The tensor product of Gorenstein-projective modules over category algebras

TL;DR

This paper characterizes when the tensor product of Gorenstein-projective modules over category algebras remains Gorenstein-projective, linking this property to the monomorphism nature of all morphisms in the category.

Contribution

It establishes a precise condition under which Gorenstein-projective modules are closed under tensor product over category algebras, specifically for finite free and projective EI categories.

Findings

Gorenstein-projective modules are closed under tensor product iff all morphisms are monomorphisms.

Provides a characterization linking category morphisms to module properties.

Advances understanding of module behavior over category algebras.

Abstract

For a finite free and projective EI category, we prove that Gorenstein-projective modules over its category algebra are closed under the tensor product if and only if each morphism in the given category is a monomorphism.

Equations

$$(M\hat{\otimes }N)(x)=M(x){\otimes }_{k}N(x)$$

$$M_{is}\otimes_{R_{s}}X_{s}\overset{g_{is}}{\longrightarrow}\left(\begin{array}[]{c}0,\cdots,0,M_{is},\cdots,M_{in}\end{array}\right)\otimes_{\Gamma^{\prime}_{n-i}}X^{\prime\prime}\overset{\iota\otimes{\rm Id}}{\longrightarrow}M_{i}^{**}\otimes_{\Gamma^{\prime}_{n-i}}X^{\prime\prime},$$

$$\alpha*\beta=\left\{\begin{array}[]{ll}\alpha\circ\beta,&\text{ if }\text{$\alpha$ and $\beta$ can be composed in $\mathscr{C}$};\\ 0,&\text{otherwise.}\end{array}\right.$$

$$(M\hat{\otimes }N)(x)=M(x){\otimes }_{k}N(x)$$

$$0\to M\to P_0\to P_1\to \cdots \to P_d\to Y\to 0$$

$$0\to M\hat{\otimes }N\to P_0\hat{\otimes }N\to P_1\hat{\otimes }N\to \cdots \to P_d\hat{\otimes }N\to Y\hat{\otimes }N\to 0,$$

$$F:\Gamma \text{-}\underline{\mathrm{Gproj}}\stackrel{\sim }{⟶}{\mathrm{D}}_{sg}(\Gamma )$$

$$\phi :kX{\otimes }_{kG}kY\stackrel{\sim }{⟶}k(X{\times }_{G}Y),x\otimes y\mapsto (x,y),$$

$$\phi :k{Y}_1{\otimes }_{k}k{Y}_2\stackrel{\sim }{⟶}k(Y_1\times Y_2),y_1\otimes y_2\mapsto (y_1,y_2),$$

$${\phi }_{ip}:M_{ip}{\otimes }_{R_p}(R_p{\otimes }_k{M}_{pq})\to M_{ip}{\otimes }_k{M}_{iq}$$

$$\phi :{\mathrm{Hom}}_{\mathcal{C}}(x_p,x_i){\times }_{{\mathrm{Aut}}_{\mathcal{C}}(x_p)}({\mathrm{Aut}}_{\mathcal{C}}(x_p)\times {\mathrm{Hom}}_{\mathcal{C}}(x_q,x_p))\to {\mathrm{Hom}}_{\mathcal{C}}(x_p,x_i)\times {\mathrm{Hom}}_{\mathcal{C}}(x_q,x_i)$$

$$x=x_0\stackrel{{\alpha }_1}{\to }{x}_1\stackrel{{\alpha }_2}{\to }\cdots \stackrel{{\alpha }_m}{\to }{x}_m=y$$

$$x=y_0\stackrel{{\beta }_1}{\to }{y}_1\stackrel{{\beta }_2}{\to }\cdots \stackrel{{\beta }_n}{\to }{y}_n=y,$$

$$x=x_0\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\begin{split}M_{t}^{**}\otimes_{\Gamma^{\prime}_{n-t}}\left(\begin{array}[]{c}X_{t+1}\\ \vdots\\ X_{n}\end{array}\right)&\simeq(M_{t,t+1}^{0},M_{t,t+2}^{0},\cdots,M_{tn}^{0})\otimes_{\Gamma^{\prime}_{D,n-t}}\Gamma_{n-t}^{\prime}\otimes_{\Gamma^{\prime}_{n-t}}\left(\begin{array}[]{c}X_{t+1}\\ \vdots\\ X_{n}\end{array}\right)\\ &\simeq(M_{t,t+1}^{0},M_{t,t+2}^{0},\cdots,M_{tn}^{0})\otimes_{\Gamma^{\prime}_{D,n-t}}\left(\begin{array}[]{c}X_{t+1}\\ \vdots\\ X_{n}\end{array}\right)\\ &\simeq\bigoplus_{j=t+1}^{n}M_{tj}^{0}\otimes_{R_{j}}X_{j}.\\ \end{split}$$

$${\phi }_t^{\ast \ast }:\underset{j=t+1}{\overset{n}{⨁}}{M}_{tj}^0{\otimes }_{R_j}{X}_j\to X_t,\sum _{j=t+1}^n(m_j\otimes x_j)\mapsto \sum _{j=t+1}^n{\phi }_{tj}(m_j\otimes x_j).$$

$${\phi }_t^{\ast \ast }:\underset{j=t+1}{\overset{p}{⨁}}{M}_{tj}^0{\otimes }_{R_j}(M_{jp}{\otimes }_k{M}_{jq})\to M_{tp}{\otimes }_k{M}_{tq}$$

$$\underset{j=t+1}{\overset{p}{⨆}}{\mathrm{Hom}}_{\mathcal{C}}^0(x_j,x_t){\times }_{{\mathrm{Aut}}_{\mathcal{C}}(x_j)}\left({\mathrm{Hom}}_{\mathcal{C}}(x_p,x_j)\times {\mathrm{Hom}}_{\mathcal{C}}(x_q,x_j)\right)=:B$$

$$B\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

Full text

1. Introduction

Let $k$ be a field. Let $\mathscr{C}$ be a finite category, that is, it has only finitely many morphisms, and consequently it has only finitely many objects. Denote by $k$-mod the category of finite dimensional $k$-vector spaces and $(k\text{-mod})^{\mathscr{C}}$ the category of covariant functors from $\mathscr{C}$ to $k$-mod.

Recall that the category $k\mathscr{C}$-mod of left modules over the category algebra $k\mathscr{C}$ is identified with $(k\text{-mod})^{\mathscr{C}}$; see [7, Proposition 2.1]. The category $k\mathscr{C}$-mod is a symmetric monoidal category. More precisely, the tensor product -$\hat{\otimes}$- is defined by

[TABLE]

for any $M,N\in(k\text{-mod})^{\mathscr{C}}$ and $x\in{\rm Obj}\mathscr{C}$, and $\alpha.(m\otimes n)=\alpha.m\otimes\alpha.n$ for any $\alpha\in\mathrm{Mor}\mathscr{C},m\in M(x),n\in N(x)$; see [8, 9].

Let $\mathscr{C}$ be a finite EI category. Here, the EI condition means that all endomorphisms in $\mathscr{C}$ are isomorphisms. In particular, $\mathrm{Hom}_{\mathscr{C}}(x,x)=\mathrm{Aut}_{\mathscr{C}}(x)$ is a finite group for each object $x$. Denote by $k\mathrm{Aut}_{\mathscr{C}}(x)$ the group algebra. Recall that a finite EI category $\mathscr{C}$ is projective over $k$ if each $k{\rm Aut}_{\mathscr{C}}(y)$-$k{\rm Aut}_{\mathscr{C}}(x)$-bimodule $k{\rm Hom}_{\mathscr{C}}(x,y)$ is projective on both sides; see [5, Definition 4.2].

Denote by $k\mathscr{C}$-Gproj the full subcategory of $k\mathscr{C}$-mod consisting of Gorenstein-projective $k\mathscr{C}$-modules. We say that $\mathscr{C}$ is GPT-closed, if $X,Y\in k\mathscr{C}$-Gproj implies $X\hat{\otimes}Y\in k\mathscr{C}$-Gproj.

Let us explain the motivation to study GPT-closed categories. Recall from [6] that for a finite projective EI category $\mathscr{C}$, the stable category $k\mathscr{C}$-$\underline{\rm Gproj}$ modulo projective modules has a natural tensor triangulated structure such that it is tensor triangle equivalent to the singularity category of $k\mathscr{C}$. In general, its tensor product is not explicitly given. However, if $\mathscr{C}$ is GPT-closed, then the tensor product -$\hat{\otimes}$- on Gorenstein-projective modules induces the one on $k\mathscr{C}$-$\underline{\rm Gproj}$; see Proposition 3.4. In this case, we have a better understanding of the tensor triangulated category $k\mathscr{C}$-$\underline{\rm Gproj}$.

Proposition 1.1**.**

Let $\mathscr{C}$ be a finite projective EI category. Assume that $\mathscr{C}$ is GPT-closed. Then each morphism in $\mathscr{C}$ is a monomorphism.

The concept of a finite free EI category is introduced in [3].

Theorem 1.2**.**

Let $\mathscr{C}$ be a finite projective and free EI category. Then the category $\mathscr{C}$ is $GPT$-closed if and only if each morphism in $\mathscr{C}$ is a monomorphism.

2. Gorenstein triangular matrix algebras

In this section, we recall some necessary preliminaries on triangular matrix algebras and Gorenstein-projective modules.

Recall an $n\times n$ upper triangular matrix algebra \Gamma=\left(\begin{array}[]{cccc}R_{1}&M_{12}&\cdots&M_{1n}\\ &R_{2}&\cdots&M_{2n}\\ &&\ddots&\vdots\\ &&&R_{n}\\ \end{array}\right), where each $R_{i}$ is an algebra for $1\leq i\leq n$, each $M_{ij}$ is an $R_{i}$-$R_{j}$-bimodule for $1\leq i<j\leq n$, and the matrix algebra map is denoted by $\psi_{ilj}:M_{il}\otimes_{R_{l}}M_{lj}\rightarrow M_{ij}$ for $1\leq i<l<j<t\leq n$; see [5].

Recall that a left $\Gamma$-module X=\left(\begin{array}[]{c}X_{1}\\ \vdots\\ X_{n}\\ \end{array}\right) is described by a column vector, where each $X_{i}$ is a left $R_{i}$-module for $1\leq i\leq n$, and the left $\Gamma$-module structure map is denoted by $\varphi_{jl}:M_{jl}\otimes_{R_{l}}X_{l}\rightarrow X_{j}$ for $1\leq j<l\leq n$; see [5]. Dually, we have the description of right $\Gamma$-modules via row vectors.

Notation 2.1**.**

Let $\Gamma_{t}$ be the algebra given by the $t\times t$ leading principal submatrix of $\Gamma$ and $\Gamma^{\prime}_{n-t}$ be the algebra given by cutting the first $t$ rows and the first $t$ columns of $\Gamma$. Denote the left $\Gamma_{t}$-module \left(\begin{array}[]{c}M_{1,t+1}\\ \vdots\\ M_{t,t+1}\\ \end{array}\right) by $M_{t}^{*}$ and the right $\Gamma^{\prime}_{n-t}$-module \left(\begin{array}[]{c}M_{t,t+1},\cdots,M_{tn}\end{array}\right) by $M_{t}^{**}$, for $1\leq t\leq n-1$. Denote by $\Gamma_{t}^{D}$= diag\left(\begin{array}[]{c}R_{1},\cdots R_{t}\end{array}\right) the sub-algebra of $\Gamma_{t}$ consisting of diagonal matrices, and $\Gamma^{\prime}_{D,n-t}$= diag\left(\begin{array}[]{c}R_{t+1},\cdots R_{n}\end{array}\right) the sub-algebra of $\Gamma^{\prime}_{n-t}$ consisting of diagonal matrices.

Let \Gamma=\left(\begin{array}[]{cc}R_{1}&M_{12}\\ 0&R_{2}\end{array}\right) be an upper triangular matrix algebra. Recall that the $R_{1}$-$R_{2}$-bimodule $M_{12}$ is compatible, if the following two conditions hold; see[10, Definition 1.1]:

1. (C1)

If $Q^{\bullet}$ is an exact sequence of projective $R_{2}$-modules, then $M_{12}\otimes_{R_{2}}Q^{\bullet}$ is exact; 2. (C2)

If $P^{\bullet}$ is a complete $R_{1}$-projective resolution, then $\mathrm{Hom}_{R_{1}}(P^{\bullet},M_{12})$ is exact.

We use $\mathrm{pd}$ and $\mathrm{id}$ to denote the projective dimension and the injective dimension of a module, respectively.

Lemma 2.2**.**

Let $\Gamma$ be an upper triangular matrix algebra satisfying all $R_{i}$ Gorenstein. If $\Gamma$ is Gorenstein, then each $\Gamma_{t}$-$R_{t+1}$-bimodule $M^{*}_{t}$ is compatible and each $R_{t}$-$\Gamma^{\prime}_{n-t}$-bimodule $M^{**}_{t}$ is compatible for $1\leq t\leq n-1$.

Proof.

Let \Lambda=\left(\begin{array}[]{cc}S_{1}&N_{12}\\ 0&S_{2}\end{array}\right) be an upper triangular matrix algebra. Recall the fact that if $\mathrm{pd}_{S_{1}}(N_{12})<\infty$ and $\mathrm{pd}(N_{12})_{S_{2}}<\infty$, then $N_{12}$ is compatible; see [10, Proposition 1.3]. Recall that $\Gamma$ is Gorenstein if and only if all bimodules $M_{ij}$ are finitely generated and have finite projective dimension on both sides; see [5, Proposition 3.4]. Then we have $\mathrm{pd}_{R_{t}}(M^{**}_{t})<\infty$ and $\mathrm{pd}(M^{*}_{t})_{R_{t+1}}<\infty$ for $1\leq t\leq n-1$. By [5, Lemma 3.1], we have $\mathrm{pd}(M^{**}_{t})_{\Gamma^{\prime}_{n-t}}<\infty$ and $\mathrm{pd}_{\Gamma_{t}}(M^{*}_{t})<\infty$ for $1\leq t\leq n-1$. Then we are done. ∎

Let $A$ be a finite dimensional algebra over a field $k$. Denote by $A$-mod the category of finite dimensional left $A$-modules. The opposite algebra of $A$ is denoted by $A^{\rm op}$. We identify right $A$-modules with left $A^{\rm op}$-modules.

Denote by $(-)^{*}$ the contravariant functor ${\rm Hom}_{A}(-,A)$ or ${\rm Hom}_{A^{\rm op}}(-,A)$. 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 ${\rm ev}_{X}:X\rightarrow X^{**}$ given by ${\rm ev}_{X}(x)(f)=f(x)$ for $x\in X$ and $f\in X^{*}$. Recall that an $A$-module $G$ is Gorenstein-projective provided that ${\rm Ext}^{i}_{A}(G,A)=0={\rm Ext}^{i}_{A^{\rm op}}(G^{*},A)$ for $i\geq 1$ and the evaluation map ${\rm ev}_{G}$ is bijective; see [1, Proposition 3.8].

The algebra $A$ is Gorenstein if ${\rm id}_{A}A<\infty$ and ${\rm id}A_{A}<\infty$. It is well known that for a Gorenstein algebra $A$ we have ${\rm id}_{A}A={\rm id}A_{A}$; see [11, Lemma A]. For $m\geq 0$, a Gorenstein algebra $A$ is $m$-Gorenstein if ${\rm id}_{A}A={\rm id}A_{A}\leq m$. 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 $m\geq 0$. Let $A$ be an $m$-Gorenstein algebra. Then we have the following statements.

1. (1)

An $A$-module $M\in A$-Gproj if and only if ${\rm Ext}^{i}_{A}(M,A)=0$ for all $i>0$. 2. (2)

If $M\in A$-Gproj and $L$ is a right $A$-module with finite projective dimension, then ${\rm Tor}^{A}_{i}(L,M)=0$ for all $i>0$. 3. (3)

If there is an exact sequence $0\rightarrow M\rightarrow G_{0}\rightarrow G_{1}\rightarrow\cdots\rightarrow G_{m}$ with $G_{i}$ Gorenstein-projective, then $M\in A$-Gproj. $\square$

Lemma 2.4**.**

[10, Theorem 1.4]** Let $M_{12}$ be a compatible $R_{1}$-$R_{2}$-bimodule, and \Gamma=\left(\begin{array}[]{cc}R_{1}&M_{12}\\ 0&R_{2}\end{array}\right). Then X=\left(\begin{array}[]{c}X_{1}\\ X_{2}\end{array}\right)\in\Gamma-Gproj if and only if $\varphi_{12}:M_{12}\otimes_{R_{2}}X_{2}\rightarrow X_{1}$ is an injective $R_{1}$-map, $\rm{Coker}\varphi_{12}\in R_{1}$-Gproj, and $X_{2}\in R_{2}$-Gproj. $\square$

We have a slight generalization of Lemma 2.4.

Lemma 2.5**.**

Let $\Gamma$ be a Gorenstein upper triangular matrix algebra with each $R_{i}$ a group algebra. Then X=\left(\begin{array}[]{c}X_{1}\\ \vdots\\ X_{n}\end{array}\right)\in\Gamma-Gproj if and only if each $R_{t}$-map \varphi_{t}^{**}=\sum\limits_{j=t+1}^{n}\varphi_{tj}:M_{t}^{**}\otimes_{\Gamma^{\prime}_{n-t}}\left(\begin{array}[]{c}X_{t+1}\\ \vdots\\ X_{n}\end{array}\right)\rightarrow X_{t} sending \left(\begin{array}[]{c}m_{t,t+1},\cdots,m_{tn}\end{array}\right)\otimes\left(\begin{array}[]{c}x_{t+1}\\ \vdots\\ x_{n}\end{array}\right) to $\sum\limits_{j=t+1}^{n}\varphi_{tj}(m_{tj}\otimes x_{j})$ is injective for $1\leq t\leq n-1$.

Proof.

We have that each $R_{t}$-$\Gamma^{\prime}_{n-t}$-bimodule $M^{**}_{t}$ is compatible for $1\leq t\leq n-1$ by Lemma 2.2.

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 \Gamma=\left(\begin{array}[]{cc}R_{1}&M^{**}_{1}\\ 0&\Gamma^{\prime}_{n-1}\end{array}\right), and X=\left(\begin{array}[]{c}X_{1}\\ X^{\prime}\end{array}\right). Since $X\in\Gamma$-Gproj, by Lemma 2.4, we have that the $R_{1}$-map $\varphi_{1}^{**}:M_{1}^{**}\otimes_{\Gamma^{\prime}_{n-1}}X^{\prime}\rightarrow X_{1}$ is injective and $X^{\prime}\in\Gamma^{\prime}_{n-1}$-Gproj. By induction, we have that each $R_{t}$-map \varphi_{t}^{**}:M_{t}^{**}\otimes_{\Gamma^{\prime}_{n-t}}\left(\begin{array}[]{c}X_{t+1}\\ \vdots\\ X_{n}\end{array}\right)\rightarrow X_{t} is injective for $1\leq t\leq n-1$.

For the “if” part, we use induction on $n$. The case $n=2$ is due to Lemma 2.4. Assume that $n>2$. Write \Gamma=\left(\begin{array}[]{cc}R_{1}&M^{**}_{1}\\ 0&\Gamma^{\prime}_{n-1}\end{array}\right), and X=\left(\begin{array}[]{c}X_{1}\\ X^{\prime}\end{array}\right). By induction, we have $X^{\prime}\in\Gamma^{\prime}_{n-1}$-Gproj. Since the $R_{1}$-map $\varphi_{1}^{**}:M_{1}^{**}\otimes_{\Gamma^{\prime}_{n-1}}X^{\prime}\rightarrow X_{1}$ is injective and its cokernel belongs to $R_{1}$-Gproj for $R_{1}$ a group algebra, we have $X\in\Gamma$-Gproj by Lemma 2.4. ∎

Corollary 2.6**.**

Let $\Gamma$ be a Gorenstein upper triangular matrix algebra with each $R_{i}$ a group algebra. Assume that X=\left(\begin{array}[]{c}X_{1}\\ \vdots\\ X_{s}\\ 0\\ \vdots\\ 0\end{array}\right)\in\Gamma-Gproj. Then each $R_{i}$-map $\varphi_{is}:M_{is}\otimes_{R_{s}}X_{s}\rightarrow X_{i}$ is injective for $1\leq i<s\leq n$.

Proof.

We write X=\left(\begin{array}[]{c}X^{\prime}\\ X^{\prime\prime}\end{array}\right), where X^{\prime\prime}=\left(\begin{array}[]{c}X_{i+1}\\ \vdots\\ X_{s}\\ \vdots\\ 0\end{array}\right) for each $1\leq i<s\leq n$. We claim that each $R_{i}$-map $f_{is}:M_{is}\otimes_{R_{s}}X_{s}\rightarrow M_{i}^{**}\otimes_{\Gamma^{\prime}_{n-i}}X^{\prime\prime}$ sending $m_{is}\otimes x_{s}$ to \left(\begin{array}[]{c}0,\cdots,m_{is},\cdots,0\end{array}\right)\otimes\left(\begin{array}[]{c}0,\cdots,x_{s},\cdots,0\end{array}\right)^{t} is injective, where $(-)^{t}$ is the transpose. Since $\varphi_{is}=\varphi_{i}^{**}\circ f_{is}$ for $1\leq i<s\leq n$, then we are done by Lemma 2.5.

For the claim, we observe that for each $1\leq i<s\leq n$, the $R_{i}$-map $f_{is}$ is a composition of the following

[TABLE]

where the right $\Gamma^{\prime}_{n-i}$-map \left(\begin{array}[]{c}0,\cdots,M_{is},\cdots,M_{in}\end{array}\right)\overset{\iota}{\longrightarrow}M_{i}^{**} is the inclusion map and $g_{is}$ sends $m_{is}\otimes x_{s}$ to \left(\begin{array}[]{c}0,\cdots,m_{is},\cdots,0\end{array}\right)\otimes\left(\begin{array}[]{c}0,\cdots,x_{s},\cdots,0\end{array}\right)^{t}. We observe a $R_{i}$-map \left(\begin{array}[]{c}0,\cdots,M_{is},\cdots,M_{in}\end{array}\right)\otimes_{\Gamma^{\prime}_{n-i}}X^{\prime\prime}\overset{g^{\prime}_{is}}{\longrightarrow}M_{is}\otimes_{R_{s}}X_{s}, \left(\begin{array}[]{c}0,\cdots,m_{is},\cdots,m_{in}\end{array}\right)\otimes\left(\begin{array}[]{c}0,\cdots,x_{s},\cdots,0\end{array}\right)^{t}\mapsto m_{is}\otimes x_{s} satisfying $g^{\prime}_{is}\circ g_{is}={\mathrm{Id}_{M_{is}\otimes_{R_{s}}X_{s}}}$. Hence the $R_{i}$-map $g_{is}$ is injective. We observe that the right $\Gamma^{\prime}_{n-i}$-modules \left(\begin{array}[]{c}0,\cdots,M_{is},\cdots,M_{in}\end{array}\right) and $M_{i}^{**}$ have finite projective dimensions; see [5, Lemma 3.1], and $X^{\prime\prime}\in\Gamma^{\prime}_{n-i}$-Gproj by Lemma 2.4. Then the $R_{i}$-map $\iota\otimes{\rm Id}$ is injective by Lemma 2.3 (2). ∎

3. Proof of Proposition 1.1

Let $k$ be a field. Let $\mathscr{C}$ be a finite category, that is, it has only finitely many morphisms, and consequently it has only finitely many objects. Denote by $\mathrm{Mor}\mathscr{C}$ the finite set of all morphisms in $\mathscr{C}$. The category algebra k$\mathscr{C}$ of $\mathscr{C}$ is defined as follows: $\emph{k}\mathscr{C}=\bigoplus\limits_{\alpha\in\mathrm{Mor}\mathscr{C}}k\alpha$ as a $k$-vector space and the product $*$ is given by the rule

[TABLE]

The unit is given by $1_{k\mathscr{C}}=\sum\limits_{x\in\mathrm{Obj}\mathscr{C}}\mathrm{Id}_{x}$, where $\mathrm{Id}_{x}$ is the identity endomorphism of an object $x$ in $\mathscr{C}$.

If $\mathscr{C}$ and $\mathscr{D}$ are two equivalent finite categories, then k$\mathscr{C}$ and k$\mathscr{D}$ are Morita equivalent; see [7, Proposition 2.2]. In particular, $k\mathscr{C}$ is Morita equivalent to $k\mathscr{C}_{0}$, where $\mathscr{C}_{0}$ is any skeleton of $\mathscr{C}$. So we may assume that $\mathscr{C}$ is skeletal, that is, for any two distinct objects $x$ and $y$ in $\mathscr{C}$, $x$ is not isomorphic to $y$.

The category $\mathscr{C}$ is called a finite EI category provided that all endomorphisms in $\mathscr{C}$ are isomorphisms. In particular, $\mathrm{Hom}_{\mathscr{C}}(x,x)=\mathrm{Aut}_{\mathscr{C}}(x)$ is a finite group for any object $x$ in $\mathscr{C}$. Denote by $k\mathrm{Aut}_{\mathscr{C}}(x)$ the group algebra.

Throughout this paper, we assume that $\mathscr{C}$ is a finite EI category which is skeletal, and $\mathrm{Obj}\mathscr{C}=\{x_{1},x_{2},\cdots,x_{n}\}$, $n\geq 2$, satisfying $\mathrm{Hom}_{\mathscr{C}}(x_{i},x_{j})=\emptyset$ if $i<j$.

Let $M_{ij}=k\mathrm{Hom}_{\mathscr{C}}(x_{j},x_{i})$. Write $R_{i}=M_{ii}$, which is a group algebra. Recall that the category algebra $k\mathscr{C}$ is isomorphic to the corresponding upper triangular matrix algebra \Gamma_{\mathscr{C}}=\left(\begin{array}[]{cccc}R_{1}&M_{12}&\cdots&M_{1n}\\ &R_{2}&\cdots&M_{2n}\\ &&\ddots&\vdots\\ &&&R_{n}\\ \end{array}\right); see [5, Section 4].

Let us fix a field $k$. Denote by $k$-mod the category of finite dimensional $k$-vector spaces and $(k\text{-mod})^{\mathscr{C}}$ the category of covariant functors from $\mathscr{C}$ to $k$-mod.

Recall that the category $k\mathscr{C}$-mod of left modules over the category algebra $k\mathscr{C}$, is identified with $(k\text{-mod})^{\mathscr{C}}$; see [7, Proposition 2.1]. The category $k\mathscr{C}$-mod is a symmetric monoidal category. More precisely, the tensor product -$\hat{\otimes}$- is defined by

[TABLE]

for any $M,N\in(k\text{-mod})^{\mathscr{C}}$ and $x\in{\rm Obj}\mathscr{C}$, and $\alpha.(m\otimes n)=\alpha.m\otimes\alpha.n$ for any $\alpha\in\mathrm{Mor}\mathscr{C},m\in M(x),n\in N(x)$; see [8, 9].

Let $\mathscr{C}$ be a finite EI category, and \Gamma=\Gamma_{\mathscr{C}}=\left(\begin{array}[]{cccc}R_{1}&M_{12}&\cdots&M_{1n}\\ &R_{2}&\cdots&M_{2n}\\ &&\ddots&\vdots\\ &&&R_{n}\\ \end{array}\right) be the corresponding upper triangular matrix algebra. Let X=\left(\begin{array}[]{c}X_{1}\\ \vdots\\ X_{n}\\ \end{array}\right) and Y=\left(\begin{array}[]{c}Y_{1}\\ \vdots\\ Y_{n}\\ \end{array}\right) be two $\Gamma$-modules, where the left $\Gamma$-module structure maps are denoted by $\varphi^{X}_{ij}$ and $\varphi^{Y}_{ij}$, respectively. We observe that X\hat{\otimes}Y=\left(\begin{array}[]{c}X_{1}\otimes_{k}Y_{1}\\ \vdots\\ X_{n}\otimes_{k}Y_{n}\\ \end{array}\right), where the module structure map $\varphi_{ij}:M_{ij}\otimes_{R_{j}}(X_{j}\otimes_{k}Y_{j})\rightarrow X_{i}\otimes_{k}Y_{i}$ is induced by the following: $\varphi_{ij}(\alpha_{ij}\otimes(a_{j}\otimes b_{j}))=\varphi^{X}_{ij}(\alpha_{ij}\otimes a_{j})\otimes\varphi^{Y}_{ij}(\alpha_{ij}\otimes b_{j})$, where $\alpha_{ij}\in{\rm Hom}_{\mathscr{C}}(x_{j},x_{i})$, $a_{j}\in X_{j}$ and $b_{j}\in Y_{j}$.

Definition 3.1**.**

Let $\mathscr{C}$ be a finite EI category, and $\Gamma$ be the corresponding upper triangular matrix algebra. We say that $\mathscr{C}$ is* GPT-closed, if $X,Y\in\Gamma$-Gproj implies $X\hat{\otimes}Y\in\Gamma$-Gproj.*

Recall that a finite EI category $\mathscr{C}$ is projective over $k$ if each $k{\rm Aut}_{\mathscr{C}}(y)$-$k{\rm Aut}_{\mathscr{C}}(x)$-bimodule $k{\rm Hom}_{\mathscr{C}}(x,y)$ is projective on both sides; see [5, Definition 4.2]. We recall the fact that the category algebra $k\mathscr{C}$ is Gorenstein if and only if $\mathscr{C}$ is projective over $k$; see  [5, Proposition 5.1].

Let $\mathscr{C}$ be a finite projective EI category, and $\Gamma$ be the corresponding upper triangular matrix algebra. Denote by $C_{i}$ the $i$-th column of $\Gamma$ which is a $\Gamma$-$R_{i}$-bimodule and projective on both sides.

Proposition 3.2**.**

Let $\mathscr{C}$ be a finite projective EI category, and $\Gamma$ be the corresponding upper triangular matrix algebra. Then the following statements are equivalent.

1. (1)

The category $\mathscr{C}$ is GPT-closed. 2. (2)

For any $1\leq p\leq q\leq n$, $C_{p}\hat{\otimes}C_{q}\in\Gamma$-Gproj. 3. (3)

For any $1\leq p\leq q\leq n$, $C_{p}\hat{\otimes}C_{q}\in\Gamma$-proj.

Proof.

“(1)$\Rightarrow$ (2)” and “(3)$\Rightarrow$ (2)” are obvious.

“(2)$\Rightarrow$ (3)” We only need to prove that the $\Gamma$-module $C_{p}\hat{\otimes}C_{q}$ has finite projective dimension, since a Gorenstein-projective module with finite projective dimension is projective. We have C_{p}\hat{\otimes}C_{q}=\left(\begin{array}[]{c}M_{1p}\otimes_{k}M_{1q}\\ \vdots\\ M_{p-1,p}\otimes_{k}M_{p-1,q}\\ R_{p}\otimes_{k}M_{pq}\\ 0\\ \vdots\\ 0\end{array}\right). Since $\mathscr{C}$ is projective, we have that each $M_{ip}$ is a projective $R_{i}$-module for $1\leq i\leq p$. Then each $M_{ip}\otimes_{k}M_{iq}$ is a projective $R_{i}$-module since $R_{i}$ is a group algebra for $1\leq i\leq p$. Hence the $\Gamma$-module $C_{p}\hat{\otimes}C_{q}$ has finite projective dimension by [5, Corollary 3.6]. Then we are done.

“(2)$\Rightarrow$ (1)” We have that $\Gamma$ is a Gorenstein algebra by [5, Proposition 5.1]. Then there is $d\geq 0$ such that $\Gamma$ is a $d$-Gorenstein algebra.

For any $M\in\Gamma$-Gproj, consider the following exact sequence

[TABLE]

with $P_{i}$ projective, $0\leq i\leq d$. Applying -$\hat{\otimes}N$ on the above exact sequence, we have an exact sequence

[TABLE]

since the tensor product -$\hat{\otimes}$- is exact in both variables. If $N$ is projective, we have that each $P_{i}\hat{\otimes}N$ is Gorenstein-projective for $0\leq i\leq d$ by (2). Then we have $M\hat{\otimes}N\in\Gamma$-Gproj by Lemma 2.3 (3). If $N$ is Gorenstein-projective, we have that each $P_{i}\hat{\otimes}N$ is Gorenstein-projective for $0\leq i\leq d$ in exact sequence (3.1) by the above process. Then we have $M\hat{\otimes}N\in\Gamma$-Gproj by Lemma 2.3 (3). Then we are done. ∎

The argument in “(2)$\Rightarrow$ (3)” of Proposition 3.2 implies the following result. It follows that the tensor product -$\hat{\otimes}$- on $\Gamma$-Gproj induces the one on $\Gamma$-$\underline{\rm Gproj}$, still denoted by -$\hat{\otimes}$-.

Lemma 3.3**.**

Assume that $\mathscr{C}$ is GPT-closed. Let $M\in\Gamma$-Gproj and $P\in\Gamma$-proj. Then $M\hat{\otimes}P\in\Gamma$-proj. $\square$

Let $\mathscr{C}$ be a finite projective EI category, and $\Gamma$ be the corresponding upper triangular matrix algebra of $\mathscr{C}$. Recall that a complex in ${\rm D}^{b}(\Gamma\text{-mod})$, the bounded derived category of finitely generated left $\Gamma$-modules, is called a perfect complex if it is isomorphic to a bounded complex of finitely generated projective modules. Recall from [2] that the singularity category of $\Gamma$, denoted by ${\rm D}_{sg}(\Gamma)$, is the Verdier quotient category ${\rm D}^{b}(\Gamma\text{-mod})/{\rm perf}(\Gamma)$, where ${\rm perf}(\Gamma)$ is a thick subcategory of ${\rm D}^{b}(\Gamma\text{-mod})$ consisting of all perfect complexes.

Recall from [6] that there is a triangle equivalence

[TABLE]

sending a Gorenstein-projective module to the corresponding stalk complex concentrated on degree zero. The functor $F$ transports the tensor product on ${\rm D}_{sg}(\Gamma)$ to $\Gamma\text{-}\underline{\rm Gproj}$ such that the category $\Gamma\text{-}\underline{\rm Gproj}$ becomes a tensor triangulated category.

Proposition 3.4**.**

Let $\mathscr{C}$ be a finite projective EI category, and $\Gamma$ be the corresponding upper triangular matrix algebra of $\mathscr{C}$. Assume that the category $\mathscr{C}$ is GPT-closed. Then the tensor product -$\hat{\otimes}$- on $\Gamma$-$\underline{\rm Gproj}$ induced by the tensor product on $\Gamma$-Gproj coincide with the one transported from ${\rm D}_{sg}(\Gamma)$, up to natural isomorphism.

Proof.

Consider the functor $F$ in (3.2). Recall that the tensor product on ${\rm D}_{sg}(\Gamma)$ is induced by the tensor product -$\hat{\otimes}$- on ${\rm D}^{b}(\Gamma\text{-mod})$, where the later is given by -$\hat{\otimes}$- on $\Gamma$-mod. We have $F(M)\hat{\otimes}F(N)=F(M\hat{\otimes}N)$ in ${\rm D}_{sg}(\Gamma)$ for any $M,N\in\Gamma$-$\underline{\rm Gproj}$. This implies that $F$ is a tensor triangle equivalence. Then we are done. ∎

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 “$\sim$” on the product $X\times Y$ as follows: $(x,y)\sim(x^{\prime},y^{\prime})$ if and only if there is an element $g\in G$ such that $x=x^{\prime}g$ and $y=g^{-1}y^{\prime}$ for $x,x^{\prime}\in X$ and $y,y^{\prime}\in Y$. Write the quotient set $X\times Y/\sim$ as $X\times_{G}Y$.

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

[TABLE]

where $x\in X$ and $y\in Y$. $\square$

Lemma 3.6**.**

Let $Y_{1}$ and $Y_{2}$ be two left $G$-sets. Then we have an isomorphism of left $kG$-modules

[TABLE]

where $y_{1}\in Y_{1},y_{2}\in Y_{2}$. $\square$

Lemma 3.7**.**

Let $\mathscr{C}$ be a finite projective EI category, and $\Gamma$ be the corresponding upper triangular matrix algebra. Assume $1\leq p\leq q\leq n$. Then $C_{p}\hat{\otimes}C_{q}\in\Gamma$-proj implies that each morphism in $\bigsqcup_{y\in{\rm Obj}\mathscr{C}}\mathrm{Hom}_{\mathscr{C}}(x_{p},y)$ is a monomorphism.

Proof.

We have C_{p}\hat{\otimes}C_{q}=\left(\begin{array}[]{c}M_{1p}\otimes_{k}M_{1q}\\ \vdots\\ M_{p-1,p}\otimes_{k}M_{p-1,q}\\ R_{p}\otimes_{k}M_{pq}\\ 0\\ \vdots\\ 0\end{array}\right). Then each $R_{i}$-map

[TABLE]

sending $\alpha\otimes(g\otimes\beta)$ to $\alpha\circ g\otimes\alpha\circ\beta$, where $\alpha\in\mathrm{Hom}_{\mathscr{C}}(x_{p},x_{i}),g\in\mathrm{Aut}_{\mathscr{C}}(x_{p}),\beta\in\mathrm{Hom}_{\mathscr{C}}(x_{q},x_{p})$, is injective for $1\leq i<p\leq q\leq n$ by Corollary 2.6. We have that the sets $\mathrm{Hom}_{\mathscr{C}}(x_{p},x_{i})\times_{\mathrm{Aut}_{\mathscr{C}}(x_{p})}(\mathrm{Aut}_{\mathscr{C}}(x_{p})\times\mathrm{Hom}_{\mathscr{C}}(x_{q},x_{p}))$ and $\mathrm{Hom}_{\mathscr{C}}(x_{p},x_{i})\times\mathrm{Hom}_{\mathscr{C}}(x_{q},x_{i})$ are $k$-basis of $M_{ip}\otimes_{R_{p}}(R_{p}\otimes_{k}M_{pq})$ and $M_{ip}\otimes_{k}M_{iq}$, respectively by Lemma 3.5 and Lemma 3.6. For each $1\leq i<p$, since $\varphi_{ip}$ is injective, we have an injective map

[TABLE]

sending $(\alpha,(g,\beta))$ to $(\alpha\circ g,\alpha\circ\beta)$, for $\alpha\in\mathrm{Hom}_{\mathscr{C}}(x_{p},x_{i}),g\in\mathrm{Aut}_{\mathscr{C}}(x_{p}),\beta\in\mathrm{Hom}_{\mathscr{C}}(x_{q},x_{p})$.

For each $1\leq i<p$, and $\alpha\in\mathrm{Hom}_{\mathscr{C}}(x_{p},x_{i})$, let $\beta,\beta^{\prime}\in\mathrm{Hom}_{\mathscr{C}}(x_{q},x_{p})$ satisfy $\alpha\circ\beta=\alpha\circ\beta^{\prime}$. Then we have $(\alpha,\alpha\circ\beta)=(\alpha,\alpha\circ\beta^{\prime})$, that is, $\varphi(\alpha,({\rm Id}_{x_{p}},\beta))=\varphi(\alpha,({\rm Id}_{x_{p}},\beta^{\prime}))$. Since $\varphi$ is injective, we have $(\alpha,({\rm Id}_{x_{p}},\beta))=(\alpha,({\rm Id}_{x_{p}},\beta^{\prime}))$ in $\mathrm{Hom}_{\mathscr{C}}(x_{p},x_{i})\times_{\mathrm{Aut}_{\mathscr{C}}(x_{p})}(\mathrm{Aut}_{\mathscr{C}}(x_{p})\times\mathrm{Hom}_{\mathscr{C}}(x_{q},x_{p}))$. Hence $\beta=\beta^{\prime}$. Then we have that $\alpha$ is a monomorphism. ∎

Proposition 3.8**.**

Let $\mathscr{C}$ be a finite projective EI category. Assume that $\mathscr{C}$ is GPT-closed. Then that each morphism in $\mathscr{C}$ is a monomorphism.

Proof.

It follows from Proposition 3.2 and Lemma 3.7. ∎

Let $\mathcal{P}$ be a finite poset. We assume that ${\rm Obj}\mathcal{P}=\{x_{1},\cdots,x_{n}\}$ satisfying $x_{i}\nleq x_{j}$ if $i<j$, and $\Gamma$ is the corresponding upper triangular matrix algebra. We observe that each entry of $\Gamma$ is [math] or $k$, and each projective $\Gamma$-module is a direct sum of some $C_{i}$ for $1\leq i\leq n$. For any $a,b\in{\rm Obj}\mathcal{P}$ satisfying $a\nleq b$ and $b\nleq a$, denote by $L_{a,b}=\{x\in{\rm Obj}\mathcal{P}\mid a<x,b<x\}$.

Example 3.9**.**

Let $\mathcal{P}$ be a finite poset. Then $\mathcal{P}$ is $GPT$-closed if and only if any two distinct minimal elements in $L_{a,b}$ has no common upper bound for $a,b\in{\rm Obj}\mathcal{P}$ satisfying $a\nleq b$ and $b\nleq a$.

For the “if” part, assume that any two distinct minimal elements in $L_{a,b}$ has no common upper bound. By Proposition 3.2, we only need to prove that $C_{t}\hat{\otimes}C_{n}$ is projective for $1\leq t\leq n$, since the general case of $C_{t}\hat{\otimes}C_{j}$ can be considered in $\Gamma_{max\{t,j\}}$.

For each $1\leq t\leq n$, if $(C_{n})_{t}=k$, that is, $x_{n}\leq x_{t}$, then $(C_{t})_{i}=k$ implies $(C_{n})_{i}=k$ for $1\leq i\leq t$. Hence we have $C_{t}\hat{\otimes}C_{n}\simeq C_{t}$. Assume that $(C_{n})_{t}=0$, that is, $x_{n}\nleq x_{t}$. Let $L^{\prime}_{x_{t},x_{n}}=\{x_{s_{1}},\cdots,x_{s_{r}}\}$ be all distinct minimal elements in $L_{x_{t},x_{n}}$. For each $1\leq i<t$, if $(C_{t})_{i}=k=(C_{n})_{i}$, that is, $x_{n}\leq x_{i},x_{t}\leq x_{i}$, then there is a unique $x_{s_{l}}\in L^{\prime}_{x_{t},x_{n}}$ satisfying $x_{s_{l}}\leq x_{i}$, that is, there is a unique $x_{s_{l}}\in L^{\prime}_{x_{t},x_{n}}$ satisfying $(C_{s_{l}})_{i}=k$, since any two distinct elements in $L^{\prime}_{x_{t},x_{n}}$ has no common upper bound. Then we have $C_{t}\hat{\otimes}C_{n}\simeq\bigoplus\limits_{l=1}^{r}C_{s_{l}}$.

For the “only if” part, assume that $x_{t},x_{j}\in{\rm Obj}\mathcal{P}$ satisfying $x_{t}\nleq x_{j}$ and $x_{j}\nleq x_{t}$ and $C_{t}\hat{\otimes}C_{j}\simeq\bigoplus\limits_{l=1}^{r}C_{s_{l}}$. Then each $x_{s_{l}}\in L_{x_{t},x_{j}}$. Assume that $x_{s_{1}}$ and $x_{s_{2}}$ be two distinct minimal elements in $L_{x_{t},x_{j}}$ having a common upper bound $x_{i}$. Then $(C_{t}\hat{\otimes}C_{j})_{i}=k$ and $(C_{s_{1}}\oplus C_{s_{2}})_{i}=k\oplus k$, which is a contradiction.* $\square$*

4. Proof of Theorem 1.2

Let $\mathscr{C}$ be a finite EI category. Recall from  [3, Definition 2.3] that a morphism $x\overset{\alpha}{\rightarrow}y$ in $\mathscr{C}$ is unfactorizable if $\alpha$ is not an isomorphism and whenever it has a factorization as a composite $x\overset{\beta}{\rightarrow}z\overset{\gamma}{\rightarrow}y$, then either $\beta$ or $\gamma$ is an isomorphism. Let $x\overset{\alpha}{\rightarrow}y$ in $\mathscr{C}$ be an unfactorizable morphism. Then $h\circ\alpha\circ g$ is also unfactorizable for every $h\in\mathrm{Aut}_{\mathscr{C}}(y)$ and every $g\in\mathrm{Aut}_{\mathscr{C}}(x)$; see [3, Proposition 2.5]. Let $x\overset{\alpha}{\rightarrow}y$ in $\mathscr{C}$ be a morphism with $x\neq y$. Then it has a decomposition $x=x_{0}\overset{\alpha_{1}}{\rightarrow}x_{1}\overset{\alpha_{2}}{\rightarrow}\cdots\overset{\alpha_{n}}{\rightarrow}x_{n}=y$ with all $\alpha_{i}$ unfactorizable; see [3, Proposition 2.6].

Following [3, Definition 2.7], we say that a finite EI category $\mathscr{C}$ satisfies the Unique Factorization Property (UFP), if whenever a non-isomorphism $\alpha$ has two decompositions into unfactorizable morphisms:

[TABLE]

and

[TABLE]

then $m=n$, $x_{i}=y_{i}$, and there are $h_{i}\in\mathrm{Aut}_{\mathscr{C}}(x_{i})$, $1\leq i\leq m-1$, such that the following diagram commutes :

[TABLE]

Let $\mathscr{C}$ be a finite EI category. Following [4, Section 6], we say that $\mathscr{C}$ is 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 $n\geq 2$. Let $\mathscr{C}$ be a finite projective and free EI category with $\mathrm{Obj}\mathscr{C}=\{x_{1},x_{2},\cdots,x_{n}\}$ satisfying $\mathrm{Hom}_{\mathscr{C}}(x_{i},x_{j})=\emptyset$ if $i<j$ and $\Gamma$ be the corresponding upper triangular matrix algebra of $\mathscr{C}$. Then $\Gamma$ is $1$-Gorenstein; see [5, Theorem 5.3].

Set $\mathrm{Hom}_{\mathscr{C}}^{0}(x_{j},x_{i})=\{\alpha\in\mathrm{Hom}_{\mathscr{C}}(x_{j},x_{i})\mid\alpha\text{ is unfactorizable}\}$. Denote by $M^{0}_{ij}=k\mathrm{Hom}_{\mathscr{C}}^{0}(x_{j},x_{i})$, which is an $R_{i}$-$R_{j}$-sub-bimodule of $M_{ij}$; see [5, Notation 4.8]. Recall the left $\Gamma_{t}$-module $M_{t}^{*}$ and the right $\Gamma^{\prime}_{n-t}$-module $M_{t}^{**}$ in Notation 2.1, for $1\leq t\leq n-1$. Observe that $M_{t}^{**}\simeq(M_{t,t+1}^{0},M_{t,t+2}^{0},\cdots,M_{tn}^{0})\otimes_{\Gamma^{\prime}_{D,n-t}}\Gamma_{n-t}^{\prime}$; compare [5, Lemmas 4.10 and 4.11], which implies that M_{t}^{*}\simeq\Gamma_{t}\otimes_{\Gamma_{t}^{D}}\left(\begin{array}[]{c}M_{1,t+1}^{0}\\ \vdots\\ M_{t,t+1}^{0}\end{array}\right).

Let X=\left(\begin{array}[]{c}X_{1}\\ \vdots\\ X_{n}\end{array}\right) be a left $\Gamma$-module. For each $1\leq t\leq n-1$, we have

[TABLE]

Recall the $R_{t}$-map $\varphi_{t}^{**}$ in Lemma 2.5. Here, we observe that

[TABLE]

Lemma 4.1**.**

Let $\mathscr{C}$ be a finite projective and free EI category and $\Gamma$ be the corresponding upper triangular matrix algebra. Assume $1\leq p\leq q\leq n$. If each morphism in $\bigsqcup_{y\in{\rm Obj}\mathscr{C}}\bigsqcup_{j=1}^{p}\mathrm{Hom}_{\mathscr{C}}(x_{j},y)$ is a monomorphism, then $C_{p}\hat{\otimes}C_{q}\in\Gamma$-proj.

Proof.

We only need to prove that each $R_{t}$-map

[TABLE]

is injective for $1\leq t<p$ by Lemma 2.5 and Proposition 3.2.

By Lemmas  3.5 and 3.6, we have that the set $\mathrm{Hom}_{\mathscr{C}}(x_{p},x_{t})\times\mathrm{Hom}_{\mathscr{C}}(x_{q},x_{t})$ is a $k$-basis of $M_{tp}\otimes_{k}M_{tq}$, and the set

[TABLE]

is a $k$-basis of $\bigoplus\limits_{j=t+1}^{p}M_{tj}^{0}\otimes_{R_{j}}(M_{jp}\otimes_{k}M_{jq})$.

We have the following commutative diagram

[TABLE]

Observe that $\varphi_{t}^{**}$ is injective if and only if $\varphi_{t}^{**}\mid_{B}$ is injective for each $1\leq t<p$.

Assume that $\varphi_{t}^{**}(\alpha,(\beta,\theta))=\varphi_{t}^{**}(\alpha^{\prime},(\beta^{\prime},\theta^{\prime}))$, where $\alpha\in\mathrm{Hom}^{0}_{\mathscr{C}}(x_{j},x_{t})$, $\beta\in\mathrm{Hom}_{\mathscr{C}}(x_{p},x_{j})$, $\theta\in\mathrm{Hom}_{\mathscr{C}}(x_{q},x_{j})$ and $\alpha^{\prime}\in\mathrm{Hom}^{0}_{\mathscr{C}}(x_{j^{\prime}},x_{t})$, $\beta^{\prime}\in\mathrm{Hom}_{\mathscr{C}}(x_{p},x_{j^{\prime}})$, $\theta^{\prime}\in\mathrm{Hom}_{\mathscr{C}}(x_{q},x_{j^{\prime}})$. Then we have $\alpha\beta=\alpha^{\prime}\beta^{\prime}$ in $\mathrm{Hom}_{\mathscr{C}}(x_{p},x_{t})$ and $\alpha\theta=\alpha^{\prime}\theta^{\prime}$ in $\mathrm{Hom}_{\mathscr{C}}(x_{q},x_{t})$. Since $\mathscr{C}$ is free and $\alpha,\alpha^{\prime}$ are unfactorizable, we have that $j=j^{\prime}$ and there is $g\in\mathrm{Aut}_{\mathscr{C}}(x_{j})$ such that $\alpha=\alpha^{\prime}g$ and $\beta=g^{-1}\beta^{\prime}$. Since $\alpha\theta=\alpha^{\prime}\theta^{\prime}=\alpha g^{-1}\theta^{\prime}$ and $\alpha$ is a monomorphism, we have that $\theta=g^{-1}\theta^{\prime}$. Then we have that $(\alpha,(\beta,\theta))=(\alpha^{\prime}g,(g^{-1}\beta^{\prime},g^{-1}\theta^{\prime}))=(\alpha^{\prime},(\beta^{\prime},\theta^{\prime}))$, which implies that the map $\varphi_{t}^{**}\mid_{B}$ is injective. ∎

Theorem 4.2**.**

Let $\mathscr{C}$ be a finite projective and free EI category. Then the category $\mathscr{C}$ is $GPT$-closed if and only if each morphism in $\mathscr{C}$ is 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).