Abstract
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 ).
Citation
Kohji YANAGAWA. "Derived category of squarefree modules and local cohomology with monomial ideal support." J. Math. Soc. Japan 56 (1) 289 - 308, January, 2004. https://doi.org/10.2969/jmsj/1191418707
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