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Register a new userA New Outlook on Cofiniteness
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Let $mathfrak{a}$ be an ideal of a commutative noetherian (not necessarily local) ring $R$. In the case $cd(mathfrak{a},R)leq 1$, we show that the subcategory of $mathfrak{a}$-cofinite $R$-modules is abelian. Using this and the technique of way-out functors, we show that if $cd(mathfrak{a},R)leq 1$, or $dim(R/mathfrak{a}) leq 1$, or $dim(R) leq 2$, then the local cohomology module $H^{i}_{mathfrak{a}}(X)$ is $mathfrak{a}$-cofinite for every $R$-complex $X$ with finitely generated homology modules and every $i in mathbb{Z}$. We further answer Question 1.3 in the three aforementioned cases, and reveal a correlation between Questions 1.1, 1.2, and 1.3.
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Let $R$ be a commutative noetherian ring, and $mathcal{Z}$ a stable under specialization subset of $Spec(R)$. We introduce a notion of $mathcal{Z}$-cofiniteness and study its main properties. In the case $dim(mathcal{Z})leq 1$, or $dim(R)leq 2$, or $R$ is semilocal with $cd(mathcal{Z},R) leq 1$, we show that the category of $mathcal{Z}$-cofinite $R$-modules is abelian. Also, in each of these cases, we prove that the local cohomology module $H^{i}_{mathcal{Z}}(X)$ is $mathcal{Z}$-cofinite for every homologically left-bounded $R$-complex $X$ whose homology modules are finitely generated and every $i in mathbb{Z}$.
Let $frak a$ be an ideal of a commutative noetherian ring $R$ with unity and $M$ an $R$-module supported at $V(fa)$. Let $n$ be the supermum of the integers $i$ for which $H^{fa}_i(M) eq 0$. We show that $M$ is $fa$-cofinite if and only if the $R$-module $Tor^R_i(R/fa,M)$ is finitely generated for every $0leq ileq n$. This provides a hands-on and computable finitely-many-steps criterion to examine $mathfrak{a}$-confiniteness. Our approach relies heavily on the theory of local homology which demonstrates the effectiveness and indispensability of this tool.
In this paper, we introduce a new graph whose vertices are the nonzero zero-divisors of commutative ring $R$ and for distincts elements $x$ and $y$ in the set $Z(R)^{star}$ of the nonzero zero-divisors of $R$, $x$ and $y$ are adjacent if and only if $xy=0$ or $x+yin Z(R)$. we present some properties and examples of this graph and we study his relation with the zero-divisor graph and with a subgraph of total graph of a commutative ring.
We continue our study of the new extension of zero-divisor graph. We give a complete characterization for the possible diameters of $widetilde{Gamma}(R)$ and $widetilde{Gamma}(R[x_1,dots,x_n])$, we investigate the relation between the zero-divisor graph, the subgraph of total graph on $Z(R)^{star}$ and $widetilde{Gamma}(R)$ and we present some other properties of $widetilde{Gamma}(R)$.
Let (R,m,k) be a local ring. We establish a totally reflexive analogue of the New Intersection Theorem, provided for every totally reflexive R-module M, there is a big Cohen-Macaulay R-module B_M such that the socle of B_Motimes_RM is zero. When R is a quasi-specialization of a G-regular local ring or when M has complete intersection dimension zero, we show the existence of such a big Cohen-Macaulay R-module. It is conjectured that if R admits a non-zero Cohen-Macaulay module of finite Gorenstein dimension, then it is Cohen-Macaulay. We prove this conjecture if either R is a quasi-specialization of a G-regular local ring or a quasi-Buchsbaum local ring.
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