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arXiv:1701.04169v1 [math.RT] 16 Jan 2017
THE TENSOR PRODUCT OF GORENSTEIN-PROJECTIVE
MODULES OVER CATEGORY ALGEBRAS
REN WANG
Abstract. For a finite free and projective EI category, we prove that Gorenstein-
projective modules over its category algebra are closed under the tensor prod-
uct if and only if each morphism in the given category is a monomorphism.
1. Introduction
Let kbe a field. Let Cbe a finite category, that is, it has only finitely many
morphisms, and consequently it has only finitely many objects. Denote by k-mod
the category of finite dimensional k-vector spaces and (k-mod)Cthe category of
covariant functors from Cto 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 monoidal category. More precisely, the tensor product - ˆ
⊗- is defined by
(Mˆ
⊗N)(x) = M(x)⊗kN(x)
for any M, N ∈(k-mod)Cand x∈ObjC, and α.(m⊗n) = α.m ⊗α.n for any
α∈MorC, m ∈M(x), n ∈N(x); see [8,9].
Let Cbe a finite EI category. Here, the EI condition means that all endomor-
phisms in Care isomorphisms. In particular, HomC(x, x) = AutC(x) is a finite
group for each object x. Denote by kAutC(x) the group algebra. Recall that
a finite EI category Cis projective over kif each kAutC(y)-kAutC(x)-bimodule
kHomC(x, y) is pro jective on both sides; see [5, Definition 4.2].
Denote by kC-Gproj the full subcategory of kC-mod consisting of Gorenstein-
projective kC-modules. We say that Cis GPT-closed, if X, Y ∈kC-Gproj implies
Xˆ
⊗Y∈kC-Gproj.
Let us explain the motivation to study GPT-closed categories. Recall from [6]
that for a finite projective EI category C, the stable category kC-Gproj modulo
projective modules has a natural tensor triangulated structure such that it is tensor
triangle equivalent to the singularity category of kC. In general, its tensor product
is not explicitly given. However, if Cis GPT-closed, then the tensor pro duct - 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 Cbe a finite projective EI category. Assume that Cis
GPT-closed. Then each morphism in Cis a monomorphism.
Date: September 4, 2018.
2010 Mathematics Subject Classification. Primary 16G10; Secondary 16D90, 18E30.
Key words and phrases. finite EI category, category algebra, Gorenstein-projective module,
tensor product, tensor triangulated category.
1
2 REN WANG
The concept of a finite free EI category is introduced in [3].
Theorem 1.2. Let Cbe a finite projective and free EI category. Then the category
Cis GP T -closed if and only if each morphism in Cis a monomorphism.
2. Gorenstein triangular matrix algebras
In this section, we recall some necessary preliminaries on triangular matrix alge-
bras and Gorenstein-projective modules.
Recall an n×nupper triangular matrix algebra Γ =
R1M12 ···M1n
R2···M2n
....
.
.
Rn
, where
each Riis an algebra for 1 ≤i≤n, each Mij is an Ri-Rj-bimodule for 1 ≤i <
j≤n, and the matrix algebra map is denoted by ψilj :Mil ⊗RlMlj →Mij for
1≤i < l < j < t ≤n; see [5].
Recall that a left Γ-module X=
X1
.
.
.
Xn
is described by a column vector, where
each Xiis a left Ri-module for 1 ≤i≤n, and the left Γ-module structure map is
denoted by ϕjl :Mjl ⊗RlXl→Xjfor 1 ≤j < l ≤n; see [5]. Dually, we have the
description of right Γ-modules via row vectors.
Notation 2.1. Let Γtbe the algebra given by the t×tleading principal submatrix
of Γand Γ′
n−tbe the algebra given by cutting the first trows and the first tcolumns
of Γ. Denote the left Γt-module
M1,t+1
.
.
.
Mt,t+1
by M∗
tand the right Γ′
n−t-module
Mt,t+1,··· , Mtn by M∗∗
t, for 1≤t≤n−1. Denote by ΓD
t=diagR1,···Rtthe
sub-algebra of Γtconsisting of diagonal matrices, and Γ′
D,n−t=diagRt+1 ,···Rn
the sub-algebra of Γ′
n−tconsisting of diagonal matrices.
Let Γ = R1M12
0R2be an upper triangular matrix algebra. Recall that the R1-R2-
bimodule M12 is compatible, if the following two conditions hold; see[10, Definition
1.1]:
(C1) If Q•is an exact sequence of pro jective R2-modules, then M12 ⊗R2Q•is
exact;
(C2) If P•is a complete R1-projective resolution, then HomR1(P•, M12) is exact.
We use pd and id to denote the projective dimension and the injective dimension
of a module, respectively.
Lemma 2.2. Let Γbe an upper triangular matrix algebra satisfying all RiGoren-
stein. If Γis Gorenstein, then each Γt-Rt+1-bimodule M∗
tis compatible and each
Rt-Γ′
n−t-bimodule M∗∗
tis compatible for 1≤t≤n−1.
Proof. Let Λ = S1N12
0S2be an upper triangular matrix algebra. Recall the fact
that if pdS1(N12)<∞and pd(N12)S2<∞, then N12 is compatible; see [10,
Proposition 1.3]. Recall that Γ is Gorenstein if and only if all bimodules Mij
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 pdRt(M∗∗
t)<∞and pd(M∗
t)Rt+1 <∞for 1 ≤
t≤n−1. By [5, Lemma 3.1], we have pd(M∗∗
t)Γ′
n−t<∞and pdΓt(M∗
t)<∞for
1≤t≤n−1. Then we are done.
Let Abe a finite dimensional algebra over a field k. Denote by A-mod the category
of finite dimensional left A-modules. The opposite algebra of Ais denoted by Aop.
We identify right A-modules with left Aop -modules.
Denote by (−)∗the contravariant functor 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 evX:X→X∗∗ given by evX(x)(f) = f(x) for x∈X
and f∈X∗. Recall that an A-module Gis Gorenstein-projective provided that
Exti
A(G, A) = 0 = Exti
Aop (G∗, A) for i≥1 and the evaluation map evGis bijective;
see [1, Proposition 3.8].
The algebra Ais Gorenstein if idAA < ∞and idAA<∞. It is well known that
for a Gorenstein algebra Awe have idAA= idAA; see [11, Lemma A]. For m≥0,
a Gorenstein algebra Ais m-Gorenstein if idAA= idAA≤m. Denote by A-Gproj
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.
The following lemma is well known; see [1, Propositions 3.8 and 4.12 and Theorem
3.13].
Lemma 2.3. Let m≥0. Let Abe an m-Gorenstein algebra. Then we have the
following statements.
(1) An A-module M∈A-Gproj if and only if Exti
A(M, A) = 0 for all i > 0.
(2) If M∈A-Gproj and Lis a right A-module with finite projective dimension,
then TorA
i(L, M ) = 0 for all i > 0.
(3) If there is an exact sequence 0→M→G0→G1→ · · · → Gmwith Gi
Gorenstein-projective, then M∈A-Gproj.
Lemma 2.4. [10, Theorem 1.4] Let M12 be a compatible R1-R2-bimodule, and
Γ = R1M12
0R2. Then X=X1
X2∈Γ-Gproj if and only if ϕ12 :M12 ⊗R2X2→X1
is an injective R1-map, Cokerϕ12 ∈R1-Gproj, and X2∈R2-Gproj.
We have a slight generalization of Lemma 2.4.
Lemma 2.5. Let Γbe a Gorenstein upper triangular matrix algebra with each
Ria group algebra. Then X=
X1
.
.
.
Xn
∈Γ-Gproj if and only if each Rt-map
ϕ∗∗
t=
n
P
j=t+1
ϕtj :M∗∗
t⊗Γ′
n−t
Xt+1
.
.
.
Xn
→Xtsending mt,t+1,··· , mtn ⊗
xt+1
.
.
.
xn
to
n
P
j=t+1
ϕtj (mtj ⊗xj)is injective for 1≤t≤n−1.
Proof. We have that each Rt-Γ′
n−t-bimodule M∗∗
tis compatible for 1 ≤t≤n−1
by Lemma 2.2.
4 REN WANG
For the “only if” part, we use induction on n. The case n= 2 is due to Lemma 2.4.
Assume that n > 2. Write Γ = R1M∗∗
1
0 Γ′
n−1, and X=X1
X′. Since X∈Γ-
Gproj, by Lemma 2.4, we have that the R1-map ϕ∗∗
1:M∗∗
1⊗Γ′
n−1X′→X1is
injective and X′∈Γ′
n−1-Gproj. By induction, we have that each Rt-map ϕ∗∗
t:
M∗∗
t⊗Γ′
n−t
Xt+1
.
.
.
Xn
→Xtis injective for 1 ≤t≤n−1.
For the “if” part, we use induction on n. The case n= 2 is due to Lemma 2.4.
Assume that n > 2. Write Γ = R1M∗∗
1
0 Γ′
n−1, and X=X1
X′. By induction, we
have X′∈Γ′
n−1-Gproj. Since the R1-map ϕ∗∗
1:M∗∗
1⊗Γ′
n−1X′→X1is injective
and its cokernel belongs to R1-Gproj for R1a group algebra, we have X∈Γ-Gproj
by Lemma 2.4.
Corollary 2.6. Let Γbe a Gorenstein upper triangular matrix algebra with each
Ria group algebra. Assume that X=
X1
.
.
.
Xs
0
.
.
.
0
∈Γ-Gproj. Then each Ri-map
ϕis :Mis ⊗RsXs→Xiis injective for 1≤i < s ≤n.
Proof. We write X=X′
X′′ , where X′′ =
Xi+1
.
.
.
Xs
.
.
.
0
for each 1 ≤i < s ≤n. We
claim that each Ri-map fis :Mis ⊗RsXs→M∗∗
i⊗Γ′
n−iX′′ sending mis ⊗xsto
0,··· , mis,··· ,0⊗0,··· , xs,··· ,0tis injective, where (−)tis the transpose.
Since ϕis =ϕ∗∗
i◦fis for 1 ≤i < s ≤n, then we are done by Lemma 2.5.
For the claim, we observe that for each 1 ≤i < s ≤n, the Ri-map fis is a
composition of the following
Mis ⊗RsXs
gis
−→ 0,··· ,0, Mis,··· , Min ⊗Γ′
n−iX′′ ι⊗Id
−→ M∗∗
i⊗Γ′
n−iX′′,
where the right Γ′
n−i-map 0,··· , Mis,··· , Min ι
−→ M∗∗
iis the inclusion map
and gis sends mis ⊗xsto 0,··· , mis,··· ,0⊗0,··· , xs,··· ,0t. We observe a
Ri-map 0,··· , Mis,··· , Min ⊗Γ′
n−iX′′ g′
is
−→ Mis ⊗RsXs,0,··· , mis,··· , min ⊗
0,··· , xs,··· ,0t7→ mis ⊗xssatisfying g′
is ◦gis = IdMis⊗RsXs. Hence the Ri-map
gis is injective. We observe that the right Γ′
n−i-modules 0,··· , Mis,··· , Min and
M∗∗
ihave finite projective dimensions; see [5, Lemma 3.1], and X′′ ∈Γ′
n−i-Gproj
by Lemma 2.4. Then the Ri-map ι⊗Id is injective by Lemma 2.3 (2).
THE TENSOR PRODUCT OF GORENSTEIN-PROJECTIVE MODULES 5
3. Proof of Proposition 1.1
Let kbe a field. Let Cbe 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 k Cof Cis defined as
follows: kC=L
α∈MorC
kα as a k-vector space and the product ∗is given by the rule
α∗β=α◦β, if αand βcan be composed in C;
0,otherwise.
The unit is given by 1kC=P
x∈ObjC
Idx, where Idxis the identity endomorphism of
an object xin C.
If Cand Dare two equivalent finite categories, then kCand kDare Morita
equivalent; see [7, Proposition 2.2]. In particular, kCis Morita equivalent to kC0,
where C0is any skeleton of C. So we may assume that Cis skeletal, that is, for
any two distinct objects xand yin C,xis not isomorphic to y.
The category Cis called a finite EI category provided that all endomorphisms
in Care isomorphisms. In particular, HomC(x, x) = AutC(x) is a finite group for
any object xin C. Denote by kAutC(x) the group algebra.
Throughout this paper, we assume that Cis a finite EI category which is skeletal,
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 kCis isomorphic to the corresponding upper triangular
matrix algebra ΓC=
R1M12 ···M1n
R2···M2n
....
.
.
Rn
; see [5, Section 4].
Let us fix a field k. Denote by k-mod the category of finite dimensional k-vector
spaces and (k-mod)Cthe category of covariant functors from Cto 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 monoidal category. More precisely, the tensor product - ˆ
⊗- is defined by
(Mˆ
⊗N)(x) = M(x)⊗kN(x)
for any M, N ∈(k-mod)Cand x∈ObjC, and α.(m⊗n) = α.m ⊗α.n for any
α∈MorC, m ∈M(x), n ∈N(x); see [8,9].
Let Cbe a finite EI category, and Γ = ΓC=
R1M12 ···M1n
R2···M2n
....
.
.
Rn
be the cor-
responding upper triangular matrix algebra. Let X=
X1
.
.
.
Xn
and Y=
Y1
.
.
.
Yn
be two Γ-modules, where the left Γ-module structure maps are denoted by ϕX
ij
and ϕY
ij , respectively. We observe that Xˆ
⊗Y=
X1⊗kY1
.
.
.
Xn⊗kYn
, where the module
6 REN WANG
structure map ϕij :Mij ⊗Rj(Xj⊗kYj)→Xi⊗kYiis induced by the following:
ϕij (αij ⊗(aj⊗bj)) = ϕX
ij (αij ⊗aj)⊗ϕY
ij (αij ⊗bj), where αij ∈HomC(xj, xi),
aj∈Xjand bj∈Yj.
Definition 3.1. Let Cbe a finite EI category, and Γ be the corresponding upper
triangular matrix algebra. We say that Cis GPT-closed, if X, Y ∈Γ-Gproj implies
Xˆ
⊗Y∈Γ-Gproj.
Recall that a finite EI category Cis projective over kif each kAutC(y)-kAutC(x)-
bimodule kHomC(x, y) is projective on both sides; see [5, Definition 4.2]. We recall
the fact that the category algebra kCis Gorenstein if and only if Cis projective
over k; see [5, Proposition 5.1].
Let Cbe a finite projective EI category, and Γ be the corresponding upper trian-
gular matrix algebra. Denote by Cithe i-th column of Γ which is a Γ-Ri-bimodule
and projective on both sides.
Proposition 3.2. Let Cbe a finite projective EI category, and Γbe the correspond-
ing upper triangular matrix algebra. Then the following statements are equivalent.
(1) The category Cis GPT-closed.
(2) For any 1≤p≤q≤n,Cpˆ
⊗Cq∈Γ-Gproj.
(3) For any 1≤p≤q≤n,Cpˆ
⊗Cq∈Γ-proj.
Proof. “(1)⇒(2)” and “(3)⇒(2)” are obvious.
“(2)⇒(3)” We only need to prove that the Γ-module Cpˆ
⊗Cqhas finite projective
dimension, since a Gorenstein-projective module with finite pro jective dimension
is projective. We have Cpˆ
⊗Cq=
M1p⊗kM1q
.
.
.
Mp−1,p ⊗kMp−1,q
Rp⊗kMpq
0
.
.
.
0
. Since Cis projective, we
have that each Mip is a projective Ri-module for 1 ≤i≤p. Then each Mip ⊗kMiq
is a projective Ri-module since Riis a group algebra for 1 ≤i≤p. Hence the
Γ-module Cpˆ
⊗Cqhas finite projective dimension by [5, Corollary 3.6]. Then we are
done.
“(2)⇒(1)” We have that Γ is a Gorenstein algebra by [5, Proposition 5.1]. 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→P0→P1→ · · · → Pd→Y→0
with Piprojective, 0 ≤i≤d. Applying - ˆ
⊗Non the above exact sequence, we have
an exact sequence
0→Mˆ
⊗N→P0ˆ
⊗N→P1ˆ
⊗N→ · · · → Pdˆ
⊗N→Yˆ
⊗N→0,(3.1)
since the tensor product - ˆ
⊗- is exact in both variables. If Nis projective, we have
that each Piˆ
⊗Nis Gorenstein-projective for 0 ≤i≤dby (2). Then we have
Mˆ
⊗N∈Γ-Gproj by Lemma 2.3 (3). If Nis Gorenstein-projective, we have that
each Piˆ
⊗Nis Gorenstein-pro jective for 0 ≤i≤din exact sequence (3.1) by the
above process. Then we have Mˆ
⊗N∈Γ-Gproj by Lemma 2.3 (3). Then we are
done.
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-Gpro j induces the one on Γ-Gproj, still
denoted by - ˆ
⊗-.
Lemma 3.3. Assume that Cis GPT-closed. Let M∈Γ-Gproj and P∈Γ-pro j.
Then Mˆ
⊗P∈Γ-proj.
Let Cbe a finite projective EI category, and Γ be the corresponding upper triangu-
lar matrix algebra of C. Recall that a complex in Db(Γ-mod), the bounded derived
category of finitely generated left Γ-modules, is called a perfect complex if it is iso-
morphic to a bounded complex of finitely generated projective modules. Recall from
[2] that the singularity category of Γ, denoted by Dsg (Γ), is the Verdier quotient
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 ∼
−→ Dsg(Γ) (3.2)
sending a Gorenstein-projective module to the corresponding stalk complex con-
centrated on degree zero. The functor Ftransports the tensor product on Dsg (Γ)
to Γ-Gproj such that the category Γ-Gproj becomes a tensor triangulated category.
Proposition 3.4. Let Cbe a finite projective EI category, and Γbe the corre-
sponding upper triangular matrix algebra of C. Assume that the category Cis
GPT-closed. Then the tensor product - ˆ
⊗- on Γ-Gproj induced by the tensor product
on Γ-Gproj coincide with the one transported from Dsg(Γ), up to natural isomor-
phism.
Proof. Consider the functor Fin (3.2). Recall that the tensor product on Dsg (Γ)
is induced by the tensor product - ˆ
⊗- on Db(Γ-mod), where the later is given by - ˆ
⊗-
on Γ-mod. We have F(M)ˆ
⊗F(N) = F(Mˆ
⊗N) in Dsg(Γ) for any M , N ∈Γ-Gpro j.
This implies that Fis a tensor triangle equivalence. Then we are done.
Let kbe a field and Gbe a finite group. Recall that a left (resp. right) G-set
is a set with a left (resp. right) G-action. Let Ybe a left G-set and Xbe a
right G-set. Recall an equivalence relation “∼” on the product X×Yas follows:
(x, y)∼(x′, y′) if and only if there is an element g∈Gsuch that x=x′gand
y=g−1y′for x, x′∈Xand y, y′∈Y. Write the quotient set X×Y/ ∼as X×GY.
The following two lemmas are well known.
Lemma 3.5. Let Ybe a left G-set and Xbe a right G-set. Then there is an
isomorphism of k-vector spaces
ϕ:kX ⊗kG kY ∼
−→ k(X×GY), x ⊗y7→ (x, y),
where x∈Xand y∈Y.
Lemma 3.6. Let Y1and Y2be two left G-sets. Then we have an isomorphism of
left kG-modules
ϕ:kY1⊗kkY2
∼
−→ k(Y1×Y2), y1⊗y27→ (y1, y2),
where y1∈Y1, y2∈Y2.
Lemma 3.7. Let Cbe a finite projective EI category, and Γbe the corresponding
upper triangular matrix algebra. Assume 1≤p≤q≤n. Then Cpˆ
⊗Cq∈Γ-proj
implies that each morphism in Fy∈Ob jCHomC(xp, y)is a monomorphism.
8 REN WANG
Proof. We have Cpˆ
⊗Cq=
M1p⊗kM1q
.
.
.
Mp−1,p ⊗kMp−1,q
Rp⊗kMpq
0
.
.
.
0
. Then each Ri-map
ϕip :Mip ⊗Rp(Rp⊗kMpq)→Mip ⊗kMiq
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≤nby 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-basis of Mip ⊗Rp(Rp⊗kMpq ) and Mip ⊗kMiq , respectively by
Lemma 3.5 and Lemma 3.6. For each 1 ≤i < p, since ϕip is injective, we have an
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, (Idxp, β)) =
ϕ(α, (Idxp, β′)). Since ϕis injective, we have (α, (Idxp, β)) = (α, (Idxp, β ′)) in
HomC(xp, xi)×AutC(xp)(AutC(xp)×HomC(xq, xp)). Hence β=β′. Then we
have that αis a monomorphism.
Proposition 3.8. Let Cbe a finite projective EI category. Assume that Cis
GPT-closed. Then that each morphism in Cis a monomorphism.
Proof. It follows from Proposition 3.2 and Lemma 3.7.
Let Pbe a finite poset. We assume that ObjP={x1,··· , xn}satisfying xixj
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 Cifor 1 ≤i≤n. For any a, b ∈ObjPsatisfying aband ba, denote by
La,b ={x∈ObjP | a < x, b < x}.
Example 3.9. Let Pbe a finite poset. Then Pis GP T -closed if and only if any
two distinct minimal elements in La,b has no common upper bound for a, b ∈ObjP
satisfying aband ba.
For the “if” part, assume that any two distinct minimal elements in La,b has no
common upper bound. By Proposition 3.2, we only need to prove that Ctˆ
⊗Cnis
projective for 1 ≤t≤n, since the general case of Ctˆ
⊗Cjcan be considered in
Γmax{t,j}.
For each 1 ≤t≤n, if (Cn)t=k, that is, xn≤xt, then (Ct)i=kimplies
(Cn)i=kfor 1 ≤i≤t. Hence we have Ctˆ
⊗Cn≃Ct. Assume that (Cn)t= 0, that
is, xnxt. Let L′
xt,xn={xs1,··· , xsr}be all distinct minimal elements in Lxt,xn.
For each 1 ≤i < t, if (Ct)i=k= (Cn)i, that is, xn≤xi, xt≤xi, then there is
a unique xsl∈L′
xt,xnsatisfying xsl≤xi, that is, there is a unique xsl∈L′
xt,xn
satisfying (Csl)i=k, since any two distinct elements in L′
xt,xnhas no common
upper bound. Then we have Ctˆ
⊗Cn≃
r
L
l=1
Csl.
THE TENSOR PRODUCT OF GORENSTEIN-PROJECTIVE MODULES 9
For the “only if” part, assume that xt, xj∈ObjPsatisfying xtxjand xjxt
and Ctˆ
⊗Cj≃
r
L
l=1
Csl. Then each xsl∈Lxt,xj. Assume that xs1and xs2be
two distinct minimal elements in Lxt,xjhaving a common upper bound xi. Then
(Ctˆ
⊗Cj)i=kand (Cs1⊕Cs2)i=k⊕k, which is a contradiction.
4. Proof of Theorem 1.2
Let Cbe a finite EI category. Recall from [3, Definition 2.3] that a morphism
xα
→yin Cis unfactorizable if αis not an isomorphism and whenever it has a
factorization as a composite xβ
→zγ
→y, then either βor γis an isomorphism. Let
xα
→yin Cbe an unfactorizable morphism. Then h◦α◦gis also unfactorizable for
every h∈AutC(y) and every g∈AutC(x); see [3, Proposition 2.5]. Let xα
→yin
Cbe a morphism with x6=y. Then it has a decomposition x=x0
α1
→x1
α2
→ · · · αn
→
xn=ywith all αiunfactorizable; see [3, Proposition 2.6].
Following [3, Definition 2.7], we say that a finite EI category Csatisfies the
Unique Factorization Property (UFP), if whenever a non-isomorphism αhas two
decompositions into unfactorizable morphisms:
x=x0
α1
→x1
α2
→ · · · αm
→xm=y
and
x=y0
β1
→y1
β2
→ · · · βn
→yn=y,
then m=n,xi=yi, and there are hi∈AutC(xi), 1 ≤i≤m−1, such that the
following diagram commutes :
x=x0
α1//x1
α2//
h1
x2
α3//
h2
···αm−1
//xm−1
αm//
hm−1
xm=y
x=x0
β1//x1
β2//x2
β3//··· βm−1
//xm−1
βm//xm=y
Let Cbe a finite EI category. Following [4, Section 6], we say that Cis a finite
free EI category if it satisfies the UFP. By [3, Proposition 2.8], this is equivalent to
the original definition [3, Definition 2.2].
Let n≥2. Let Cbe a finite projective and free EI category with Ob jC=
{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 Hom0
C(xj, xi) = {α∈HomC(xj, xi)|αis unfactorizable}. Denote by M0
ij =
kHom0
C(xj, xi), which is an Ri-Rj-sub-bimodule of Mij ; see [5, Notation 4.8]. Recall
the left Γt-module M∗
tand the right Γ′
n−t-module M∗∗
tin Notation 2.1, for 1 ≤t≤
n−1. Observe that M∗∗
t≃(M0
t,t+1, M 0
t,t+2,··· , M 0
tn)⊗Γ′
D,n−tΓ′
n−t; compare [5,
Lemmas 4.10 and 4.11], which implies that M∗
t≃Γt⊗ΓD
t
M0
1,t+1
.
.
.
M0
t,t+1
.
10 REN WANG
Let X=
X1
.
.
.
Xn
be a left Γ-module. For each 1 ≤t≤n−1, we have
M∗∗
t⊗Γ′
n−t
Xt+1
.
.
.
Xn
≃(M0
t,t+1, M 0
t,t+2,··· , M 0
tn)⊗Γ′
D,n−tΓ′
n−t⊗Γ′
n−t
Xt+1
.
.
.
Xn
≃(M0
t,t+1, M 0
t,t+2,··· , M 0
tn)⊗Γ′
D,n−t
Xt+1
.
.
.
Xn
≃
n
M
j=t+1
M0
tj ⊗RjXj.
Recall the Rt-map ϕ∗∗
tin Lemma 2.5. Here, we observe that
ϕ∗∗
t:
n
M
j=t+1
M0
tj ⊗RjXj→Xt,
n
X
j=t+1
(mj⊗xj)7→
n
X
j=t+1
ϕtj (mj⊗xj).
Lemma 4.1. Let Cbe a finite projective and free EI category and Γbe the corre-
sponding upper triangular matrix algebra. Assume 1≤p≤q≤n. If each morphism
in Fy∈ObjCFp
j=1 HomC(xj, y)is a monomorphism, then Cpˆ
⊗Cq∈Γ-proj.
Proof. We only need to prove that each Rt-map
ϕ∗∗
t:
p
M
j=t+1
M0
tj ⊗Rj(Mjp ⊗kMjq )→Mtp ⊗kMtq
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 Mtp ⊗kMtq , and the set
p
G
j=t+1
Hom0
C(xj, xt)×AutC(xj)(HomC(xp, xj)×HomC(xq, xj)) =: B
is a k-basis of
p
L
j=t+1
M0
tj ⊗Rj(Mjp ⊗kMjq ).
We have the following commutative diagram
B⊆//
ϕ∗∗
t|B
p
L
j=t+1
M0
tj ⊗Rj(Mjp ⊗kMjq )
ϕ∗∗
t
HomC(xp, xt)×HomC(xq, xt)⊆//Mtp ⊗kMtq
Observe that ϕ∗∗
tis injective if and only if ϕ∗∗
t|Bis injective for each 1 ≤t < p.
Assume that ϕ∗∗
t(α, (β, θ)) = ϕ∗∗
t(α′,(β′, θ′)), where α∈Hom0
C(xj, xt), β∈
HomC(xp, xj), θ∈HomC(xq, xj) and α′∈Hom0
C(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 Cis free and α, α′are unfactorizable, we have that j=j′
and there is g∈AutC(xj) such that α=α′gand β=g−1β′. Since αθ =α′θ′=
THE TENSOR PRODUCT OF GORENSTEIN-PROJECTIVE MODULES 11
αg−1θ′and αis a monomorphism, we have that θ=g−1θ′. Then we have that
(α, (β, θ)) = (α′g, (g−1β′, g −1θ′)) = (α′,(β′, θ′)), which implies that the map ϕ∗∗
t|B
is injective.
Theorem 4.2. Let Cbe a finite projective and free EI category. Then the category
Cis GP T -closed if and only if each morphism in Cis a monomorphism.
Proof. The “if” part is just by Proposition 3.2 and Lemma 4.1. The “only if” part
is just by Proposition 3.8.
Acknowledgements
The author is grateful to her supervisor Professor Xiao-Wu Chen for his guidance.
This work is supported by the National Natural Science Foundation of China (No.s
11522113 and 11571329).
References
[1] M. Auslander, M. Bridger, Stable module theory, Memoirs of the American Mathematical
Society, No. 94 (1969).
[2] R.-O. Buchweitz, Maximal Cohen-Macaulay modules and Tate-cohomology over Gorenstein
rings, Unpublished Manuscript, Available at: http://hdl.handle.net/1807/16682, 1987.
[3] L. P. Li, A characterization of finite EI categories with hereditary category algebras, J. Al-
gebra 345 (2011), 213–241.
[4] L. P. Li, A generalized Koszul theory and its application, Trans. Amer. Math. Soc. 366
(2014), 931–977.
[5] R. Wang, Gorenstein triangular matrix rings and category algebras, J. Pure Appl. Algebra
220 (2016), no. 2, 666–682.
[6] R. Wang, The MCM-approximation of the trivial module over a category algebra, J. Algebra
Appl. 16 (2017), no. 5. DOI:10.1142/S0219498817501092.
[7] P. Webb, An introduction to the representations and cohomology of categories, Group repre-
sentation theory, 149–173, EPFL Press, Lausanne, 2007.
[8] F. Xu, Spectra of tensor triangulated categories over category algebras, Arch. Math. (Basel)
103 (2014), no. 3, 235–253.
[9] F. Xu, Tensor structure on kC-mod and cohomology, Proc. Edinb. Math. Soc. (2) 56 (2013),
no. 1, 349–370.
[10] P. Zhang, Gorenstein-projective modules and symmetric recollements, J. Algebra 388 (2013),
65–80.
[11] A. Zaks, Injective dimensions of semi-primary rings, J. Algebra 13 (1969), 73–86.
School of Mathematical Sciences, University of Science and Technology of China,
Hefei, Anhui 230026, P. R. China
E-mail address:renw@mail.ustc.edu.cn
