Derived Category of Squarefree Modules and Local Cohomology with Monomial Ideal Support

A “squarefree module” over a polynomial ring $S=k[x_1,\mathrm{.}\mathrm{.},x_n]$ is a generalization of a Stanley-Reisner ring, and allows us to apply homological methods to the study of monomial ideals systematically. Let $Sq$ be the category of squarefree modules. Then the derived category $D^b(Sq)$ of $Sq$ has three duality functors which act on $D^b(Sq)$ just like three transpositions of the symmetric group $S_3$ (up to translation). This phenomenon is closely related to the Koszul dulaity (in particular, the Bernstein-Gel’fand-Gel’fand correspondence). We also study the local cohomology module $H_{I_{\mathrm{\Delta }}}^i(S)$ at a Stanley-Reisner ideal $I_{\mathrm{\Delta }}$ using squarefree modules. Among other things, we see that Hochster’s formula on the Hilbert function of $H_m^i(S\mathrm{/}{I}_{\mathrm{\Delta }})$ is also a formula on the characteristic cycle of $H_{I_{\mathrm{\Delta }}}^{n-i}(S)$ as a module over the Weyl algebra $S<\partial_1, …, \partial_n >$(if$chara(k)=0$).