Questions tagged [local-cohomology]

For questions related to local cohomology theory.

66 questions
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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 theorem which shows the connection between UFDs and…
10
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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 Noetherian and sober topological space) and $P\in X$ 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: $$\Gamma_I(M)=\bigcup_{n\geq0}\operatorname{Ann}_M(I^n)$$ in which for each natural $n\geq 0$: …
6
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1 answer

Castelnuovo-Mumford regularity over different rings

Let $S = k[x_1, \ldots, x_n, t]$ be the polynomial ring in $n+1$ variables over a field $k$ and let $R = k[x_1, \ldots, x_n]$. I have stumbled upon the following definition/result. Let $\{f_1, \ldots, f_r \} \subset S$ be a set of homogeneous…
6
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2 answers

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 about the numbers of holes in topological spaces.…
6
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1 answer

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 by Mayer-Vietoris sequence but I can't,…
Angel
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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. Denote by $j:U \to \mathbb{P}^n$ the natural…
5
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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 from the structure of $\Gamma_m(E^\bullet(M))$…
Chris
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4
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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 $K_\bullet(x_1,\dots,x_n)$ is defined to be $K_\bullet(x_1)…
Manos
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4
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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 $R$ be a Noetherian ring, $I,J$ $R$-ideals, and $M$…
4
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On the proof of a result of Bayer and Stillman

I'm reading through the paper A criterion for detecting m-regularity of Bayer and Stillmann and came across a proof, where I don't understand an implication. The following things may need to be mentioned: $S = k[x_1,\ldots,x_n]$, $I \subset S$ is a…
4
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Grothendieck type vanishing result for Local Cohomology over not necessarily affine schemes?

Let $(X,\mathcal O_X)$ be a Noetherian, affine Scheme and $\mathcal F$ be a quasi-coherent Sheaf of $\mathcal O_X$-modules on $X$. Let $\dim \mathcal F$ be the Krull dimension of $\{x\in X| \mathcal F_x\ne0\}$. Let $Z$ be a closed subset of $X$,…
4
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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 equal to cohomological dimension of $M$ with…
4
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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,y)}\frac{\Bbb Z[x,y]}{(5x+4y)})\subseteq…
3
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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 should not be true, but i don't have a…
user114539
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