Questions tagged [local-cohomology]

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Cohomological dimension and height of ideals

Let $I$ be an ideal in a Noetherian ring $R$. We define the cohomological dimension of $I$ to be $\operatorname{cd}(I)=\operatorname{sup}\{i\in \mathbb N:\operatorname{H}_I^i(R)\neq0\}$ and it is ...
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Does a ring homomorphism induce a morphism in local cohomology?

Let $\rho:R\longrightarrow S$ be a homomorphism of Noetherian rings and, for the ideals $I\subset R$ and $J\subset S$, we have $\rho(I)\subseteq J$. Does this induce a morphism in local cohomology ...
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Mayer-Vietoris sequence from a bicartesian square of commutative rings

An article that I am reading quotes the following theorem (5.3 p.481, reformulated to focus on the commutative case) from Algebraic K-Theory by Hyman Bass: Let $\require{AMScd}$ \begin{CD} A @>p_2&...
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When is the following a formula for local cohomology?

Suppose $R$ is a Noetherian local ring, and $\kappa$ its residue field. For $R$ module $M$, we can consider the module $$N:=\kappa \otimes_S RHom(\kappa,M)$$ where $S$ is the derived ring of ...
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Künneth formula for local cohomology with support

In "Differential operators on the flag varieties" (http://www.numdam.org/article/AST_1981__87-88__43_0.pdf) by Brylinski, he uses on page 53 a Künneth formula for local cohomology with ...
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For a Cohen-Macaulay module $M$ of dimension $t$ over a local CM ring of dimension $n$, is $\text{Ext}^{n-t}_R(M,\omega)$ Cohen-Macaulay?

Let $(R,\mathfrak m)$ be a local Cohen-Macaulay ring of dimension $n$ with a canonical module $\omega$. Let $M$ be a finitely generated $R$-module with $\text{depth } M=\dim M=t$. Using Bruns&...
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Trivial morphism between local cohomology groups

I have two questions concerning morphism between local cohomology groups which I think are related. Let $G$ be a reductive group with Weyl group $W$ and $B \subset G$ a Borel. Let $X=G/B$ be the flag ...
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Sequence in local cohomology for multiple closed subsets

Let $X$ be topological space with closed subsets $A,B,C \subset X$ and $\mathcal{F} \in Sh(X)$. I'm trying to understand \begin{equation*} H^i_{A\cap B}(X,\mathcal{F}) \oplus H^i_{A\cap C}(X,\mathcal{...
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Comparing long exact sequences for local cohomology

Let $X$ be a topological space, $Z_1,Z_2 \subset X$ closed subsets and $\mathcal{F} \in Sh(X)$. Then we have, for example by Hartshorne Excercise III 2.4, the Mayer Vietoris sequence for local ...
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Does local cohomology commute with pullback?

Let $Y$ be a topological spaces and $Z \subset Y$ be locally closed, i.e. $Z=V \cap U^c$ for $U,V \subset Y$ open. For any abelian sheaf $\mathcal{F}$ on $Y$ let $\Gamma_Z(Y,\mathcal{F}):=\ker(\...
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Asymptotic stability of associated primes of graded local cohomology modules

This question concerns the asymptotic behaviour of associated primes of graded components of local cohomology modules. A survey of this can be found in M. Brodmann, Asymptotic behaviour of cohomology: ...
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Smallest non-vanishing index of Local-Cohomology and Ext functor, as in the definition of depth, 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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Difference between local cohomology and cohomology with support in a family

Let $X$ be a topological space. A collection of closed subsets of $X$ is called a family of supports (in the sense of Cartan) if: (1) the union of any two elements of $\Phi$ is an element of $\Phi$, ...
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Local cohomology with supports in a constructible set

Let $X$ be a topological space (I'm interested in the case of $X$ being a complex algebraic variety with the Zariski topology) and $Z$ a constructible subset (i.e. a finite union of locally closed ...
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Independence of embedding for higher sheaf cohomology of local cohomology on projective space

Suppose $Y$ is a projective variety over a field $k$. Fix an embedding $\iota: Y \hookrightarrow \mathbb{P}^n_k$ for some $n$, and consider the local cohomology sheaves $\mathcal{H}^j_Y(\omega_{\...
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Relation between local cohomology and open immersions

Let $X$ be a noetherian scheme $U \subset X$ an open subset with complement $Z = X- U$. Assume $Z$ is cut out by the ideal sheaf $\mathcal{I} \subset \mathcal{O}_X$. We have exact sequences: $$0 \to \...
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Does local cohomology commute with taking the degree-zero component?

Let $S = \oplus_{d \geq 0} S_d$ be a graded (Noetherian) ring, let $I \subset S$ be a homogeneous ideal, and let $f \in S$ be a homogeneous element. Denote by $S_{(f)}$ the subring of degree-$0$ ...
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invariants that can be measured by Local Cohomology

What invariants can be measured by Local Cohomology (and what application it has)? As an example of what I mean: Local Cohomology can measure invariants like depth and dim. So in some cases Local ...
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Local cohomology commuting with fiber

Let $A$ be a nice commutative ring (say, $A=k[t_1,\ldots , t_n]$, ring of polynomials over an algebraically closed field $k$). Let $M$ be an $A[x]$-module, which is finitely generated as an $A$-...
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2 votes
1 answer
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On conflicting descriptions for tor of a local cohomology group

Let $X$ be a smooth projective surface and $C$ a Cartier divisor on $X$. Denote by $\mathcal{H}^1_C(\mathcal{O}_X)$ the sheaf associated to the presheaf $U \mapsto H^1_{C \cap U}(\mathcal{O}_X|_U)$. ...
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A basic question on local cohomology

I had posted this question on stackexchange but did not get any response, hence putting it up on mathoverflow. Let $X$ be a smooth, projective variety, $i:X \hookrightarrow \mathbb{P}^n$ a closed ...
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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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Local-cohomology and Hom

Let $f:R\to S$ be a flat homomorphism of commutative Noetherian rings. "Flat Base Change Theorem", compares the local cohomology modules $H^i_a(M) \otimes_R S$ and $H^i_{aS} (M\otimes_R S)$ for $i ∈ ...
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2 votes
1 answer
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Decide two indices of Ext functor

This question is from the proof of Theorem 11.34 in the book: Twenty-four Hours of Local Cohomology. Let $R$ and $S$ be CM local ring and $R\to S$ a local homomorphism such that $S$ is a finite ...
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How far finiteness dimension can be from edges? Example for $f_m(S/I)\ge depth S/I+2$

Let $ (R,m) $ be a commutative unital noetherian local ring (with $m$ as its maximal ideal), $ I $ an ideal of $ R $, and $ M $ a finite $R$-module with $\dim M\gt 0$. $f_I(M) = \inf\ \{i : H_I^i(M)\ ...
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2 votes
1 answer
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finiteness dimension

$R$ is a local Noetherian ring. $f_I(M)$, the finiteness dimension of a module $M$ relative to $I$, is defined in ...
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3 votes
2 answers
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local cohomology mayer-vietoris sequence

(I originally asked this question on Math.SE here. As suggested on meta.MathOverflow (posting an unanswered Math.SE question on MathOverflow), I've waited about a week before reposting it here. Note ...
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1 vote
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local cohomology of Buchsbaum ring

Let $(R,m)$ be a Buchsbaum ring of dimension d. Can we say that $d$-th local cohomology $H_{m}^d(R)$ has finite length?
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syzygy of a generalized cohen-macaulay module

Let $R$ be a local, noetherian ring of dimension $d$ and suppose it is generalized cohen-macaulay. Is it true that For any finitely generated $ R $-module $ M $, which is maximal generalized cohen-...
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What is your expectation of the depth?

Let $S=k[x_1,...,x_9]$ be a polynomial ring over field $k$. Set $q_1=(x_1,x_2,x_5,x_6)$, $q_2=(x_1,x_2,x_6,x_7)$, $q_3=(x_2,x_3,x_7,x_8)$, $q_4=(x_1,x_5,x_6,x_7)$, $q_5=(x_1,x_6,x_7,x_8)$, $q_6=(x_2,...
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8 votes
1 answer
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example of Local cohomology

Let $S=k[x_1,...,x_n]$ be a polynomial ring over field $k$ with maximal ideal $m=(x_1,...,x_n)$. I wanna make a $3$-dimensional $S$-module $M$ such that $H^0_m(M)=H^1_m(M)=0$ and $H^2_m(M)\neq 0$ be ...
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4 votes
1 answer
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finiteness of local cohomology

Well-known Theorem: Let $a$ be an ideal of the noetherian ring $R$ and let $M$ be a finitely generated $R$-module. Let $i \in \Bbb N_0$ be such that $H^j_a (M)$ is finitely generated for all $j < ...
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  • 225
0 votes
1 answer
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$H_{I}^{n}(M)\cong H_{I}^{n}(R)\otimes_R M.$

Let $R$ be a Noetherian ring and $I$ an ideal of $R$. If $n$ is the cohomological dimension of $I$, then why is the following isomorphism true: $$H_{I}^{n}(M)\cong H_{I}^{n}(R)\otimes_R M.$$ The ...
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1 answer
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Is there a prime of height $i$ in support of $H^i_I(R)$?

$I$ is an ideal of a local Noetherian ring $R$ and $i>0$ . Clearly the height of primes in support of $H^i_I(R)$ is at least $i$ The question is if it contains a prime of height $i$, specially ...
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1 answer
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Relation between $H^i_I(-)$ and $H^i_J(-)$ when $I\subset J$

What is the relation between $H^i_I(-)$ and $H^i_J(-)$ (cohomological functors) when $I\subset J$ are ideals of a (local) noetherian ring?
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4 votes
1 answer
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Comparison for formal local cohomology

Let $(R, \mathfrak{m})$ be a local ring and $X = Spec(R)$. Let $Y = V(I)$ be a closed subscheme of $X$, defined by an ideal $I \subset R$, and let $P \in X$ (in fact, $P \in Y$) be the closed point. ...
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3 votes
0 answers
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On comparison of various linear topologies on a noetherian local ring

In what follows we will always use this notation: $R$ will be a commutative noetherian ring with unity, $X=\mathrm{Spec}\:R$, $f\colon X\rightarrow X$ a self-morphism of schemes, $\varphi\colon R\...
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3 votes
1 answer
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Local Cohomology and Maximal-Cohen-Macaulay modules

Checking a recent article [this one, specifically section 3.1] I found the following claim (I'm paraphrasing, of course): Let $A$ be a graded connected noetherian algebra (not necessarily ...
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