Simple applications of higher derived functors

Question

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.

Answer 1

One example that immediately comes to my mind are various cohomology functors. Let me therefore talk about some of them (hoping that these general examples are in some sense what you wanted to see otherwise I maybe need to choose more specific applications).

Singular cohomology (you can also think of simplicial or cellular cohomology if you prefer):

If $X$ is a "sufficiently nice" topological space (I think we need locally contractible plus paracompact, but would need to look it up again to be completely sure), then singular cohomology coincides with a certain sheaf cohomology, namely the sheaf cohomology $H^i(X,\underline{\mathbb{Z}})$ of the constant $\mathbb{Z}$-sheaf $\underline{\mathbb{Z}}$ (the sheafification of the constant $\mathbb{Z}$-presheaf ). As sheaf cohomology is defined as the derived functors of the global section functor we thus realized the usual topological cohomology as a derived functor.

That means your question in this context is about the use of cohomology. This question certainly has many answers in books and on the internet, in particular here on SE. One aspect is that cohomology is a nice topological invariant and as such helps us to distinguish topological spaces which is certainly something we want to do and often it is just not enough to only consider cohomology in degree $0$ and $1$.

The derived functor point of view here tells you at least that singular cohomology arises as an example of such, but I am not sure about the active use of this point of view. I would guess that the comparison with sheaf cohomology in general is the more useful observation.

Sheaf cohomology:

We could have of course also skipped the comparison with singular cohomology and only considered sheaf cohomology to begin with, but I guess most people feel more comfortable with singular cohomology.

I don't really want to speak about sheaf cohomology here, but if you happen to know algebraic geometry to some extend you have probably seen how much sheaf cohomology is used in general and you will also find some applications or examples of the higher sheaf cohomology groups being used.

To give one quite basic example, the higher sheaf cohomology groups allow us to define the Euler characteristic of a line bundle $\chi(\mathscr{L})$. The Euler characteristic has very nice applications in the theory of algebraic surfaces for example, where you use it to define certain intersection numbers. These are used constantly in the realm of the classification of surfaces with applications to many structure theorems, classifications of certain singularities on surfaces as rational double points etc. Moreover, these applications are not too difficult to understand (you can find them in Badescu'sbook on algebraic surfaces).

Group cohomology:

Group cohomology of a group $G$ is defined as the derived functors of the invariant functor $M \mapsto M^G$ for $G$-modules $M$. Here I would like to talk about the second derived functor, i.e. the second group cohomology $H^2(G,M)$. The cohomology group $H^2(G,M)$ can be identified with classes of certain types of extensions of $G$ by $M$, namely with isomorphism classes of extensions $M \rightarrow E \rightarrow G$ that give rise to the action of $G$ on $M$. Since you mentioned homological algebra I am sure that you are well aware of the usefulness of exact sequences and more specifically extensions (and also why there is interest in extensions).

Also here we have that cohomology is an invariant of course.

I am not sure whether the derived point of view helps with establishing the upper correspondence (i.e. whether there is a nicer proof using it), but at least explicit computations as for example the cohomology of cyclic groups benefits from the long exact sequence and hence from the derived point fo view. Working with the cocycles explicitly is just really messy.

Not talking about cohomology anymore:

I hope that you will get some additional answers with more examples (maybe more concrete ones or picking some of my examples in a more concrete context) as well.

Answer 2

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.