For questions related to local cohomology theory.
Questions tagged [local-cohomology]
72 questions
13
votes
1 answer
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 theorem which shows the connection between UFDs and…
messi
- 1,317
10
votes
2 answers
Local cohomology with respect to a point. (Hartshorne III Ex 2.5)
I'm trying to do Hartshorne's exercises on local cohomology at the moment and seem to be stuck in Exercise III 2.5. The problem goes as follows:
$X$ is supposed to be a Zariski space (i.e a Noetherian and sober topological space) and $P\in X$ a…
Andreas
- 173
8
votes
2 answers
Prove that $\Gamma_I(\frac{M}{\Gamma_I(M)})=0$
I was trying to prove this theorem (problem):
Suppose that $R$ is a commutative ring with identity, $I\unlhd R$, and $M$ an $R$-module. We define: $$\Gamma_I(M)=\bigcup_{n\geq0}\operatorname{Ann}_M(I^n)$$
in which for each natural $n\geq 0$:
…
RSh
- 601
6
votes
1 answer
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 $\{f_1, \ldots, f_r \} \subset S$ be a set of homogeneous…
horus189
- 311
6
votes
1 answer
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 by Mayer-Vietoris sequence but I can't,…
Angel
- 812
6
votes
0 answers
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. Denote by $j:U \to \mathbb{P}^n$ the natural…
user46578
- 669
5
votes
1 answer
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 $K_\bullet(x_1,\dots,x_n)$ is defined to be $K_\bullet(x_1)…
Manos
- 26,949
5
votes
1 answer
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 from the structure of $\Gamma_m(E^\bullet(M))$…
Chris
- 1,539
4
votes
1 answer
Mayer-Vietoris sequence for local cohomology
Update 7:35pm UTC 3/23/14: I've reposted this quesion on MathOverflow here.
As an assignment in my commutative algebra class, I need to prove the Mayer-Vietoris sequence for local cohomology:
Let $R$ be a Noetherian ring, $I,J$ $R$-ideals, and $M$…
Avi Steiner
- 4,309
4
votes
0 answers
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 Craig Huneke. And then It said that the local…
Well
- 390
- 1
- 9
4
votes
1 answer
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 mentioned:
$S = k[x_1,\ldots,x_n]$, $I \subset S$ is a…
Tylwyth
- 435
4
votes
1 answer
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 spectrum (i.e. $M_P, N_P$ are free over $R_P$ for…
user521337
- 3,735
4
votes
0 answers
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 $\{x\in X| \mathcal F_x\ne0\}$. Let $Z$ be a closed subset of $X$,…
user102248
- 1,473
4
votes
0 answers
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 equal to cohomological dimension of $M$ with…
Sang Cheol Lee
- 3,259
4
votes
1 answer
Vanishing of a local cohomology module
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,y)}\frac{\Bbb Z[x,y]}{(5x+4y)})\subseteq…
Angel
- 812