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Register a new userModules whose finiteness dimensions coincide with their cohomological dimensions
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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.
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Let R be a commutative noetherian ring. In this paper, we study specialization-closed subsets of Spec R. More precisely, we first characterize the specialization-closed subsets in terms of various closure properties of subcategories of modules. Then, for each nonnegative integer n we introduce the notion of n-wide subcategories of R-modules to consider the question asking when a given specialization-closed subset has cohomological dimension at most n.
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.
Let $mathfrak{a}$ be an ideal of a noetherian (not necessarily local) ring $R$ and $M$ an $R$-module with $mathrm{Supp}_RMsubseteqmathrm{V}(mathfrak{a})$. We show that if $mathrm{dim}_RMleq2$, then $M$ is $mathfrak{a}$-cofinite if and only if $mathrm{Ext}^i_R(R/mathfrak{a},M)$ are finitely generated for all $ileq 2$, which generalizes one of the main results in [Algebr. Represent. Theory 18 (2015) 369--379]. Some new results concerning cofiniteness of local cohomology modules $mathrm{H}^i_mathfrak{a}(M)$ for any finitely generated $R$-module $M$ are obtained.
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.
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