Stanley-Reisner rings, sheaves, and Poincare-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 $H^i(B,M^{+})=[H_m^{i+1}(M){]}_0$ 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 $k$.

15 pages. In the newest version, I have modified some minor places. To appear in Mathematical Research Letters