$I$ 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

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