This is the first in a series of posts whose goal is to compute the cohomology of the (several times) punctured closed -adic disk as well as the cohomology of smooth (and some non-smooth) algebraic curves.
The original goal for this post was to discuss the calculation of the etale cohomology for the adic space where are classical points. Moreover, I wanted to illustrate some interesting theory (the theory of cohomology with supports for adic spaces) that helps with such computations and some thorny details which make the computation different than one might expect if one copied verbatim the algebraic analogue.
Because of this last sentence, but also to give the (essentially correct) blueprint for how we would compute this cohomology I wanted to discuss computing the cohomology of smooth connected curves or, equivalently, the cohomology of smooth connected projective curves which are punctured (several times). So, this post has grown quite a bit longer than originally intended.
A model: the cohomology of curves
Before we begin actually computing the cohomology of we discuss how to compute the cohomology of an integral smooth curve over an algebraically closed field . This will be useful since, in some sense, it contains the exact blueprint necessary to compute the cohomology of the multiply punctured disk.
We begin by recalling the following basic result that will allow us to essentially reduce the computation from general smooth connected curves to smooth connected projective curves.
Lemma 1: Let be an algebraically closed field and a smooth connected curve over . There exists a unique smooth projective connected curve over such that where .
Proof: See the discussion in the subsection titled Affineness or projectiveness in this post.
So, let us now fix a prime invertible in . We then wish to compute the etale cohomology for all . Namely, we show the following:
Proposition 2: Let be an algebraically closed field and let be a smooth projective connected curve over of genus . Let and set . Then, for every prime invertible in and we have the following:
Here and are simple integers depending on :
There are several ways upon which one can base their approach to this computation. One of these essentially treats the proper case (i.e. the case ) and the affine case on equal footing and computes them by essentially by the same method (e.g. see [Stacks, Tag03RR]). The method we elect to take here is to use machinery which, in general, helps one relate the cohomology of a scheme and an open subset of the scheme. I speak, of course, of the notion of cohomology with supports. I prefer this method since I think it’s a) more fun, and b) containing ideas that are more generally applicable.
The proper case
We handle the proper case as an initial base case from which we will inductively remove points (and see how the cohomology changes) to reach the general case.
Remark 3: The approach we take here is, essentially, the canonical/consummate one. All presentations of the proof of this result are, as far as I know, essentially equivalent with small tweaks here or there depending on taste. If one would like more details than what I write here (including a nice introduction to the Brauer group of a scheme) I highly recommend Dan Litt’s excellently written notes [Lit].
Let us simplify Proposition 2 to the proper case:
Proposition 4: Let be an algebraically closed field and let be a smooth proper integral curve over of genus . Then, for every prime invertible in and integer , we have the following:
The idea of proof is to use the ubiquitous and powerful Kummer sequence. Namely, we recall the following fundamental fact:
Lemma 5 (the Kummer sequence): Let be any scheme and let be a positive integer invertible in . Then, there is exact sequence of sheaves on the (small or big) etale site of given by
Proof: It’s evident that this sequence is exact save the fact that the map of etale sheaves is surjective. To do this, let be an etale morphism. Then, we need to show that for every that there exists some etale cover such that is an -power. But, such a covering is given by
where, here, is the obvious quasi-coherent -algebra and stands for the relative spectrum. The reason that this map is etale is precisely because is invertible in and thus in .
As a corollary we get the following:
Corollary 6: Let be any scheme and let be a positive integer invertible in . Then, there is a long exact sequence of abelian groups
where denotes the multiplication-by- map.
This seems somewhat irrelevant for our computation of since it seems to be useful for computing only the cohomology of the sheaf . Of course, this feeling only persists until one makes the following incredibly simple, and incredibly basic observation:
Lemma 7: Let be -scheme (where is a primitive -root of unity). Then, there is an isomorphism of sheaves on the (big or small) etale site of .
Proof: One needs only check that the group schemes and the constant group scheme are isomorphic on the small big site of . But, note that and are the pullbacks from the group schemes and where . This should allow one to feel confident that the basic argument one wants to make is kosher (e.g. you don’t have to worry about idiosyncracies of the constant sheaf on non-Noetherian spaces).
Thus, we see that, in particular, if we have one has that
In particular, since is algebraically closed and is invertible on we know that is actually a -scheme and thus we can profitably use the Kummer sequence to attack Proposition 4 (at least hopefully).
In particular, we see that to compute we must compute for all . This, as it turns out, is not too bad since the function fields of curves are particularly simple.
Now, since we begin the computation of in earnest at degree . We do this by citing a very general result:
Lemma 8 (Hilbert’s Theorem 90): Let be any scheme. Then, there is a canonical isomorphism of abelian groups
Proof: We will not give a full proof of this result here. Let us only indicate what an immoderately high-level idea of proof might be. Namely, what is classifying for any site ? The answer is -torsors. But, note that (these are automorphisms of -modules) and by the ‘theory of twists’ (e.g. see Theorem 5.1 of the notes from this post) this means that is classifying ‘-modules on locally free of rank in the -topology’. So, this theorem comes down to a statement that whether -module is locally free of rank in a topology doesn’t actually depend on the choice. An even more general useful version of this is seen as [Ols, Proposition 4.3.8].
In particular, we see that . This leaves us to try and compute for . This is where we will have to take advantage of the aforementioned niceness of the function field of . We will do this by appealing to yet another useful exact sequence.
To do this, let be the inclusion of the generic point of in to . We then have the following pivotal sequence:
Lemma 9(the divisor sequence): Let be any Noetherian, integral, separated, and locally factorial scheme. Let be the generic point of and let be the natural inclusion. Then, there is a short exact sequence of sheaves on the (small) etale site of :
where denotes the points of of codimension and is the natural inclusion.
Proof: See [Lit, Lemma 2] or [Stacks, Tag03RI] for a rigorous proof. Let me just say that, roughly, this is just a souped up version of the equivalence of ‘Weil divisors’ and ‘Cartier divisors’ on a locally factorial scheme.
This lemma tells us that to compute the cohomology of we really only have to compute the cohomology of and the cohomology of . One of these is decidedly easier than the other:
Lemma 10: Let be an algebraically closed field and let be a connected smooth projective curve over . Then,
Proof: Since cohomology commutes with direct sums (e.g. see [Stacks,Tag0F11 (2)]) and is acyclic (e.g. see [Fu, Proposition 5.7.4]) one reduces this computation to the claim that for all one has that for all that . But, note that since is a closed point, since is a curve, that is finite and thus is algebraically closed. Thus, the claim follows by standard theory (e.g. see [Fu, Proposition 5.7.8]).
To compute we would like to somehow reduce this to computing since this (again by [Fu, Proposition 5.7.8]) is just computing Galois cohomology which one might be hopeful is doable. In fact, let’s put our money where our math is and compute (or at least cite a reference for the computation of) :
Lemma 11: Let be an algebraically closed field and let be an extension of transcendence degree . Then, for .
Proof: As usual, see [Lit] or [Stacks,Tag03RG] but also take a look at [Star] and [Poo, Chapter 1].
Remark 12: We don’t wish to downlplay the above result. While it’s fairly standard fare (it’s mostly just somewhat ‘standard’ Galois cohomology with the exception of the very powerful theorem of Tsen) it is the main technical result needed for this computation. So, one should take it seriously. You will learn a lot of mathematics reading the above references.
Now, a prioiri, we cannot directly use this compute to since, of course, there is no reason that . To see why this vanishing of higher pushforwards is relevant we fall back on the general method by which one compares the cohomology of a sheaf and its pushforward:
Lemma 13 (the Leray spectral sequence):Let be a map of schemes and an abelian sheaf on the small etale site of . Then, there is an spectral sequence:
Proof: See [Stacks, Tag03QB].
Applying this for we get the spectral sequence
So, if we can prove that we’ll essentially be done since this spectral sequence will collapse showing that
and the latter is zero (for ) by Lemma since .
So, how can we show that ? Well, as per usual, it suffices to show that for every geometric point of . But, (see [Stacks, Tag03Q9] and/or [Stacks, Tag03Q6]) one has the following
where is the local ring of at in the etale topology. But, one can check (e.g. see [Stacks, Tag03RJ]) that is always the spectrum a field extension of of transcendence degree at most . Thus, we deduce the vanishing of also by Lemma 11.
So, we deduce the the following:
Lemma 14: Let be an algebraically closed field and let be a smooth connected projective curve over . Then, for .
Thus, returning to the divisor sequence we see that we get a long exact sequence
which, putting together Lemma 10 and Lemma 14, easily gives us the following:
Lemma 15: Let be an algebraically closed field and let be a smooth connected projective curve over . Then, the following holds:
Then, finally, returning to the Kummer sequence we deduce the following exact sequence
(where we have cut off prior terms in the long exact sequence since is surjective and we cut off successive terms since vanishes). Thus, we deduce the following:
Lemma 16: Let be an algebraically closed field and let be a smooth connected projective curve over . Then for all primes invertible on and integers the following holds:
Thus, to prove Proposition 4 it suffices to understand the group . Recall that since is a smooth connected proper curve there is a surjective group map and we denote by the kernel of this map. Evidently there is a non-canonical splitting
so it suffices to understand the structure of . This is the so-called theory of the Jacobian. Namely, we have the following well-known proposition:
Lemma 17: Let be an algebraically closed field and let be a smooth connected projective curve over of genus . Then, the group functor
is representable by a -dimensional abelian variety.
Proof: See [Poo, §5.7.1] and the references therein.
Remark 18: I can’t help but roughly explain why Lemma 17 isn’t crazy if one thinks of the complex picture. Namely, let’s imagine that . In this case know by Serre’s GAGA theorem that and . But, note that we have a short exact sequence of sheaves on (the so-called exponential sequence)
In particular, one gets an exact sequence which contains the terms
Now, using the identifications
one can identify
and so we see that we can identify
Now, we have identifications
and Hodge theory shows that the image of
is a full lattice so that
is a -dimensional complex torus. Moreover, Hodge theory also provides a polarization on
and thus is really a -dimensional abelian variety. Hopefully this makes Lemma 17 seem not so farfetched. It also must indicate its depth since this proof was extremely analytic (it used the exponential sequence, singular cohomology, Hodge theory,…).
Thus, we see that Proposition 4 follows from Lemma 16, Lemma 17 and the following:
Lemma 19: Let be an algebraically closed field and a -dimensional abelian variety over . Then, for a prime invertible in one has that is surjective and has kernel isomorphic to .
Proof: See [Con, Proposition 4.2.2].
Remark 20: There are two improvements upon Proposition 4 that should be noted. First, one is able to replace the condition that is algebraically closed by the condition that is separably closed. This follows from [Stacks, Tag0DDG]. This is funny because one really needs something like the proper base change theorem since the above proof really does actually need algebraically closed. Indeed, in the proof of the pivotal Lemma 11 the key result is Tsen’s theorem which says that the Brauer group of a finite extension of is trivial if is algebraically closed. This is actually false if is just assumed separably closed (e.g. see this comment of Marguax on mathoverflow).
The second improvement is that one wants to know what happens if is an arbitrary field by which we mean one would like to know what the action of on is. For this one can see Theorem 7 of this post.
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