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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When Ideal transform functor is Artinian?

Let $R$ be a Noetherian ring and $I$ an ideal. For a finitely generated module $M$ over $R$ we define $$D_I(M)=\varinjlim_\limits{n\geq1}\operatorname{Hom}_R(I^n,M).$$ A variety of nice results about $...
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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,...
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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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Depth of tensor product of modules which are locally free on the punctured spectrum of regular local ring

Let $M,N$ be (non-zero) finitely generated modules over a regular local ring $(R, \mathfrak m)$ of dimension $d$ such that $M_P, N_P$ are free (non-zero) $R_P$-modules for every prime ideal $P\ne \...
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On the first Local Cohomology module of a complete local ring of depth $1$

Let $(R,\mathfrak m)$ be $\mathfrak m$-adically complete Noetherian local ring of depth $1$. Thus the local cohomology module $H^1_{\mathfrak m}(R)$ is a non-zero Artinian module. My question is: How ...
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Increasing the number of ideals in an exact sequence

In Broadmann and Sharp's book, Local Cohomology: An Algebraic Introduction with Geometric Applications, the exercise $3.2.4$ is about an exact sequence of the form $\DeclareMathOperator{\Hom}{Hom}$ $...
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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 ...
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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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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 ...
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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 $\...
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$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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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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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 ...
Arpit Kansal's user avatar
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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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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 ...
user 1's user avatar
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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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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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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. ...
user 1's user avatar
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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 ...
user 1's user avatar
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2 votes
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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 ...
user155489's user avatar
5 votes
1 answer
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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,\...
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