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

Recently, I defined a squarefree module over a polynomial ring $S = k[x_1,

…, x_n]$generalizingtheStanley-Reisnerring$k[\Delta] = S/I_\Delta$ of a simplicial complex $\mathrm{\Delta }\subset 2^{1,\mathrm{.}\mathrm{.}\mathrm{.},n}$. 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 $2^{1,\mathrm{.}\mathrm{.}\mathrm{.},n}$. For example, $k[\mathrm{\Delta }{]}^{+}$ is (the direct image to $B$ of) the constant sheaf on the geometric realization $\mathrm{\mid }\mathrm{\Delta }\mathrm{\mid }\subset B$. 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 $\mathrm{\mid }\mathrm{\Delta }\mathrm{\mid }$ is a manifold, then $k[\mathrm{\Delta }]$ is a Buchsbaum ring whose canonical module is a squarefree module giving the orientation sheaf of $\mathrm{\mid }\mathrm{\Delta }\mathrm{\mid }$ with the coefficients in $k$.