I'm not an expert in local cohomology, but the following problems have come up in my work, and I'd like to get a sense of where things stand.
Let $\mathbb{A}^n=\operatorname{Spec} \mathbb{C}[x_1, \dots x_n]$ be affine space. Let $I$ be an ideal given by $(x_1, \dots x_k)$ for some $0 \leq k \leq n$, and let $M$ be a similar ideal, with generators $(x_{i_1}, \dots x_{i_j})$. The first problem: compute the local cohomology spaces $H^i_{I}(\mathbb{A}^n, M)$.
Just from poking around with Cech complexes, this seems pretty straightforward and combinatorial. In particular:
$H^0_{I}(\mathbb{A}^n, M)=0$ when $I$ is not 0, since the generators (in any order) of $I$ form a regular sequence on $\mathbb{C}[x_1, \dots x_n]$ and a fortiriori no elements of $M \subset \mathbb{C}[x_1, \dots x_n]$ can be $I-$torsion.
$H^1_{I}(\mathbb{A}^n, M)= 0$ if $I \not\subset M$. If $I \subset M$, then $H^1_{I}(\mathbb{A}^n,M)$ should be "cyclic" (in the D-module sense), with generator given by Cech cocycle $(\frac{x_1}{x_1}, \dots \frac{x_k}{x_k}) \in \bigoplus_{i} M_{(x_i)}$.
More generally, the higher terms seem to be given by some kind of inclusion/exclusion. Example: a Cech cocycle in $Z^2_{(x_1, x_2, x_3,x_4)}(\mathbb{A}^5, (x_2, x_3,x_4,x_5))$ is $(\frac{x_2}{x_1x_2},\frac{x_3}{x_1x_3}, \frac{-x_4}{x_1x_4},\frac{0}{x_2x_3}, \frac{0}{x_2x_4},\frac{0}{x_3x_4})$.
Do all generators (again in a D-module sense) appear in this combinatorial fashion? Do concrete descriptions of these generators appear in the literature?
Lastly, if this is all well known, does it generalize to the setup where $X$ is a reduced Cohen-Macaulay scheme, with $I$ and $M$ both being ideals generated by (subsets of) systems of parameters?
