Geometric Meaning of higher pushforward for open immersions

Question

Here by higher pushforward I mean the right derived functor of the pushforward functor.

I'm wondering if higher pushforward of open immersions between schemes have some geometric meaning. For a proper flat map, the higher pushforward is like a sheaf which contains information of the cohomology of the fibers. For the higher pushforward of open immersions, the sheaves are supported on the complement of the open, so there are no such thing like "fibers", what's a better way to understand higher pushforward in this case?

Answer 1

I think a common way people understand this is via local cohomology and cohomology with support.

Suppose that $Z \subseteq X$ is a closed subset with $\alpha : U = X \setminus Z \to X$ the open immersion. Since you are interested in the pushforward, I'm going to assume that $X = \text{Spec}R $ is affine (this is harmless for your purposes I think). I'm going to assume that $M$ is a coherent $O_X$-module and we are going to pushforward $M \mid_U$ (this is also harmless). Then as it sounds like you already know, there are isomorphisms for all $i > 0$: $$ R^i \alpha_* (M\mid_U) = \underline H^{i+1}_Z(M) $$ where by the underlined $\underline{H}$ I just mean the sheafy local cohomology module.

And so now you are basically asking, what can we say about local cohomology modules?

Well, of course it depends on $M$, but people study this from a number of perspectives. Unfortunately for you, the simple geometric interpretation I typically hear involves as sheaf cohomology of complements of closed sets... There are notable results in this direction though. For example, see the paper of Barth in AJM in 1970, Hartshorne and Speiser in the Annals 1977, or the work of Peskine and Szpiro (1973), or just SGA2.

However, a large percentage of modern commutative algebra is interested in the study of these modules.

Let me highlight a couple recent important problems/questions/areas of work, especially those with geometric connections.

1. Depth and Serre's conditions, you probably already know that these are connected if you've read Hartshorne, but there are lots of other deeper connections as well. You could for instance see the book Cohen-Macaulay Rings by Bruns and Herzog. You can also look up various forms of local duality. Of course, depth conditions also appear in the study of Moduli spaces, see for example recent work of Koll\'ar.

2. $M = O_X$. This is the most common situation that I see studied. In characteristic $p > 0$, there are literally dozens of papers that study the action of Frobenius on these modules (see for example the recent paper of Enescu-Hochster in ANT or Lyubeznik's paper on $F$-modules in Crelle). In any characteristic, people certainly also study these as things which $\mathcal{D}$-modules act upon.

3. Associated primes. A lot of attention has been paid to what associated primes can occur for local cohomology modules. For example, it was a longstanding open conjecture that there are always finitely many such associated primes. Anurag Singh disproved this, MRL (2000) (although its true in some cases, see the work of Huneke and Sharp and Lyubeznik). Later Moty Katzman gave an equicharacteristic example.

4. Hodge Theory: You can put mixed Hodge structures on local cohomology modules, and these can have various interpretations. For singularities, see the work of Steenbrink in the early 1980s.

5. Singularities and section rings. If $Z$ is a closed point and $M = O_X$, then the these modules are Matlis dual to canonical modules. In particular, things like rational singularities can be detected that way. Even more specifically, if $R$ is standard graded (ie a cone), then these local cohomology modules have graded pieces that correspond to sheaf cohomology on the variety you took the cone over. Because of this, many questions about projective geometry are studied this way by people with a more algebraic bent. Especially I'd say stuff about Castelnuovo-Mumford regularity.

Answer 2

Let $i:U\rightarrow X$ be an open immersion and let ${\cal F}$ be a coherent sheaf on $U$.

Let $z$ be a point in $X$. For any affine open set $V\subset X$ containing $V$, you can compute $H^n(V\cap U,{\cal F})$. The fiber of $R^ni_*{\cal F}$ is a limit of these cohomology groups over all such $V$.

Note that if $z$ is contained in $U$, then it has an affine open neighborhood $V$ contained in $U$, and $H^n(V,{\cal F})$ must be zero; therefore the fiber at $z$ is zero.

More generally, the group $H^n(V\cap U,{\cal F})$ can be non-zero only if $V\cap U$ is non-affine, which can happen only if $V\cap U\neq V$, which can happen only if $U\neq X$. So it's the fact that $U$ is ``missing some points'' that allows these cohomology groups to be non-zero. The fiber at $z$ is, therefore (and speaking very roughly) a measure of the extent to which $z$'s absence from $U$ is allowing ${\cal F}$ to have non-zero cohomology on the restrictions of affine open sets.