On modules $M$ with $\tau(M) \cong \nu \Omega^{d+2}(M)$ for isolated singularities of Krull dimension $d$

TL;DR

This paper generalizes a classical formula for the Auslander-Reiten translate to certain isolated singularities, linking it to vanishing Ext conditions and providing new insights into Cohen-Macaulay modules.

Contribution

It extends the classical Auslander-Reiten formula to $2d$-periodic isolated singularities, connecting $ au(M)$ with $ u \, \, \\Omega^{d+2}(M)$ under Ext vanishing conditions.

Findings

Generalized Auslander-Reiten formula for isolated singularities

Characterized when $ au(M) \\cong u \\Omega^{d+2}(M)$ holds

Provided applications for Artin algebras.

Abstract

A classical formula for the Auslander-Reiten translate $\tau$ says that $\tau(M)\cong \nu \Omega^2(M)$ for every indecomposable module $M$ of a selfinjective Artin algebra. We generalise this by showing that for a $2d$-periodic isolated singularity $A$ of Krull dimension $d$, we have for the Auslander-Reiten translate of an indecomposable non-projective Cohen-Macaulay $A$-module $M$, $\tau(M)\cong \nu \Omega^{d+2}(M)$ if and only if $Ext_A^{d+1}(M,A)=Ext_A^{d+2}(M,A)=0$. We give several applications for Artin algebras.

Equations

$$P_1\stackrel{g}{\to }{P}_0\to X\to 0$$

$$P_1\to P_0\to X\to 0$$

$$X^{\ast }\to (P_0{)}^{\ast }\to (P_1{)}^{\ast }\to \mathrm{Tr}(X)\to 0.$$

$$({\Omega }^d\mathrm{Tr}{\Omega }^d\mathrm{Tr})\mathrm{Tr}{\Omega }^d(X)\cong {\Omega }^d\mathrm{Tr}(X)$$

$$\mathrm{Tr}{\Omega }^d(X)\cong {\Omega }^d\mathrm{Tr}(X).$$

$${\mathrm{Ext}}_A^{d+1}(X,A)={\mathrm{Ext}}_A^{d+2}(X,A)=0⟺$$

$${\mathrm{Ext}}_A^1({\Omega }^d(X),A)={\mathrm{Ext}}_A^2({\Omega }^d(X),A)=0⟺$$

$${\mathrm{Ext}}_A^1(\mathrm{Tr}(\mathrm{Tr}({\Omega }^d(X))),A)={\mathrm{Ext}}_A^2(\mathrm{Tr}(\mathrm{Tr}({\Omega }^d(X))),A)=0⟺$$

$${\mathrm{Ext}}_A^1(\mathrm{Tr}({\Omega }^d(\mathrm{Tr}(X))),A)={\mathrm{Ext}}_A^2(\mathrm{Tr}({\Omega }^d(\mathrm{Tr}(X))),A)=0⟺$$

$${\mathrm{Ext}}_A^1(\mathrm{Tr}({\mathrm{Tr}}_d(X)),A)={\mathrm{Ext}}_A^2(\mathrm{Tr}({\mathrm{Tr}}_d(X)),A)=0⟺$$

$${\Omega }^2\mathrm{Tr}{\Omega }^{2+d}(X)\cong {\Omega }^2\mathrm{Tr}{\Omega }^2\mathrm{Tr}\mathrm{Tr}{\Omega }^d(X)$$

$$\cong (\mathrm{Tr}{\Omega }^d(X){)}^{\ast \ast }\cong ({\Omega }^d\mathrm{Tr}(X){)}^{\ast \ast }\cong ({\mathrm{Tr}}_d(X){)}^{\ast \ast }.$$

$$\mathrm{mod}-fAf/(\mathrm{add}(Af))\to \overline{{\mathrm{Per}}_{{\tau }^{-1}}(A)}.$$

Full text

Introduction

A classical result in Auslander-Reiten theory for Artin algebras is that for every indecomposable non-projective module $X$ over a selfinjective Artin algebra one has $\tau(X)\cong\nu\Omega^{2}(X)$, see for example proposition 3.7. in [ARS]. This isomorphism has several important applications for the Auslander-Reiten theory of selfinjective algebras, we refer for example to [DP] and [KZ].

In section 2 of [RZ] it was recently noted that for a general Artin algebra $A$, every indecomposable non-projective module $X$ satisfies $\tau(X)\cong\nu\Omega^{2}(X)$ in case $\operatorname{Ext}_{A}^{1}(X,A)=\operatorname{Ext}_{A}^{2}(X,A)=0$. As a main result of this article, we sharpen their result by showing that also the converse is true.

Theorem**.**

Let $A$ be an Artin algebra with an indecomposable non-projective module $M$. Then the following are equivalent:

1. (1)

$\operatorname{Ext}_{A}^{1}(X,A)=\operatorname{Ext}_{A}^{2}(X,A)=0$. 2. (2)

$\operatorname{Tr}(X)$* is reflexive.* 3. (3)

$\tau(X)\cong\nu\Omega^{2}(X)$.

We actually prove a much more general result for isolated singularities and obtain the previous theorem as a special case by setting $d=0$. We refer to the next section for the relevant definitions.

Theorem**.**

Let $A$ be a $2d-periodic$ isolated singularity of Krull dimension $d$ and $X\in\operatorname{CM}(A)$ indecomposable non-projective. Then the following are equivalent:

1. (1)

$\operatorname{Ext}_{A}^{d+1}(X,A)=\operatorname{Ext}_{A}^{d+2}(X,A)=0$. 2. (2)

$\operatorname{Tr}_{d}(X)$* is reflexive.* 3. (3)

$\tau(X)\cong\nu\Omega^{d+2}(X)$.

In the final section we give several applications of our main result for Artin algebras. I thank Jeremy Rickard for allowing me to use his proof in 1.2.

1. A characterisation of $\tau(M)\cong\nu\Omega^{d+2}(M)$

We assume that $A$ is always a (not necessarily commutative) noetherian ring and modules are finitely generated right modules unless otherwise stated. We write $(-)^{*}$ short for the functor $Hom_{A}(-,A)$. Recall that a module $X$ is called torsionless in case the natural evaluation map $\operatorname{ev}_{M}:M\rightarrow M^{**}$ is injective and it is called reflexive in case $\operatorname{ev}_{M}$ is an isomorphism. For a general full subcategory $\mathcal{C}$ of $\operatorname{mod}-A$, we denote by $\underline{\mathcal{C}}$ the subcategory $\mathcal{C}$ modulo projective objects (called the stable category of $\mathcal{C}$) and by $\overline{\mathcal{C}}$ we denote the subcategory $\mathcal{C}$ modulo injective objects (called the costable category of $\mathcal{C}$). We now define the Auslander-Bridger transpose $\operatorname{Tr}(X)$ of a module $X$: Let

[TABLE]

be a projective presentation of $X$. Then define $\operatorname{Tr}(X)$ as the cokernel of the map $g^{*}$. The following theorem is classical:

Theorem 1.1**.**

Let $A$ be a noetherian ring. Then $\operatorname{Tr}$ induces a duality $\underline{mod}-A\rightarrow\underline{mod}-A^{op}$ with $\operatorname{Tr}\operatorname{Tr}\cong id_{\underline{mod}-A}$.

Proof.

See proposition 2.6. in [AB]. ∎

We will need the following equivalent characterisations of reflexive modules. The proof that (2) implies (1) is due to Jeremy Rickard.

Proposition 1.2**.**

Let $A$ be a noetherian algebra and $M$ an $A$-module. The following are equivalent:

1. (1)

$M$* is reflexive.* 2. (2)

$M\cong M^{**}$. 3. (3)

$\operatorname{Ext}_{A}^{1}(\operatorname{Tr}(M),A)=\operatorname{Ext}_{A}^{2}(\operatorname{Tr}(M),A)=0$.

Proof.

The equivalence between (1) and (3) is well known, see formula (0.1) in the introduction of [AB]. (1) implies (2) by definition. We now show that (2) implies (1): Assume $M\cong M^{**}$. Then $M$ is a dual and for any dual the canonical evaluation map $M\rightarrow M^{**}$ is a split monomorphism, see for example Proposition 1.1.9. of [C] (note that the book [C] is about commutative rings but the same proof applies to non-commutative rings). Thus $M$ is a direct summand of $M^{**}=M$ and therefore $M\cong M\oplus N$ for some non-zero module $N$ in case $M$ is not reflexive. We show that this leads to a contradiction and thus $M$ has to be reflexive. In case $N\neq 0$, there is a strictly increasing chain of submodules of $M$ as follows: $N<N^{2}<N^{3}<....$, which contradicts that $M$ is finitely generated and thus noetherian.

∎

We now give the relevant definitions on isolated singularities, where we closely follow section 3 of [Iya]. We refer to [Iya] for more information and examples. Let $R$ be a commutative noetherian complete local ring of Krull dimension $d$ and assume furthermore that $R$ is a Gorenstein ring. Recall that an $R$-module $X$ is called Cohen-Macaulay in case $X=0$ or $\dim(X)=d=0pt(X)$.

We always assume that an $R$-algebra $A$ is module-finite and semiperfect so that projective covers exist. We define the full subcategory $\operatorname{CM}(A):=\{X\in\operatorname{mod}-A|X$ is Cohen-Macaulay as an $R$-module $\}$. We call $A$ an $R$-order in case $A\in\operatorname{CM}(A)$. We call an $R$-order $A$ an isolated singularity in case the Hom-spaces modulo projectives $\underline{\operatorname{Hom}}(X,Y)$ have finite length as $R$-modules for all $X,Y\in\operatorname{CM}(A)$. For example for $d=0$ the isolated singularities are exactly the Artin algebras. We assume in the following that $A$ is always an isolated singularity of Krull dimension $d$. We have a duality $D_{d}:=Hom_{R}(-,R):\operatorname{CM}(A)\rightarrow\operatorname{CM}(A^{op})$ and use this to define the Nakayama functors as $\nu:=D_{d}(-)^{*}$ and $\nu^{-1}:=(-)^{*}D_{d}$. The following theorem motivates the definition of the Auslander-Reiten translates for isolated singularities:

Theorem 1.3**.**

Let $A$ be an isolated singularity of Krull dimension $d$. Then there is a duality $\Omega^{d}\operatorname{Tr}:\underline{\operatorname{CM}}(A)\rightarrow\underline{\operatorname{CM}}(A^{op})$ with $(\Omega^{d}\operatorname{Tr})^{2}\cong id_{\underline{\operatorname{CM}}(A)}$.

Proof.

See for example [Iya], theorem 3.4.

∎

We set $\operatorname{Tr}_{d}:=\Omega^{d}\operatorname{Tr}$ and define the Auslander-Reiten translate $\tau$ of an isolated singularity of Krull dimension $d$ as $\tau:=D_{d}\operatorname{Tr}_{d}$ and the inverse Auslander-Reiten translate $\tau^{-1}$ as $\tau^{-1}:=\operatorname{Tr}_{d}D_{d}$. In this text we are interested in a homological characterisation for indecomposable non-projective modules $M\in\operatorname{CM}(A)$ with $\tau(M)\cong\nu\Omega^{d+2}(M)$ for isolated singularities.

Lemma 1.4**.**

Let $A$ be an isolated singularity. Let $X$ be an indecomposable non-projective $A$-module. Then $X^{*}\cong\Omega^{2}Tr(X)$.

Proof.

Let

[TABLE]

be a minimal projective presentation of $X$ and apply the functor $(-)^{*}$ to it. Since for a projective module $P$ also $P^{*}$ is projective, we obtain by the definition of the Auslander-Bridger transpose the following exact sequence, where $(P_{0})^{*}$ and $(P_{1})^{*}$ are projective and give a minimal projective presentation of $\operatorname{Tr}(X)$:

[TABLE]

Thus $X^{*}\cong\Omega^{2}\operatorname{Tr}(X)$, since we assumed that $X$ as no projective summands. ∎

Corollary 1.5**.**

Let $A$ be an isolated singularity with an indecomposable non-projective module $X$. Then $X$ is reflexive if and only if $\Omega^{2}Tr\Omega^{2}Tr(X)\cong X$.

Proof.

By 1.2, $X$ is reflexive if and only if $X\cong X^{**}$. Assume first that $X$ is indecomposable non-projective and reflexive. Note that $X^{*}$ can not be projective, or else also $X^{**}\cong X$ would be projective, which contradicts our assumptions on $X$.

But by 1.4, we have $X^{*}\cong\Omega^{2}Tr(X)$ and then $X^{**}\cong\Omega^{2}Tr\Omega^{2}Tr(X)$. Now assume that $X$ is indecomposable non-projective with $\Omega^{2}Tr\Omega^{2}Tr(X)\cong X$. Then also $\Omega^{2}(\operatorname{Tr}(X))$ is non-projective and $\Omega^{2}Tr\Omega^{2}Tr(X)\cong X^{**}$ forces $X\cong X^{**}$ and thus $X$ is reflexive.

∎

We call an isolated singularity $A$ of Krull dimension $d$ $l$-periodic in case the functor $\Omega^{l}:\underline{\operatorname{CM}-}(A)\rightarrow\underline{\operatorname{CM}-}(A)$ is isomorphic to the identity functor.

Lemma 1.6**.**

Let $A$ be a $2d$-periodic isolated singularity with Krull dimension $d$ and $X\in\operatorname{CM}(A)$ indecomposable non-projective. Then $\operatorname{Tr}(\Omega^{d}(X))\cong\Omega^{d}(\operatorname{Tr}(X))$ in the stable category.

Proof.

We use that $\Omega^{d}Tr\Omega^{d}Tr\cong id$ in the stable category by 1.3. We have for every $X\in\operatorname{CM}(A)$: $\Omega^{2d}(X)\cong X$. This is equivalent to $\Omega^{d}\operatorname{Tr}\operatorname{Tr}\Omega^{d}(X)\cong X$. Note that when for $Y\in\operatorname{mod}-A$: $\Omega^{d}\operatorname{Tr}(Y)=Z\in\operatorname{CM}(A^{op})$, then also $Y=\Omega^{d}\operatorname{Tr}(Z)\in\operatorname{CM}(A)$. This shows that $Y=\operatorname{Tr}\Omega^{d}(X)\in\operatorname{CM}(A)$.

We now apply the functor $\Omega^{d}\operatorname{Tr}$ to $\Omega^{d}\operatorname{Tr}\operatorname{Tr}\Omega^{d}(X)\cong X$ and use that $(\Omega^{d}\operatorname{Tr})^{2}\cong id$ to obtain:

[TABLE]

which is gives

[TABLE]

∎

Theorem 1.7**.**

Let $A$ be a $2d-periodic$ isolated singularity of Krull dimension $d$ and $X\in\operatorname{CM}(A)$ indecomposable non-projective. Then the following are equivalent:

1. (1)

$\operatorname{Ext}_{A}^{d+1}(X,A)=\operatorname{Ext}_{A}^{d+2}(X,A)=0$. 2. (2)

$\operatorname{Tr}_{d}(X)$* is reflexive.* 3. (3)

$\tau(X)\cong\nu\Omega^{d+2}(X)$.

Proof.

We have

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

$\operatorname{Tr}_{d}(X)$ is reflexive, by our characterision (3) of reflexive modules in 1.2.

Here we used that $\operatorname{Ext}_{A}^{i}(\operatorname{Tr}(\operatorname{Tr}(\Omega^{d}(X))),A)\cong\operatorname{Ext}_{A}^{i}(\operatorname{Tr}(\Omega^{d}(\operatorname{Tr}(X))),A)$ for $i=1,2$ in the stable category since we have $\operatorname{Tr}(\Omega^{d}(X))\cong\Omega^{d}(\operatorname{Tr}(X))$ in the stable category by the previous lemma 1.6. This gives the equivalence between (1) and (2).

Now assume that $\tau(X)\cong\nu\Omega^{2+d}(X)$. Applying the duality $D_{d}$ on both sides and using that $(-)^{*}\cong\Omega^{2}\operatorname{Tr}(-)$, by 1.4 this is equivalent to $\operatorname{Tr}_{d}(X)\cong\Omega^{2}\operatorname{Tr}\Omega^{2+d}(X)$. We simplify this using our assumption that $\operatorname{Tr}(\Omega^{d}(X))\cong\Omega^{d}(\operatorname{Tr}(X))$ and $(-)^{**}\cong\Omega^{2}\operatorname{Tr}\Omega^{2}\operatorname{Tr}$ in the stable category:

[TABLE]

[TABLE]

Thus the isomorphism $\tau(X)\cong\nu\Omega^{2+d}(X)$ is equivalent to $\operatorname{Tr}_{d}(X)\cong(\operatorname{Tr}_{d}(X))^{**}$, which is equivalent to $\operatorname{Tr}_{d}(X)$ being reflexive by (2) of 1.2. ∎

Example 1.8**.**

Let $d\geq 1$. Let $S=K[[x_{0},...,x_{d}]]$ the formal power series ring in $(d+1)$-variables and $f\in(x_{0},...,x_{d})$. Set $A=S/(f)$ and assume that $A$ is an isolated singularity. In this case $A$ is local (and thus semiperfect) and has Krull dimension $d$. $A$ is called an isolated hypersurface singularity. We refer to [LW] and [Y] for more information on such algebras and several classification results. According to theorem 4.9. of [Iya], we have $\Omega^{2}\cong id$ in $\underline{\operatorname{CM}-}(A)$ and thus $A$ is $2d$-periodic and our main result 1.7 applies to such algebras to give $\tau(M)\cong\nu(M)$ in case $d$ is even and $\tau(M)\cong\nu\Omega^{1}(M)$ in case $d$ is odd.

2. Applications for Artin algebras

We restrict in this section to Artin algebras (that is to the case of Krull dimension $d=0$). We assume that the reader is familiar with the basic representation theory of Artin algebras and refer for example to [ARS]. We denote the duality of an Artin algebra by $D$. We call a module $M$ $\tau$-perfect in case $\tau(M)\cong\nu\Omega^{2}(M)$ and $\tau^{-1}$-perfect in case $\tau^{-1}(M)\cong\nu^{-1}\Omega^{-2}(M)$. $\operatorname{Per}_{\tau}(A)$ is defined as the full subcategory of $\tau$-perfect modules and $\operatorname{Per}_{\tau^{-1}}(A)$ is defined as the full subcategory of $\tau^{-1}$-perfect modules. We denote by $\ref{(}A)$ the full subcategory of reflexive modules. Furthermore, we call a module $N$ coreflexive in case $D(N)$ is reflexive and we define the subcategory $\operatorname{Coref}(A)$ as the full subcategory of coreflexive $A$-modules. We state the case $d=0$ of our main result here explicitly due to its importance:

Theorem 2.1**.**

Let $A$ be an Artin algebra with an indecomposable non-projective module $X$. Then the following are equivalent:

1. (1)

$\operatorname{Ext}_{A}^{1}(X,A)=\operatorname{Ext}_{A}^{2}(X,A)=0$. 2. (2)

$\operatorname{Tr}(X)$* is reflexive.* 3. (3)

$X$* is $\tau$-perfect, that is $\tau(X)\cong\nu\Omega^{2}(X)$.*

We also state the dual of the previous theorem:.

Theorem 2.2**.**

Let $A$ be an Artin algebra with an indecomposable non-injective module $X$. Then the following are equivalent:

1. (1)

$\operatorname{Ext}_{A}^{1}(D(A),X)=\operatorname{Ext}_{A}^{2}(D(A),X)=0$. 2. (2)

$\operatorname{Tr}(D(X))$* is reflexive.* 3. (3)

$X$* is $\tau^{-1}$-perfect, that is $\tau(X)\cong\nu^{-1}\Omega^{-2}(X)$.*

In selfinjective Artin algebras it is in fact true that for every module $M$ with $\tau(M)\cong\nu\Omega^{2}(M)$, we even have $\tau(M)\cong\nu\Omega^{2}(M)\cong\Omega^{2}\nu(M)$. But this fails in general as we show in the next example:

Example 2.3**.**

Let $A$ be the Nakayama algebra, given by quiver and relations, with Kupisch series $[2,2,2,1]$ with simple modules numbered from 0 to 3. Then the simple module $S_{0}$ satisfies $\tau(S_{0})\cong\nu\Omega^{2}(S_{0})\cong S_{1}$, but $\Omega^{2}\nu(S_{0})\cong 0$.

The next proposition shows that the classification of $\tau^{-1}$-perfect modules is essentially equivalent to the classification of reflexive modules for general Artin algebras.

Proposition 2.4**.**

Let $A$ be an Artin algebra.

1. (1)

$\tau$* induces an equivalence of categories between $\underline{\operatorname{Per}_{\tau}}(A)$ and $\overline{\operatorname{Coref}}(A)$.* 2. (2)

$\tau^{-1}$* induces an equivalence of categories between $\overline{\operatorname{Per}_{\tau^{-1}}}(A)$ and $\underline{\ref{missing}}(A)$.*

Proof.

We only prove (1), since the proof of (2) is dual. $\tau$ is an equivalence between $\underline{\operatorname{mod}}-A$ and $\overline{\operatorname{mod}}-A$. Thus when we restricted to the subcategory $\underline{\operatorname{Per}_{\tau}}(A)$, $\tau$ induces an equivalence to the image $\tau(\underline{\operatorname{Per}_{\tau}}(A))$. By 2.1, $X\in\operatorname{Per}_{\tau}(A)$ implies that $\operatorname{Tr}(X)$ is reflexive and thus $\tau(X)=DTr(X)$ is coreflexive. Thus the image is contained in $\overline{\operatorname{Coref}}(A)$. Now $\tau:\underline{\operatorname{Per}_{\tau}}(A)\rightarrow\overline{\operatorname{Coref}}(A)$ is also dense since when $Y\in\overline{\operatorname{Coref}}(A)$ is given, we have that $\tau^{-1}(Y)$ is $\tau$-perfect. This is true since by 2.1, $\tau^{-1}(Y)$ is $\tau$-perfect if and only if $\operatorname{Tr}(\tau^{-1}(Y))$ is reflexive, which is equivalent to $DTr(\tau^{-1}(Y))=\tau(\tau^{-1}(Y))=Y$ being coreflexive.

∎

We can also obtain a new characterisation of selfinjective Artin algebras from our main theorem:

Corollary 2.5**.**

Let $A$ be an Artin algebra. Then the following are equivalent:

1. (1)

$A$* is selfinjective.* 2. (2)

Every simple module is $\tau$-perfect.

Proof.

Since $\tau\cong\nu\Omega^{2}$ in a selfinjective Artin algebra, it is clear that (1) implies (2). Now assume that (2) holds. Then we have $\operatorname{Ext}_{A}^{1}(S,A)=0$ for every simple modules $S$ by 2.1. But by corollary 2.5.4. of [Ben], we have $\operatorname{Hom}_{A}(S,\Omega^{-1}(A))\cong\operatorname{Ext}_{A}^{1}(S,A)=0$ for every simple module $S$. Thus the socle of $\Omega^{-1}(A)$ is zero and thus also $\Omega^{-1}(A)$ is zero. This means that $A$ is isomorphic to its injective envelope. But this shows that every projective module is injective and thus $A$ is selfinjective.

∎

We give two applications of this result to algebras with dominant dimension at least 2 and algebras that are 2-Iwanaga-Gorenstein. Recall that the dominant dimension of a module $M$ with minimal injective coresolution $(I_{i})$ is defined as the smallest number $k$ such that $I_{k}$ is not projective. The dominant dimension of an algebra is by definition the dominant dimension of the regular module. It is well known that an algebra $A$ has dominant dimension at least one if and only if there is a minimal faithful projective-injective left $A$-module of the form $Af$ for some idempotent $f$. Furthermore, such an algebra has dominant dimension at least two if and only if $A\cong End_{fAf}(Af)$ in addition.

Corollary 2.6**.**

Let $A$ be an algebra with a minimal faithful projective-injective left module $Af$ with dominant dimension at least two. There is an equivalence of categories

[TABLE]

Proof.

Since $A$ has dominant dimension at least two and by a result of Morita in [M] theorem 3.3., an $A$-module $X$ has dominant dimension at least two if and only if $X$ is reflexive. Now the full subcategory of modules with dominant dimension at least two $\operatorname{Dom}_{2}$ is equivalent to $\operatorname{mod}-fAf$ and the projective modules are sent to the modules in $\operatorname{\mathrm{add}}-Af$ (this is a special case of lemma 3.1. in [APT]) . This means that there is an equivalence $\underline{Ref}(A)=\underline{Dom_{2}}\cong\operatorname{mod}-fAf/(\operatorname{\mathrm{add}}(Af))$. But by 2.4, $\underline{Ref}(A)$ is equivalent to $\overline{\operatorname{Per}_{\tau^{-1}}(A)}$, which finishes the proof. ∎

Example 2.7**.**

Let $K$ be an infinite field and $V$ an $n$-dimensional vector space. Let $A$ be the Schur algebra $S(n,r)$ for $n\geq r$. Then $A$ has dominant dimenion at least two and $fAf$ is isomorphic to the symmetric group algebra $KS_{r}$ and $Af$ to $V^{\otimes n}$, see for example theorem 1.2. in [KSX]. The previous corollary gives us in this case an equivalence between the module category of the group algebra of the symmetric group modulo the subcategory $\operatorname{\mathrm{add}}(V^{\otimes n})$ to the costable category of $\tau^{-1}$-perfect modules over the Schur algebra.

Recall that an Artin algebra $A$ is called $n$-Iwanaga-Gorenstein in case the left and right injective dimensions of the regular modules $A$ are equal to $n$ for a natural number $n$. An $A$-module $X$ over an $n$-Iwanga-Gorenstein algebra is called Gorenstein projective in case $\operatorname{Ext}_{A}^{i}(X,A)=0$ for all $i\geq 1$. All projective modules are Gorenstein projective by definition and one of the main goals of Gorenstein homological algebra is to find a classification of all Gorenstein projective modules for a given Artin algebra. In the next corollary we give a new characterisation of Gorenstein projective modules for 2-Iwanaga-Gorenstein algebras. Note that this is a large class of algebras that contain for example all cluster-tilted algebras by [KR] and all algebras considered by [GLS].

Corollary 2.8**.**

Let $A$ be a 2-Iwanaga-Gorenstein algebra. Then an indecomposable module $X$ is Gorenstein projective if and only if $X$ is $\tau$-perfect, that is $\tau(X)\cong\nu\Omega^{2}(X)$.

Proof.

Since $A$ has injective dimension at most 2, we have that for an arbitrary module $X$: $\operatorname{Ext}_{A}^{i}(X,A)=0$ for all $i\geq 1$ if and only if $\operatorname{Ext}_{A}^{1}(X,A)=\operatorname{Ext}_{A}^{2}(X,A)=0$. But this condition is by 2.1 equivalent to $X$ being $\tau$-perfect.

∎

Example 2.9**.**

Let $A$ be the Nakayama algebra with Kupisch series [3,3,4] and primitive idempotents $e_{0},e_{1}$ and $e_{2}$. $A$ is 2-Iwanaga-Gorenstein. The indecomposable non-projective Gorenstein projective $A$-modules are $e_{0}A/e_{0}J^{1}$ and $e_{1}A/e_{1}J^{2}$.