On the Notion of Derived Tameness

Geiß, Christof; Krause, Henning

doi:10.1142/s0219498802000112

Cited by 29 publications

(10 citation statements)

References 14 publications

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“…A is called derived tame (see Geiss and Krause, 2000) if, for each cohomology dimension vector (d i ) i2Z , there exist a localization R ¼ k[x] f with respect to some f 2 k[x] and a finite number of bounded complexes of A-R-bimodules C 1 , . Let k be an algebraically closed field and A be a finite-dimensional k-algebra.…”

mentioning

confidence: 99%

Indecomposables in Derived Categories of Skewed-Gentle Algebras

Bekkert

Marcos

Merklen

2003

Communications in Algebra

171

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confidence: 99%

Indecomposables in Derived Categories of Skewed-Gentle Algebras

Bekkert

Marcos

Merklen

2003

Communications in Algebra

171

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“…Remark 2.2. As in [7], every Â-module X has a maximal injective submodule X inj since direct limits of injective modules are injective and Zorn's lemma. Let X red = X/X inj , two Â-modules X and Y are isomorphic as objects in Mod Â if and only if X red and Y red are isomorphic in Mod Â.…”

Section: Preliminariesmentioning

confidence: 97%

“…For any algebra A, there is some useful results about the image of Happel's functor as following [7,Lemma 3.4,3.5].…”

Section: Proposition 23 [14 Theorem 72]mentioning

confidence: 99%

Generically Trivial Derived Categories

Han

2015

Communications in Algebra

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We study generic objects in triangulated categories and give a characterization of the finite dimensional algebras A such that the derived categories D(Mod A) are generically trivial. This characterization is an analogue of a result of Crawley-Boevey for module categories.As a consequence, we show that D(Mod A) is generically trivial if and only if the category of perfect complexes K b (proj A) is locally finite.

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“…Note that some previous work does exist on the representation type of derived categories, see [8], but it does not apply to the categories considered in this paper.…”

Section: Open Problemsmentioning

confidence: 99%

Calabi–Yau categories and Poincaré duality

Jørgensen

2008

Trends in Representation Theory of Algebras and Related Topics

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The singular cochain complex of a topological space is a classical object. It is a Differential Graded algebra which has been studied intensively with a range of methods, not least within rational homotopy theory.More recently, the tools of Auslander-Reiten theory have also been applied to the singular cochain complex. One of the highlights is that by these methods, each Poincaré duality space gives rise to a Calabi-Yau category. This paper is a review of the theory.has the full subcategory D c (C * (X; k)) consisting of compact DG modules; these play the role of finitely generated representations. Theorem 6.4 now says that if k has characteristic 0, then D c (C * (X; k)) is an n-Calabi-Yau category (1.1) ⇔ X has n-dimensional Poincaré duality over k.Let me briefly explain the terminology. A triangulated category T, such as for instance D c (C * (X; k)), is called n-Calabi-Yau if n is the smallest non-negative integer for which Σ n , the nth power of the suspension functor, is a Serre functor, that is, permits natural isomorphismsThe topological space X is said to have n-dimensional Poincaré duality over k if there is an isomorphismExamples of n-Calabi-Yau categories are higher cluster categories, see [15, sec. 4], and examples of spaces with n-dimensional Poincaré duality are compact ndimensional manifolds. Equation (1.1) provides a link between the currently popular theory of Calabi-Yau categories and algebraic topology. It also gives a new class of examples of Calabi-Yau categories which, so far, typically have been exemplified by higher cluster categories. The new categories appear to behave very differently from higher cluster categories, cf. Section 7, Problem 7.8.A number of other results are also obtained, not least on the structure of the AR quiver of D c (C * (X; k)) which, for a space with Poincaré duality, consists of copies of the repetitive quiver ZA ∞ , see Theorem 6.5.In a speculative vein, the theory presented here ties in with the version of noncommutative geometry in which a DG algebra, or more generally a DG category, is viewed as a non-commutative scheme. The idea is to think of the derived category of the DG algebra or DG category as being the derived category of quasi-coherent sheaves on a non-commutative scheme (which does not actually exist). There appear so far to be no published references for this viewpoint which has been brought forward by Drinfeld and Kontsevich, but it does seem to call for a detailed study of the derived categories of DG algebras and DG categories. Auslander-Reiten theory is an obvious tool to try, and [11], [12], and [20] along with this paper can, perhaps, be viewed as a first, modest step.As indicated, the paper is a review. The results were known previously, the main references being [11], [12], and [20]; more details of the origin of individual results are given in the introductions to the sections. There is no claim to originality, except that some of the proofs are new. It is also the first time this material has appeared together.Most of the paper is phras...

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On the Notion of Derived Tameness

Abstract: The notion of tameness for the derived category of a finite dimensional algebra is introduced and standard properties are established. This is based on classical tameness definitions of Drozd and Crawley-Boevey for the category of finite dimensional representations.

Cited by 29 publications

References 14 publications

Indecomposables in Derived Categories of Skewed-Gentle Algebras

Indecomposables in Derived Categories of Skewed-Gentle Algebras

Generically Trivial Derived Categories

Calabi–Yau categories and Poincaré duality

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