Annals of Representation Theory

Bardy-Panse, Nicole

Hébert, Auguste

Rousseau, Guy

Twin masures associated with Kac–Moody groups over Laurent polynomials

Daugherty, Zajj

Ram, Arun

Calibrated representations of two boundary Temperley–Lieb algebras

Kimura, Yuta

Koshio, Ryotaro

Kozakai, Yuta

Minamoto, Hiroyuki

Mizuno, Yuya

$\tau $-tilting theory and silting theory of skew group algebra extensions

Asadollahi, Javad

Buan, Aslak Bakke

Mazorchuk, Walter

Schroll, Sibylle

Javad Asadollahi, Aslak Bakke Buan, Walter Mazorchuk & Sibylle
Schroll
Introduction from Editors-in-Chief
Volume 1, issue 1 (2024), p. I-II
https://doi.org/10.5802/art.11
© The authors, 2024
This article is licensed under the
Creative Commons Attribution (CC-BY) 4.0 License.
http://creativecommons.org/licenses/by/4.0/
Annals of Representation Theory is published by the
Norwegian University of Science and Technology
and a member of the
Centre Mersenne for Open Scientific Publishing
e-ISSN: pending
C EN T R E
MER S ENN E

Annals of Representation Theory
Volume 1, issue 1 (2024), p. I–II
https://doi.org/10.5802/art.11
Introduction from Editors-in-Chief
Javad Asadollahi, Aslak Bakke Buan, Walter Mazorchuk and
Sibylle Schroll
Welcome to the inaugural edition of
Annals of Representation Theory - ART
This new fully digital journal is dedicated to publishing top quality original research
articles in the general area of contemporary representation theory, including its algebraic,
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field of representation theory.
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charges for authors.
We are very grateful for all the support and encouragement we have received in the
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who have accepted to be editors of the journal and the authors who have contributed
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ART was established in 2022 and is owned and operated by NTNU - Norwegian Univer-
sity of Technology and Science. We thank Nina Andersen, Sigurd Eriksson, Ståle Kalkvik,
Sølvi Karlsen and Lene Johansen Løkkhaug, all at NTNU – University library, for their
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Our web-publishing partner, Centre Mersenne, merits our kind appreciation, with spe-
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The realization of this journal was made possible through a generous donation from
Prof. Emerita Idun Reiten of NTNU. The first volume of Annals of Representation Theory
is dedicated to her in acknowledgment of her invaluable contributions to representation
theory and mathematics in general.
I
II Javad Asadollahi et al.
Editorial board
Editors-in-Chief:
- Javad Asadollahi (University of Isfahan)
- Aslak Bakke Buan (Norwegian University of Science and Technology)
- Walter Mazorchuk (Uppsala University)
- Sibylle Schroll (University of Cologne/Norwegian University of Science and Tech-
nology)
Production Editor:
- Sondre Kvamme (Norwegian University of Science and Technology)
Editors:
- Lidia Angeleri Hügel (University of Verona)
- Kevin Coulembier (University of Sydney)
- Ben Elias (University of Oregon)
- Osamu Iyama (University of Tokyo)
- Bernhard Keller (University of Paris)
- Radha Kessar (University of Manchester)
- Henning Krause (University of Bielefeld)
- Peter Littelmann (University of Cologne)
- Wendy Lowen (University of Antwerpen)
- Anne Schilling (University of California, Davis)
- Andrea Solotar (University of Buenos Aires)
- Hugh Thomas (Université du Québec à Montréal)
- Jerzy Weyman (Jagiellonian University)
Ann. Repr. Th. 1 (2024), 1, p. I–II https://doi.org/10.5802/art.11


Introduction from Editors-in-Chief

Brundan, Jonathan

Wang, Weiqiang

Webster, Benjamin

The nil-Brauer category

oai:centre-mersenne.org:ART_2024__1_2_193_

Evseev, Anton

Mathas, Andrew

Content systems and deformations of cyclotomic KLR algebras of type $A$ and $C$

Koshitani, Shigeo

Lassueur, Caroline

Sambale, Benjamin

Principal $2$-blocks with wreathed defect groups up to splendid Morita equivalence

Marczinzik, René

René Marczinzik
On self-orthogonal modules in Iwanaga–Gorenstein rings
Volume 1, issue 1 (2024), p. 67-70
https://doi.org/10.5802/art.4
Communicated by Lidia Angeleri Hügel.
© The author, 2024
This article is licensed under the
Creative Commons Attribution (CC-BY) 4.0 License.
http://creativecommons.org/licenses/by/4.0/
Annals of Representation Theory is published by the
Norwegian University of Science and Technology
and a member of the
Centre Mersenne for Open Scientific Publishing
e-ISSN: pending
C EN T R E
MER S ENN E

Annals of Representation Theory
Volume 1, issue 1 (2024), p. 67–70
https://doi.org/10.5802/art.4
On self-orthogonal modules in Iwanaga–Gorenstein
rings
René Marczinzik
Dedicated to the memory of Jimmy
Abstract. 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.
1. 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 nth syzygy modules Ωn(mod A)
consisting of modules X that are projective or direct summands of a module of the form
Ωn(M) for some M ∈ mod A. A module M is called self-orthogonal if ExtiA(M, M) = 0
for all i ≥ 1. 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.
Manuscript received 2023-03-22, revised 2023-10-20 and accepted 2023-11-27.
Keywords. self-orthogonal module, Iwanaga–Gorenstein ring.
2010 Mathematics Subject Classification. 16G10, 16E10.
67
68 René Marczinzik
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 projec-
tive. 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 exam-
ple [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 indecompos-
able non-projective module X ∈ CM A is periodic. Assume M has the property that
ExtuA(M, M) = 0 for all u > n. Then M has finite projective dimension. In particu-
lar, 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.
2. Proof of the main results
In this section CMA will denote the category of maximal Cohen–Macaulay modules
modulo projective modules.
Lemma 2.1. Assume A is an n-Iwanaga–Gorenstein ring and M ∈ CM A and N ∈
mod A.
(1) HomA(M, N) ∼= HomA(Ωs(M), Ωs(N)) for all s ≥ 0.
(2) ExtpA(M, N) ∼= HomA(Ωp(M), N) for all p ≥ 1.
(3) If M is indecomposable, then also Ωi(M) ∈ CM A is indecomposable for all i ≥ 1.
(4) If M, N ∈ CM A are indecomposable and satisfy Ω1(M) ∼= Ω1(N) then M ∼= N .
Proof. Note that every module X ∈ CM A satisfies ExtiA(X, A) = 0 for all i > 0, since X is
a direct summand of a module of the form Ωn(Y ) and ExtiA(Ωn(Y ), A) = Extn+iA (Y, A) = 0
for all i > 0 since A has injective dimension n. Then (1) and (2) are a special case of [4,
§ 2.1]. (3) follows from [6, Corollary 3.3] and (4) is a consequence [5, Theorem 5.5]. □
Recall that a module X ∈ mod A is called periodic if Ωl(X) ∼= X for some l ≥ 1.
Theorem 2.2. Let A be an n-Iwanaga–Gorenstein ring such that every indecompos-
able non-projective module X ∈ CM A is periodic. Assume M has the property that
ExtuA(M, M) = 0 for all u > n. Then M has finite projective dimension. In particu-
lar, all self-orthogonal modules have finite projective dimension.
Proof. Assume M has the property that Extn+lA (M, M) = 0 for all l ≥ 1. Then
Extn+lA (M, M) ∼= Ext
l
A(Ωn(M), M) ∼= HomA(Ωl+n(M), M)
Ann. Repr. Th. 1 (2024), 1, p. 67–70 https://doi.org/10.5802/art.4
On self-orthogonal modules in Iwanaga–Gorenstein rings 69
by dimension shifting and Lemma 2.1(2), which we are allowed to use since Ωn(M) ∈
CM A. Using 2.1(1) with s = n and then (2) again we obtain:
HomA
(
Ωl+n(M), M
)
∼= HomA
(
Ωl+n+n(M), Ωn(M)
)
∼= Extn+lA (Ω
n(M), Ωn(M)) .
Thus Extn+lA (Ωn(M), Ωn(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 Ωn(M). Then X ∈ CM A
with Extn+lA (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 ∼= Ωq(X) for some q ≥ 1. Note
that this also implies that X ∼= Ωqm(X) for all m ≥ 1. Then for all p ≥ 1 and m ≥ 1 we
obtain:
ExtpA(X, X) ∼= Ext
p
A (Ω
qm(X), X) ∼= Extp+qmA (X, X).
Now choose m big enough so that p + qm > n. Then
ExtpA(X, X) ∼= Ext
p+qm
A (X, X) = 0.
Thus X is self-orthogonal. But we also have by Lemma 2.1(2)
ExtqA(X, X) ∼= HomA (Ω
q(X), X) ∼= HomA(X, X) ̸= 0,
since the identity map in HomA(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 Ωn(M) is the zero module in the stable module category and thus Ωn(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 Ωi(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 Ωi(X) ∼= Ωi+l(X) for some i ≥ 0 and
l ≥ 1. This implies that X ∼= Ωl(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 mono-
mial ideal I. Then A has the property that Ω2(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 hypersur-
face 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.
Ann. Repr. Th. 1 (2024), 1, p. 67–70 https://doi.org/10.5802/art.4
70 René Marczinzik
References
[1] D. Eisenbud, Homological algebra on a complete intersection, with an Application to Group Represen-
tations, Trans. Am. Math. Soc. 260 (1980), no. 1, 35–64.
[2] H. Enomoto, Maximal self-orthogonal modules and a new generalization of tilting modules, 2023, https:
//arxiv.org/abs/2301.13498.
[3] C. Geiss and I. Reiten, Gentle algebras are Gorenstein, Representations of algebras and related topics.
Proceedings from the 10th international conference, ICRA X, Toronto, Canada, 2002, Fields Institute
Communications, vol. 45, American Mathematical Society, 2005, pp. 129–133.
[4] O. Iyama, Auslander correspondence, Adv. Math. 210 (2007), no. 1, 51–82.
[5] N. Kameyama, Y. Kimura, and K. Nishida, On stable equivalences of module subcategories over a
semiperfect noetherian ring, Colloq. Math. 137 (2014), no. 1, 7–26.
[6] C. M. Ringel and P. Zhang, Gorenstein-projective and semi-Gorenstein-projective modules, Algebra
Number Theory 14 (2020), no. 1, 1–36.
[7] K. Yamagata, Frobenius algebras, in Handbook of algebra. Vol. 1, North-Holland, 1996, pp. 841–887.
[8] B. Zimmermann-Huisgen, Predicting syzygies over monomial relations algebras, Manuscr. Math. 70
(1991), no. 2, 157–182.
— René Marczinzik —
Mathematical Institute of the University of Bonn, Endenicher Allee 60, 53115 Bonn,
Germany
E-mail address: marczire@math.uni-bonn.de
URL: https://sites.google.com/view/rene-marczinzik/home
Ann. Repr. Th. 1 (2024), 1, p. 67–70 https://doi.org/10.5802/art.4


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Twin masures associated with Kac–Moody groups over Laurent polynomials

Calibrated representations of two boundary Temperley–Lieb algebras

$\tau$ -tilting theory and silting theory of skew group algebra extensions

Introduction from Editors-in-Chief

The nil-Brauer category

Content systems and deformations of cyclotomic KLR algebras of type $A$ and $C$

Principal $2$ -blocks with wreathed defect groups up to splendid Morita equivalence

On self-orthogonal modules in Iwanaga–Gorenstein rings

Fibrewise stratification of group representations

Rigid integral representations of quivers over arbitrary commutative rings

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