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?
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1$\begingroup$ You can define local cohomology for topological spaces (and singular cohomology), or for algebraic varieties (and $\ell$-adic/de Rham/... cohomlogy). In all cases, the local cohomology of closed subspace $i:Z\hookrightarrow X$ defined in terms of the six functors to be $i^!k_X$, where $k_X$ is the constant sheaf on $X$. When both spaces are smooth or $i$ is a regular embedding, this is just (a shift of) the constant sheaf. Thus, the local cohomology (and presumably the top degree part too) measures the singular behaviour "in the normal direction". $\endgroup$– PulcinellaCommented Oct 16, 2022 at 21:30
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1$\begingroup$ I presume you have looked at Grothedieck's Local Cohomology, pages 30 and 64, where these groups are related to Ext groups and dualizing modules. Apologies if this is obvious, as I have only consulted this one source. $\endgroup$– roy smithCommented Oct 17, 2022 at 0:22
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1$\begingroup$ A common structure in commutative algebra is to consider $H^d_{\mathfrak{m}}(R)$ where $R$ is local Noetherian. That local cohomology module is the Matlis dual of $\omega_R$, the canonical module of $R$ (assuming one exists, for instance if $R$ is complete or finite type over a field or $Z$). There are lots of other things one might say too. In particular, one way to study anything about the canonical module is to instead study the local cohomology module. $\endgroup$– Karl SchwedeCommented Oct 17, 2022 at 4:40
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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.
answered Oct 16, 2022 at 21:21