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 \bm{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 Db(\bm{Sq}) has two duality functors \bm{D} and \bm{A}. The functor \bm{D} is a common one with Hi(\bm{D}(M·))=ExtSn+i (M·, omega s), while the Alexander duality functor \bm{A} is rather combinatorial. We have a strange relation \bm{D}\circ \bm{A}\circ \bm{D}\circ \bm{A}\circ \bm{D}\circ \bm{A}≅ \bm{T}2n, where \bm{T} is the translation functor. The functors \bm{A}\circ \bm{D} and \bm{D}\circ \bm{A} give a non-trivial autoequi-valence of Db(\bm{Sq}). This equivalence corresponds to the Koszul duality for Λ, which is a Koszul algebra with Λ1≅Λ. Our \bm{D} and \bm{A} are also related to the Bernstein-Gelfand-Gelfand correspondence.The local cohomology H_{IΛ}i(S) at a Stanley-Reisner ideal IΔ can be constructed from the squarefree module ExtSi(S/IΔ, omegaS). We see that Hochsters formula on the \bm{Z}n-graded Hilbert function of H_{\mathfrak{m}}i(S/IΔ) is also a formula on the characteristic cycle of H_{\mathit{1}Δ}n-i(S) as a module over the Weyl algebra A=k‹ x1, ..., xn, ∂1, ..., ∂n› (if char(k)=0).
社団法人日本数学会の論文
2004-01-01

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

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