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

A squarefree module over a polynomial ring S = k[x1, . . . , xn] is a generalization of a Stanley-Reisner ring, and allows us to apply homological methods to the study of monomial ideals more systematically. The category Sq of squarefree modules is equivalent to the category of finitely generated left Λ-modules, where Λ is the incidence algebra of the Boolean lattice 2. The derived category D(Sq) has two duality functors D and A. The functor D is a common one with H(D(M)) = Ext S (M , ωS), while the Alexander duality functor A is rather combinatorial. We have a strange relation D ◦ A ◦ D ◦ A ◦ D ◦ A ∼= T, where T is the translation functor. The functors A ◦D and D ◦A give a non-trivial autoequivalence of D(Sq). This equivalence corresponds to the Koszul duality for Λ, which is a Koszul algebra with Λ ∼= Λ. Our D and A are also related to the Bernstein-Gel’fand-Gel’fand correspondence. The local cohomology H I∆(S) at a Stanley-Reisner ideal I∆ can be constructed from the squarefree module ExtiS(S/I∆, ωS). We see that Hochster’s formula on the Z-graded Hilbert function of H m (S/I∆) is also a formula on the characteristic cycle of H I∆ (S) as a module over the Weyl algebra A = k〈x1, . . . , xn, ∂1, . . . , ∂n〉 (if char(k) = 0).