On self-orthogonal modules in Iwanaga-Gorenstein rings

Let $A$ be an Iwanaga-Gorenstein ring. Enomoto conjectured that a self-orthogonal $A$-module has finite projective dimension. We prove this conjecture for $A$ having the property that every indecomposable non-projective maximal Cohen-Macaulay module is periodic. This answers a question of Enomoto and shows the conjecture for monomial quiver algebras and hypersurface rings.


Introduction
We assume always that A is a two-sided noetherian semiperfect ring and all modules are finitely generated right modules unless otherwise stated.Recall that A is called n-Iwanaga-Gorenstein if the injective dimensions of A as a left and right module are equal to n.If the n does not matter we will often just say Iwanaga-Gorenstein ring instead of n-Iwanaga-Gorenstein ring.The category of maximal Cohen-Macaulay modules CM A of a n-Iwanaga-Gorenstein ring is defined as the category of n th syzygy modules Ω n (mod A) consisting of modules X that are projective or direct summands of a module of the form The definition of Iwanaga-Gorenstein rings includes the classical cases of Iwanaga-Gorenstein rings, namely the commutative local Gorenstein rings and the Iwanaga-Gorenstein-Artin algebras.
We are interested in the following problem that was stated in [2, Conjecture 4.8] for Artin algebras.

Problem 1. Let A be Iwanaga-Gorenstein and let M be self-orthogonal. Then M has finite projective dimension.
A positive solution of this problem would have important consequences for the theory of tilting modules for Iwanaga-Gorenstein Artin algebras, see [2,Sections 3 and 4].For Artin algebras the conjecture of Enomoto is a generalisation of the classical Tachikawa conjecture that states that a self-orthogonal module over a selfinjective algebra is projective.The Tachikawa conjecture can be seen as one of the most important homological conjectures for Artin algebras since a counterexample to the Tachikawa conjecture would give counterexamples to other homological conjectures such as the Nakayama conjecture, the Auslander-Reiten conjecture and the finitistic dimension conjecture, see for example [7] for a survey on those conjectures.Our main result gives a positive answer to the above problem for an important class of Iwanaga-Gorenstein algebras: Theorem 1.1.Let A be an n-Iwanaga-Gorenstein ring such that every indecomposable non-projective module X ∈ CM A is periodic.Assume M has the property that Ext u A (M, M ) = 0 for all u > n.Then M has finite projective dimension.In particular, all self-orthogonal modules have finite projective dimension.
In [2] a positive solution to the above problem was proven for representation-finite Iwanaga-Gorenstein Artin algebras using the theory of generalised tilting modules.In [2, Question 4.7] a more direct proof for this case is asked for and our main result gives such a direct proof for a much larger class of Iwanaga-Gorenstein Artin algebras, which contain all CM-finite Iwanaga-Gorenstein rings and in particular the subclass of all such representation-finite algebras.

Proof of the main results
In this section CMA will denote the category of maximal Cohen-Macaulay modules modulo projective modules.Recall that a module X ∈ mod A is called periodic if Ω l (X) ∼ = X for some l ≥ 1.

Theorem 2.2. Let A be an n-Iwanaga-Gorenstein ring such that every indecomposable non-projective module X ∈ CM A is periodic. Assume M has the property that
Ext u A (M, M ) = 0 for all u > n.Then M has finite projective dimension.In particular, all self-orthogonal modules have finite projective dimension.
Proof.Assume M has the property that Ext n+l A (M, M ) = 0 for all l ≥ 1.Then by dimension shifting and Lemma 2.1 (2), which we are allowed to use since Ω n (M ) ∈ CM A. Using 2.1 (1) with s = n and then (2) again we obtain: Thus Ext n+l A (Ω n (M ), Ω n (M )) = 0 for all l ≥ 1, since we assume that Ext n+l A (M, M ) = 0 for all l ≥ 1.Let X be an indecomposable direct summand of Ω n (M ).Then X ∈ CM A with Ext n+l A (X, X) = 0 for all l ≥ 1. Assume that X is non-zero in the stable module category.By assumption X is periodic.So assume that X ∼ = Ω q (X) for some q ≥ 1.Note that this also implies that X ∼ = Ω qm (X) for all m ≥ 1.Then for all p ≥ 1 and m ≥ 1 we obtain: Now choose m big enough so that p + qm > n.Then Thus X is self-orthogonal.But we also have by Lemma 2.1 (2) since the identity map in Hom A (X, X) is certainly non-zero.This is a contradiction and thus X must be zero in the stable module category.Thus every indecomposable direct summand of Ω n (M ) is the zero module in the stable module category and thus Ω n (M ) must be the a projective module, which implies that M has finite projective dimension.□ Recall that an Iwanaga-Gorenstein ring is called CM-finite if there are only finitely many indecomposable modules in CM A.

Corollary 2.3.
Let A be a CM-finite Iwanaga-Gorenstein ring.Then every self-orthogonal module has finite projective dimension.
Proof.We show that every indecomposable module X ∈ CM A is periodic.Then the result follows from 2.2.Let X be indecomposable.Since with X also Ω i (X) ∈ CM A is indecomposable for all i ≥ 0 by Lemma 2.1 (3) and since there are only finitely many indecomposable modules in CM A, we have that Ω i (X) ∼ = Ω i+l (X) for some i ≥ 0 and l ≥ 1.This implies that X ∼ = Ω l (X) by Lemma 2.1 (4) and thus X is periodic.□ We give two important examples.The first is for finite dimensional algebras and the second for commutative local rings.
Example 2.4.Let A be a finite dimensional quiver algebra KQ/I with admissible monomial ideal I. Then A has the property that Ω 2 (mod −A) is representation-finite, see [8], and thus A is CM-finite if A is Iwanaga-Gorenstein.In particular, all gentle algebras are CM-finite Iwanaga-Gorenstein algebras as gentle quiver algebras are always Iwanaga-Gorenstein by [3].Thus for the class of monomial Iwanaga-Gorenstein algebras, every self-orthogonal module has finite projective dimension by our main result.
Example 2.5.Let R be a regular commutative local ring and f ̸ = 0. Then the hypersurface ring A = R/(f ) is Iwanaga-Gorenstein and every module X ∈ CM A is periodic of period at most 2 by the classical result about matrix factorisations by Eisenbud, see [1].By our main result, every self-orthogonal A-module has finite projective dimension.