TITLE: Simple applications of higher derived functors
QUESTION [8 upvotes]: I was surprised to not have found this question discussed before on this or any other forum, hence I am posting it myself.
From an abstract point of view of homological algebra derived functors are very appealing: given a right exact functor $F$, they provide a canonical way of "completing" the image of a short exact sequence $0\to A\to B\to C\to 0$ to a long exact sequence
$$\dots\to L^2F(C)\to L^1F(A)\to L^1F(B)\to L^1F(C)\to F(A)\to F(B)\to F(C)\to 0.$$
This makes potential applications of such functors relatively clear: $L^1F(C)$ can be used to help measure the failure of exactness of $F$, specifically injectivity of the map $F(A)\to F(B)$. Hence if we have a way of finding $L^1F$ (and computing the connecting homomorphisms), then we gain information about the way $F$ acts.
This only explains the usefulness of $L^1F$. The way the use of higher derived functors was explained to me is by "kicking the can down the road" - to understand $L^1F$ we may want to use $L^2F$ which from the long exact sequence somehow measures how non-left-exact $L^1F$ is, and so on. Having learned some more advanced mathematics (group cohomology in CFT) I can see this being the case, but discussing the topics recently with more novice friends I have realized I don't really know any examples which I could briefly explain to them!
In contrast, I can think of plenty of cases where first derived functor shows up its usefulness. My favorite is probably the following result:

Prop: Suppose $C$ is a flat $R$-module. Then for any exact sequence $0\to A\to B\to C\to 0$ and $R$-module $M$ the sequence $0\to A\otimes M\to B\otimes M\to C\otimes M\to 0$ is exact.
Proof: since $C$ is flat, we have $\newcommand{\Tor}{\operatorname{Tor}}\Tor^R_1(C,M)=0$. The result is now clear from the long exact sequence.

Of course, fully explaning this result would take some work (specifically balancing of $\Tor$) but at the very least we see $\Tor_1$ is a nice device which makes such results straightforward (and makes the result easy to remember!). Other simple ways in which $\Tor_1$ comes up are the Kunneth's Theorem and Universal Coefficient Theorem.
However, as already mentioned, I am not aware of any simple such applications of any higher derived functors. Hence I ask:

What are some nice uses of higher derived functors $L^nF,n\geq 2$ (or $R^nF,n\geq 2$) which could be explained to, say, an advanced undergraduate as a motivation or example of how they are useful?

Some kinds uses I can think of would be a use in computation of lower derived functors (including the starting functor) or as coming up in a proof or at least a statement of some result. I am interested in any of these, though for the latter I would appreciate some indication of how the higher functors come up in the result.

REPLY [2 votes]: Here are two natural, elementary problems:
Given a smooth, compact, closed $n$ dimensional manifold $M$, how many open sets to we need to cover $M$ with open sets $U_i$, such that each each $U_i$ is contractible, with all finite intersections $U_{i_1}\cap U_{i_2}...\cap U_{i_k}$ are finite unions of contractible spaces.
Similarly, given a smooth projective variety $X$, whats the minimal cardinality of an open affine cover of $X$?
For both these questions, we can easily obtain a lower bound of $\dim(M)$ and $\dim(X)$ respectively, using some higher derived functors. In the manifold case, we observe that the constant sheaf $\underline {\mathbb{Z}}$ has $H^n(M,\underline {\mathbb{Z}})\neq 0$, by Poincare duality, so we have a (high) nonvanishing cohomology group (derived functor of global sections). But then if we had a cover of open sets of that nice form with less than $n+1$ opens, this cohomology group would vanish by direct computation, via Cech cohomology.
In the projective variety case, we do the same thing, we can find a quasicoherent sheaf with nonvanishing cohomology in degree $\dim(X)$ (canonical sheaf, by Serre duality), so if we had such an affine cover, computing again with Cech cohomology, this group must vanish by the acyclicity of quasicoherent sheaves over affines, so this cover can't exist.
The ideas behind these proofs goes beyond these examples too, in lots of situations, one can detect complexity via the nonvanishing of certain derived functors.
For a final commutative algebra example, the nonvanishing of the local cohomology group $H_I^n(R)$ implies that the ideal $I$ cannot be generated by $n-1$ elements, and that we cannot even find another ideal $J$ generated by $n-1$ elements with the same radical as $I$. To my understanding, these derived functors are one of the few tools we actually have for proving assertions about generation of ideals up to radical in this manner, which is another entirely elementary problem.