The tensor product of Gorenstein-projective modules over category algebras

0.18 MB·

by  Ren Wang

THE TENSOR PRODUCT OF GORENSTEIN-PROJECTIVE
MODULES OVER CATEGORY ALGEBRAS
RENWANG
7 Abstract. ForafinitefreeandprojectiveEIcategory,weprovethatGorenstein-
1 projectivemodulesoveritscategoryalgebraareclosedunderthetensorprod-
0 uctifandonlyifeachmorphisminthegivencategory isamonomorphism.
2

n
a
J
1. Introduction
6
1
Let k be a field. Let C be a finite category, that is, it has only finitely many

] morphisms, and consequently it has only finitely many objects. Denote by k-mod
T C
the category of finite dimensional k-vector spaces and (k-mod) the category of
R covariant functors from C to k-mod.
.
h Recall that the category kC-mod of left modules over the category algebra kC
at is identified with (k-mod)C; see [7, Proposition 2.1]. The category kC-mod is a
m symmetric monoidalcategory. More precisely,the tensor product -⊗ˆ- is defined by
[ (M⊗ˆN)(x)=M(x)⊗kN(x)

1 for any M,N ∈ (k-mod)C and x ∈ ObjC, and α.(m⊗n) = α.m⊗α.n for any
v
α∈MorC,m∈M(x),n∈N(x); see [8, 9].
9
6 Let C be a finite EI category. Here, the EI condition means that all endomor-
1
4 phisms in C are isomorphisms. In particular, HomC(x,x) = AutC(x) is a finite
0 group for each object x. Denote by kAutC(x) the group algebra. Recall that
. a finite EI category C is projective over k if each kAutC(y)-kAutC(x)-bimodule
1
0 kHomC(x,y) is projective on both sides; see [5, Definition 4.2].
7 Denote by kC-Gproj the full subcategory of kC-mod consisting of Gorenstein-
1
projective kC-modules. We say that C is GPT-closed, if X,Y ∈kC-Gproj implies
:
v X⊗ˆY ∈kC-Gproj.
i
X
Let us explain the motivation to study GPT-closed categories. Recall from [6]
r that for a finite projective EI category C, the stable category kC-Gproj modulo
a
projectivemoduleshasanaturaltensortriangulatedstructuresuchthatitistensor
triangleequivalentto the singularitycategoryofkC. Ingeneral,its tensorproduct
isnotexplicitlygiven. However,ifC isGPT-closed,thenthetensorproduct-⊗ˆ-on
Gorenstein-projective modules induces the one on kC-Gproj; see Proposition 3.4.
In this case, we have a better understanding of the tensor triangulated category
kC-Gproj.
Proposition 1.1. Let C be a finite projective EI category. Assume that C is
GPT-closed. Then each morphism in C is a monomorphism.
Date:February12,2017.
2010 Mathematics Subject Classification. Primary16G10; Secondary16D90,18E30.
Key words and phrases. finite EI category, category algebra, Gorenstein-projective module,
tensorproduct,tensor triangulatedcategory.
1
2 RENWANG
The concept of a finite free EI category is introduced in [3].
Theorem 1.2. Let C be a finiteprojective and free EI category. Then the category
C is GPT-closed if and only if each morphism in C is a monomorphism.
2. Gorenstein triangular matrix algebras
In this section, we recall some necessary preliminaries on triangular matrix alge-
bras and Gorenstein-projective modules.
R M ···M
1 12 1n
 R2 ···M2n
Recall an n×n upper triangular matrix algebra Γ =  ... ... , where
 
 R 
n
 
each R is an algebra for 1 ≤ i ≤ n, each M is an R -R -bimodule for 1 ≤ i <
i ij i j
j ≤ n, and the matrix algebra map is denoted by ψ : M ⊗ M → M for
ilj il Rl lj ij
1≤i<l <j <t≤n; see [5].
X
1
Recall that a left Γ-module X =  ..  is described by a column vector, where
.
X 
n
 
each X is a left R -module for 1 ≤ i ≤ n, and the left Γ-module structure map is
i i
denoted by ϕ : M ⊗ X → X for 1 ≤ j < l ≤ n; see [5]. Dually, we have the
jl jl Rl l j
description of right Γ-modules via row vectors.
Notation 2.1. Let Γ be the algebra given by the t×t leading principal submatrix
t
of Γ and Γ′ be the algebra given by cuttingthe first t rows and the first t columns
n−t
M
1,t+1
.
of Γ. Denote the left Γt-module  ..  by Mt∗ and the right Γ′n−t-module
M 
t,t+1
 
M ,··· ,M by M∗∗, for 1≤t≤n−1. Denote by ΓD= diag R ,···R the
t,t+1 tn t t 1 t
s(cid:0)ub-algebra of Γ (cid:1)consisting of diagonal matrices, and Γ′ = dia(cid:0)g R ,···(cid:1)R
t D,n−t t+1 n
the sub-algebra of Γ′ consisting of diagonal matrices. (cid:0) (cid:1)
n−t
R M
LetΓ= 1 12 beanuppertriangularmatrixalgebra. RecallthattheR -R -
(cid:18) 0 R2 (cid:19) 1 2
bimodule M is compatible, if the following two conditions hold; see[10, Definition
12
1.1]:
(C1) If Q• is an exact sequence of projective R -modules, then M ⊗ Q• is
2 12 R2
exact;
(C2) IfP• isacompleteR -projectiveresolution,thenHom (P•,M )isexact.
1 R1 12
We use pdandidto denote the projectivedimensionandthe injective dimension
of a module, respectively.
Lemma 2.2. Let Γ be an upper triangular matrix algebra satisfying all R Goren-
i
stein. If Γ is Gorenstein, then each Γ -R -bimodule M∗ is compatible and each
t t+1 t
R -Γ′ -bimodule M∗∗ is compatible for 1≤t≤n−1.
t n−t t
S N
Proof. Let Λ = 1 12 be an upper triangular matrix algebra. Recall the fact
(cid:18) 0 S2 (cid:19)
that if pd (N ) < ∞ and pd(N ) < ∞, then N is compatible; see [10,
S1 12 12 S2 12
Proposition 1.3]. Recall that Γ is Gorenstein if and only if all bimodules M
ij
THE TENSOR PRODUCT OF GORENSTEIN-PROJECTIVE MODULES 3
are finitely generated and have finite projective dimension on both sides; see [5,
Proposition 3.4]. Then we have pd (M∗∗) < ∞ and pd(M∗) < ∞ for 1 ≤
Rt t t Rt+1
t1≤≤nt≤−n1.−B1y.[T5,hLenemwmeaar3e.1d]o,nwee. have pd(Mt∗∗)Γ′n−t <∞ and pdΓt(Mt∗)<∞ fo(cid:3)r
LetAbeafinitedimensionalalgebraoverafieldk. DenotebyA-modthecategory
of finite dimensional left A-modules. The opposite algebraof A is denoted by Aop.
We identify right A-modules with left Aop-modules.
Denote by (−)∗ the contravariantfunctor HomA(−,A) or HomAop(−,A). Let X
be a left A-module. Then X∗ is a right A-module and X∗∗ is a left A-module.
There is an evaluation map ev : X →X∗∗ given by ev (x)(f)=f(x) for x∈X
X X
and f ∈ X∗. Recall that an A-module G is Gorenstein-projective provided that
Exti (G,A)=0=Exti (G∗,A)fori≥1andthe evaluationmapev isbijective;
A Aop G
see [1, Proposition 3.8].
The algebra A is Gorenstein if id A<∞ and idA <∞. It is well known that
A A
for a Gorenstein algebra A we have id A= idA ; see [11, Lemma A]. For m ≥0,
A A
a Gorenstein algebra A is m-Gorenstein if id A=idA ≤m. Denote by A-Gproj
A A
the full subcategory of A-mod consisting of Gorenstein-projective A-modules, and
A-proj the full subcategory of A-mod consisting of projective A-modules.
Thefollowinglemmaiswellknown;see[1,Propositions3.8and4.12andTheorem
3.13].
Lemma 2.3. Let m ≥ 0. Let A be an m-Gorenstein algebra. Then we have the
following statements.
(1) An A-module M ∈A-Gproj if and only if Exti (M,A)=0 for all i>0.
A
(2) IfM ∈A-GprojandLis aright A-modulewithfiniteprojective dimension,
then TorA(L,M)=0 for all i>0.
i
(3) If there is an exact sequence 0 → M → G → G → ··· → G with G
0 1 m i
Gorenstein-projective, then M ∈A-Gproj. (cid:3)
Lemma 2.4. [10, Theorem 1.4] Let M be a compatible R -R -bimodule, and
12 1 2
R M X
Γ= 1 12 . ThenX = 1 ∈Γ-Gprojifandonlyifϕ :M ⊗ X →X
(cid:18) 0 R2 (cid:19) (cid:18)X2(cid:19) 12 12 R2 2 1
is an injective R -map, Cokerϕ ∈R -Gproj, and X ∈R -Gproj. (cid:3)
1 12 1 2 2
We have a slight generalization of Lemma 2.4.
Lemma 2.5. Let Γ be a Gorenstein upper triangular matrix algebra with each
X
1
.
Ri a group algebra. Then X =  ..  ∈ Γ-Gproj if and only if each Rt-map
X 
n
 
X x
t+1 t+1
n . .
ϕ∗t∗ = ϕtj :Mt∗∗⊗Γ′n−t .. →Xt sending mt,t+1,··· ,mtn ⊗ .. 
j=Pt+1  X  (cid:0) (cid:1)  x 
n n
   
n
to ϕ (m ⊗x ) is injective for 1≤t≤n−1.
tj tj j
j=Pt+1
Proof. We have that each R -Γ′ -bimodule M∗∗ is compatible for 1 ≤ t ≤ n−1
t n−t t
by Lemma 2.2.
4 RENWANG
Forthe“onlyif”part,weuseinductiononn. Thecasen=2isduetoLemma2.4.
R M∗∗ X
Assume that n > 2. Write Γ = 1 1 , and X = 1 . Since X ∈ Γ-
(cid:18) 0 Γ′ (cid:19) (cid:18)X′(cid:19)
n−1
Gproj, by Lemma 2.4, we have that the R1-map ϕ∗1∗ : M1∗∗ ⊗Γ′n−1 X′ → X1 is
injective and X′ ∈ Γ′ -Gproj. By induction, we have that each R -map ϕ∗∗ :
n−1 t t
X
t+1
Mt∗∗⊗Γ′n−t  ... →Xt is injective for 1≤t≤n−1.
 X 
n
 
For the “if” part, we use induction on n. The case n = 2 is due to Lemma 2.4.
R M∗∗ X
Assume that n > 2. Write Γ = 1 1 , and X = 1 . By induction, we
(cid:18) 0 Γ′ (cid:19) (cid:18)X′(cid:19)
n−1
have X′ ∈ Γ′n−1-Gproj. Since the R1-map ϕ∗1∗ : M1∗∗⊗Γ′n−1 X′ → X1 is injective
andits cokernelbelongsto R -GprojforR agroupalgebra,wehaveX ∈Γ-Gproj
1 1
by Lemma 2.4. (cid:3)
Corollary 2.6. Let Γ be a Gorenstein upper triangular matrix algebra with each
X
1
.
 . 
.
Ri a group algebra. Assume that X = X0s ∈ Γ-Gproj. Then each Ri-map
 
 .. 
 . 
 
 0 
 
ϕ :M ⊗ X →X is injective for 1≤i<s≤n.
is is Rs s i
X
i+1
.
 . 
.
X′
Proof. We write X = (cid:18)X′′(cid:19), where X′′ =  Xs  for each 1 ≤ i < s ≤ n. We
 .. 
 . 
 
 0 
 
claim that each Ri-map fis : Mis ⊗Rs Xs → Mi∗∗ ⊗Γ′n−i X′′ sending mis ⊗xs to
0,··· ,m ,··· ,0 ⊗ 0,··· ,x ,··· ,0 t is injective, where (−)t is the transpose.
is s
(cid:0)Since ϕis =ϕ∗i∗◦f(cid:1)is fo(cid:0)r 1≤i<s≤n,(cid:1)then we are done by Lemma 2.5.
For the claim, we observe that for each 1 ≤ i < s ≤ n, the R -map f is a
i is
composition of the following
Mis⊗Rs Xs −g→is 0,··· ,0,Mis,··· ,Min ⊗Γ′n−i X′′ ι−⊗→IdMi∗∗⊗Γ′n−i X′′,
(cid:0) (cid:1)
where the right Γ′ -map 0,··· ,M ,··· ,M −→ι M∗∗ is the inclusion map
n−i is in i
and g sends m ⊗x to (cid:0)0,··· ,m ,··· ,0 ⊗ (cid:1)0,··· ,x ,··· ,0 t. We observe a
is is s is s
(cid:0) g′ (cid:1) (cid:0) (cid:1)
Ri-map 0,··· ,Mis,··· ,Min ⊗Γ′n−iX′′ −→is Mis⊗RsXs, 0,··· ,mis,··· ,min ⊗
0,··· ,x(cid:0) ,··· ,0 t 7→m ⊗x(cid:1)satisfyingg′ ◦g =Id (cid:0) . HencetheR -m(cid:1)ap
s is s is is Mis⊗RsXs i
(cid:0)g isinjective. W(cid:1)eobservethattherightΓ′ -modules 0,··· ,M ,··· ,M and
is n−i is in
M∗∗ have finite projective dimensions; see [5, Lemma 3(cid:0).1], and X′′ ∈ Γ′ -G(cid:1)proj
i n−i
by Lemma 2.4. Then the R -map ι⊗Id is injective by Lemma 2.3 (2). (cid:3)
i
THE TENSOR PRODUCT OF GORENSTEIN-PROJECTIVE MODULES 5
3. Proof of Proposition 1.1
Let k be a field. Let C be a finite category, that is, it has only finitely many
morphisms, and consequently it has only finitely many objects. Denote by MorC
the finite set of all morphisms in C. The category algebra kC of C is defined as
follows: kC = kα asa k-vectorspaceandthe product∗ is givenby the rule
α∈LMorC
α◦β, if α and β can be composed in C;
α∗β =
(cid:26)0, otherwise.
The unit is given by 1kC = Idx, where Idx is the identity endomorphism of
x∈PObjC
an object x in C.
If C and D are two equivalent finite categories, then kC and kD are Morita
equivalent; see [7, Proposition 2.2]. In particular, kC is Morita equivalent to kC ,
0
where C is any skeleton of C. So we may assume that C is skeletal, that is, for
0
any two distinct objects x and y in C, x is not isomorphic to y.
The category C is called a finite EI category provided that all endomorphisms
in C are isomorphisms. In particular, HomC(x,x) =AutC(x) is a finite group for
any object x in C. Denote by kAutC(x) the group algebra.
Throughoutthispaper,weassumethatC isafiniteEIcategorywhichisskeletal,
and ObjC ={x1,x2,··· ,xn}, n≥2, satisfying HomC(xi,xj)=∅ if i<j.
Let Mij = kHomC(xj,xi). Write Ri = Mii, which is a group algebra. Recall
that the category algebra kC is isomorphic to the corresponding upper triangular
R M ···M
1 12 1n
 R2 ···M2n
matrix algebra ΓC = ... ... ; see [5, Section 4].
 
 R 
n
 
Let us fix a field k. Denote by k-mod the category of finite dimensional k-vector
spaces and (k-mod)C the category of covariant functors from C to k-mod.
Recall that the category kC-mod of left modules over the category algebra kC,
is identified with (k-mod)C; see [7, Proposition 2.1]. The category kC-mod is a
symmetric monoidalcategory. More precisely,the tensor product -⊗ˆ- is defined by
(M⊗ˆN)(x)=M(x)⊗ N(x)
k
for any M,N ∈ (k-mod)C and x ∈ ObjC, and α.(m⊗n) = α.m⊗α.n for any
α∈MorC,m∈M(x),n∈N(x); see [8, 9].
R M ···M
1 12 1n
 R2 ···M2n
Let C be a finite EI category, and Γ = ΓC =  ... ...  be the cor-
 
 R 
n
 
X Y
1 1
responding upper triangular matrix algebra. Let X =  ..  and Y =  .. 
. .
X  Y 
n n
   
be two Γ-modules, where the left Γ-module structure maps are denoted by ϕX
ij
X ⊗ Y
1 k 1
and ϕY, respectively. We observe that X⊗ˆY =  .. , where the module
ij .
X ⊗ Y 
n k n
 
6 RENWANG
structure map ϕ : M ⊗ (X ⊗ Y ) → X ⊗ Y is induced by the following:
ij ij Rj j k j i k i
ϕij(αij ⊗(aj ⊗bj)) = ϕXij(αij ⊗aj)⊗ϕYij(αij ⊗bj), where αij ∈ HomC(xj,xi),
a ∈X and b ∈Y .
j j j j
Definition 3.1. Let C be a finite EI category, and Γ be the corresponding upper
triangularmatrixalgebra. WesaythatC is GPT-closed, ifX,Y ∈Γ-Gprojimplies
X⊗ˆY ∈Γ-Gproj.
RecallthatafiniteEIcategoryC isprojective over kifeachkAutC(y)-kAutC(x)-
bimodule kHomC(x,y)isprojectiveonbothsides; see[5,Definition4.2]. We recall
the fact that the category algebra kC is Gorenstein if and only if C is projective
over k; see [5, Proposition 5.1].
LetC be a finite projectiveEIcategory,andΓ be the correspondingupper trian-
gular matrix algebra. Denote by C the i-thcolumn of Γ whichis a Γ-R -bimodule
i i
and projective on both sides.
Proposition3.2. LetC beafiniteprojectiveEIcategory, andΓbethecorrespond-
ing upper triangular matrix algebra. Then the following statements are equivalent.
(1) The category C is GPT-closed.
(2) For any 1≤p≤q ≤n, C ⊗ˆC ∈Γ-Gproj.
p q
(3) For any 1≤p≤q ≤n, C ⊗ˆC ∈Γ-proj.
p q
Proof. “(1)⇒ (2)” and “(3)⇒ (2)” are obvious.
“(2)⇒(3)”WeonlyneedtoprovethattheΓ-moduleC ⊗ˆC hasfiniteprojective
p q
dimension, since a Gorenstein-projective module with finite projective dimension
M ⊗ M
1p k 1q
.
 . 
.
Mp−1,p⊗kMp−1,q
is projective. We have Cp⊗ˆCq = Rp⊗kMpq . Since C is projective, we
 
 0 
 
 .. 
 . 
 
 0 
 
havethateachM is a projectiveR -module for1≤i≤p. TheneachM ⊗ M
ip i ip k iq
is a projective R -module since R is a group algebra for 1 ≤ i ≤ p. Hence the
i i
Γ-moduleC ⊗ˆC hasfinite projectivedimensionby[5,Corollary3.6]. Thenweare
p q
done.
“(2)⇒ (1)” We have that Γ is a Gorensteinalgebraby [5, Proposition5.1]. Then
there is d≥0 such that Γ is a d-Gorenstein algebra.
For any M ∈Γ-Gproj, consider the following exact sequence
0→M →P →P →···→P →Y →0
0 1 d
withP projective,0≤i≤d. Applying -⊗ˆN onthe aboveexactsequence,wehave
i
an exact sequence
0→M⊗ˆN →P ⊗ˆN →P ⊗ˆN →···→P ⊗ˆN →Y⊗ˆN →0, (3.1)
0 1 d
since the tensor product -⊗ˆ- is exact in both variables. If N is projective, we have
that each P ⊗ˆN is Gorenstein-projective for 0 ≤ i ≤ d by (2). Then we have
i
M⊗ˆN ∈ Γ-Gproj by Lemma 2.3 (3). If N is Gorenstein-projective, we have that
each P ⊗ˆN is Gorenstein-projective for 0 ≤ i ≤ d in exact sequence (3.1) by the
i
above process. Then we have M⊗ˆN ∈ Γ-Gproj by Lemma 2.3 (3). Then we are
done. (cid:3)
THE TENSOR PRODUCT OF GORENSTEIN-PROJECTIVE MODULES 7
The argument in “(2)⇒ (3)” of Proposition 3.2 implies the following result. It
follows that the tensor product -⊗ˆ- on Γ-Gproj induces the one on Γ-Gproj, still
denoted by -⊗ˆ-.
Lemma 3.3. Assume that C is GPT-closed. Let M ∈ Γ-Gproj and P ∈ Γ-proj.
Then M⊗ˆP ∈Γ-proj. (cid:3)
LetC beafiniteprojectiveEIcategory,andΓbethecorrespondinguppertriangu-
larmatrixalgebraofC. RecallthatacomplexinDb(Γ-mod), the boundedderived
categoryof finitely generatedleft Γ-modules, is called a perfect complex if it is iso-
morphictoaboundedcomplexoffinitelygeneratedprojectivemodules. Recallfrom
[2] that the singularity category of Γ, denoted by D (Γ), is the Verdier quotient
sg
category Db(Γ-mod)/perf(Γ), where perf(Γ) is a thick subcategory of Db(Γ-mod)
consisting of all perfect complexes.
Recall from [6] that there is a triangle equivalence
∼
F :Γ-Gproj−→D (Γ) (3.2)
sg
sending a Gorenstein-projective module to the corresponding stalk complex con-
centrated on degree zero. The functor F transports the tensor product on D (Γ)
sg
toΓ-GprojsuchthatthecategoryΓ-Gprojbecomesatensortriangulatedcategory.
Proposition 3.4. Let C be a finite projective EI category, and Γ be the corre-
sponding upper triangular matrix algebra of C. Assume that the category C is
GPT-closed. Then thetensorproduct -⊗ˆ-on Γ-Gprojinducedbythetensor product
on Γ-Gproj coincide with the one transported from D (Γ), up to natural isomor-
sg
phism.
Proof. Consider the functor F in (3.2). Recall that the tensor product on D (Γ)
sg
isinducedbythetensorproduct-⊗ˆ-onDb(Γ-mod),wherethe laterisgivenby-⊗ˆ-
onΓ-mod. We haveF(M)⊗ˆF(N)=F(M⊗ˆN)inD (Γ) forany M,N ∈Γ-Gproj.
sg
This implies that F is a tensor triangle equivalence. Then we are done. (cid:3)
Let k be a field and G be a finite group. Recall that a left (resp. right) G-set
is a set with a left (resp. right) G-action. Let Y be a left G-set and X be a
right G-set. Recall an equivalence relation “∼” on the product X ×Y as follows:
(x,y) ∼ (x′,y′) if and only if there is an element g ∈ G such that x = x′g and
y =g−1y′ forx,x′ ∈X andy,y′ ∈Y. Writethe quotientsetX×Y/∼asX× Y.
G
The following two lemmas are well known.
Lemma 3.5. Let Y be a left G-set and X be a right G-set. Then there is an
isomorphism of k-vector spaces
∼
ϕ:kX⊗ kY −→k(X × Y), x⊗y 7→(x,y),
kG G
where x∈X and y ∈Y. (cid:3)
Lemma 3.6. Let Y and Y be two left G-sets. Then we have an isomorphism of
1 2
left kG-modules
∼
ϕ:kY ⊗ kY −→k(Y ×Y ), y ⊗y 7→(y ,y ),
1 k 2 1 2 1 2 1 2
where y ∈Y ,y ∈Y . (cid:3)
1 1 2 2
Lemma 3.7. Let C be a finite projective EI category, and Γ be the corresponding
upper triangular matrix algebra. Assume 1 ≤ p ≤ q ≤ n. Then C ⊗ˆC ∈ Γ-proj
p q
implies that each morphism in y∈ObjC HomC(xp,y) is a monomorphism.
F
8 RENWANG
M ⊗ M
1p k 1q
.
 . 
.
Mp−1,p⊗kMp−1,q
Proof. We have Cp⊗ˆCq = Rp⊗kMpq . Then each Ri-map
 
 0 
 
 .. 
 . 
 
 0 
 
ϕ :M ⊗ (R ⊗ M )→M ⊗ M
ip ip Rp p k pq ip k iq
sending α⊗(g⊗β) to α◦g⊗α◦β, where α ∈ HomC(xp,xi),g ∈ AutC(xp),β ∈
HomC(xq,xp), is injective for 1 ≤ i < p ≤ q ≤ n by Corollary 2.6. We have that
the sets HomC(xp,xi)×AutC(xp)(AutC(xp)×HomC(xq,xp)) and HomC(xp,xi)×
HomC(xq,xi) are k-basisof Mip⊗Rp(Rp⊗kMpq) andMip⊗kMiq, respectively by
Lemma 3.5 and Lemma 3.6. For each 1 ≤ i < p, since ϕ is injective, we have an
ip
injective map
ϕ:HomC(xp,xi)×AutC(xp)(AutC(xp)×HomC(xq,xp))→HomC(xp,xi)×HomC(xq,xi)
sending (α,(g,β)) to (α ◦ g,α ◦ β), for α ∈ HomC(xp,xi),g ∈ AutC(xp),β ∈
HomC(xq,xp).
For each 1 ≤ i < p, and α ∈ HomC(xp,xi), let β,β′ ∈ HomC(xq,xp) satisfy
α◦β = α◦β′. Then we have (α,α◦β) = (α,α◦β′), that is, ϕ(α,(Id ,β)) =
xp
ϕ(α,(Id ,β′)). Since ϕ is injective, we have (α,(Id ,β)) = (α,(Id ,β′)) in
xp xp xp
HomC(xp,xi) ×AutC(xp) (AutC(xp) × HomC(xq,xp)). Hence β = β′. Then we
have that α is a monomorphism. (cid:3)
Proposition 3.8. Let C be a finite projective EI category. Assume that C is
GPT-closed. Then that each morphism in C is a monomorphism.
Proof. It follows from Proposition 3.2 and Lemma 3.7. (cid:3)
Let P be a finite poset. We assume that ObjP ={x1,··· ,xn} satisfying xi (cid:2)xj
if i < j, and Γ is the corresponding upper triangular matrix algebra. We observe
that each entry of Γ is 0 or k, and each projective Γ-module is a direct sum of
some Ci for 1 ≤i ≤n. For any a,b∈ObjP satisfying a (cid:2)b and b (cid:2)a, denote by
L ={x∈ObjP |a<x,b<x}.
a,b
Example 3.9. Let P be a finite poset. Then P is GPT-closed if and only if any
twodistinctminimalelementsinL hasnocommonupperboundfora,b∈ObjP
a,b
satisfying a(cid:2)b and b(cid:2)a.
For the “if” part, assume that any two distinct minimal elements in L has no
a,b
common upper bound. By Proposition 3.2, we only need to prove that C ⊗ˆC is
t n
projective for 1 ≤ t ≤ n, since the general case of C ⊗ˆC can be considered in
t j
Γ .
max{t,j}
For each 1 ≤ t ≤ n, if (C ) = k, that is, x ≤ x , then (C ) = k implies
n t n t t i
(C ) =k for1≤i≤t. Hence we haveC ⊗ˆC ≃C . Assume that (C ) =0, that
n i t n t n t
is, xn (cid:2)xt. Let L′xt,xn ={xs1,··· ,xsr} be all distinct minimal elements in Lxt,xn.
For each 1 ≤ i < t, if (C ) = k = (C ) , that is, x ≤ x ,x ≤ x , then there is
t i n i n i t i
a unique x ∈ L′ satisfying x ≤ x , that is, there is a unique x ∈ L′
sl xt,xn sl i sl xt,xn
satisfying (C ) = k, since any two distinct elements in L′ has no common
sl i xt,xn
r
upper bound. Then we have C ⊗ˆC ≃ C .
t n sl
lL=1
THE TENSOR PRODUCT OF GORENSTEIN-PROJECTIVE MODULES 9
For the “only if” part, assume that xt,xj ∈ObjP satisfying xt (cid:2)xj and xj (cid:2)xt
r
and C ⊗ˆC ≃ C . Then each x ∈ L . Assume that x and x be
t j sl sl xt,xj s1 s2
lL=1
two distinct minimal elements in L having a common upper bound x . Then
xt,xj i
(C ⊗ˆC ) =k and (C ⊕C ) =k⊕k, which is a contradiction. (cid:3)
t j i s1 s2 i
4. Proof of Theorem 1.2
Let C be a finite EI category. Recall from [3, Definition 2.3] that a morphism
x →α y in C is unfactorizable if α is not an isomorphism and whenever it has a
β γ
factorizationas a composite x→z →y, then either β or γ is an isomorphism. Let
x→α y inC beanunfactorizablemorphism. Thenh◦α◦g isalsounfactorizablefor
α
every h ∈ AutC(y) and every g ∈ AutC(x); see [3, Proposition 2.5]. Let x → y in
C be a morphism with x6=y. Then it has a decomposition x=x →α1 x →α2 ···α→n
0 1
x =y with all α unfactorizable; see [3, Proposition 2.6].
n i
Following [3, Definition 2.7], we say that a finite EI category C satisfies the
Unique Factorization Property (UFP), if whenever a non-isomorphism α has two
decompositions into unfactorizable morphisms:
x=x →α1 x →α2 ···α→m x =y
0 1 m
and
x=y →β1 y →β2 ···→βn y =y,
0 1 n
then m = n, xi = yi, and there are hi ∈ AutC(xi), 1 ≤ i ≤ m−1, such that the
following diagram commutes :
x=x α1 //x α2 //x α3// ···αm−//1x αm//x =y
0 1 2 m−1 m
h1 h2 hm−1
x=x β1 // x(cid:15)(cid:15) β2 //x(cid:15)(cid:15) β3// ··· βm−//1x (cid:15)(cid:15) βm//x =y
0 1 2 m−1 m
Let C be a finite EI category. Following [4, Section 6], we say that C is a finite
free EI categoryif it satisfiesthe UFP. By [3, Proposition2.8],this is equivalentto
the original definition [3, Definition 2.2].
Let n ≥ 2. Let C be a finite projective and free EI category with ObjC =
{x1,x2,··· ,xn} satisfying HomC(xi,xj) = ∅ if i < j and Γ be the corresponding
upper triangular matrix algebra of C. Then Γ is 1-Gorenstein; see [5, Theorem
5.3].
Set Hom0C(xj,xi)= {α∈HomC(xj,xi)| α is unfactorizable}. Denote by Mi0j =
kHom0 (x ,x ),whichisanR -R -sub-bimoduleofM ;see[5,Notation4.8]. Recall
C j i i j ij
theleftΓ -moduleM∗ andthe rightΓ′ -moduleM∗∗ inNotation2.1,for1≤t≤
t t n−t t
n−1. Observe that Mt∗∗ ≃ (Mt0,t+1,Mt0,t+2,··· ,Mt0n)⊗Γ′D,n−t Γ′n−t; compare [5,
M0
1,t+1
Lemmas 4.10 and 4.11], which implies that M∗ ≃Γ ⊗  .. .
t t ΓD .
t
M0 
 t,t+1
10 RENWANG
X
1
Let X = ..  be a left Γ-module. For each 1≤t≤n−1, we have
.
X 
n
 
X X
t+1 t+1
Mt∗∗⊗Γ′n−t  ... ≃(Mt0,t+1,Mt0,t+2,··· ,Mt0n)⊗Γ′D,n−t Γ′n−t⊗Γ′n−t  ... 
 X   X 
n n
   
X
t+1
≃(Mt0,t+1,Mt0,t+2,··· ,Mt0n)⊗Γ′D,n−t  ... 
 X 
n
 
n
≃ M0 ⊗ X .
tj Rj j
j=Mt+1
Recall the R -map ϕ∗∗ in Lemma 2.5. Here, we observe that
t t
n n n
ϕ∗∗ : M0 ⊗ X →X , (m ⊗x )7→ ϕ (m ⊗x ).
t tj Rj j t j j tj j j
j=Mt+1 j=Xt+1 j=Xt+1
Lemma 4.1. Let C be a finite projective and free EI category and Γ be the corre-
spondinguppertriangularmatrixalgebra. Assume1≤p≤q ≤n. Ifeachmorphism
in y∈ObjC pj=1HomC(xj,y) is a monomorphism, then Cp⊗ˆCq ∈Γ-proj.
F F
Proof. We only need to prove that each R -map
t
p
ϕ∗∗ : M0 ⊗ (M ⊗ M )→M ⊗ M
t tj Rj jp k jq tp k tq
j=Mt+1
is injective for 1≤t<p by Lemma 2.5 and Proposition 3.2.
By Lemmas 3.5 and 3.6, we have that the set HomC(xp,xt)×HomC(xq,xt) is a
k-basis of M ⊗ M , and the set
tp k tq
p
Hom0C(xj,xt)×AutC(xj)(HomC(xp,xj)×HomC(xq,xj))=:B
j=Gt+1
p
is a k-basis of M0 ⊗ (M ⊗ M ).
tj Rj jp k jq
j=Lt+1
We have the following commutative diagram
p
B ⊆ // M0 ⊗ (M ⊗ M )
tj Rj jp k jq
j=Lt+1
ϕ∗t∗|B ϕ∗t∗
(cid:15)(cid:15) (cid:15)(cid:15)
⊆ //
HomC(xp,xt)×HomC(xq,xt) Mtp⊗kMtq
Observe that ϕ∗∗ is injective if and only if ϕ∗∗ | is injective for each 1≤t<p.
t t B
Assume that ϕ∗∗(α,(β,θ)) = ϕ∗∗(α′,(β′,θ′)), where α ∈ Hom0 (x ,x ), β ∈
t t C j t
HomC(xp,xj), θ ∈ HomC(xq,xj) and α′ ∈ Hom0C(xj′,xt), β′ ∈ HomC(xp,xj′),
θ′ ∈ HomC(xq,xj′). Then we have αβ = α′β′ in HomC(xp,xt) and αθ = α′θ′ in
HomC(xq,xt). Since C is free and α,α′ are unfactorizable, we have that j = j′
and there is g ∈ AutC(xj) such that α = α′g and β = g−1β′. Since αθ = α′θ′ =