$I$ 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


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



Is there a prime of height $i$ in support of $H^i_I(R)$?
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 $k$.


