2 00 3 Stanley - Reisner Rings , Sheaves , and Poincaré - Verdier Duality

A few years ago, I defined a squarefree module over a polynomial ring S = k[x1, . . . , xn] generalizing the Stanley-Reisner ring k[∆] = S/I∆ of a simplicial complex ∆ ⊂ 2. This notion is very useful in the StanleyReisner 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. For example, k[∆] is (the direct image to B of) the constant sheaf on the geometric realization |∆| ⊂ B. We have H(B, M) ∼= [H m (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 |∆| is a manifold, then k[∆] is a Buchsbaum ring and its canonical module Kk[∆] is a squarefree module which gives the orientation sheaf of |∆| with the coefficients in k.