Stanley-Reisner rings, sheaves, and Poincaré-Verdier duality

Abstract

A few years ago, I defined a squarefree module over a polynomial ring S = k[x1, . . . , xn] generalizing the Stanley-Reisner ring k[Δ] = S/IΔ of a simplicial complex Δ ⊂ 2 {1, ... , n}. This notion is very useful in the Stanley-Reisner ring theory. In this paper, from a squarefree S-module M, we construct the k-sheaf M+ on an (n - 1) simplex B which is the geometric realization of 2{1, ... , n}. For example k[Δ]+ is (the direct image to B of) the constant sheaf on the geometric realization |Δ| ⊂ B. We have Hi (B, M+) ≅ [Hmi+1 (M)]0 for all i ≥ 1. The Poincaré-Verdier duality for sheaves M+ on B corresponds to the local duality for squarefree modules over S. For example, if |Δ| is a manifold, then k[Δ] is a Buchsbaum ring and its canonical module Kk[Δ] is a squarefree module which gives the orientation sheaf of |Delta;| with the coefficients in k.