The tag has no usage guidance.

learn more… | top users | synonyms

1
vote
1answer
129 views

local cohomology and radical of ideal

Let $R$ be commutative ring with identity, $M$ an $R$-module, and $I$ an ideal of $R$ . One defines $I$-torsion functor $Γ_I$ as: $\Gamma_I(M)=\bigcup_{n\in N} (0:_MI^n).$ When $R$ is Noetherian, ...
0
votes
0answers
113 views

Local-cohomology and Hom

Let $f:R\to S$ be a flat homomorphism of commutative Noetherian rings. "Flat Base Change Theorem", compares the local cohomology modules $H^i_a(M) \otimes_R S$ and $H^i_{aS} (M\otimes_R S)$ for $i ∈ ...
2
votes
1answer
146 views

Decide two indices of Ext functor

This question is from the proof of Theorem 11.34 in the book: Twenty-four Hours of Local Cohomology. Let $R$ and $S$ be CM local ring and $R\to S$ a local homomorphism such that $S$ is a finite ...
0
votes
0answers
59 views

Vanishing of top local cohomology when $R$ is domain

Let ${\rm R}$ be a Noetherian domain, $\frak a$ an ideal of ${\rm R}$ and $c:=\operatorname{cd}(\frak a, {\rm R})$ is finite. Is it true that $\operatorname{Ann}_R(H^c_{\frak a}({\rm R}))=0$. Note ...
0
votes
0answers
55 views

annihilators of top local cohomology modules

Let $R$ be a commutative Noetherian ring. Let $\frak a$ be an ideal of $R$ and ${\rm M}$ be a f.g $R$- module such that $c:=cd(\frak{a},{\rm M})$ is finite and $x\in R$. Is it true that $xH^c_{\frak ...
5
votes
0answers
71 views

How far finiteness dimension can be from edges? Example for $f_m(S/I)\ge depth S/I+2$

Let $ (R,m) $ be a commutative unital noetherian local ring (with $m$ as its maximal ideal), $ I $ an ideal of $ R $, and $ M $ a finite $R$-module with $\dim M\gt 0$. $f_I(M) = \inf\ \{i : H_I^i(M)\ ...
2
votes
1answer
199 views

finiteness dimension

$R$ is a local Noetherian ring. $f_I(M)$, the finiteness dimension of a module $M$ relative to $I$, is defined in ...
2
votes
2answers
289 views

local cohomology mayer-vietoris sequence

(I originally asked this question on Math.SE here. As suggested on meta.MathOverflow (posting an unanswered Math.SE question on MathOverflow), I've waited about a week before reposting it here. Note ...
1
vote
1answer
148 views

local cohomology of Buchsbaum ring

Let $(R,m)$ be a Buchsbaum ring of dimension d. Can we say that $d$-th local cohomology $H_{m}^d(R)$ has finite length?
2
votes
0answers
113 views

syzygy of a generalized cohen-macaulay module

Let $R$ be a local, noetherian ring of dimension $d$ and suppose it is generalized cohen-macaulay. Is it true that For any finitely generated $ R $-module $ M $, which is maximal generalized ...
1
vote
0answers
207 views

What is your expectation of the depth?

Let $S=k[x_1,...,x_9]$ be a polynomial ring over field $k$. Set $q_1=(x_1,x_2,x_5,x_6)$, $q_2=(x_1,x_2,x_6,x_7)$, $q_3=(x_2,x_3,x_7,x_8)$, $q_4=(x_1,x_5,x_6,x_7)$, $q_5=(x_1,x_6,x_7,x_8)$, ...
5
votes
1answer
406 views

example of Local cohomology

Let $S=k[x_1,...,x_n]$ be a polynomial ring over field $k$ with maximal ideal $m=(x_1,...,x_n)$. I wanna make a $3$-dimensional $S$-module $M$ such that $H^0_m(M)=H^1_m(M)=0$ and $H^2_m(M)\neq 0$ be ...
4
votes
1answer
236 views

finiteness of local cohomology

Well-known Theorem: Let $a$ be an ideal of the noetherian ring $R$ and let $M$ be a finitely generated $R$-module. Let $i \in \Bbb N_0$ be such that $H^j_a (M)$ is finitely generated for all $j < ...
-1
votes
1answer
220 views

$H_{I}^{n}(M)\cong H_{I}^{n}(R)\otimes_R M.$

Let $R$ be a Noetherian ring and $I$ an ideal of $R$. If $n$ is the cohomological dimension of $I$, then why is the following isomorphism true: $$H_{I}^{n}(M)\cong H_{I}^{n}(R)\otimes_R M.$$ The ...
1
vote
1answer
162 views

Is there a prime of height $i$ in support of $H^i_I(R)$?

$I$ is an ideal of a local Noetherian ring $R$ and $i>0$ . Clearly the height of primes in support of $H^i_I(R)$ is at least $i$ The question is if it contains a prime of height $i$, specially ...
0
votes
1answer
169 views

Relation between $H^i_I(-)$ and $H^i_J(-)$ when $I\subset J$

What is the relation between $H^i_I(-)$ and $H^i_J(-)$ (cohomological functors) when $I\subset J$ are ideals of a (local) noetherian ring?
3
votes
1answer
304 views

Comparison for formal local cohomology

Let $(R, \mathfrak{m})$ be a local ring and $X = Spec(R)$. Let $Y = V(I)$ be a closed subscheme of $X$, defined by an ideal $I \subset R$, and let $P \in X$ (in fact, $P \in Y$) be the closed point. ...
2
votes
0answers
220 views

On comparison of various linear topologies on a noetherian local ring

In what follows we will always use this notation: $R$ will be a commutative noetherian ring with unity, $X=\mathrm{Spec}\:R$, $f\colon X\rightarrow X$ a self-morphism of schemes, $\varphi\colon ...
3
votes
1answer
440 views

Local Cohomology and Maximal-Cohen-Macaulay modules

Checking a recent article [this one, specifically section 3.1] I found the following claim (I'm paraphrasing, of course): Let $A$ be a graded connected noetherian algebra (not necessarily ...