Perverse sheaves on a triangulated space
type: Item
instance of: scholarly article
title: Perverse sheaves on a triangulated space
zbMATH Open document ID: 0905.18007
published in: Mathematical Research Letters
publication date: 1997-10-06T00:00:00Z
review text: Let \(X\) be a stratified space such that any stratum (cell) \(C\) is homeomorphic to an open ball and its closure \(\overline C\) is homeomorphic to a closed ball in a compatible way. Let \(p: \mathbb{Z}_{\geq 0} \to\mathbb{Z}\) be a perversity and \({\mathcal C}\) the category of perverse sheaves on \(X\) with respect to \(p\). The simple objects of \({\mathcal C}\) are Goresky-MacPherson extensions \(P_C\), where \(C\) is a cell in \(X\). Assume that the cell decomposition of \(X\) satisfies the following condition: for every pair of cells \(C\) and \(C'\), the intersection \(\overline C\cap \overline C'\) is either empty or homeomorphic to a closed ball. The first result asserts that the graded algebra \(A= \oplus_{C,C'} \text{Ext}^* (P_C,P_{C'})\) is generated by the component of degree 1. With the assumption that the cell decomposition of \(X\) is a simplicial triangulation, the author proves that the algebra \(A\) is quadratic and Koszul. Some consequences of the above results are given (for example, the category \({\mathcal C}\) is equivalent to the category of finite-dimensional representations of a quadratic algebra \(B)\).
reviewed by: Vasile Brînzănescu
Mathematics Subject Classification ID: 18F20
Mathematics Subject Classification ID: 14F10
Mathematics Subject Classification ID: 54B40
Mathematics Subject Classification ID: 57Q15
Mathematics Subject Classification ID: 17A45
Mathematics Subject Classification ID: 14F05
zbMATH DE Number: 1053502
zbMATH Keywords: quadratic algebras
zbMATH Keywords: Koszul algebras
zbMATH Keywords: stratified spaces
zbMATH Keywords: perverse sheaves
zbMATH Keywords: Goresky-MacPherson extensions
author: Alexander Polishchuk
MaRDI profile type: Publication
DOI: 10.4310/MRL.1997.V4.N2.A2
description: scientific article; zbMATH DE number 1053502