Questions tagged [local-cohomology]

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On the natural isomorphism between $I$-torsion functor and direct limit of $\mathrm{Hom}$ functor

Let $R$ be a commutative ring with unity with and let $I$ be a proper ideal. (I'm not assuming $R$ is Noetherian.) For every $M \in R$-Mod, let $\Gamma_I(M):=\{m \in M : I^n m=0$ for some $n\ge 1\}$....
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$H^1_I(R) \cong S/R$ as $R$-modules.

Let $R$ be a domain with quotient field $K$, let $I\neq 0$ be an $R$-ideal, and write $I^{-i}=R\underset{K}{:} I^i$. Then $S=\bigcup\limits_{i\geq 0} I^{-i}$ is a subring of $K$ containing $R$. Show $$...
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Show that $H_I^1(R)=0$ if and only if $I$ is not contained in an associated prime of any principal ideal.

Let $R$ be a Noetherian domain and $I\neq 0$ an ideal. Show that $H_I^1(R)=0$ if and only if $I$ is not contained in an associated prime of any principal ideal. Proof: $\Rightarrow$ Suppose that $H^...
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Local cohomology and primary decomposition

Let $R$ be a Noetherian ring, $I$ an ideal and $M$ a finite $R$-module. Let $0=N_1\cap \dots \cap N_s$ in $M$ be a primary decomposition with $N_i$ $p_i$-primary submodules of $M$. Show that $\...
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1answer
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$I$-torsion functor and $\sqrt{I}$

Let $R$ be a Noetherian ring and $I,J\subseteq R$ ideals. For any $R$-module $M$, define $\Gamma_I(M)=\{x\in M\mid I^n x=0\text{ for some }n\in\mathbb{N}\}$. Suppose that $\Gamma_I=\Gamma_J$. I want ...
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Motivation for local cohomology and local homotopy theories in Algebraic topology.

In general topology, I know about the local topological properties. In algebraic topology homotopy and cohomology theories is also easily understandable. For examples Betti numbers gives information ...
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Castelnuovo-Mumford regularity of a polynomial ring

I am reading from Brodmann, Sharp Local Cohomology about Castelnuovo-Mumford (CM)-regularity, and although I think had understood some things about it I got stuck in an exercise which wants to prove ...
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239 views

Saturation of a module and local cohomology

Let $R=k[x_1,\dots,x_n]$ be a polynomial ring over some field $k$, let $M$ be an $R$-module and $M^{\text{sat}}$ the saturation of $M$. Denote by $m$ the maximal ideal of $R$. Is it true that: ...
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Cohen-Macaulayness and short exact sequence

Suppose $R$ is a polynomial ring of dimension $n$ and let $M$ be a finitely generated module of projective dimension $m-1$. Suppose we have a resolution $$ 0\rightarrow F_m\rightarrow F_{m-1}\...
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Twisted Cohomology of BO

I'm interested in the cohomology of the classifying space $BO(n)$ of real vector bundles of fiber dimension $n$ with coefficients in the integers, twisted by the $O(1)$ bundle defined by the class $...
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Nonvanishing of local cohomology and associated primes

First, here is my primary question: Is there an example of a connected graded Noetherian $k$-algebra $R$ such that there is a $\mathfrak{p} \in \mathrm{Ass}_R R$ with $\mathrm{dim}\; R/\mathfrak{p}...
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A version of Peskine and Szpiro's theorem in vanishing of local cohomology.

C. Peskine and L. Szpiro in "Dimension projective finie et cohomologie locale",(Proposition 4.1) proved the following vanishing theorem for local cohomology: Let$R$ be a regular local ring of ...
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Trying to understand Corollary $4.7 $ (page $60$) from Eisenbud's Geometry of Syzygies

Corollary: If $X$ is a set of $n$ points in $\mathbb P^r$, then the regularity of $S_X$ is the smallest integer $d$ such that the space of forms vanishing on the points $X$ has codimension $n$ in the ...
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a little “paradox” in local cohomology of zero-dimensional ideals

Let $S = k[x_1,x_2,x_3]$ be a polynomial ring of dimension $3$ over an infinite field, and let $I$ be a homogeneous ideal of height $3$. Since $S$ has no zero divisors, the Krull dimension of $I$ is $...
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Need some suggestion for an introductory talk on 'Local Cohomology'?

Next week i am to give a talk on 'Local Cohomology' and i am writing to request suggestions for some basic interesting results for the talk.The relevant information is as follows: (1) The audience ...
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Show that the $i$th local cohomology functor is zero for $i > 0$

Let $I$ be an ideal of a Noetherian ring $R$, and let $M$ be a module over $R$. Let $\Gamma_I(M)$ be the set of all elements $m$ of $M$ for which $I^n m = 0$ for some $n \geq 1$. Then $\Gamma_I(-)$ ...
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659 views

How to show that a locally exact form is globally closed

Every closed differential form is locally exact. That's quite obvious, as it follows (Poincare lemma) from the vanishing $k$-th cohomology groups ($k>0$) of contractible open subsets of $\mathbb R^...
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A basic question on local cohomology

Let $X$ be a smooth, projective variety, $i:X \hookrightarrow \mathbb{P}^n$ a closed immersion for some $n>0$, $U \subset X$ an open subset and $Z \subset X$ a local complete intersection subscheme....
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Cohomological dimension of an arbitrary module.

In the paper, [P, Schenzel, On formal local cohomology and connectedness, J of Alg, 315 (2007), 894--923], he proves the following statement. (Corollary 2.2) Let $M$ be a finitely generated $R$-...
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The last nonzero local cohomology module is not finitely generated. [closed]

Let $R$ be a Noetherian ring and $I$ an ideal of $R$. If $M$ is a finitely generated $R$-module and $i\neq 0$ is the greatest integer such that $H^i_I(M)$ is nonzero, then $H^i_I(M)$ is not a finitely ...
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cohomology ring of cross-section space of fibre-bundles

Given an $m$-dimensional manifold $M$, let $TM$ be the tangent bundle of $M$ and $SM$ be the $m$-sphere bundle over $M$ obtained by fibre-wise one point compactification of $TM$. Let $\Gamma(SM)$ be ...
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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, it'...
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$\Gamma_I(E)$ is an injective $R$-module? $H^i_I(E)=0;\forall i\gt 0$

1.Let $R$ be a commutative ring, $M$ an $R$-module, $I$ an ideal in $R$, and $E$ an injective $R$-module. Can one claim that $H^i_I(E)=0;\forall i\gt 0$? 2.In the case of noetherian rings we know ...
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definitions of $I$-torsion functor $\Gamma_I$

Let $R$ be a commutative ring, $M$ an $R$-module and $I$ be an ideal in $R$. Bruns-Herzog, Brodmann-Sharp and many other authors define $I$-torsion functor $\Gamma_I$ as: $$\Gamma_I(M)=\bigcup_{n\in ...
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1answer
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Local cohomology killed by a power of I

Notations:: $H^i_I(M)$ is $i^{th}$ local cohomology of $M$ with support in $I$ and $H^i_I(M)=R^i\Gamma_I(M)$ where $R^i\Gamma_I(M)$ is the right derived functor of a covariant left exact functor, ...
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1answer
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Is the converse of Proposition 3.5.4 (c) of Bruns_Herzog true?

Question 1. Is the converse of Proposition $3.5.4 (c)$ of Bruns_Herzog true? I can see that $R$ is cohen-macaulay. so if one can prove that $r(R)=1$ , $R$ will be Gorenstein. Question 2. ...
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$H^i_I(M)$ is finitely generated iff the support of $Ext^{d-i}_S(M, S)$ has dimension zero

$(R,m)$ is a local Noetherian ring. $M$ is a finite $R$-module. Here, using dualizing complex, Karl Schwede says that if $R=S/I$ where $S$ is regular of dimension $d$, then we have: "$H^i_m(M)$ is ...
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376 views

Direct-Sum Decomposition of an Artinian module

Let $R$ be a commutative Noetherian ring. Suppose $M$ is a finitely-generated non-zero Artinian $R$-module. Question: How can we prove that there are maximal ideals $m_1 , m_2 , \ldots , m_n$ such ...
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581 views

Koszul Homology vs Koszul Cohomology

Let $R$ be a ring and $x \in R$. The Koszul complex $K_\bullet(x)$ is then $0 \rightarrow R \stackrel{x}{\rightarrow} R \rightarrow 0$. Given $x_1,\dots,x_n \in R$ the Koszul complex $K_\bullet(x_1,\...
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1answer
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Does the $I$-torsion functor commute with inverse limit?

Let $I$ be an ideal of a commutative ring with unit. Is $\Gamma_I(\varprojlim M_j)\cong \varprojlim(\Gamma_I M_j)$? Any reference of the proof or a counterexample is appreciated. It seems this ...
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1answer
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Vanishing of local cohomology and primary decomposition

Let $R$ be an $n$-dimensional Noetherian ring with proper ideal $I$. If $I = \mathfrak{a} \cap \mathfrak{b}$ and $H^n _\mathfrak{a}(M) = H^n _\mathfrak{b}(M) = 0$, for some $R$-module $M$, show $H^n_I(...
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1answer
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$\Gamma_{\mathfrak a}(I)$ is an injective $R$-module for every injective $R$-module $I$

Is there a proof for Proposition 2.1.4 of Local Cohomology book by Brodmann-Sharp not using Artin–Rees Lemma? Proposition 2.1.4: Let $I$ be an injective $R$-module. Then $\Gamma_{\mathfrak a}(I)$ ...
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$\operatorname{supp}(M) \subseteq \operatorname{supp}(N) \iff f_I(M)\subseteq f_I(N) $?

Let $ R $ be a commutative unital ring, $ I $ an ideal of $ R $, and $ M $ an $ R $-module. It has proven (here) that if $\operatorname{supp}(M) \subseteq \operatorname{supp}(N)$ then $\operatorname{...
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The Relationship Between Cohomological Dimension and Support

Let $ R $ be a commutative unital ring, $ I $ an ideal of $ R $, and $ M $ an $ R $-module. The cohomological dimension of $ M $ with respect to $ I $ is defined as $$ \operatorname{cd}(I,M) \stackrel{...
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Vanishing of local cohomology groups

Let $k$ be a field and let $X$ be a smooth separated $k$-variety. Let $T$ be a closed integral subscheme of $X$ of generic point $\eta$. The object of interest here is the local cohomology group $$ H^{...
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proving a property of Castelnuovo-Mumford regularity

Let $k$ be a field, $S = k[x_1,\dots,x_n]$ the polynomial ring, $m = (x_1,\dots,x_n)$ and $I$ a homogeneous ideal contained in $m^2$. Define $R = S/I$. For $p \in \mathbb{N}$ we say that $R$ is $p$-...
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Let $I= (X_1,X_2) \cap (X_3,X_4)$. Is $ara(I)≥3$? Is $ara(I)≥4$?

This question is related to Can $(X_1,X_2) \cap (X_3,X_4)$ be generated with two elements from $k[X_1,X_2,X_3,X_4]$? Let $R=k[X_1,X_2,X_3,X_4]$ and $I= (X_1,X_2) \cap (X_3,X_4)$. I know that $4≥\...
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1answer
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Proposition 3.5.1 in Bruns and Herzog, Cohen-Macaulay Rings

Bruns and Herzog in their book Cohen-Macaulay Rings, page 128 consider a local Noetherian ring $(R,m,k)$, an $R$-module $M$ and they define the functor $\Gamma_m(\cdot)$ as $\Gamma_m(M) = \varinjlim \...
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Mayer-Vietoris sequence for local cohomology

Update 7:35pm UTC 3/23/14: I've reposted this quesion on MathOverflow here. As an assignment in my commutative algebra class, I need to prove the Mayer-Vietoris sequence for local cohomology: Let $...
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Local cohomology with respect to a point. (Hartshorne III Ex 2.5)

I'm trying to do Hartshorne's exercises on local cohomology at the moment and seem to be stuck in Exercise III 2.5. The problem goes as follows: $X$ is supposed to be a Zariski space (i.e a ...
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Prove that $\Gamma_I(\frac{M}{\Gamma_I(M)})=0$

I was trying to prove this theorem (problem): Suppose that $R$ is a commutative ring with identity, $I\unlhd R$, and $M$ an $R$-module. We define: $$\Gamma_I(M)=\bigcup_{n\geq0}\operatorname{Ann}_M(...
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A corollary of Grothendieck’s Finiteness Theorem

Well-known Theorem: Grothendieck’s Finiteness Theorem. Assume that $R$ is a homomorphic image of a regular (commutative Noetherian) ring. Let $\mathfrak a$ be an ideal of $R$, and let $M$ be a ...
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All local cohomology modules being zero

Let $R$ be a Noetherian ring with unit, $I$ be an ideal of $R$ and let $M$ be a finitely generated $R$-module. Suppose $H_{I}^j(M)=0$ for all $j$, then how can one show that $M=IM$? The converse of ...
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How does Local Cohomology detect UFD?

I read that Grothendieck developed Local Cohomology to answer a question of Pierre Samuel about when certain type of rings are UFDs. I know the basics of local cohomology but I have not seen a ...
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1answer
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The local cohomology modules are Artinian

Let $(R,m,k)$ be Noetherian local ring and $M$ a finitely generated $R$-module. Lemma 3.5.4 of Bruns-Herzog states that the local cohomology modules $H^i_m(M)$ are Artinian and that this follows ...
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1answer
222 views

Vanishing of local cohomology $\operatorname{H}^1_J(\Gamma_I(M))=0$

Let $M$ be a module over Noetherian ring $R$ such that $\operatorname{H}^1_I(M)=0$ for every ideal $I$ of $R$. Show that $\operatorname{H}^1_J(\Gamma_I(M))=0$ for every ideal $J$. I tried to prove it ...
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296 views

Cohomological dimension, dimension of modules and arithmetic rank

Let $R$ be a noetherian ring, $I$ an ideal of $R$ and $M$ a finitely generated $R$- module. I know two facts: first, dimension of $M$ (i.e. Krull dimension of $R/{\rm ann}(M)$) is greater than or ...
4
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1answer
363 views

Vanishing of a local cohomology module

I guess $$H^2_{(x,y)}\left(\frac{\Bbb Z[x,y]}{(5x+4y)}\right)=0$$ It is well known $\operatorname{Supp} H^i_I(M)‎\subseteq V(I)\cap \operatorname{Supp}(M)$, therefore $$\operatorname{Supp} H^2_{(x,...
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1answer
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Example of not right exactness of local cohomology functor

Let $M$ be a module over a commutative ring $R$, $\mathfrak a$ is an ideal of $R$. Define $\Gamma_\mathfrak a(M)=\lbrace m\in M\mid\mathfrak a^tm=0 \text{ for some } t\in \mathbb{N}\rbrace$. Then $\...