Dissertation•
Functorial Filtrations for Semiperfect generalisations of Gentle Algebras
01 Aug 2017-
TL;DR: In this article, the authors introduce a class of semiperfect rings which generalise the class of finite-dimensional gentle algebras and consider complexes of modules over these rings, which have finitely generated projective homogeneous components.
Abstract: In this thesis we introduce a class of semiperfect rings which generalise the class of
finite-dimensional gentle algebras. We consider complexes of modules over these rings
which have finitely generated projective homogeneous components. We then classify them
up to homotopy equivalence. The method we use to solve this classification problem is
called the functorial filtrations method. The said method was previously only used to
classify modules.
Citations
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TL;DR: In this paper, a text on rings, fields and algebras is intended for graduate students in mathematics, aiming the level of writing at the novice rather than at the expert, and by stressing the role of examples and motivation.
Abstract: This text, drawn from the author's lectures at the University of California at Berkeley, is intended as a textbook for a one-term course in basic ring theory. The material covered includes the Wedderburn-Artin theory of semi-simple rings, Jacobson's theory of the radical representation theory of groups and algebras, prime and semi-prime rings, primitive and semi-primitive rings, division rings, ordered rings, local and semi-local rings, and perfect and semi-perfect rings. By aiming the level of writing at the novice rather than at the expert, and by stressing the role of examples and motivation, the author has produced a text which is suitable not only for use in a graduate course, but also for self-study by other interested graduate students. Numerous exercises are also included. This graduate textbook on rings, fields and algebras is intended for graduate students in mathematics.
1,479 citations
Journal Article•
TL;DR: In this article, a class of n-complete algebras by endomorphism ring of tilting modules was constructed and described by other way, which is the same way that we describe n-completeness of the present paper.
Abstract: Iyama constructs a class of n-complete algebras and absolute n-complete algebras by cone from finite representation type and hereditary algebras,this paper we construct a class of n-complete algebras and absolute n-complete algebras by endomorphism ring of tilting modules,so we describe n-complete algebras by other way
17 citations
Posted Content•
TL;DR: In this article, the unbounded homotopy category of complexes of projective modules is considered, and the corresponding isoclasses are described and the Krull-Remak-Schmidt-Azumaya property is verified.
Abstract: We consider algebras defined over a complete, local and noetherian ground ring. They are gentle algebras in case the ground ring is a field. The unbounded homotopy category of complexes of projective modules is considered. Complexes with finitely-generated homogeneous components are shown to be isomorphic to direct sums of indecomposable string and band complexes. The corresponding isoclasses are described, and the Krull-Remak-Schmidt-Azumaya property is verified. This classification problem is solved using the idea of functorial filtrations.
5 citations
Cites result from "Functorial Filtrations for Semiperf..."
...The results in this article form part of the author’s PhD thesis [6]....
[...]
08 Mar 2023
TL;DR: In this paper , a class of Gorenstein algebras over commutative noetherian rings was introduced, and duality theorems for various categories of representations were established.
Abstract: In the recent paper"The Nakayama functor and its completion for Gorenstein algebras", a class of Gorenstein algebras over commutative noetherian rings was introduced, and duality theorems for various categories of representations were established. The manuscript on hand provides more context to the results presented in the aforementioned work, identifies new classes of Gorenstein algebras, and explores their behaviour under standard operations like taking tensor products and tilting.
2 citations
Posted Content•
TL;DR: In this article, the authors characterise cosilting modules in terms of their indecomposable summands, providing a new approach to the classification of torsion pairs over cluster-tilted algebras.
Abstract: Torsion pairs in the category of finitely presented modules over a noetherian ring can be parametrised by the class of cosilting modules. In this paper, we characterise such modules in terms of their indecomposable summands, providing a new approach to the classification of torsion pairs. In particular, we classify cosilting modules over cluster-tilted algebras of type $\tilde{A}$. We do this by using a geometric model for finite- and infinite-dimensional modules over such algebras.
2 citations
References
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Book•
01 Jan 1969
TL;DR: It is shown here how the Noetherian Rings and Dedekind Domains can be transformed into rings and Modules of Fractions using the following structures:
Abstract: * Introduction * Rings and Ideals * Modules * Rings and Modules of Fractions * Primary Decomposition * Integral Dependence and Valuations * Chain Conditions * Noetherian Rings * Artin Rings * Discrete Valuation Rings and Dedekind Domains * Completions * Dimension Theory
4,168 citations
TL;DR: In this paper, a text on rings, fields and algebras is intended for graduate students in mathematics, aiming the level of writing at the novice rather than at the expert, and by stressing the role of examples and motivation.
Abstract: This text, drawn from the author's lectures at the University of California at Berkeley, is intended as a textbook for a one-term course in basic ring theory. The material covered includes the Wedderburn-Artin theory of semi-simple rings, Jacobson's theory of the radical representation theory of groups and algebras, prime and semi-prime rings, primitive and semi-primitive rings, division rings, ordered rings, local and semi-local rings, and perfect and semi-perfect rings. By aiming the level of writing at the novice rather than at the expert, and by stressing the role of examples and motivation, the author has produced a text which is suitable not only for use in a graduate course, but also for self-study by other interested graduate students. Numerous exercises are also included. This graduate textbook on rings, fields and algebras is intended for graduate students in mathematics.
1,479 citations
11 May 2018
TL;DR: In this paper, a comprehensive introduction to module theory and the related part of ring theory, including original results as well as the most recent work, is presented and accompanied by complete proofs, where the smallest Grothendieck category containing it is denoted by o[M] and module theory is developed in this category.
Abstract: This volume provides a comprehensive introduction to module theory and the related part of ring theory, including original results as well as the most recent work. It is a useful and stimulating study for those new to the subject as well as for researchers and serves as a reference volume. Starting form a basic understanding of linear algebra, the theory is presented and accompanied by complete proofs. For a module M, the smallest Grothendieck category containing it is denoted by o[M] and module theory is developed in this category. Developing the techniques in o[M] is no more complicated than in full module categories and the higher generality yields significant advantages: for example, module theory may be developed for rings without units and also for non-associative rings. Numerous exercises are included in this volume to give further insight into the topics covered and to draw attention to related results in the literature.
1,141 citations
"Functorial Filtrations for Semiperf..." refers methods in this paper
...In the proof below we freely use results about module categories from the books by Anderson and Fuller [2] and Wisbauer [65]....
[...]
Book•
15 Jan 1996
TL;DR: In this article, the authors introduce homotopic algebra and define the notion of simplicial sets, derived categories and derived functors, and triangulated categories for homotopy algebra.
Abstract: I. Simplicial Sets.- II. Main Notions of the Category Theory.- III. Derived Categories and Derived Functors.- IV. Triangulated Categories.- V. Introduction to Homotopic Algebra.- References.
1,043 citations
"Functorial Filtrations for Semiperf..." refers background or methods in this paper
...We follow: books by Aluffi [1] and Freyd [26] and papers by Bergman [10] and Krause [45] for ideas about abelian categories; and the books by Gelfand and Manin [33], Neeman [52], Weibel [64] and Zimmerman [67] for ideas about triangulated categories....
[...]
...V X s oo f //W (4)What we call multiplicative systems, Gelfand and Manin [33] call localisation classes....
[...]
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