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

Derived category of squarefree modules and local cohomology with monomial ideal support