Perverse sheaves on a triangulated space
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 \(B)\).
scientific article; zbMATH DE number 1053502