Journal of the Mathematical Society of Japan
- J. Math. Soc. Japan
- Volume 56, Number 1 (2004), 289-308.
Derived category of squarefree modules and local cohomology with monomial ideal support
A squarefree module over a polynomial ring 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 of squarefree modules is equivalent to the category of finitely generated left -modules, where is the incidence algebra of the Boolean lattice . The derived category has two duality functors and . The functor is a common one with , while the Alexander duality functor is rather combinatorial. We have a strange relation , where is the translation functor. The functors and give a non-trivial autoequivalence of . This equivalence corresponds to the Koszul duality for , which is a Koszul algebra with . Our and are also related to the Bernstein-Gel'fand-Gel'fand correspondence.
The local cohomology at a Stanley-Reisner ideal can be constructed from the squarefree module . We see that Hochster's formula on the -graded Hilbert function of is also a formula on the characteristic cycle of as a module over the Weyl algebra (if ).
J. Math. Soc. Japan, Volume 56, Number 1 (2004), 289-308.
First available in Project Euclid: 3 October 2007
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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
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