This is a continuation of this post.
The affine case
Restating and applying purity
Let us now finally return to computing to proving Proposition 2 in the affine case. Let us first write it more concretely:
Proposition 31: Let be an algebraically closed field and a smooth connected projective curve over of genus . Let and . Set . Then, for every prime invertible in and integer
Given our discussion of cohomology with supports this is fairly easy. In fact, it’s laughably easy if we take the Purity Theorem for granted. Namely, let us begin by noting that by Proposition 23 we have a long exact sequence
and thus from Proposition 4 it really suffices to compute . If we assume the Purity Theorem (in the form of Corollary 30) this is trivial since is a disjoint union of points. Namely, we see
from which deducing Proposition 31 is a triviality.
Giving a more hands-on proof
But, the above feels a little bit like cheating to me. Namely, we don’t need the elephant gun that is the Purity Theorem to do this computation. In fact, this computation really is just a basic case of the Purity Theorem that we can do by hand. So, let’s do it.
Note that by iteration it suffices to deal with the case when is a single point (but is now allowed to be affine—-this won’t change the computation by the Excision Theorem). Or, if this iteration argument doesn’t make you happy convince yourself of the easy fact that since that
which, either way, reduces us to the computation of for a single closed point .
Now, the key point for this computation is then that by Lemma 26 we can reduce this to a really explicit computation. Namely, Lemma 26 tells us that (since our schemes are Noetherian) that
This may not seem very helpful but, in fact, the scheme is something incredibly explicit!
Namely, we have the following
Lemma 32: Let the notation be as above. Then, there is an isomorphism of schemes
and can be described as the integral closure of inside (NB: since is algebraically closed there is no difference here between henselization and strict henselization).
Proof: The first claim follows from the fact that there exists an open subset of in and an etale map with . Indeed, just take any etale map . Then, we know that will map to some point in which, by etaleness, and the fact that is algebraically closed closed implies that . One can then just postcompose with the translation isomorphism . We now claim that this pointed etale map induces an isomorphism
but since is an etale neighborhood this is almost by the definition of the henselization/strict henselization.
It remains to justify why has the claimed form. There are several ways to justify this. For example, one can see [Mil, Example I.4.10 (a)] and/or [Mil, Example I.4.10 (b)] and the references therein. One can also see [Nag, Page 180].
So, we see that (as hinted at in the idea of proof for the Purity Theorem) one can use the fact that our calculation is etale local nature and the fact that smooth things all locally look the same to reduce this to an explicit computation with a specific scheme. Namely, by the above it’s clear that we need to compute the group . But, how do we do this?
Well, using Proposition 23 and the fact that strictly Henselian local rings have vanishing higher cohomology (e.g. see [Fu, Proposition 5.7.3] or [Stacks, Tag09AY]) it’s easy to see that
where here we are denoting by the scheme (the punctured spectrum).
Now has pretty simple structure in that it’s a Henselian DVR (e.g. combine [Stacks, Tag06DI] with [Stacks, Tag06LK]). In particular, it’s easy to see that where . From this it’s easy to conclude that
Thus, we are reduced to understanding the Galois cohomology of . But, this is surprisingly simple in our case:
Lemma 33: Let be a Henselian DVR of residue characteristic . Let be a uniformizer of and set and . Assume that is algebraically closed. Finally, let be a finite Galois extension of degree with . Then, .
Note that the reason this Lemma is of interest to us is that it evidently implies that
Corollary 34: In particular, if denotes the pro-prime-to- completion of then (non-canonically)
This corollary also makes complete geometric sense. We are to think of as being something like a ‘small etale disk around in ‘ Then, of course, we should think of as being something like a ‘punctured disk’ around and, consequently, it would make sense that
should be something like . Corollary 34 validates this belief up to the ever-present concerns about wild ramification.
We also see that Corollary 34 immediately implies that
So, summarizing the above discussion we get that for that
which, combining with our trivial result that gives us
This is as already told to us by the Purity Theorem and which, we’ve already observed, easily implies Proposition 31.
Thus, all that remains to finish our hands-on proof of Proposition 31 is the proof of Lemma 33:
Proof (Lemma 33): Let be any finite Galois extension of of order which is prime . Since is Henselian there is at a unique extension of the valuation on to a valuation on (e.g. see [EP, Lemma 4.1.1] and [EP, Theorem 4.1.3]). Let be the valuation ring of and let denote a uniformizer of . Note that since (e.g. see [EP, Theorem 3.3.5]) and (since the residue field of is algebraically closed) we must have that is totally ramified. Thus, where .
But, note that has a solution in the residue field of (since it’s algebraically closed) and since is coprime to the residue characteristic of we see from the fact that is Henselian (e.g. this follows from [EP, Lemma 4.1.1]) that has an -root in . Thus, we see that, by replacing with an -root of , we can assume . Then by standard theory (e.g. see [Sut, Theorem 11.5] and note that the proof doesn’t use completeness, but only Henselianess).
To summarize the proof in the affine case : we used the theory of cohomology with supports to break the cohomology into two parts. The first part was the classes on that come from . The second part was the classses on introduced by the punctures we made at the points . Then, using that cohomology with supports is insensitive to shrinking in the etale topology we were able to focus in on the loops around each puncture individually and, using the fact that smooth curves all etale locally look , we were able to essentially understand this as a computation involving the punctured spectrum of the henselian local ring at which, again, we intuited as a punctured disk. This space was concrete enough/well-behaved enough that we could compute its cohomology by hand and got (ignoring wild ramification issues) that its cohomology agrees with what we expected (a punctured disk).
Two bonus computations
Even though our main goal (at least locally) was to prove Proposition 2 (which we have now done) our computations relied so heavily on smoothness hypotheses that one might be scared that the methods we have developed are ill-equipped to deal with singular cases. This is not at all the case. Namely, using our theory of cohomology with supports we can compute this cohomology much like we did for smooth curves, but in reverse.
Namely, let be our singular curve and let be its smooth locus and its finite discrete singular locus. We are now interested in not in the cohomology of but in the cohomology of . But, again, by Proposition 23 this comes down to studying the cohomology of (which we know by Proposition 2/Proposition 31) and the cohomology of with supports at . The latter can, as in the smooth case, be studied by studying etale local geometry of at which, again, allows us to make explicit computations involving the singularity type of .
The two examples we compute arefortunately/unfortunately (depending on your bend) the obvious ones: the cuspidal and nodal cubics.
The nodal cubic
As before let us fix an algebraically closed field of characteristic not or (for sanity’s sake) and a prime invertible in . Let us define to be the projective nodal cubic curve given by
this has a singular locus (where we write ) and smooth locus . The structure of itself is quite simple. Namely, we claim that . Indeed, this follows from the fact that the normalization of is the map
(e.g. see [Mac]) which has the property that consists of two points so that isomorphically maps to .
Now, by Proposition 23 we get a long exact sequence
Now, since we know from Proposition 2/Proposition 31 that
and thus we really only need to compute .
To compute this we use Lemma 26 (since our schemes are Noetherian) to say that
But, the point is that while the global geometry of is somewhat daunting the etale local geometry at is quite simple. Namely, we have the following:
Lemma 35: Consider . Then, there is an isomorphism
Proof: It suffices to find an etale neighborhood of such that . To do this we let be the unique double cover of .
Abstractly one can describe it as follows. As is well-known (e.g. see [Har, Exercise 6.9]). Let be the line bundle on corresponding to . Then, (it’s the unique such line bundle). Then, using this isomorphism one can put an algebra structure on and we can set .
A less highfalutin way to build is to have it be the projectivization of the map
corresponding to the -algebra map with
Then, one can show that this map is a finite etale cover (of degree ). Moreover, the point of maps to . Moreover one can explicitly check that from where the conclusion follows.
In words, if we’re only interested in etale local geometry (which is the case due the fact that cohomology with supports is insensitive to shrinking in the etale topoogy) we can eschew the global complicated nature of the nodal cubic when focusing on its singular point and see it as, essentially, the intersection of two lines.
Thus, to compute
we need really only compute
But, just as before in the smooth curve case we can use Proposition 23 and the vanishing of higher cohomology of strictly local rings to see that if we set then
But, what is in this case? It is no longer the spectrum of the fraction field of a Henselian DVR. Indeed, it’s actually the disjoint union of two such objects. The idea is that
(where means the non-vanishing locus of in ) and, as one can check, and are both (essentially canonically) the spectra of the fraction field of .
We can then use the results of the previous section (namely Corollary 34) to thus deduce that
(don’t forget that that has two components!). In fact we can geometrically intepret this non-vanishing. Remember that (roughly) classifies -torsors on that become trivialized on . If is normal this can never hapen (as we saw by the vanishing of ) essentially because if you have a connected finite etale cover one has that is actually integral (e.g. see [Stacks, Tag0BQL]) and so is still connected so that cannot be trivial. But, for the nodal cubic the covers can fail to have this property. In fact note that, geometrically, the nodal curve looks like a pinched torus. Its covers then look like necklaces of wedges of spheres (e.g. see the discussion here and here) where its exactly the wedge points (the ‘kissing points’) that map to the node . So, for such a cover one has that is certainly not trivial, but just looks like a bunch of disjoint copies of –the trivial torsor over !
Let us return to the rigorous computation. For
(where we have used that is a disjoint union of the spectra of two Henselian discretely valued fields and Corollary 34). From this we then deduce that
as we would expect topologically.
The cuspidal cubic
Finally, let’s flip our method on its head by computing the etale cohomology of an directly and then using this to say something non-trivial about the cohomology groups .
First, let’s fix our set up. Again, let be an algebraically closed field of characteristic not or and let be a prime invertible in . Let us consider the projective cuspidal cubic:
this has singular locus (where we abbreviate to ) and smooth locus .
The cheeky way to compute the cohomology of is then to employ the following observation:
Lemma 36: Let notation be as above. Then, the normalization of is and the normalization map is a universal homeomorphism.
Proof: Note that is evidently finite (or even easier is that it’s proper) and surjective and thus universally closed and universally surjective. Thus, it suffices to check that is universally injective. It suffices by standard theory (e.g. see the discussion in [Stacks, Tag01S2]) to show that for every point that is purely inseparably. For the generic point this is clear since is birational so it suffices to check for closed points of or, equivalently, for the points lying over the cusp. This is done in [Mac] explicitly for the cusp.
Now, one deduces from the topological invariance of the etale site (e.g. see [Stacks, Tag03SI]) that and have the same cohomology. In particular, we see that
Note though that from Lemma 36 we also deduce that and thus its cohomology is known from Proposition 2/Proposition 31:
Using Proposition 23 we deduce the following:
Let us now consider the objects and and see what the above computation tells us.
To start, interestingly enough, from the equality (again we’re implicitly using Lemma 26 and thus Noetherianity, but I assume this needn’t be said at this point)we actually deduce that
is surjective and thus is connected. Surprisingly, this is essentially equivalent to Lemma 36. Namely, Lemma 36 roughly says that is (geometrically) unibranch (e.g. see [Stacks Tag0BPZ]), which since is normal away from , is really saying that is geometrically unibranch. But, being geometrically unibranch is actually equivalent to the claim that is connected (e.g. see [Stacks, Tag0BQ4]).
Let us in fact notice that being connected implies that, in fact, is a field. Indeed, note that is reduced (e.g. see [Stacks, Tag06DH]) and thus it will be domainif and only if it’s irreducible. But, note that is dimension (e.g. see [Stacks, Tag06LK]) and thus there are no intermediary primes lying between the minimal primes of and . Thus, is a discrete set on the minimal primes of and since is connected, this implies that there is a unique minimal prime of . Thus, is a domain. Moreover, since it’s of dimension it’s easy to see that where .
Note that this field is the fraction field of a Henselian local domain but one which is not a DVR. Thus, the results like Corollary 34 don’t apply. In fact, I don’t actually know how to compute directly. That said, from our sideways calculation of . Indeed, using the usual isomorphism for
we deduce that
Again, this is quite different from our other calculation and quite strange. In our previous calculations we calculated a global group by studying the local group. Here we had a local group that, a priori, we didn’t know how to explicitly compute but once we globalized it we were able to exploit the global picture to get the desired result.
I have a gut feeling that there is some topological invariance trick that works locally for computing the cohomology of similar to how we computed the cohomology for the cuspidal cubic. I don’t currently see it though. Please feel free to let me know if you see an approach!
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