  A "squarefree module" over a polynomial ring $S = k[x_1, .., x_n]$ 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 $H_{I_\Delta}^i(S)$
at a Stanley-Reisner ideal $I_\Delta$ using squarefree modules. Among other
things, 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 $chara(k)=0$).
