈Kerαi+1. Since βi+1=0, it follows from (GP4) (by taking xi+1=0) that pi+1∈Imαi, i.e., Kerαi+1⊆Imαi. Together with (GP1), we see that the sequence P• in (37) is exact.
Sinceβi=0, (GP4) means that M⊗BQ• is an exact sequence.
Sinceβi=0, (GP5) means that HomAP•,A, HomAP•,M, and HomBQ•,B are exact sequences.
SinceP• and HomAP•,A are exact sequences, it follows that P• is a complete A-projective resolution. Similarly, Q• is a complete B-projective resolution.
Conversely, assume thatP• is a complete A-projective resolution, Q• is a complete B-projective resolution, and M⊗BQ• and HomAP•,M are exact sequences. Then, conditions (GP1), (GP2), (GP4), and (GP5) are satisfied, where βi=0,∀i∈ℤ. It follows from Theorem 1 that (36) is a complete Λ-projective resolution.


## 4. Gorenstein-Projective Modules

In this section, we describe all the Gorenstein-projective modules overΛ=AMBA0B.


### 4.1. Main Result

Keeping the notations in Section3, we put(39)N⊥=L∈A−mod|ExtAiL,N=0,∀i≥1.In the next result, we give sufficient and necessary conditions for all Gorenstein-projective modules overΛ=AMBA0B.Theorem 2.
LetΛ=AMBA0B be an Artin algebra and N be an Λ-module. Then, N is a Gorenstein-projective Λ-module if and only if one of the following cases holds:(1)
N=L0 with L∈GprojA and the sequence HomAP•,M is exact, where L admits a complete A-projective resolution(40)⋯⟶α−2P−1⟶α−1P0⟶α0P1⟶α1⋯.(2)
N=M⊗BKK with K∈GprojB, where K admits a complete B-projective resolution(41)⋯⟶g−2Q−1⟶g−1Q0⟶g0Q1⟶g1⋯,such thatKerIdM⊗gi=M⊗BKergi∀i∈ℤ.(3)
N is the Λ-module:(42)N=Kerf0=p0,x0∈P0⊕M⊗BQ0|α0p0=0,β0p0+IdM⊗g0x0=0Kerg0,where Pi,Qi,αi,gi,βi is as in (9) with Pi≠0≠Qi for some i and fi is given as in (10), such that conditions GP1–GP5 are satisfied.Proof.
First, we prove the sufficiency.
Suppose that case1 holds. So, N=L0 with L∈GprojA with L∈GprojA, where L=Kerα0∈GprojA with a complete A-projective resolution(43)P•=⋯⟶α−2P−1⟶α−1P0⟶α0P1⟶α1⋯,such that the sequence HomAP•,M is exact.
Thus, the sequence(44)U•=⋯⟶α−20P−10⟶α−10P00⟶α00P10⟶α10⋯is exact.
We know thatHomΛU•,MB≅HomAP•,M. By assumption, the sequence HomAP•,M is exact. Thus, HomΛU•,MB is exact sequence.
Note thatHomΛU•,A0≅HomAP•,A. Since P• is complete A-projective resolution, we have that the sequence HomAP•,A is exact. Thus, HomΛU•,A0 is exact sequence.
SinceΛ=AM0B≅A0⊕MB as left Λ-modules, HomΛU•,AM0B is exact. Thus, U• is a complete Λ-projective resolution. Thus,(45)Kerα00=Kerα00=L0=Nis a Gorenstein-projective Λ-module.
Suppose that case (2) holds. That is,N=M⊗BKK with K∈GprojB, where K admits a complete B-projective resolution(46)⋯⟶g−2Q−1⟶g−1Q0⟶g0Q1⟶g1⋯,such that KerIdM⊗gi=M⊗BKergi∀i∈ℤ.
SinceKerIdM⊗gi=M⊗BKergi∀i∈ℤ, the sequence(47)⋯⟶IdN⊗g−2M⊗BQ−1⟶IdM⊗g−1M⊗BQ0⟶IdM⊗g0M⊗BQ1⟶IdM⊗g1⋯is exact. Thus, the sequence(48)W•=⋯⟶IdN⊗g−2g−2M⊗BQ−1Q−1⟶IdM⊗g−1g−1N⊗BQ0Q0⟶IdM⊗g0g0N⊗BQ1Q1⟶IdM⊗g1g1⋯is exact.
Note thatHomΛW•,A0=0. We have that(49)HomΛW•,MB≅HomBQ•,Bis exact, i.e., the sequence HomΛW•,MB is exact.
SinceΛ=AM0B≅A0⊕MB as left Λ-module, we have that W• is a complete Λ-projective resolution. Thus,(50)KerIdM⊗g0g0=KerIdM⊗g0Kerg0=M⊗BKK=Nis a Gorenstein-projective Λ-module.
Suppose that case (3) holds. Then, by Theorem1, the sequence(51)X•≔⋯⟶f−2X−1⟶f−1X0⟶f0X1⟶f1⋯is a complete Λ-projective resolution, where Xi is as given in (8). Thus, G=Kerf0 is a Gorenstein-projective Λ-module.
Now, we show the necessity. LetN be a Gorenstein-projective Λ-module. Then, N=Kerf0, where fi is the Λ-map occurring in a complete Λ-projective resolution of X•. By Theorem 1, X• is of form (33), where Xi and fi are given in (8) and (10), respectively, satisfying conditions GP1–GP5.
IfQi=0,∀i∈ℤ, in (8), then βi=0 and gi=0, ∀i∈ℤ. By GP1 and GP4, we see that(52)P•:⋯⟶α−2P−1⟶α−1P0⟶α0P1⟶α1⋯is a complete A-projective resolution, and GP5 means that HomAP•,M is exact. Thus, N=Kerf0=Kerα00 is of form (1).
IfPi=0,∀i∈ℤ, in (8), then αi=0 and βi=0, ∀i∈ℤ. By GP2 and GP5, we have that(53)Q•:⋯⟶g−2Q−1⟶g−1Q0⟶g0Q1⟶g1⋯is a complete B-projective resolution, and GP5 means that KerIdN⊗gi=N⊗Kergi∀i∈ℤ. Thus, Kerg0∈GprojB, and N=Kerf0=N⊗BKerg0Kerg0 is of form (2).
IfPi≠0 and Qi≠0 for some i∈ℤ in (9), then N=Kerf0 is of form (3) by Theorem 1. This completes the proof.


### 4.2. Special Gorenstein-Projective Modules

Corollary 1.
Suppose thatL is a Gorenstein-projective A-module with respect to the complete A-projective resolution(54)P•=⋯⟶α−2P−1⟶α−1P0⟶α0P1⟶α1⋯,and suppose that K is a Gorenstein-projective B-module with respect to the complete B-projective resolution(55)Q•=⋯⟶g−2Q−1⟶g−1Q0⟶g0Q1⟶g1⋯.Assume thatM⊗BQ• and HomAP•,M are exact sequences. Then, Lo⊕N⊗BKK is a Gorenstein-projective Λ-module with respect to the complete Λ-projective resolution(56)X•=⋯⟶f−1P00⊕M⊗Q0Q0⟶f0P10⊕M⊗Q1Q1⟶f1⋯,where Λ-map is fi=αi000IdM⊗gigi:Pi0⊕M⊗BQiQi⟶Pi+10⊕M⊗BQi+1Qi+1.Proof.
SinceM⊗BQ• is exact, we have that X• is exact.
Since each term in the complexX• is projective, applying HomΛX•,− to the exact sequence(57)0⟶M0⟶MB⟶0B⟶0,we get following exact sequence of complexes:(58)0⟶HomΛX•,M0⟶HomΛX•,MB⟶HomΛX•,0B⟶0.
SinceQ• is a complete B-projective resolution, it follows that HomΛX•,0B≅HomBQ•,B is exact. By assumption, HomAQ•,M is exact. Thus, it follows that HomΛX•,MB is also exact.
SinceHomΛX•,A0≅HomAP•,A and HomAP•,A is exact, it follows that HomΛX•,A0 is exact.
All together, we see thatX• is a complete Λ-projective resolution. Thus, Kerf0=L0⊕M⊗BKK is a Gorenstein-projective Λ-module.


## 4.1. Main Result

Keeping the notations in Section3, we put(39)N⊥=L∈A−mod|ExtAiL,N=0,∀i≥1.In the next result, we give sufficient and necessary conditions for all Gorenstein-projective modules overΛ=AMBA0B.Theorem 2.
LetΛ=AMBA0B be an Artin algebra and N be an Λ-module. Then, N is a Gorenstein-projective Λ-module if and only if one of the following cases holds:(1)
N=L0 with L∈GprojA and the sequence HomAP•,M is exact, where L admits a complete A-projective resolution(40)⋯⟶α−2P−1⟶α−1P0⟶α0P1⟶α1⋯.(2)
N=M⊗BKK with K∈GprojB, where K admits a complete B-projective resolution(41)⋯⟶g−2Q−1⟶g−1Q0⟶g0Q1⟶g1⋯,such thatKerIdM⊗gi=M⊗BKergi∀i∈ℤ.(3)
N is the Λ-module:(42)N=Kerf0=p0,x0∈P0⊕M⊗BQ0|α0p0=0,β0p0+IdM⊗g0x0=0Kerg0,where Pi,Qi,αi,gi,βi is as in (9) with Pi≠0≠Qi for some i and fi is given as in (10), such that conditions GP1–GP5 are satisfied.Proof.
First, we prove the sufficiency.
Suppose that case1 holds. So, N=L0 with L∈GprojA with L∈GprojA, where L=Kerα0∈GprojA with a complete A-projective resolution(43)P•=⋯⟶α−2P−1⟶α−1P0⟶α0P1⟶α1⋯,such that the sequence HomAP•,M is exact.
Thus, the sequence(44)U•=⋯⟶α−20P−10⟶α−10P00⟶α00P10⟶α10⋯is exact.
We know thatHomΛU•,MB≅HomAP•,M. By assumption, the sequence HomAP•,M is exact. Thus, HomΛU•,MB is exact sequence.
Note thatHomΛU•,A0≅HomAP•,A. Since P• is complete A-projective resolution, we have that the sequence HomAP•,A is exact. Thus, HomΛU•,A0 is exact sequence.
SinceΛ=AM0B≅A0⊕MB as left Λ-modules, HomΛU•,AM0B is exact. Thus, U• is a complete Λ-projective resolution. Thus,(45)Kerα00=Kerα00=L0=Nis a Gorenstein-projective Λ-module.
Suppose that case (2) holds. That is,N=M⊗BKK with K∈GprojB, where K admits a complete B-projective resolution(46)⋯⟶g−2Q−1⟶g−1Q0⟶g0Q1⟶g1⋯,such that KerIdM⊗gi=M⊗BKergi∀i∈ℤ.
SinceKerIdM⊗gi=M⊗BKergi∀i∈ℤ, the sequence(47)⋯⟶IdN⊗g−2M⊗BQ−1⟶IdM⊗g−1M⊗BQ0⟶IdM⊗g0M⊗BQ1⟶IdM⊗g1⋯is exact. Thus, the sequence(48)W•=⋯⟶IdN⊗g−2g−2M⊗BQ−1Q−1⟶IdM⊗g−1g−1N⊗BQ0Q0⟶IdM⊗g0g0N⊗BQ1Q1⟶IdM⊗g1g1⋯is exact.
Note thatHomΛW•,A0=0. We have that(49)HomΛW•,MB≅HomBQ•,Bis exact, i.e., the sequence HomΛW•,MB is exact.
SinceΛ=AM0B≅A0⊕MB as left Λ-module, we have that W• is a complete Λ-projective resolution. Thus,(50)KerIdM⊗g0g0=KerIdM⊗g0Kerg0=M⊗BKK=Nis a Gorenstein-projective Λ-module.
Suppose that case (3) holds. Then, by Theorem1, the sequence(51)X•≔⋯⟶f−2X−1⟶f−1X0⟶f0X1⟶f1⋯is a complete Λ-projective resolution, where Xi is as given in (8). Thus, G=Kerf0 is a Gorenstein-projective Λ-module.
Now, we show the necessity. LetN be a Gorenstein-projective Λ-module. Then, N=Kerf0, where fi is the Λ-map occurring in a complete Λ-projective resolution of X•. By Theorem 1, X• is of form (33), where Xi and fi are given in (8) and (10), respectively, satisfying conditions GP1–GP5.
IfQi=0,∀i∈ℤ, in (8), then βi=0 and gi=0, ∀i∈ℤ. By GP1 and GP4, we see that(52)P•:⋯⟶α−2P−1⟶α−1P0⟶α0P1⟶α1⋯is a complete A-projective resolution, and GP5 means that HomAP•,M is exact. Thus, N=Kerf0=Kerα00 is of form (1).
IfPi=0,∀i∈ℤ, in (8), then αi=0 and βi=0, ∀i∈ℤ. By GP2 and GP5, we have that(53)Q•:⋯⟶g−2Q−1⟶g−1Q0⟶g0Q1⟶g1⋯is a complete B-projective resolution, and GP5 means that KerIdN⊗gi=N⊗Kergi∀i∈ℤ. Thus, Kerg0∈GprojB, and N=Kerf0=N⊗BKerg0Kerg0 is of form (2).
IfPi≠0 and Qi≠0 for some i∈ℤ in (9), then N=Kerf0 is of form (3) by Theorem 1. This completes the proof.


## 4.2. Special Gorenstein-Projective Modules

Corollary 1.
Suppose thatL is a Gorenstein-projective A-module with respect to the complete A-projective resolution(54)P•=⋯⟶α−2P−1⟶α−1P0⟶α0P1⟶α1⋯,and suppose that K is a Gorenstein-projective B-module with respect to the complete B-projective resolution(55)Q•=⋯⟶g−2Q−1⟶g−1Q0⟶g0Q1⟶g1⋯.Assume thatM⊗BQ• and HomAP•,M are exact sequences. Then, Lo⊕N⊗BKK is a Gorenstein-projective Λ-module with respect to the complete Λ-projective resolution(56)X•=⋯⟶f−1P00⊕M⊗Q0Q0⟶f0P10⊕M⊗Q1Q1⟶f1⋯,where Λ-map is fi=αi000IdM⊗gigi:Pi0⊕M⊗BQiQi⟶Pi+10⊕M⊗BQi+1Qi+1.Proof.
SinceM⊗BQ• is exact, we have that X• is exact.
Since each term in the complexX• is projective, applying HomΛX•,− to the exact sequence(57)0⟶M0⟶MB⟶0B⟶0,we get following exact sequence of complexes:(58)0⟶HomΛX•,M0⟶HomΛX•,MB⟶HomΛX•,0B⟶0.
SinceQ• is a complete B-projective resolution, it follows that HomΛX•,0B≅HomBQ•,B is exact. By assumption, HomAQ•,M is exact. Thus, it follows that HomΛX•,MB is also exact.
SinceHomΛX•,A0≅HomAP•,A and HomAP•,A is exact, it follows that HomΛX•,A0 is exact.
All together, we see thatX• is a complete Λ-projective resolution. Thus, Kerf0=L0⊕M⊗BKK is a Gorenstein-projective Λ-module.



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