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

Abstract

A squarefree module over a polynomial ring $S=k[x_1,\text{...},x_n]$ 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 $\mathbf{Sq}$ of squarefree modules is equivalent to the category of finitely generated left $\Lambda$-modules, where $\Lambda$ is the incidence algebra of the Boolean lattice $2^{\{1,\text{...},n\}}$. The derived category $D^b\left(\mathbf{Sq}\right)$ has two duality functors $\mathbf{D}$ and $\mathbf{A}$. The functor $\mathbf{D}$ is a common one with $H^i\left(\mathbf{D},\left(M^{\bullet }\right)\right)={\mathrm{Ext}}_S^{n+i}\left(M^{\bullet },{\omega }_S\right)$, while the Alexander duality functor $\mathbf{A}$ is rather combinatorial. We have a strange relation $\mathbf{D}\circ \mathbf{A}\circ \mathbf{D}\circ \mathbf{A}\circ \mathbf{D}\circ \mathbf{A}\cong {\mathbf{T}}^{2n}$, where $\mathbf{T}$ is the translation functor. The functors $\mathbf{A}\circ \mathbf{D}$ and $\mathbf{D}\circ \mathbf{A}$ give a non-trivial autoequivalence of $D^b\left(\mathbf{Sq}\right)$. This equivalence corresponds to the Koszul duality for $\Lambda$, which is a Koszul algebra with ${\Lambda }^{!}\cong \Lambda$. Our $\mathbf{D}$ and $\mathbf{A}$ are also related to the Bernstein-Gel'fand-Gel'fand correspondence.

The local cohomology $H_{I_{\Delta }}^i\left(S\right)$at a Stanley-Reisner ideal $I_{\Delta }$ can be constructed from the squarefree module $Ex{t}_S^i(S/I_{\Delta },{\omega }_S)$. We see that Hochster's formula on the ${\mathbf{Z}}^n$-graded Hilbert function of $H_{\mathfrak{m}}^i(S/I_{\Delta })$ is also a formula on the characteristic cycle of $H_{I_{\Delta }}^{n-i}\left(S\right)$ as a module over the Weyl algebra $A=k\langle x_1,\text{...},x_n,{\partial }_1,\text{...},{\partial }_n\rangle$(if $\mathrm{char}\left(k\right)=0$).