Top local cohomology - recommendations I need some background in local cohomology to read a certain paper, which exploits the structure of the top local cohomology. Even after acquainting myself to local cohomology, I fail to understand the significance of the top local cohomology group. What would be a nice resource to understand the intuition behind the information that is held by the top local cohomology group?
 A: Associated to a ring $R$ and ideal $I\leq R$ we have a scheme $X=\text{spec}(R)$ and a closed subscheme $Z=\text{spec}(R/I)$ with complement $U=X\setminus Z$ (which will typically not be an affine scheme, unless $I$ is principal).  Now the local cohomology $H^*_I(R)$ is just the relative sheaf cohomology $H^*(X,U;\mathcal{O}_X)$.
A typical case is when $X=\mathbb{A}^n$ and $Z=\{0\}$ so $U=\mathbb{A}^n\setminus\{0\}$.  The local cohomology $H^{*}(\mathbb{A}^n,\mathbb{A}^n\setminus{\{0\}};\mathcal{O})$ is naturally compared with the ordinary singular cohomology $H^{*}(\mathbb{R}^n,\mathbb{R}^n\setminus\{0\})$.  The pair $(\mathbb{R}^n,\mathbb{R}^n\setminus\{0\})$ is homotopy equivalent to the pair $(B^n,S^{n-1})$, so the cohomology is the same as the reduced cohomology of the quotient $B^n/S^{n-1}\simeq S^n$.  In other words, we have $H^{n}(\mathbb{R}^n,\mathbb{R}^n\setminus\{0\})\simeq\mathbb{Z}$, and the other cohomology groups are zero.  This makes it unsurprising that $H^{k}(\mathbb{A}^n,\mathbb{A}^n\setminus{\{0\}};\mathcal{O})$ is zero for $k\neq n$, and that the interesting case is when $k=n$.  The actual value of the $n$'th local cohomology is not visible from this line of argument, however.
