Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

$I$ 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 when $R$ is complete and unmixed?

PS What about the case $i$ equals the cohomological dimension of $I$, ie

$$i=\sup\lbrace j: H^j_I(R)\neq 0\rbrace $$

share|improve this question
    
If $R$ is Cohen-Macaulay and say $I$ is a maximal ideal, then most of these are zero except for $H^{\dim R}_I(R)$, which has support equal to $\text{Spec }R$. In particular, $H^{\dim R}_I(R)$ has primes of height zero, height one, etc. So I don't see why primes in the support of $H^i_I(R)$ have height at least $i$ either... Do you have a particular ring / ideal combination in mind? –  Karl Schwede Jun 4 '13 at 19:19
    
@ Karl Schwede: If $p\in {\rm Supp}H^i_I(R)$ then $H^i_{IR_{P}}(R_p)\neq 0$ so by Vanishing theorem we have$i \leq \dim R_p={\rm ht}(p)$ –  QED Jun 5 '13 at 9:06
    
@ Karl Schwede By the way, if $I$ is maximal then the support is contained in $V(I)$ hence equal to $\{I\}$. Please let me know if I'm wrong. –  QED Jun 5 '13 at 9:13
    
I'm sorry, I was thinking the vanishing locus of the annihilator of the module, not the support. –  Karl Schwede Jun 5 '13 at 15:12

1 Answer 1

up vote 1 down vote accepted

Let $(R, \mathfrak{m}) = k[[a, b, c, d]]$ where $k$ a field. Set $I_1 = (a, b),I_2 = (c, d)$ and $I = I_1 \cap I_2$. Consider $H^3_I(R)$ (use Mayer-Vietoris exact sequence, Hartshorne-Lichtenbaum vanising theorem) we can prove that $\text{supp}H^3_I(R) = \{\mathfrak{m}\}$. Notice that, by Lyubeznik (D-modules, F-modules) we have $H^3_I(R) \cong E(k)$ the injective hull of $k$.

share|improve this answer
    
@Pham Hung Quy, Thanks a lot for good counter-example. I've edited the question. Please take a look because you may solve it again.:-) –  QED Jun 8 '13 at 10:30
    
in my example $cd(I) = 3$. –  Pham Hung Quy Jun 8 '13 at 11:08

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.