Sheaves on Triangulated Spaces and Koszul Duality

Let X be a finite connected simplicial complex, and let δ be a perversity (i.e., some function from integers to integers). One can consider two categories: (1) the category of perverse sheaves cohomologically constructible with respect to the triangulation, and (2) the category of sheaves constant along the perverse simplices (δ-sheaves). We interpret the categories (1) and (2) as categories of modules over certain quadratic (and even Koszul) algebras A(X, δ) and B(X,δ) respectively, and we prove that A(X, δ) and B(X, δ) are Koszul dual to each other. We define the δ-perverse topology on X and prove that the category of sheaves on perverse topology is equivalent to the category of δ sheaves. Finally, we study the relationship between the Koszul duality functor and the Verdier duality functor for simplicial sheaves and cosheaves. Introduction. 1. The study of constructible sheaves on a cell complex X leads to the notion of a cellular sheaf, which was developed by W. Fulton, M. Goresky, R. MacPherson, and C. McCrory in a seminar at Brown University in 1977-78. The systematic exposition of the theory of cellular sheaves has been presented in the A. Shepard’s Doctoral Thesis [Shep] (cf. [Kash]). A cellular sheaf is a gadget which assigns vector spaces to cells in X and linear maps to pairs of incident cells. (Cellular sheaves can also be interpreted as sheaves on the finite topology generated by open stars of cells.) It is easy to interpret such linear algebra gadgets as modules over an associative algebra B(X). In this paper we will work with a finite connected simplicial complex X . We can consider simplicial complexes without the loss of generality since any reasonable stratified space can be triangulated [Gor]. The category of constructible sheaves of F-vector spaces on X is denoted by SHc(X). We formulate here the basic result of the cellular sheaf theory. Theorem A. The following categories SHc(X) ≃ mod-B(X) are equivalent. Typeset by AMS-TEX 1