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Register a new userGeneralized local cohomology modules and homological Gorenstein dimensions
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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.
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Let (R,m) be a commutative Noetherian local ring. It is known that R is Cohen-Macaulay if there exists either a nonzero finitely generated R-module of finite injective dimension or a nonzero Cohen-Macaulay R-module of finite projective dimension. In this paper, we investigate the Gorenstein analogues of these facts.
The commutative and homological algebra of modules over posets is developed, as closely parallel as possible to the algebra of finitely generated modules over noetherian commutative rings, in the direction of finite presentations, primary decompositions, and resolutions. Interpreting this finiteness in the language of derived categories of subanalytically constructible sheaves proves two conjectures due to Kashiwara and Schapira concerning sheaves with microsupport in a given cone. The motivating case is persistent homology of arbitrary filtered topological spaces, especially the case of multiple real parameters. The algebraic theory yields computationally feasible, topologically interpretable data structures, in terms of birth and death of homology classes, for persistent homology indexed by arbitrary posets. The exposition focuses on the nature and ramifications of a suitable finiteness condition to replace the noetherian hypothesis. The tameness condition introduced for this purpose captures finiteness for variation in families of vector spaces indexed by posets in a way that is characterized equivalently by distinct topological, algebraic, combinatorial, and homological manifestations. Tameness serves both the theoretical and computational purposes: it guarantees finite primary decompositions, as well as various finite presentations and resolutions all related by a syzygy theorem, and the data structures thus produced are computable in addition to being interpretable. The tameness condition and its resulting theory are new even in the finitely generated discrete setting, where being tame is materially weaker than being noetherian.
We prove a duality theorem for graded algebras over a field that implies several known duality results : graded local dualit
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).
Let $A$ and $B$ be rings, $U$ a $(B, A)$-bimodule and $T=left(begin{smallmatrix} A & 0 U & B end{smallmatrix}right)$ be the triangular matrix ring. In this paper, we characterize the Gorenstein homological dimensions of modules over $T$, and discuss when a left $T$-module is strongly Gorenstein projective or strongly Gorenstein injective module.
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