Journal of the Mathematical Society of Japan

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


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A squarefree module over a polynomial ring S=k[x1,...,xn] 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 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 DbSq has two duality functors D and A. The functor D is a common one with HiDM= ExtSn+iMωS, while the Alexander duality functor A is rather combinatorial. We have a strange relation DADADAT2n, where T is the translation functor. The functors AD and DA give a non-trivial autoequivalence of DbSq. This equivalence corresponds to the Koszul duality for Λ, which is a Koszul algebra with Λ!Λ. Our D and A are also related to the Bernstein-Gel'fand-Gel'fand correspondence.

The local cohomology HIΔi Sat a Stanley-Reisner ideal IΔ can be constructed from the squarefree module ExtSi(S/IΔ,ωS). We see that Hochster's formula on the Zn-graded Hilbert function of Hmi(S/IΔ) is also a formula on the characteristic cycle of HIΔn-iS as a module over the Weyl algebra A=kx1,...,xn,1,...,n(if chark=0).

Article information

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

First available in Project Euclid: 3 October 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 13D25
Secondary: 13D02: Syzygies, resolutions, complexes 13D45: Local cohomology [See also 14B15] 13F55: Stanley-Reisner face rings; simplicial complexes [See also 55U10] 13N10: Rings of differential operators and their modules [See also 16S32, 32C38] 18E30: Derived categories, triangulated categories

Stanley-Reisner ring local duality Alexander duality local cohomology Bernstein-Gel'fand-Gel'fand correspondence Koszul duality


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.

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