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


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).


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Kohji YANAGAWA. "Derived category of squarefree modules and local cohomology with monomial ideal support." J. Math. Soc. Japan 56 (1) 289 - 308, January, 2004.


Published: January, 2004
First available in Project Euclid: 3 October 2007

zbMATH: 1064.13010
MathSciNet: MR2028674
Digital Object Identifier: 10.2969/jmsj/1191418707

Primary: 13D25
Secondary: 13D02 , 13D45 , 13F55 , 13N10 , 18E30

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

Rights: Copyright © 2004 Mathematical Society of Japan


Vol.56 • No. 1 • January, 2004
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