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Auslander-Gorenstein algebras from Serre-formal algebras via replication

TL;DR: In this paper, the authors introduce a new family of algebras called Serre-formal alges, which are Iwanaga-gorenstein alges for which applying any power of the Serre functor on any indecomposable projective module, the result remains a stalk complex.
Abstract: We introduce a new family of algebras, called Serre-formal algebras. They are Iwanaga-Gorenstein algebras for which applying any power of the Serre functor on any indecomposable projective module, the result remains a stalk complex. Typical examples are given by (higher) hereditary algebras and self-injective algebras; it turns out that other interesting algebras such as (higher) canonical algebras are also Serre-formal. Starting from a Serre-formal algebra, we consider a series of algebras - called the replicated algebras - given by certain subquotients of its repetitive algebra. We calculate the self-injective dimension and dominant dimension of all such replicated algebras and determine which of them are minimal Auslander-Gorenstein, i.e. when the two dimensions are finite and equal to each other. In particular, we show that there exist infinitely many minimal Auslander-Gorenstien algebras in such a series if, and only if, the Serre-formal algebra is twisted fractionally Calabi-Yau. We apply these results to a construction of algebras from Yamagata, called SGC extensions, given by iteratively taking the endomorphism ring of the smallest generator-cogenerator. We give a sufficient condition so that the SGC extensions and replicated algebras coincide. Consequently, in such a case, we obtain explicit formulae for the self-injective dimension and dominant dimension of the SGC extension algebras.
Citations
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Journal ArticleDOI
TL;DR: In this article, the dominant dimensions of Nakayama algebras and more general algesbras A with an idempotent e such that there is a minimal faithful injective-projective module eA and such that eAe is a Nakaya algebra are given.

26 citations

Posted Content
TL;DR: In this paper, the dominant and codominant dimensions of modules are related to Gorenstein homological properties, and a new construction of non-selfinjective algebras having Auslander-Reiten components consisting only of GPs is given.
Abstract: This article relates dominant and codominant dimensions of modules to Gorenstein homological properties. We call a module Gorenstein projective-injective in case it is Gorenstein projective and Gorenstein injective. For gendo-symmetric algebras we find a characterisation of Gorenstein projective-injective modules in terms of dominant and codominant dimensions and find representation theoretic and homological properties of the category of Gorenstein projective-injective modules. In particular, we give a new construction of non-selfinjective algebras having Auslander-Reiten components consisting only of Gorenstein projective modules. We introduce the class of nearly Gorenstein algebras, where the Gorenstein projective-injective modules have especially nice properties in case the algebra is gendo-symmetric. We also show that the Nakayama conjecture holds for nearly Gorenstein algebras.

14 citations

Posted Content
TL;DR: In this article, it was shown that a finite lattice with at least two points is distributive if and only if it is an Auslander regular ring, which gives a homological characterisation of distributive lattices.
Abstract: Let $L$ denote a finite lattice with at least two points and let $A$ denote the incidence algebra of $L$. We prove that $L$ is distributive if and only if $A$ is an Auslander regular ring, which gives a homological characterisation of distributive lattices. In this case, $A$ has an explicit minimal injective coresolution, whose $i$-th term is given by the elements of $L$ covered by precisely $i$ elements. We give a combinatorial formula of the Bass numbers of $A$. We apply our results to show that the order dimension of a distributive lattice $L$ coincides with the global dimension of the incidence algebra of $L$. Also we categorify the rowmotion bijection for distributive lattices using higher Auslander-Reiten translates of the simple modules.

9 citations

Journal ArticleDOI
TL;DR: In this paper, the authors give new properties of algebras with finite self-injective dimension coinciding with the dominant dimension d ≥ 2, which are called minimal Auslander-Gorenstein algesias.

9 citations

Posted Content
TL;DR: In this article, the authors give a systematic construction of non-local algebras where the subcategory of Gorenstein projective modules does not coincide with stable modules using the theory of gendo-symmetric algesbras.
Abstract: In \cite{AB}, Auslander and Bridger introduced Gorenstein projective modules and only about 40 years after their introduction a finite dimensional algebra $A$ was found in \cite{JS} where the subcategory of Gorenstein projective modules did not coincide with $^{\perp}A$, the category of stable modules. The example in \cite{JS} is a commutative local algebra. We explain why it is of interest to find such algebras that are non-local with regard to the homological conjectures. We then give a first systematic construction of algebras where the subcategory of Gorenstein projective modules does not coincide with $^{\perp}A$ using the theory of gendo-symmetric algebras. We use Liu-Schulz algebras to show that our construction works to give examples of such non-local algebras with an arbitrary number of simple modules.

6 citations

References
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Book
11 May 2010
TL;DR: Artin rings as mentioned in this paper have been used to represent morphisms in the Auslander-Reiten-quiver and the dual transpose and almost split sequences, and they have been shown to be stable equivalence.
Abstract: 1. Artin rings 2. Artin algebras 3. Examples of algebras and modules 4. The transpose and the dual 5. Almost split sequences 6. Finite representation type 7. The Auslander-Reiten-quiver 8. Hereditary algebras 9. Short chains and cycles 10. Stable equivalence 11. Modules determining morphisms.

2,044 citations

Book
01 Dec 1984
TL;DR: In this article, the construction of stable separating tubular families and tubular algebras are discussed. But they do not discuss the relation between tubular extensions and directed algesbras.
Abstract: Integral quadratic forms.- Quivers, module categories, subspace categories (Notation, results, some proofs).- Construction of stable separating tubular families.- Tilting functors and tubular extensions (Notation, results, some proofs).- Tubular algebras.- Directed algebras.

1,581 citations

Journal ArticleDOI
TL;DR: The first of a series of papers dealing with the representation theory of artin algebras is presented in this paper, where the main purpose is to develop terminology and background material which will be used in the rest of the papers in the series.
Abstract: This is the first of a series of papers dealing with the representation theory of artin algebras, where by an artin algebra we mean an artin ring having the property that its center is an artin ring and λ is a finitely generated module over its center. The over all purpose of this paper is to develop terminology and background material which will be used in the rest of the papers in the series. While it is undoubtedly true that much of this material can be found in the literature or easily deduced from results already in the literature, the particular development presented here appears to be new and is especially well suited as a foundation for the papers to come.

1,267 citations

Journal ArticleDOI
Peter Gabriel1
TL;DR: In this paper, a linear representation of a given category is given by a map V associating with any morphism ϕ: a→e of K a linear vector space map V(ϕ): V(a)→V(e).
Abstract: LetK be the structure got by forgetting the composition law of morphisms in a given category. A linear representation ofK is given by a map V associating with any morphism ϕ: a→e ofK a linear vector space map V(ϕ): V(a)→V(e). We classify thoseK having only finitely many isomorphy classes of indecomposable linear representations. This classification is related to an old paper by Yoshii [3].

862 citations

Book ChapterDOI
01 Jan 1980

440 citations