Consistent algebras and special tilting sequences
type: Item
instance of: scholarly article
title: Consistent algebras and special tilting sequences
zbMATH Open document ID: 0841.14013
DOI: 10.1007/BF02572609
author: Lutz Hille
published in: Mathematische Zeitschrift
publication date: 1996-07-16T00:00:00Z
full work available at URL: https://eudml.org/doc/174804
review text: We consider a nonsingular projective algebraic variety \(X\) over an algebraic closed field of characteristic 0 and \(\varepsilon = (E_0, \dots, E_N)\) an exceptional sequence of coherent sheaves on \(X\) with the additional conditions, that the \(E_i\) generate the derived category \({\mathcal D}^b(X)\) and there are \(\dim X + 1\) disjoint subsequences, every sheaf \(E_i\) is in one of these subsequences and for all \(E_i\) and \(E_j\) in the same subsequence we have \(\text{Ext}^l(E_i, E_j) = 0\) for all \(l \geq 0\). Then we consider the finite dimensional algebra \({\mathcal A} = \text{End} (\bigoplus E_i)\). We show that \(\bigoplus E_i\) is a tilting sheaf on \(X\) and \({\mathcal A}\) is a Koszul and coKoszul algebra. In the main result we construct an explicit functor \(\varphi : \text{mod-} \widehat {\mathcal B} \to C^b (X)\) from the module category of finite generated modules over the repetive algebra of the priddy dual algebra of \({\mathcal A}\) to the category of bounded complexes of coherent sheaves on \(X\) which induces a triangle equivalence \(\Phi : \underline {\text{mod}}-\widehat {\mathcal B} \to {\mathcal D}^b(X)\) of the corresponding stable module category with the derived category of coherent sheaves on \(X\).
Mathematics Subject Classification ID: 14F10
Mathematics Subject Classification ID: 18E30
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 18G15
Mathematics Subject Classification ID: 18F20
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 16D90
zbMATH DE Number: 856559
zbMATH Keywords: Koszul algebra
zbMATH Keywords: mutations
zbMATH Keywords: helices
zbMATH Keywords: stable module category of a selfinjective algebra
zbMATH Keywords: exceptional sequence of coherent sheaves
zbMATH Keywords: derived category
zbMATH Keywords: tilting sheaf
MaRDI profile type: Publication
cites work: Tilting sheaves in representation theory of algebras
cites work: Coherent sheaves on \({\mathbb{P}}^n\) and problems of linear algebra
cites work: On the derived category of a finite-dimensional algebra
cites work: Vector bundles on complex projective spaces
description: scientific article; zbMATH DE number 856559