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\title{\textbf{Self-injective dimensions under}\\\textbf{stable equivalences of Morita type}}
\author{Qiyue Tang}
\date{7 November 2026}

\begin{document}
\maketitle

\begin{abstract}
Stable equivalences of Morita type preserve the left and right self-injective dimensions of Artin algebras. The defining tensor functors can send injective modules to noninjective modules.
\end{abstract}

\begingroup
\renewcommand{\thefootnote}{}
\footnotetext{\textit{Keywords.} Stable equivalence of Morita type, self-injective dimension, Artin algebra.\par
\textit{2020 Mathematics Subject Classification.} Primary 16E10. Secondary 16D50.}
\endgroup

\section{Introduction}

Brou\'e formulated stable equivalence of Morita type in the study of blocks of finite group algebras \cite[\S5.A]{Broue}. Over a field, the defining bimodules induce inverse equivalences of the stable module categories, whose morphisms are taken modulo maps factoring through projective modules. For block algebras, Brou\'e recalled that a Rickard equivalence induces a stable equivalence of Morita type \cite[Proposition~5.2]{Broue}. 

Liu and Xi constructed stable equivalences of Morita type between endomorphism algebras of generators \cite[Theorems~1.1 and~3.6]{LX}. Their construction can start with self-injective algebras and produce algebras that are no longer self-injective. The example in \cite[\S5]{LX} yields two such endomorphism algebras each having left and right self-injective dimension two. By contrast, the example in \cite[p.~581]{LX} shows that an equivalence of stable module categories can relate algebras whose left self-injective dimensions are two and infinity.

When the two defining tensor functors are adjoint in both orders, Liu and Xi proved equality of left self-injective dimensions \cite[Proposition~4.1]{LX}. Their argument uses that a right adjoint to an exact functor preserves injectives \cite[Lemma~4.2]{LX}. The note added in proof cites Dugas and Mart\'inez-Villa \cite[Corollary~3.1]{DMV}, who obtained this adjunction for indecomposable defining bimodules between finite-dimensional algebras with separable semisimple quotients and no semisimple blocks. Chen, Pan and Xi \cite[Lemma~4.1(2)]{CPX} obtained this adjunction over an arbitrary field for finite-dimensional algebras with no semisimple direct summands and defining bimodules with no projective bimodule summands. Liu and Xi's Question~1 \cite[\S5, p.~584]{LX} asks whether an arbitrary stable equivalence of Morita type between Artin algebras preserves finiteness of the left self-injective dimension.

Beligiannis \cite[Theorem~8.12(iii)]{Bel} stated the invariance of the Gorenstein property, meaning simultaneous finiteness of the left and right self-injective dimensions. Sun and Zhao proved equality of the dimensions on each side under symmetric separable equivalence, which requires the two tensor functors to be adjoint in both orders \cite[Corollary~4.3 in the preprint]{SZsym}.

We use the bimodule splittings to split the maps on Ext induced by the tensor functors. The argument uses the same direct-summand principle as Kadison's transfer of Tor \cite[Proposition~6.3]{Kadison}.

\section{Invariance of self-injective dimensions}

Let \(k\) be a commutative Artinian ring. An Artin \(k\)-algebra is a unital \(k\)-algebra finitely generated as a \(k\)-module. All bimodules considered have coincident left and right \(k\)-actions. For an algebra \(A\), write \({}_AA\) and \(A_A\) for the left and right regular modules. For a left \(A\)-module \(X\), let \(\injdim({}_AX)\) denote the least nonnegative length of an injective resolution in the category of all left \(A\)-modules, or \(\infty\) if no finite injective resolution exists. We use the same convention on the right.

For a \(k\)-algebra \(R\), write \(R^{\mathrm{op}}\) for the opposite algebra, with reversed multiplication, and \(b^{\mathrm{op}}\) for \(b\in R\) viewed in \(R^{\mathrm{op}}\). An \(R\)-bimodule is projective if it is projective over \(R\otimes_k R^{\mathrm{op}}\) for the action \((a\otimes b^{\mathrm{op}})x=axb\) on its elements \(x\), where \(a,b\in R\).

Artin \(k\)-algebras \(A\) and \(B\) are \emph{stably equivalent of Morita type} if there are bimodules \({}_AM_B\) and \({}_BN_A\), finitely generated and projective on each side, and projective bimodules \({}_AP_A\) and \({}_BQ_B\) with bimodule isomorphisms
\[
 M\otimes_B N\simeq A\oplus P,\qquad
 N\otimes_A M\simeq B\oplus Q.
\]

\begin{theorem}\label{thm:main}
Let \(A\) and \(B\) be Artin algebras stably equivalent of Morita type. Then
\[
 \injdim({}_AA)=\injdim({}_BB),\qquad
 \injdim(A_A)=\injdim(B_B).
\]
\end{theorem}

\begin{remark}
Sun and Zhao \cite[Proposition~3.2]{SZ} stated both equalities for finite-dimensional algebras over an algebraically closed field. Their proof uses the assertion in Lemma~3.1 that the defining tensor functors preserve injective modules. Section~\ref{sec:example} gives a counterexample to this assertion and exhibits the two different projections identified in its proof.
\end{remark}

\begin{proof}[Proof of Theorem~\ref{thm:main}]
Let \(F=N\otimes_A-\) be the functor from all left \(A\)-modules to left \(B\)-modules, and \(G=M\otimes_B-\) the functor in the reverse direction. The right projectivity of \(N\) and \(M\) makes \(F\) and \(G\) exact. A projective left \(A\)-module is a direct summand of a sum of copies of \({}_AA\). Since \(F\) commutes with direct sums and \(F(A)\simeq {}_BN\) is projective, \(F\) preserves projectives. The same reasoning applies to \(G\).

Fix the two bimodule isomorphisms in the definition. For a left \(A\)-module \(X\), associativity of tensor products gives
\[
 GF(X)\simeq (M\otimes_B N)\otimes_A X
       \simeq X\oplus(P\otimes_A X).
\]
The inclusion \(\iota_X\colon X\to GF(X)\) and projection \(\pi_X\colon GF(X)\to X\) onto the first summand are natural in \(X\), with \(\pi_X\iota_X=1_X\). The second bimodule isomorphism gives \(FG(Y)\simeq Y\oplus(Q\otimes_B Y)\) naturally in left \(B\)-modules \(Y\).

For a ring \(R\) and left \(R\)-modules \(X,Y\), take a projective resolution \(L_\bullet\to X\). Write \(d_{i+1}\colon L_{i+1}\to L_i\) for its differential in degree \(i+1\), where \(i\ge 0\). Write \(\Hom_R(L_i,Y)\) for the group of \(R\)-linear maps. Then \(\Ext_R^n(X,Y)\), for \(n\ge 0\), is the degree-\(n\) cohomology of the complex with terms \(\Hom_R(L_i,Y)\) and differential \(f\mapsto f\circ d_{i+1}\).

For \(R=A\), the complex \(F(L_\bullet)\) with differentials \(F(d_{i+1})\) resolves \(F(X)\) by projectives. Applying \(F\) to each homomorphism \(L_i\to Y\) gives a cochain map because \(F\) preserves composition. Define its degree-\(i\) retraction by
\[
 \Hom_B(F(L_i),F(Y))\longrightarrow\Hom_A(L_i,Y),
 \qquad h\longmapsto \pi_Y G(h)\iota_{L_i}.
\]
Naturality of \(\iota\) gives
\[
 \pi_Y G\bigl(h\circ F(d_{i+1})\bigr)\iota_{L_{i+1}}
   =\pi_Y G(h)\iota_{L_i}d_{i+1},
\]
so the retraction commutes with the cochain differentials. For \(f\colon L_i\to Y\), naturality of \(\pi\) gives
\[
 \pi_Y GF(f)\iota_{L_i}=f\pi_{L_i}\iota_{L_i}=f.
\]
Taking cohomology yields a split injection
\begin{equation}\label{eq:split}
 \Ext_A^n(X,Y)\longrightarrow\Ext_B^n(F(X),F(Y))\qquad(n\ge 0).
\end{equation}

Suppose \(d=\injdim({}_BB)\) is finite. The finitely generated projective module \({}_BN\simeq F(A)\) is a summand of a finite direct sum of copies of \({}_BB\). For each \(X\) and \(n>d\), the group \(\Ext_B^n(F(X),F(A))\) is therefore a summand of a finite sum of copies of the zero group \(\Ext_B^n(F(X),B)\). By \eqref{eq:split} with \(Y=A\), we have \(\Ext_A^{d+1}(X,A)=0\) for every left \(A\)-module \(X\).

Start with \(C^0={}_AA\) and choose short exact sequences \(0\to C^j\to J^j\to C^{j+1}\to 0\) with \(J^j\) injective for \(0\le j<d\). Since \(\Ext_A^m(X,J^j)=0\) for \(m\ge 1\), the long exact sequences give
\[
 \Ext_A^m(X,C^{j+1})\simeq \Ext_A^{m+1}(X,C^j)
 \qquad(m\ge 1,\ 0\le j<d).
\]
Iterating yields \(\Ext_A^1(X,C^d)=0\) for every \(X\), including \(d=0\) by the original vanishing. If \(C^d\) is embedded in an injective module \(J\), then \(\Ext_A^1(J/C^d,C^d)=0\) makes \(\Hom_A(J,C^d)\to\Hom_A(C^d,C^d)\) surjective. A preimage of the identity is a retraction, so \(C^d\) is injective. Splicing the short exact sequences gives an injective resolution of \({}_AA\) of length \(d\). Exchanging \(A\) and \(B\) and using the natural splitting of \(FG\) gives the reverse bound whenever \(\injdim({}_AA)\) is finite. Hence the two left dimensions are equal, whether finite or infinite.

For right modules use \(-\otimes_A M\) and \(-\otimes_B N\) in place of \(F\) and \(G\). Left projectivity makes these functors exact, and right projectivity makes them preserve projectives. For right modules \(X\) over \(A\) and \(Y\) over \(B\), the two composites split naturally as
\[
 (X\otimes_A M)\otimes_B N \simeq X\oplus(X\otimes_A P),
 \qquad
 (Y\otimes_B N)\otimes_A M \simeq Y\oplus(Y\otimes_B Q).
\] The same cochain formulas give split injections on right-module Ext groups. The functors take the regular modules to \(M_B\) and \(N_A\), respectively, which are finitely generated projective. Applying the dimension argument to these splittings proves the right equality.
\end{proof}

\section{Tensor functors and injective modules}\label{sec:example}

Let \(k\) be an algebraically closed field and let \(A\) be the algebra of upper triangular \(2\times 2\) matrices over \(k\). Write \(e_{ij}\) for the matrix unit with a \(1\) in position \((i,j)\), for \(1\le i\le j\le 2\). Right multiplication by \(e_{12}\) embeds the left ideal \(Ae_{11}=ke_{11}\) in \(Ae_{22}\), sending \(e_{11}\) to \(e_{12}\). If \(r\colon Ae_{22}\to Ae_{11}\) is \(A\)-linear, then
\[
 r(e_{12})=r(e_{12}e_{22})=e_{12}r(e_{22})=0
\]
because \(e_{12}Ae_{11}=0\). Thus the embedding has no retraction. It follows that \(Ae_{11}\) is a noninjective direct summand of \({}_AA=Ae_{11}\oplus Ae_{22}\).

Take the regular bimodule \(M=A\) and the bimodule \(N=A\oplus(A\otimes_k A)\), with multiplication on the outside factors of \(A\otimes_k A\). The map \(a\otimes b^{\mathrm{op}}\mapsto a\otimes b\) identifies this tensor summand with a free module of rank one over \(A\otimes_k A^{\mathrm{op}}\). It is also free on each side, by choosing a \(k\)-basis of the factor on which that side does not act. Both \(M\otimes_A N\) and \(N\otimes_A M\) identify with \(N\) by multiplication. Thus \(M,N\) define a stable self-equivalence of Morita type.

Let \(I=\Hom_k(e_{22}A,k)\), with left action \((a\varphi)(x)=\varphi(xa)\) for \(a\in A\), \(\varphi\in I\) and \(x\in e_{22}A\). This module is one-dimensional because \(e_{22}A=ke_{22}\). The right ideal \(e_{22}A\) is projective as a direct summand of \(A_A\). For any left \(A\)-module \(X\), the correspondence
\[
 \Hom_A(X,I)\longrightarrow\Hom_k(e_{22}A\otimes_A X,k),
 \qquad f\longmapsto [x\otimes v\mapsto f(v)(x)]
\]
is balanced since \(f(av)(x)=f(v)(xa)\) for \(a\in A\), \(v\in X\) and \(x\in e_{22}A\). Its inverse sends a functional \(\lambda\) to the map \(v\mapsto[x\mapsto\lambda(x\otimes v)]\) from \(X\) to \(I\). This map is \(A\)-linear by the tensor relation \(x\otimes av=xa\otimes v\). Thus \(\Hom_A(-,I)\) is the composite of the exact functor \(e_{22}A\otimes_A-\) and the exact \(k\)-linear dual, so \(I\) is injective. The tensor functor \(F=N\otimes_A-\) instead gives
\[
 F(I)\simeq I\oplus(A\otimes_k I)\simeq I\oplus A
\]
as left \(A\)-modules. Here the first identification uses \((a\otimes b)\otimes v\mapsto a\otimes bv\), with inverse \(a\otimes v\mapsto(a\otimes 1)\otimes v\) on the tensor summand. The second uses \(\dim_k I=1\). The summand \(Ae_{11}\) shows that \(F(I)\) is not injective. This refutes the preservation direction of \cite[Lemma~3.1(1)]{SZ} and the injective-dimension equality for arbitrary left modules in \cite[Proposition~3.2(1)]{SZ}. For the right-module assertions in parts~(2), pass to \(A^{\mathrm{op}}\) and use \(N^{\mathrm{op}}\) and \(M^{\mathrm{op}}\) as the first and second bimodules. Here an opposite bimodule has its left and right actions interchanged. Left \(A\)-modules are right \(A^{\mathrm{op}}\)-modules with \(v a^{\mathrm{op}}=av\). The map \(v\otimes n^{\mathrm{op}}\mapsto n\otimes v\) identifies \(I\otimes_{A^{\mathrm{op}}}N^{\mathrm{op}}\) with \(F(I)\) under this convention. Indeed \(n\otimes av=na\otimes v\) verifies balancing, and reversing the factors again gives its inverse. Thus the right tensor functor also sends an injective module to a noninjective module.

In the argument of \cite[Lemma~3.1(1), p.~2593]{SZ}, let \(F\) and \(G\) be the left tensor functors of the main proof. For a left \(B\)-module \(Z\), let \(\rho_Z\colon FG(Z)\to Z\) be projection onto the first summand of \(FG(Z)\simeq Z\oplus(Q\otimes_B Z)\). The proof takes a monomorphism \(f\colon X\to Y\) of left \(B\)-modules and a map \(g\colon X\to F(I)\), with \(I\) injective over \(A\). Since \(G(f)\) is a monomorphism, \(\pi_I\circ G(g)\) extends along \(G(f)\) to a map \(h\colon G(Y)\to I\). Applying \(F\) to \(\pi_I\circ G(g)=h\circ G(f)\) gives
\[
 F(\pi_I)\circ FG(g)=F(h)\circ FG(f).
\]
The published proof replaces \(F(\pi_I)\) by \(\rho_{F(I)}\) in this equality. The same substitution appears in \cite[Theorem~4.1, step~(1)]{Xi}.

For the triangular algebra above, \(G=A\otimes_A-\) identifies with the identity by multiplication. Under \(F(Z)\simeq Z\oplus(A\otimes_k Z)\), both \(\pi_Z\) and \(\rho_Z\) are projection onto \(Z\). On the summand \(A\otimes_k F(I)\) of \(F(F(I))\), the projection \(\rho_{F(I)}\) is zero, whereas \(F(\pi_I)\) restricts to \(\operatorname{id}_A\otimes\pi_I\) onto the nonzero summand \(A\otimes_k I\) of \(F(I)\).

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\end{thebibliography}

\noindent Zhili College, Tsinghua University, Beijing 100084, China\par
\noindent \textit{Email address:}\enspace
\mbox{\href{mailto:tangqy24@mails.tsinghua.edu.cn}{\texttt{tangqy24@mails.tsinghua.edu.cn}}}
\end{document}