Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userStable Under Specialization Sets and Cofiniteness
232
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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 $i in mathbb{Z}$.
rate research
Read More
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.
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.
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.
Let $R$ be a commutative ring. We show that any complete duality pair gives rise to a theory of relative homological algebra, analogous to Gorenstein homological algebra. Indeed Gorenstein homological algebra over a commutative Noetherian ring of finite Krull dimension can be recovered from the duality pair $(mathcal{F},mathcal{I})$ where $mathcal{F}$ is the class of flat $R$-modules and $mathcal{I}$ is the class of injective $R$-modules. For a general $R$, the AC-Gorenstein homological algebra of Bravo-Gillespie-Hovey is the one coming from the duality pair $(mathcal{L},mathcal{A})$ where $mathcal{L}$ is the class of level $R$-modules and $mathcal{A}$ is class of absolutely clean $R$-modules. Indeed we show here that the work of Bravo-Gillespie-Hovey can be extended to obtain similar abelian model structures on $R$-Mod from any a complete duality pair $(mathcal{L},mathcal{A})$. It applies in particular to the original duality pairs constructed by Holm-J{o} rgensen.
We define the notion of a power stable ideal in a polynomial ring $ R[X]$ over an integral domain $ R $. It is proved that a maximal ideal $chi$ $ M $ in $ R[X]$ is power stable if and only if $ P^t $ is $ P$- primary for all $ tgeq 1 $ for the prime ideal $ P = M cap R $. Using this we prove that for a Hilbert domain $R$ any radical ideal in $R[X]$ which is a finite intersection G-ideals is power stable. Further, we prove that if $ R $ is a Noetherian integral domain of dimension 1 then any radical ideal in $ R[X] $ is power stable. Finally, it is proved that if every ideal in $ R[X]$ is power stable then $ R $ is a field.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
