C. Peskine and L. Szpiro in "Dimension projective finie et cohomologie locale",(Proposition 4.1) proved the following vanishing theorem for local cohomology: Let$R$ be a regular local ring of characteristic $p\gneqq 0$. Let $I$ be an ideal of $R$ such that $R/I$ is a CM(i.e. Cohen-Macaulay) ring; then: $$H_I^i(M) = 0~~ \text{for each}~ i> \text{dim} R-\text{dim} R/I.$$ It is to be noted that this theorem does not extend to characteristic $0$.
But I think to a restricted version, when the characteristic is zero. My question is that if $R= K [[ x_1, x_2, ...x_n]]$, where $K$ is a field with $\text{char}~K=0$ and $R/I$ is a CM ring, is it true that:
$$\text{inj}~\text{dim}~H_I^i(R) =\text{dim}~ H_I^i(R)? $$
Michael Hellus in http://arxiv.org/pdf/math/0703126v1.pdf has attacked the question and in this case has established the following facts:
Let $R$ be a noetherian local regular ring containing a field, $I$ an ideal of $R$ and $i$ a natural number. Here is a summary of results:
(i) If $H^i_I(R)$ is $I$-cofinite then $\text{inj dim} _R(H^i_I(R))=\dim _R(H^i_I(R))$ holds.
(ii) There are examples where $H^i_I(R)$ is not $I$-cofinite, but $\text{inj dim} _R(H^i_I(R))=\dim _R(H^i_I(R))$ holds.
(iii) There are examples where $\text{inj dim} _R(H^i_I(R))=\dim _R(H^i_I(R))$ does not hold.
