## Journal of the Mathematical Society of Japan

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

Kohji YANAGAWA

#### Abstract

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

The local cohomology $H_{I_\Delta}^{i}(S)$at a Stanley-Reisner ideal $I_{\Delta}$ can be constructed from the squarefree module $\mathrm{Ext}_{S}^{i}(S/I_{\Delta},\omega_{S})$. We see that Hochster's formula on the $\mathbold{Z}^{n}$-graded Hilbert function of $H_{\mathfrak{m}}^{i}(S/I_{\Delta})$ is also a formula on the characteristic cycle of $H_{I_\Delta}^{n-i}(S)$ as a module over the Weyl algebra $A=k\langle x_{1},\ldots,x_{n},\partial_{1},\ldots,\partial_{n}\rangle$(if $\mathrm{char}(k)=0$).

#### Article information

Source
J. Math. Soc. Japan, Volume 56, Number 1 (2004), 289-308.

Dates
First available in Project Euclid: 3 October 2007

https://projecteuclid.org/euclid.jmsj/1191418707

Digital Object Identifier
doi:10.2969/jmsj/1191418707

Mathematical Reviews number (MathSciNet)
MR2028674

Zentralblatt MATH identifier
1064.13010

#### Citation

YANAGAWA, Kohji. Derived category of squarefree modules and local cohomology with monomial ideal support. J. Math. Soc. Japan 56 (2004), no. 1, 289--308. doi:10.2969/jmsj/1191418707. https://projecteuclid.org/euclid.jmsj/1191418707