Methods of Homological Algebra

Abstract. We study the role of the Serre functor in the theory of derived equivalences. Let A be an abelian category and let (U , V) be a t-structure on the bounded derived category D b A with heart H. We investigate when the natural embedding H → D b A can be extended to a triangle equivalence D b H → D b A. Our focus of study is the case where A is the category of finite-dimensional modules over a finite-dimensional hereditary algebra. In this case, we prove that such an extension exists if and only if the t-structure is bounded and the aisle U of the t-structure is closed under the Serre functor.

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“…It has been shown in [8, Théorème 1.3.6] (see also [18,Theorem IV.4.4]) that the heart is an abelian category.…”

Section: Torsion Pairs Weight Structures and T-structuresmentioning

confidence: 99%

Derived equivalences for hereditary Artin algebras

Stanley

Roosmalen

2016

Advances in Mathematics

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“…We amplify and complement this crash course in the appendices in §5 and in section §2 on perverse sheaves. Standard references are [19,82,87,115,107].…”

Section: Crash Course On Sheaves and Derived Categoriesmentioning

confidence: 99%

“…Standard references for what follows are [115,82,19,107,9]; see also [162]. In what follows, we freely refer to our crash-course in §1.5 and to the complete references given above.…”

Section: The Formalism Of the Constructible Derived Categorymentioning

confidence: 99%

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The decomposition theorem, perverse sheaves and the topology of algebraic maps

Andrea

Migliorini

2009

Bull. Amer. Math. Soc.

163

152

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We give a motivated introduction to the theory of perverse sheaves, culminating in the decomposition theorem of Beilinson, Bernstein, Deligne and Gabber. A goal of this survey is to show how the theory develops naturally from classical constructions used in the study of topological properties of algebraic varieties. While most proofs are omitted, we discuss several approaches to the decomposition theorem, indicate some important applications and examples.

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“…En effet, la situation est la suivante : les groupes H j c (U n (δ, r) Fo , Q ) forment un système de coefficients (au sens de [GM,chap. I,§ 2.4]) contravariant sur le complexe simplicial ∆ × Z ; le complexe résultant est le complexe deČechČ • c (U, Q ), où U désigne le recouvrement dě Cech {U n (δ, r)}.…”

Section: Annales De L'institut Fourierunclassified

Uniformisation des variétés de Laumon-Rapoport-Stuhler et conjecture de Drinfeld-Carayol

Hausberger¹

2005

Annales De L’institut Fourier

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Methods of Homological Algebra

Cited by 381 publications

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Derived equivalences for hereditary Artin algebras

Derived equivalences for hereditary Artin algebras

The decomposition theorem, perverse sheaves and the topology of algebraic maps

Uniformisation des variétés de Laumon-Rapoport-Stuhler et conjecture de Drinfeld-Carayol

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