Mathematical Research Letters

Volume 10 (2003)

Number 5

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

Pages: 635 – 650



Kohji Yanagawa (Osaka University)


A few years ago, I defined a {\it squarefree module} over a polynomial ring $S = k[x_1, \ldots, x_n]$ generalizing the Stanley-Reisner ring $k[\Delta] = S/I_\Delta$ of a simplicial complex $\Delta \subset 2^{\{1, \ldots , n\}}$. 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 $2^{\{1, \ldots , n\}}$. For example, $k[\Delta]^+$ is (the direct image to $B$ of) the constant sheaf on the geometric realization $|\Delta| \subset B$. We have $H^i(B, M^+) \cong [H^{i+1}_\m(M)]_0$ for all $i \geq 1$. The Poincaré-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 and its canonical module $K_{k[\Delta]}$ is a squarefree module which gives the orientation sheaf of $|\Delta|$ with the coefficients in $k$.

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