# Questions tagged [local-cohomology]

For questions related to local cohomology theory.

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### Do $I$-torsion and $I$-torsion-free modules form a torsion pair

Let $R$ be a commutative ring, $I \subset R$ is an ideal. The functor of $\Gamma_I(M) = \varinjlim (R/I^t, M)$ as an endofunctor of $\operatorname{Mod}(R)$ is a subfunctor of the identity functor, ...
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### If two $\delta$-functors coincide on finitely length modules do they coincide on finitely generated modules?

Let $(R,m)$ a Noetherian local ring of dimension $d$. Suppose $F^\bullet, G^\bullet \colon \operatorname{Mod} \to \operatorname{Mod}$ are two cohomological $\delta$-endofunctors on the category of $R$-...
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### A proof that two definitions of a dualizing (canonical) module are equivalent

Let $R$ be Noetherian local ring of dimension $d$ with maximal ideal $m$, and $k=R/m$. There are (at least) two definitions of a dualizing (canonical) module over $R$. In the first definition $D$ is ...
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### Example of module on dimension $d$ with trivial local cohomology $H^d_I(M)$.

Let $R$ be a Noetherian ring, $I$ is an ideal. Are there examples of an $R$-module $M$ such that local cohomology $H^d_I(M)=0$, where $d=\dim M$. It is well-known that there are no such examples over ...
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### Prove that for arbitrary polynomials $f,g,h \in k[x,y] (k$ a field),$(fgh)^2 \in (f^3,g^3,h^3)$.

Prove that for arbitrary polynomials $f,g,h \in k[x,y] (k$ a field),$(fgh)^2 \in (f^3,g^3,h^3)$(ideal of $k[x,y]$). This question is an exercise29 from Lectures on Local Cohomology, section 5, by ...
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### Vanishing first local cohomology group

I have been reading about local cohomology from Hartshorne's notes on the same and I have the following question. Let $X$ be a topological space, and let $Z\subset X$ be closed. Then we have a long ...
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### Non-finitely generated Local cohomology modules

For $i \in N_0$, the $i$-th right derived of $Γ_I$ is denoted by $H_I ^i$ and will be referred to as the $i$-th local cohomology functor with respect to $I$. It is clear that if $(R,\mathfrak m)$ is a ...
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### Does the torsion submodule and the $0$-th local Cohomology module coincide over local Cohen-Macaulay ring?

Let $M$ be a finitely generated module over a local Cohen-Macaulay ring $(R,\mathfrak m)$. If $x\in M$ is annihilated by a non-zero-divisor $r\in \mathfrak m$ , then is it true that $\mathfrak m^n x=0$...
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### Local etale cohomology for sheaves of abelian groups.

I have seen local cohomology (cohomology supported on a closed subspace) in different contexts, like for topological spaces and for quasi-coherent sheaves on a scheme. I was wondering whether the same ...
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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 ...
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### Proving a duality between Ext and Tor for maximal Cohen-Macaulay modules over Gorenstein ring

Let $(R,\mathfrak m, k)$ be a local complete Gorenstein ring of dimension $d$. Let $M,N$ are maximal Cohen-Macaulay modules (i.e. have depth equal to $d$) that are locally free on the punctured ...
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### On modules which elements are annihilated by a power of the maximal ideal and localization

Let $R$ be a Noetherian ring and let $M$ be an $R$-module in which every element is annihilated by a power of a maximal ideal $m$. Is there a natural way to define a structure of $R_m$-module on $M$? ...
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### Local cohomology modules and direct limits

Let $(R, \frak{m})$ be a Noetherian local ring and $M$ be an $R$-module such that $H^i_{\frak m} (M)=0$ for all $i>0$ ($H^i_{\frak{m}}(M)$ denotes the $i$-th local cohomology module of $M$ with ...
272 views

### Local Cohomology of a coherent sheaf can be calculated with restricting the sheaf to the support?

Let $(X,\mathcal O_X)$ be a Noetherian Scheme . Let $\mathcal F$ be a coherent sheaf of $\mathcal O_X$-module. Let $Z$ be a closed subscheme of $X$. Let $Y:=Supp \mathcal F$, which is a closed subset ...
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### On local cohomology and canonical module

I'm studying local cohomology and the canonical module of a local Cohen-Macaulay ring $R$ is very important due to local duality and its consequences (as non-vanishing of $d$-th local cohomology of an ...
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### Local cohomology as direct limit of Ext functors, for not necessarily affine schemes?

Let $(Z,\mathcal O_Z)$ be a closed subscheme of a Noetherian scheme $(X,\mathcal O_X)$. Then there is an ideal sheaf $\mathcal J$ on $X$ such that $i_*(\mathcal O_Z) \cong \mathcal O_X/\mathcal J$ , ...
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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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