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

Abstract

Recently, I defined a squarefree module over a polynomial ring $S = k[x_1, >..., x_n]$ generalizing the Stanley-Reisner ring $k[\Delta] = S/I_\Delta$ of a simplicial complex $\Delta \subset 2^{1, ..., n}$. In this paper, from a squarefree 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, $k[\Delta]^+$ is (the direct image to $B$ of) the constant sheaf on the geometric realization $|\Delta| \subset B$. We have $H^i(B, M^+) = [H^{i+1}_m(M)]_0$ for all $i > 0$. The Poincare-Verdier duality for sheaves $M^+$ on $B$ corresponds to the local duality for squarefree modules over $S$. For example, if $|\Delta|$ is a manifold, then $k[\Delta]$ is a Buchsbaum ring whose canonical module is a squarefree module giving the orientation sheaf of $|\Delta|$ with the coefficients in $k$.