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

Abstract

A squarefree module over a polynomial ring S=k[x₁, ..., x_n] 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 \bm{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^{{1, ..., n}}. The derived category D^b(\bm{Sq}) has two duality functors \bm{D} and \bm{A}. The functor \bm{D} is a common one with Hⁱ(\bm{D}(M^·))=Ext_Sⁿ⁺ⁱ (M^·, omega s), while the Alexander duality functor \bm{A} is rather combinatorial. We have a strange relation \bm{D}\circ \bm{A}\circ \bm{D}\circ \bm{A}\circ \bm{D}\circ \bm{A}≅ \bm{T}²ⁿ, where \bm{T} is the translation functor. The functors \bm{A}\circ \bm{D} and \bm{D}\circ \bm{A} give a non-trivial autoequi-valence of D^b(\bm{Sq}). This equivalence corresponds to the Koszul duality for Λ, which is a Koszul algebra with Λ¹≅Λ. Our \bm{D} and \bm{A} are also related to the Bernstein-Gel'fand-Gel'fand correspondence.
The local cohomology H_{I_Λ}ⁱ(S) at a Stanley-Reisner ideal I_Δ can be constructed from the squarefree module Ext_Sⁱ(S/I_Δ, omega_S). We see that Hochster's formula on the \bm{Z}ⁿ-graded Hilbert function of H_{\mathfrak{m}}ⁱ(S/I_Δ) is also a formula on the characteristic cycle of H_{\mathit{1}_Δ}^n-i(S) as a module over the Weyl algebra A=k‹ x₁, ..., x_n, ∂₁, ..., ∂_n› (if char(k)=0).