Questions tagged [local-cohomology]

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49 questions
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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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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 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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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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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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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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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 ...
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,...
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 \$\...