## 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 $\mathrm{char}\left(k\right)=0$).

## Citation

Kohji YANAGAWA. "Derived category of squarefree modules and local cohomology with monomial ideal support." J. Math. Soc. Japan 56 (1) 289 - 308, January, 2004. https://doi.org/10.2969/jmsj/1191418707

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