Questions tagged [local-cohomology]

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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)\ ...
user 1's user avatar
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Local cohomology with coefficients in ideals of parameters

I'm not an expert in local cohomology, but the following problems have come up in my work, and I'd like to get a sense of where things stand. Let $\mathbb{A}^n=\operatorname{Spec} \mathbb{C}[x_1, \...
Marc Besson's user avatar
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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 ...
davik's user avatar
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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 ...
KKD's user avatar
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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 ...
KKD's user avatar
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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: ...
Fred Rohrer's user avatar
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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 $i_*(\mathcal O_Z) \cong \mathcal O_X/\mathcal J$ , ...
sdey's user avatar
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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-...
user46708's user avatar
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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\...
Mahdi Majidi-Zolbanin's user avatar
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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&...
Mickaël Montessinos's user avatar
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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 ...
Avi Steiner's user avatar
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Reference for a clear version of multigraded Serre-Grothendieck-Deligne correspondence local cohomology

The Grothendieck-Serre-Deligne correspondence states the following. Let $ R $ be a Noetherian, graded ring and let $ T $ be $ \operatorname{Proj}(R) $. If $ \mathcal{F} $ is a coherent sheaf on $ T $...
Schemer1's user avatar
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Reference for application of local cohomology to complex manifolds with points removed

Reference request - I am looking at Dolbeault cohomology on compact complex manifolds (not Riemann surfaces) with points removed. I have been told that the key to doing this is to look at Local ...
Edwin Beggs's user avatar
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Local cohomology: Polynomial ring vs Power series ring

I study algebraic topology and am currently examining the applications of local (co)homology in algebraic topology. We have the canonical inclusion of rings $\mathbb{Z}[x_1,\cdots,x_n]\subset \mathbb{...
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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 ...
user782932's user avatar
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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 ...
KKD's user avatar
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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{...
KKD's user avatar
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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 $A$-...
Sasha's user avatar
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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,...
ann's user avatar
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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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