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Register a new userOn $phi$-$w$-Flat modules and Their Homological Dimensions
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In this paper, we introduce and study the class of $phi$-$w$-flat modules which are generalizations of both $phi$-flat modules and $w$-flat modules. The $phi$-$w$-weak global dimension $phi$-$w$-w.gl.dim$(R)$ of a commutative ring $R$ is also introduced and studied. We show that, for a $phi$-ring $R$, $phi$-$w$-w.gl.dim$(R)=0$ if and only if $w$-$dim(R)=0$ if and only if $R$ is a $phi$-von Neumann ring. It is also proved that, for a strongly $phi$-ring $R$, $phi$-$w$-w.gl.dim$(R)leq 1$ if and only if each nonnil ideal of $R$ is $phi$-$w$-flat, if and only if $R$ is a $phi$-PvMR, if and only if $R$ is a PvMR.
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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.
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.
Let a be an ideal of a commutative Noetherian ring R with identity. We study finitely generated R-modules M whose a-finiteness and a-cohomological dimensions are equal. In particular, we examine relative analogues of quasi-Buchsbaum, Buchsbaum and surjective Buchsbaum modules. We reveal several interactions between these types of modules that extend some of the existing results in the classical theory to the relative one.
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.
It is proved that a module $M$ over a Noetherian local ring $R$ of prime characteristic and positive dimension has finite flat dimension if Tor$_i^R({}^e R, M)=0$ for dim $R$ consecutive positive values of $i$ and infinitely many $e$. Here ${}^e R$ denotes the ring $R$ viewed as an $R$-module via the $e$th iteration of the Frobenius endomorphism. In the case $R$ is Cohen-Macualay, it suffices that the Tor vanishing above holds for a single $egeq log_p e(R)$, where $e(R)$ is the multiplicity of the ring. This improves a result of D. Dailey, S. Iyengar, and the second author, as well as generalizing a theorem due to C. Miller from finitely generated modules to arbitrary modules. We also show that if $R$ is a complete intersection ring then the vanishing of Tor$_i^R({}^e R, M)$ for single positive values of $i$ and $e$ is sufficient to imply $M$ has finite flat dimension. This extends a result of L. Avramov and C. Miller.
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