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Register a new userA duality theorem for generalized local cohomology
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We prove a duality theorem for graded algebras over a field that implies several known duality results : graded local dualit
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Let fa be an ideal of a commutative Noetherian ring R and M and N two finitely generated R-modules. Let cd_{fa}(M,N) denote the supremum of the is such that H^i_{fa}(M,N) eq 0. First, by using the theory of Gorenstein homological dimensions, we obtain several upper bounds for cd_{fa}(M,N). Next, over a Cohen-Macaulay local ring (R,fm), we show that cd_{fm}(M,N)=dim R-grade(Ann_RN,M), provided that either projective dimension of M or injective dimension of N is finite. Finally, over such rings, we establish an analogue of the Hartshorne-Lichtenbaum Vanishing Theorem in the context of generalized local cohomology modules.
We introduce a new definition of a generalized logarithmic module of multiarrangements by uniting those of the logarithmic derivation and the differential modules. This module is realized as a logarithmic derivation module of an arrangement of hyperplanes with a multiplicity consisting of both positive and negative integers. We consider several properties of this module including Saitos criterion and reflexivity. As applications, we prove a shift isomorphism and duality of some Coxeter multiarrangements by using the primitive derivation.
We give sufficient conditions for F-injectivity to deform. We show these conditions are met in two common geometrically interesting setting, namely when the special fiber has isolated CM-locus or is F-split.
Let $R$ be a commutative Noetherian ring that is a smooth $mathbb Z$-algebra. For each ideal $I$ of $R$ and integer $k$, we prove that the local cohomology module $H^k_I(R)$ has finitely many associated prime ideals. This settles a crucial outstanding case of a conjecture of Lyubeznik asserting this finiteness for local cohomology modules of all regular rings.
Let fa be an ideal of a commutative Noetherian ring R and M a finitely generated R-module. We explore the behavior of the two notions f_{fa}(M), the finiteness dimension of M with respect to fa, and, its dual notion q_{fa}(M), the Artinianess dimension of M with respect to fa. When (R,fm) is local and r:=f_{fa}(M) is less than f_{fa}^{fm}(M), the fm-finiteness dimension of M relative to fa, we prove that H^r_{fa}(M) is not Artinian, and so the filter depth of fa on M doesnt exceeds f_{fa}(M). Also, we show that if M has finite dimension and H^i_{fa}(M) is Artinian for all i>t, where t is a given positive integer, then H^t_{fa}(M)/fa H^t_{fa}(M) is Artinian. It immediately implies that if q:=q_{fa}(M)>0, then H^q_{fa}(M) is not finitely generated, and so f_{fa}(M)leq q_{fa}(M).
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