Singularity categories of derived categories of hereditary algebras are derived categories

  title={Singularity categories of derived categories of hereditary algebras are derived categories},
  author={Yuta Kimura},
  journal={Journal of Pure and Applied Algebra},
  • Y. Kimura
  • Published 15 February 2017
  • Mathematics
  • Journal of Pure and Applied Algebra
Abstract We show that for the path algebra A of an acyclic quiver, the singularity category of the derived category D b ( mod A ) is triangle equivalent to the derived category of the functor category of mod _ A , that is, D sg ( D b ( mod A ) ) ≃ D b ( mod ( mod _ A ) ) . This extends a result in [14] for the path algebra A of a Dynkin quiver. An important step is to establish a functor category analog of Happel's triangle equivalence for repetitive algebras. 
Classifications of exact structures and Cohen–Macaulay-finite algebras
Abstract We give a classification of all exact structures on a given idempotent complete additive category. Using this, we investigate the structure of an exact category with finitely manyExpand
On deformed preprojective algebras
Deformed preprojective algebras are generalizations of the usual preprojective algebras introduced by Crawley-Boevey and Holland, which have applications to Kleinian singularities, the DeligneSimpsonExpand
Singularity categories of Gorenstein monomial algebras
In this paper, we study the singularity category $D_{sg}(\mod A)$ and the $\mathbb{Z}$-graded singularity category $D_{sg}(\mod^{\mathbb Z} A)$ of a Gorenstein monomial algebra $A$. Firstly, for aExpand
Yoneda algebras and their singularity categories.
For a finite dimensional algebra $\Lambda$ of finite representation type and an additive generator $M$ for $\mathrm{mod}\,\Lambda$, we investigate the properties of the Yoneda algebraExpand
Cohen-Macaulay modules over Yoneda algebras
For a finite dimensional algebra $\Lambda$ of finite representation type and an additive generator $M$ for $\mathrm{mod}\,\Lambda$, we investigate the properties of the Yoneda algebraExpand
Covering theory, (mono)morphism categories and stable Auslander algebras
Let $\mathcal{A}$ be a locally bounded $k$-category and $G$ a torsion-free group of $k$-linear automorphisms of $\mathcal{A}$ acting freely on the objects of $\mathcal{A},$ andExpand


Realizing stable categories as derived categories
Abstract In this paper, we discuss a relationship between representation theory of graded self-injective algebras and that of algebras of finite global dimension. For a positively gradedExpand
Exact Categories
We survey the basics of homological algebra in exact categories in the sense of Quillen. All diagram lemmas are proved directly from the axioms, notably the five lemma, the 3× 3lemma and the snakeExpand
Triangulated categories of singularities and D-branes in Landau-Ginzburg models
In spite of physics terms in the title, this paper is purely mathematical. Its purpose is to introduce triangulated categories related to singularities of algebraic varieties and establish aExpand
Classifying exact categories via Wakamatsu tilting
Abstract Using the Morita-type embedding, we show that any exact category with enough projectives has a realization as a (pre)resolving subcategory of a module category. When the exact category hasExpand
Three results on Frobenius categories
This paper consists of three results on Frobenius categories: (1) we give sufficient conditions on when a factor category of a Frobenius category is still a Frobenius category; (2) we show that anyExpand
Localization of triangulated categories and derived categories
The notion of quotient and localization of abelian categories by dense subcategories (i.e., Serre classes) was introduced by Gabriel, and plays an important role in ring theory [6, 131. The notion ofExpand
Stable categories of higher preprojective algebras
We introduce (n+1)-preprojective algebras of algebras of global dimension n. We show that if an algebra is n-representation-finite then its (n+1)-preprojective algebra is self-injective. In thisExpand
Stable Categories of Graded Maximal Cohen-Macaulay Modules over Noncommutative Quotient Singularities
Tilting objects play a key role in the study of triangulated categories. A famous result due to Iyama and Takahashi asserts that the stable categories of graded maximal Cohen-Macaulay modules overExpand
Tilting and cluster tilting for preprojective algebras and Coxeter groups
We study the stable category of the factor algebra of the preprojective algebra associated with an element $w$ of the Coxeter group of a quiver. We show that there exists a silting object $M(\bf{w})$Expand
Representation Theory of Artin Algebras I
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 ringExpand