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

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

2003-03-10

arxiv.org/abs/math/0303110v1

Mathematics

Commutative Algebra

21pages, to appear in J. Math. Soc. Japan. I distributed the earlier version of this paper in 2000, but the paper has been tot

Scientific paper

A "squarefree module" over a polynomial ring $S = k[x_1, .., x_n]$ is a generalization of a Stanley-Reisner ring, and allows us to apply homological methods to the study of monomial ideals systematically. Let $Sq$ be the category of squarefree modules. Then the derived category $D^b(Sq)$ of $Sq$ has three duality functors which act on $D^b(Sq)$ just like three transpositions of the symmetric group $S_3$ (up to translation). This phenomenon is closely related to the Koszul dulaity (in particular, the Bernstein-Gel'fand-Gel'fand correspondence). We also study the local cohomology module $H_{I_\Delta}^i(S)$ at a Stanley-Reisner ideal $I_\Delta$ using squarefree modules. Among other things, we see that Hochster's formula on the Hilbert function of $H_m^i(S/I_\Delta)$ is also a formula on the characteristic cycle of $H_{I_\Delta}^{n-i}(S)$ as a module over the Weyl algebra $S<\partial_1, ..., \partial_n >$ (if $chara(k)=0$).

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