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.

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 schemesand 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

and for

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

and 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).

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.

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 isomorphismwhere .

*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

and for

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.

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!

[Con] Conrad, B., 2015. *Abelian varieties*. https://stuff.mit.edu/people/fengt/249C.pdf

[EP] Engler, A.J. and Prestel, A., 2005. *Valued fields*. Springer Science & Business Media.

[Fu] Fu, L., 2011. *Etale cohomology theory* (Vol. 13). World Scientific.

[Har] Hartshorne, R., 2013. *Algebraic geometry* (Vol. 52). Springer Science & Business Media.

[Lit] Litt, Daniel. *Cohomology of curves.* https://www.daniellitt.com/s/cohomologyofcurves.pdf

[Mac] Mack-Crane, Sander. *Normalization of Algebraic and Arithmetic Curves*. https://algebrateahousejmath.wordpress.com/2016/11/27/normalization-of-algebraic-and-arithmetic-curves/

[Mil] Milne, J.S. and Milne, J.S., 1980. *Etale cohomology (PMS-33)* (No. 33). Princeton university press.

[Nag] Nagata, M., 1962. Local rings. *Interscience Tracts in Pure and Appl. Math.*

[Ols] Olsson, M., 2016. *Algebraic spaces and stacks* (Vol. 62). American Mathematical Soc.

[Poo] Poonen, B., 2017. *Rational points on varieties* (Vol. 186). American Mathematical Soc.

[SGA 1] Grothendieck, A. and Raynaud, M., 2002. *Revêtementset groupe fondamental (SGA 1)*. arXiv preprint math/0206203.

[Stacks] The Stacks Project Authors, *The Stacks Project*, https://stacks.math.columbia.edu

[Star] Starr, Jason. *Brauer groups and Galois cohomology of function
fields of varieties*. https://www.math.stonybrook.edu/~jstarr/papers/Escola_07_08d_nocomments.pdf

[Sut] Sutherland, Andrew. *Totally ramified extensions and Krasner’s lemma*. http://math.mit.edu/classes/18.785/2016fa/LectureNotes11.pdf

]]>

We now wish to move on from the proper case of Proposition 2 to the case of non-proper smooth connected curves . Now, Lemma 1 (and the fact that Proposition 2 is stated in terms of it) strongly suggest that we should try and compute the cohohomology of by embedding it into a proper smooth connected curve and somehow understand what the ‘difference’ between the cohomology of and is.

Namely, we want a way to explicitly compare the cohomology of a scheme and an open subset . Our intuition strongly pushes us to imagine that somehow their should be a relationship involving the cohomology for , , and (the closed complement of ). It turns out that things are slightly more complicated than one might initially imagine though.

So, before we explain the precise way in which we compare these three cohomologies, let’s discuss a model sucess in a slightly modified situation. Namely, instead of working with cohomology itself let us instead think of *compactly supported cohomology*. This should be a reasonable thing to do since (assuming that we’re interested in smooth varieties) Poincare duality (e.g. see [Mil, Corollary 11.2]) says that compactly supported cohomology is essentially dual to normal cohomology.

Now, while compactly supported cohomology has a lot of lovably annoying functorial indiosyncracies (it’s not functorial for arbitrary maps and when it is it can be covariant or contravariant depending on the property of the map) one thing that compactly supported cohomology was built to do is understand our question of how the cohomology of , , and are related. Namely, we have the following simple theorem:

Observation 21:Let be an algebraically closed field and let be proper variety over , and open subscheme, and its complement. Then, for each prime and each there is a long exact sequence in compactly supported cohomology

*Proof:* For example one can see [Mil, Remark III.1.30].

*Remark 22:* Note that as exemplified in Observation 21, and as will persist below, one can generally be pretty loose with what scheme structure one is taking on a closed subscheme. Namely, if is a closed subset of a scheme then essentially all the scheme structures on have the same underlying reduced scheme. But, the passage from to doesn’t change the etale topology by the topological invariance of the etale site (e.g. see [Stacks, Tag03SI]).

It is a result like Observation 21 that we’d like to have for normal cohomology. Note that it will necessarily need to be more complicated since the weird functorial properties of compactly supported cohomology made the above possible: namely we have maps from and both in to but maps on cohomology in different directions. This phenomenon cannot occur for normal cohomology which is always contravariantly functorial. So, the open embedding gives rise to a map

and while it wouldn’t make sense for the kernel of this map to be it could, hopefully, still be some cohomology group related to .

In particular, we already know what the answer should be at least for . Namely, for an arbitrary sheaf we can certainly set

In words one should think of as the ‘sections of with supports in ‘. One can check that the functor

is left exact and we define to be its -derived functor. We call this the *cohomology with supports in *.

As promised, and essentially as constructed, we have the following:

Proposition 23:Let be a scheme, an open subscheme, and its closed complement. Then, for any abelian sheaf on there is a long exact sequence of abelian groups

*Proof:* This is [Mil, Proposition III.1.25] or [Fu, Proposition 5.6.11].

The last thing we mention in this subsection is that there is a natural way to interpret the group in terms of the theory of torsors (e.g. see the notes from this post). Namely, one can show that

Here two pairs and are isomorphic if there is an isomorphism so that . With this interpretation the map

can be interpreted as sending to the pair where is the multiplication by map. Similarly, the map

can be thought of as the map which associates to a pair the torsor .

Even though it’s not strictly necessary for our computations, I think it’s useful to explain a somewhat confusing point (or at least it was confusing to me the first time I learned this stuff) about cohomology with supports. Namely, we explicitly built as follows:

one then might wonder what the relationship between cohomology with supports and the sheaf .

In particular, an incredibly naive guess one might have is that is the cohomology of , but this is totally wrong. For example, note that, by definition we have an exact sequence

and so we get a long exact sequence that looks like

and if we want this to match Proposition 23 we’d want to match but this is horribly, horribly wrong. For example, in the case where is an affine open subcurve of the proper connected smooth curve then one has that and so

which, as Proposition 2 tells us, is quite different from in general.

That said, there is a relationship between cohomology, , and . Namely, let us set and call it the *shriek back* of along . This scary terminology comes from the fact that is an adjoint pair (e.g. see [Fu, Proposition 5.4.2]) where, in general, for a map we define so that is an adjoint pair. One can show that is left exact (e.g. see loc. cit.) and thus we have a well-defined notion of the right-defined functors

which are often, somewhat confusingly, written as . In particular, note that is a sheaf on .

Now, as one can check one has that and so, in particular, we see that

so we see that the composition

is precisely . In particular since (as one can check) preserves injectives we obtain the following from the Grothendieck spectral sequence:

Proposition 24:Let be a scheme, an open subscheme, and the complementary closed subscheme. Let be an abelian sheaf on . Then, there is a spectral sequence

Let us note that, in particular,

Thus, we see that this spectral sequence, in particular, gives us a measure on how far is from just the naive object .

One intuition one might garner from thinking of as the ‘cohomology of with supports in ‘ is that this cohomology group perhaps shouldn’t change when one shrinks the space around . One can also roughly intuit this from Proposition 23 since if we shrink around we will also shrink and so the cokernel and kernels of the maps from should be insensitive to this simultaneous shrinking.

The excision theorem, which we presently state, is precisely the rigorous version of this intuition:

Proposition 25 (the Excision Theorem):Let be a scheme, an open subscheme, and the complementary closed subscheme. Suppose that is an etale morphism with the property that is an isomorphism. Then, there is a natural isomorphismfor all .

*Proof:* See [Mil, Proposition III.1.27].

One useful corollary of this comes from considering what it says when is just a point . Namely, for every pointed etale map (an etale neighborhood of ), such that is the only point in the fiber over and is an isomorphism, the excision theorem tells us that

We might then try to pass to the limit over all the etale neighborhoods of (i.e. ‘shrink the neighborhood to zero’) and hope that the limiting structure is more concrete.

This, in fact, can be done but only with care:

Lemma 26:Let be a Noetherian scheme and let be a closed point of . Then, there is a natural isomorphismwhere is the henselization of and is the obvious map.

*Proof:* Using Proposition 23 and the Excision Theorem one reduces to showing this as a statement roughly showing that there exists a cofinal sequence of etale neighborhoods lf such that both *and* are quasi-compact. Indeed, once one knows this one obtains the result by combining the Excision Theorem, Proposition 23, [Stacks, Tag09YQ] , the definition of Henselization (e.g. see [Fu, Page 102] and [Stacks,05KS]), and [Stacks, Tag01YX]. The reason that such a sequence exists is that one can consider only affine neighborhoods (for which ) and for such we have that is Noetherian since is etale (and so is locally Noetherian) and compact. So then, not only is quasi-compact but so then is .

This focusing on Noetherianity assumptions may seem a bit dramatic/pedantic now, but it turns out to be a pivotal difference between schemes and adic spaces — in the former category Noetherianity conditions cover essentially all reasonable examples, and in the latter this is *far *from true.

Regardless, the excision lemma (and by extension Lemma 26) allow one to often times reduce questions about cohomology with supports to much more familiar ones. Namely, the fact that cohomology with supports is insensitive to shrinking of etale neighborhoods and the fact that every smooth scheme etale locally looks like affine space makes one wonder whether etale cohomology with supports can be reduced to a calculation of just affine space.

The key, simple to prove, observation necessary to make this precise is the following:

Lemma 27:Let be a scheme, a smooth scheme of relative dimension , and a closed subscheme (fiberwise) pure of codimension , such that is smooth. Then, for every point there exists a neighborhood of and an etale map such that .

*Proof:* See [SGA 1, Théorème II.4.10].

Let us say that a pair as in Lemma 27 is a *smooth pair of codimension * over . Then, one can interpret Lemma 27 as saying that every smooth pair of codimension over etale locally looks like . Thus, by etale local nature of one can generally reduce abstract computations to ones of the form which can (hopefully) be explicitly computed)

It is precisely by this reduction to specific this specific case that the following purity result is proven:

Proposition 28 (the Purity Theorem):Let be a smooth pair of codimension over and let be a locally constant torsion sheaf whose stalks have order invertible in . Then,

*Proof: *This is [Mil, Theorem IV.5.1] and/or [Fu, Corollary 8.5.6]. The proof is, as we indicated above, a reduction to the affine case and then an explicit computation.

*Remark 29:* The kind of explicit computation needed in the above is, essentially, the one done in this post, at least in the case when .

As a corollary of the Purity Theorem and Proposition 24 we immediately get a more concrete statement:

Corollary 30:Let be a field and let a smooth -variety of dimension and a smooth closed -variety of pure codimension . Then there is an isomorphism of -modules

[Con] Conrad, B., 2015. *Abelian varieties*. https://stuff.mit.edu/people/fengt/249C.pdf

[EP] Engler, A.J. and Prestel, A., 2005. *Valued fields*. Springer Science & Business Media.

[Fu] Fu, L., 2011. *Etale cohomology theory* (Vol. 13). World Scientific.

[Har] Hartshorne, R., 2013. *Algebraic geometry* (Vol. 52). Springer Science & Business Media.

[Lit] Litt, Daniel. *Cohomology of curves.* https://www.daniellitt.com/s/cohomologyofcurves.pdf

[Mac] Mack-Crane, Sander. *Normalization of Algebraic and Arithmetic Curves*. https://algebrateahousejmath.wordpress.com/2016/11/27/normalization-of-algebraic-and-arithmetic-curves/

[Mil] Milne, J.S. and Milne, J.S., 1980. *Etale cohomology (PMS-33)* (No. 33). Princeton university press.

[Nag] Nagata, M., 1962. Local rings. *Interscience Tracts in Pure and Appl. Math.*

[Ols] Olsson, M., 2016. *Algebraic spaces and stacks* (Vol. 62). American Mathematical Soc.

[Poo] Poonen, B., 2017. *Rational points on varieties* (Vol. 186). American Mathematical Soc.

[SGA 1] Grothendieck, A. and Raynaud, M., 2002. *Revêtementset groupe fondamental (SGA 1)*. arXiv preprint math/0206203.

[Stacks] The Stacks Project Authors, *The Stacks Project*, https://stacks.math.columbia.edu

[Star] Starr, Jason. *Brauer groups and Galois cohomology of function
fields of varieties*. https://www.math.stonybrook.edu/~jstarr/papers/Escola_07_08d_nocomments.pdf

[Sut] Sutherland, Andrew. *Totally ramified extensions and Krasner’s lemma*. http://math.mit.edu/classes/18.785/2016fa/LectureNotes11.pdf

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.

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 :

and

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.

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 groupswhere 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 functorgiven by

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

with

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.

[Con] Conrad, B., 2015. *Abelian varieties*. https://stuff.mit.edu/people/fengt/249C.pdf

[EP] Engler, A.J. and Prestel, A., 2005. *Valued fields*. Springer Science & Business Media.

[Fu] Fu, L., 2011. *Etale cohomology theory* (Vol. 13). World Scientific.

[Har] Hartshorne, R., 2013. *Algebraic geometry* (Vol. 52). Springer Science & Business Media.

[Lit] Litt, Daniel. *Cohomology of curves.* https://www.daniellitt.com/s/cohomologyofcurves.pdf

[Mac] Mack-Crane, Sander. *Normalization of Algebraic and Arithmetic Curves*. https://algebrateahousejmath.wordpress.com/2016/11/27/normalization-of-algebraic-and-arithmetic-curves/

[Mil] Milne, J.S. and Milne, J.S., 1980. *Etale cohomology (PMS-33)* (No. 33). Princeton university press.

[Nag] Nagata, M., 1962. Local rings. *Interscience Tracts in Pure and Appl. Math.*

[Ols] Olsson, M., 2016. *Algebraic spaces and stacks* (Vol. 62). American Mathematical Soc.

[Poo] Poonen, B., 2017. *Rational points on varieties* (Vol. 186). American Mathematical Soc.

[SGA 1] Grothendieck, A. and Raynaud, M., 2002. *Revêtementset groupe fondamental (SGA 1)*. arXiv preprint math/0206203.

[Stacks] The Stacks Project Authors, *The Stacks Project*, https://stacks.math.columbia.edu

[Star] Starr, Jason. *Brauer groups and Galois cohomology of function
fields of varieties*. https://www.math.stonybrook.edu/~jstarr/papers/Escola_07_08d_nocomments.pdf

[Sut] Sutherland, Andrew. *Totally ramified extensions and Krasner’s lemma*. http://math.mit.edu/classes/18.785/2016fa/LectureNotes11.pdf

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We start by recalling a very well-known, and easy fact:

Proposition1:Let be a scheme and a surjective map of -schemes. Assume that is universally closed, then is universally closed. In particular, if is proper and is separated and finite type then is proper.

This is one-hundred percent intuitive if we think of proper as being a sort of ‘relative compactness’–it just says that the image of a compact set is compact. The proof is also quite simple:

*Proof (Proposition 1):* Let be a morphism. We need to show that the map is closed. But, if is closed then so is where is the base change map. But, since is closed (since is universally closed) we know that the image of is closed. But, this coincides with the image of under and the conclusion follows.

In practice though properness is not always enough for applications and so one might ask the following question:

Question 2(a):Let be a scheme and a surjective map of -schemes. If is separated and of finite type and is projective, then is projective?

Or, perhaps, a better question to first ask in the ‘non-relative’ situation:

Question 2(b):Let be a field and let be a surjective map of separated -varieties. If is projective, then is projective?

In fact, the answer to this question is *obviously no *even if you require to be projective and birational by Chow’s theorem which says that if is a proper -variety then there exists a projective birational map with a projective -variety.

Of course, this only gives a negative answer to Question 2(b) if one is able to produce a proper -variety which is not projective. Such objects are always somewhat tricky to discuss, so let’s begin by recalling when properness and projectivity do agree:

Theorem 3:Let be a field and let be a proper -variety. Then, is projective in each of the following situations:

- The dimension of is .
- (Zariski–Goodman) The dimension of is and is smooth.

Before giving references for these proofs we note the following well-known lemma:

Lemma 4:Let be a field and an extension. Suppose that is a -variety such that is projective. Then, is projective.

*Proof: *For example, one can see [EGA II, Corollaire 6.6.5]. Since one can, by the standard tricks, reduce the case when is finite we actually discuss a generalization of this later in the form of Corollary 8.

With this lemma in hand we can proceed to give references for Theorem 3:

*Proof(Theorem 3):* For 1. one can see this (note that by this the notion of ‘-projective’ and ‘projective’ do not differ). For 2. see [Bad, Theorem 1.28].

An example of a smooth proper non-projective -fold is the famous example of Hironaka (e.g. see this exposition). We will see later an example of a non-smooth surface which is proper but not projective.

Now, while Question 2(b) (and thus Question 2(a)) have negative answers, one might hope that if one imposes that is finite in Question 2(b) that one obtains an affirmative answer. Namely, one has the following:

Question 2(c):Let be a field and let be a surjective finite map of separated -varieties. If is projective, is projective?

To explain why Question 2(c) *might* have a affirmative answer we explain a proof of the following slight modification:

Theorem 5:Let be a field and let be a finite flat surjection of separated -varieties. Then, if is projective then is projective.

To explain the proof it is useful first to define the ‘norm map’. Namely, let’s suppose that and are arbitrary schemes and is a finite flat map locally of finite presentation (this last condition is superflous if is Noetherian). Then, by standard theory, the quasi-coherent -module is locally free. Then, there is naturally a norm map

which is the globalization of the notion of the norm for a finite free -algebra—-this is just the map where is the left multiplication map (where we are thinking of as a free -module and so the determinant makes sense). Note that and we prefer to use the latter notation.

We thus get an induced map

where the subscript ‘et’ denotes etale cohomology, and the latter isomorphism is the usual one. One then uses the fact that finite maps are acyclic for the etale topology (e.g. see [Fu, Proposition 5.7.4]) to see that we have a canonical isomorphism

But, since this first group is just (as we’ve already mentioned) we see that putting everything together we get a map

called the *norm map*.

In words, what this really means is that if we start with a line bundle on this can be thought of as gluing together local pieces of via transition maps in . We can then imagine as being line bundle on obtained by gluing together locally using the norm of the transition maps.

*Remark 6:* Introducing etale cohomology is not at all necessary. It just is the approach that makes the most sense to me (and shows why etale cohomology is useful even for ‘Zariski constructions’). Namely, I used in the above that is zero where is taken in etale category, but it vanishes even in the Zariski category (e.g. see this).

The reason that the norm map is useful, besides being a natural way to produce line bundles, is the following

Lemma 7:Let be a finite flat map locally of finite presentation. Let be an ample line bundle on , then is an ample line bundle on .

The proof of this lemma is not too hard, and we leave it to the reader as an excercise (cf. [EGA II, Proposition 6.6.1] or this).

In particular, we deduce the following:

Corollary 8:Let be a field and let be a finite flat surjection of separated -varieties. Then, if is projective then is projective.

*Proof:* By Proposition 1 we know that is proper. Since is projective it has an ample line bundle and then is an ample line bundle on . The conclusion then follows from standard theory (i.e. use the result from this link to say that there is a dense open embedding in to a projective -scheme but use the fact that is proper to deduce that the image is closed, and thus its an isomorphism).

One can give a more concrete proof in the case when is a Galois cover whose degree is invertible in .

Proposition9:Let be a fieldand let be a finite etale surjection of connected separated -varieties such that there is a dominating Galois cover of degree invertible in . Then, if is projective then is projective.

*Remark 10:* This assumption that there is a dominating Galois cover whose degree is invertible in is automatically true if characteristic of is zero. This is also probably non-essential, but certainly saves us some headache–there’s probably just ean easier way to check this (e.g. just prove that ampleness is etale local using the fact that affineness can be checked etale locally), but I like this proof. Note also that there is always a dominating Galois cover (e.g. see [Fu,Proposition 3.2.10]).

*Proof:* Clearly we may assume without loess of generality that is Galois with degree invertible in . Let be the Galois group of . We then claim that is an ample line bundle. Let us begin by noting that is in fact a line bundle. We can check this etale locally on (e.g. we’ve already implicity used thsi fact, but it follows, for instance from [Ols, Proposition 4.3.8]). But, since is isomorphic, as an -scheme, to this is obvious. Thus, it suffices to show that is ample.

But, by [Har, Proposition III.5.3] it suffices to show that for every coherent -module there exists some such that for each and we have that . But, by our assumption that is ample there is such an for . We claim that this also works for . Indeed, note that we have the Hochschild-Serre spectral sequence (e.g. see [Mil, Theorem 2.20])

(note that we have used Zariski and not etale cohomology, but there is no difference since our objects are quasi-coherent). But, note that since is a -space and is invertible on that

for (e.g. see Corollary 12 and the suceeding discussion in this post). Thus, we deduce that

But what is this right-hand side? Namely, note that

where we have used the equality which can be checked after finite etale cover which is easy to do, by hand, on which is isomorphic, as an -scheme, to . But, then evidently the right-hand side of vanishes by the way we chose and thus the left-hand side also vanishes. The conclusion follows.

We now explain how to produce a weird surface with the property that its proper but not projective but has normalization . Of course, since the normalization map is finite (as we will see by hand in our case) this gives us an explicit counter example to Question 2(c).

*Remark 11: *This example is taken from this mathoverflow post.

To construct this example let’s take to be an algebraically closed field (of characteristic just to be safe). We now fix two curves and in with the former having degree and the latter having degree and which intersect tangentially at a point . We know that and are abstractly isomorphic (e.g. see [Vak, §19.3]) and we can assume that maps to so we can take the quotient variety that identifies and .

More rigorously, let us fix an isomorphism and consider the pushout

where is the closed embedding and is the finite surjection which is the identity on and the isomorphism on . One can show that the scheme exists by applying [Sch, Theorem 3.4] affine locally (or one can see the above cited mathoverflow post for more details). We then, by definition, get a surjection which is finite. One can see this by the explicit construction or by pure power of thought. Indeed, it’s not hard to see that is separated (by the explicit gluing criterion) and since is proper we know that is proper. It’s also clear that is quasi-finite from which the conclusion follows (e.g. see [Vak, Theorem 29.6.2]). Finally, it’s clear that is an open embedding and so is birational. Thus, we know that is the normalization map (e.g. see [GW, Proposition 12.44]).

So, our finite normalization map is finite and, since is separated, we know that is proper (e.g. by Proposition 1). We claim though that is *not* projective. Indeed, suppose that it were. Then, evidently there would exist an ample line bundle on . Note then by standard theory (e.g. see this post) that would be an ample bundle on . But, note that is trivial for all on . Indeed, say that . Note then that by standard theory (e.g. see this and apply Bezout’s lemma) if denotes the tautological closed embeddings then and . But, note that if is our tautological closed embedding then by our definition of we have that

which implies that and so . Thus, can never be ample, and thus neither can be ample. Thus, is not projective.

[Bad] Badescu, L., 2013. *Algebraic surfaces*. Springer Science & Business Media.

[EGA II] Grothendieck, A., 1961. Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné): II. Étude globale élémentaire de quelques classes de morphismes. *Publications Mathématiques de l’IHÉS*, *8*, pp.5-222.

[Fu] Fu, L., 2011. *Etale cohomology theory* (Vol. 13). World Scientific.

[GW] Görtz, U. and Wedhorn, T., 2010. *Algebraic geometry*. Wiesbaden: Vieweg+ Teubner.

*Etale cohomology (PMS-33)* (No. 33). Princeton university press.

[Ols] Olsson, M., 2016. *Algebraic spaces and stacks* (Vol. 62). American Mathematical Soc.

[Sch] Schwede, K., 2005. Gluing schemes and a scheme without closed points. *Contemporary Mathematics*, *386*, p.157.

[Vak] Vakil, R., 2017. *The Rising Sea: Foundations of Algebraic Geometry* (Ver. Nov. 18 2017). http://math.stanford.edu/~vakil/216blog/FOAGnov1817public.pdf

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In a previous post I discussed the classification of one-dimensional connected algebraic groups over a field . There an ‘algebraic group’ over meant a smooth affine group of finite type over .

Since then people have asked me directly and/or via the searches that led them to this blog asked what happens in the non-affine case. I’ll answer that question in this post by classifying one-dimensional connected geometrically reduced group schemes over (with some mild assumptions on ).

The answer, as we will see, essentially breaks in to two cases: the affine case (as discussed in the previous post) and the elliptic curve case. The two are very separate and distinct.

Let us now clarify exactly the assumptions we want and what basic properties we are able to get from them. Namely, we are interested in classifying one-dimensional groups over but without more adjectives this will be impossible. So, let discuss what adjectives we want, and what they give us.

Throughout a *group variety* over will mean a finite type group scheme over .

The first assumption that we want to assume about is that it’s connected. The reason for this is silly. If you don’t insist on working with connected groups you have the entire theory of finite groups to contend with. Indeed, for any finite group one can obtain a one-dimensional group (which is even smooth with reductive identity component) as where is the constant group. For this reason we will want to assume that is connected.

Let us note that in general if you don’t want to assume that is connected then you have a short exact sequence of group varieties

where is the identity component of and , the *component group* of , is a finite etale group scheme. In particular, if is algebraically closed then is essentially the same just an abstract finite group. Thus, one sees that the general non-connected situation comes down to classifying connected groups (which we will essentially do), finite groups (in the form of , and extensions of the latter by the former.

Let us also make an observation about connected groups, or more generally connected varieties with -points, that will be useful later (this was also in the last point):

Observation 1:Let be a field and a finite type -scheme such that . Then, is connected if and only if is geometrically connected.

*Proof:* Evidently if is connected then so is (since we have a continuous surjection ). Suppose now that is connected. Note that for an extension one has that is disconnected if and only if contains non-trivial idempotents. In particular, since every element of lies in (note that we’re applying `flat base change’ here) for some finite extension we see that disconnected implies that is disconnected for some finite extension .

That said, note that is finite and flat (since is finite and flat) and thus a clopen map. In particular, since is connected we see that every connected component of surjects on to . In particular, fixing we see that every connected component of contains a preimage of . But, since is a -point it has only one preimage. Since connected components are disjoint this implies that has only one connected component. The conclusion follows.

As a corollary we obtain the following:

Corollary 2:Let be a field and a group variety over . Then, is connected if and only if is geometrically connected.

The second assumption we will make is that our group is geometrically reduced. Recall that if is perfect this is the same thing as being reduced. The reason that such assumptions are necessary for a reasonable classification is the prescence of infinitesimal group schemes in characteristic .

Namely, recall that a group scheme over is *infinitesimal* if is the trivial algebraic group.

*Example 3:* Define where is the (relative) Frobenius map on which on -points is the additive map . One can see that is, as a scheme, just and so evidently is trivial. So, is infinitesimal.

Infinitesimal groups can be quite complicated. Their difficulty is somewhat comparable to the theory of finite groups. So, they are something we would like to ignore. For this reason we shall want to assume our groups are reduced. Moreover, to ignore issues with base change it’s also useful to assume that our groups are actually geometrically reduced (there exists reduced groups which are not geometrically reduced–see Examples 1.57 and 1.58 of [Mil]).

Let us note that if we are only interested in characteristic then this is no real assumption:

Theorem 4(Cartier):Let be a characteristic field. Then, every group variety over is (geometrically) reduced.

*Proof:* See [Mil, Theorem 3.23].

The assumption that is one-dimensional is, of course, necessary for a concise classification. Else we have to contend with all of abelian varieties, linear algebraic groups, … Hopefully this is convincing enough.

Let be a group variety over . We see what implications we get if we assume that is a one-dimensional connected geometrically reduced group variety.

In fact, geometrically reducedness for group varieties implies smoothness:

Lemma 5:Let be a field and a group variety over . Then, is geometrically reduced if and only if it’s smooth over .

*Proof:* If is smooth over then it’s geometrically reduced by standard theory. Conversely, suppose that is geometrically reduced. We need to show that is regular. This means that we need to check that the local rings at all points of are regular. But, by standard theory it suffices to check at only the closed points. By generic smoothness for geometrically reduced varieties we know that there is some point such that is regular. But, for any other closed point the left translation by map is an automorphism of varieties which carries to . Thus, is also regular.

In fact, separatedness is true for any group:

Lemma 6:Let be a field and a group variety over . Then, is separated.

*Proof:* It suffices to show that the diagonal is closed. But, the map given on -points by is clearly a morphism and where is the identity element. Since is a closed point we see that is separated as desired.

From our assumptions that is smooth and geometrically connected we immediately get the following from general theory:

Lemma 7:Let be a field and let be geometrically connected smooth finite type -scheme. Then, is geometrically integral.

*Proof:* Evidently we may assume that is algebraically closed. Suppose that had more than one irreducible component–say that and are distinct irreducible components of . Note then that if then is not a domain since and give rise to two distinct minimal primes of . But, since is regular we know that is a domain, and this is a contradictoin. Thus, the irreducible components of are disjoint. But, since is finite type we know it has finitely many irreducible components. This implies that the irreducible components are clopen. Since we assumed that had more than one irreducible component this contradicts that is connected.

But, in fact, we can also derive this irreducibility from the theory of group schemes:

Lemma 8:Let be a field and let be a connected group variety over . Then, is geometrically irreducible.

*Proof:* From Observation 1 it suffices to assume that is algebraically closed. Suppose that were not irreducible. Then, as in the proof of Lemma 7 there must be two irreducible components and of that intersect and, in particular, contain a common -point . Note though that is not contained in the union of the other irreducible components (by irreducibility!) and thus there is some -point of which is contained within a unique irreducible component. Let be the left translation by automorphism. This takes to . But, since lies on the intersection of two irreducible components of and lies in a unique irreducible component of this is a contradiction.

One thing that we get for free from our assumptions, in fact just the fact that we’re working with a geometrically connected separated curve, is that our group is either affine or projective. Namely, we have the following:

Proposition 9:Let be a field and let be a one-dimensional geometrically connected separated variety over . Then, either is affine or projective.

The idea of the proof is simple. Namely we will show that is at least contained in a projective curve and then use the fact that since we’re in one dimensional all proper open subvarieties of a projective curve are affine.

We begin by noting the following observation:

Observation 10:In the proof of Proposition 9 it suffices to assume that is smooth over .

*Remark 11:* We are using the definition of projectiveness as in here. This is contrast to the definition that Hartshorne uses (which is called *-projective* in loc. cit). This makes no real difference when discussing projectiveness of a variety, but does have difference in general. For instance, in Hartshorne’s definition of a projective morphism it needn’t be true that finite maps are projective (e.g. see the remark here).

We will cite the following lemma for simplicity:

Lemma 12:Let be a field and let be a connected one-dimensional variety over . Then, is projective if and only if it’s proper.

*Proof:* If one assumes that is smooth and integral (which is the case we’re in for our groups) then this is somewhat easy. Namely, let be an affine open in and let be the common function field between this affine open and . Note that if we embed in to some we can take the closure, which we’ll denote , to obtain a connected projective variety. It may be true that is singular. But, after taking its normalization (see below for references for normalization) we get a smooth projective integral curve which is birational to (since is birational to and contains an open that is also an open in ). One can then use the valuative criterion for propertness to take the birational map and extend it unique to a map which is necessarily an isomorphism.

For the general proper case see this (see Remark 12 for why this phrase ‘-projective’ doesn’t matter here).

*Proof (Observation 10):* By our assumptions we may clearly assume that is algebraically closed. Then, we know from standard theory that the smoothness of is equivalent to the regularity of . But, since is one-dimensional this is equivalent to the normality of . But, note that we have the normalization map . This is a surjection, this much is clear. What lies significantly deeper is that, in fact, is finite. For example, one can see [Vak, Theorem 9.7.3]. At a deeper level what is happening is that varieties are Nagata since locally their functions are finite type over a field which implies Nagata by basic theory. It is then also well-known that Nagata schemes have finite normalization maps.

Regardless, note that is regular, thus smooth and so to finish it suffices to prove that is affine or projective if and only if is.

If is affine then , being a finite over , is also affine. If is projective, then is projective since finite maps are projective and composition of projective maps are projective.

If is affine then is affine. Indeed, this follows from non-trivial general theory about finite morphisms and affineness (it is really this tag that is most relevant). If is projective then is necessarily proper by standard theory and thus projective by Lemma 12.

So, we are now free to assume that is smooth. It is now not hard to justify our intuitive sketch of proof after the statement of Proposition 9:

*Proof(Proposition 9):* We may assume that is algebraically closed.

Let be an affine open subscheme of . As in the first paragraph of the proof of Lemma 12 we can find an open embedding where is smooth projective geometrically connected curve over . By the valuative criterion for properness we can uniquely extend to a map . We claim that is an open embedding.

Indeed, let us note that is certainly quasi-finite since it’s quasi-finite on (it’s an open embedding) and is finite (since is one-dimensional). Thus, by Zariski’s main theorem (evidently our maps and schemes are quasi-compact and separated) we can find a factorization

where is a dense open embedding and is finite. It’s clear we may assume that is integral since both and are integral. Moreover, note since the normalization map is an isomorphism on the smooth locus and finite we can, up to replacing with , assume that is smooth over . Note then that is a birational map of smooth proper curves, so an isomorphism. Indeed, since is surjective we know that it’s flat (e.g. see [Qin, Proposition 3.9]). Note then that is vector bundle on . So, to show that is an isomorphism (which is clearly all we need to do) it suffices to show it’s an isomorphism generically, but this is true since is birational. The claim that is an open embedding then immediately follows.

Now, since is an open subscheme of , a smooth projective integral curve, it suffices to show that any such variety is affine or projective. This follows from the lemma following this proof.

Lemma13:Let be a field and a smooth geometrically integral projective curve over . Then, any open subscheme is affine.

*Remark 14:* This was also contained in the previous post.

*Proof:* We may assume that is algebraically closed. If is empty, we’re done. So, assume that is not empty. Then, for some -points . Consider then the line bundle , where for .

Now, choosing the sufficiently large, we know that will have negative degree, and so we may assume that . So, by choosing the sufficiently large, we can made sufficiently large. In particular, we can find such that has poles precisely at the . Thus, we obtain a non-constant rational map off of the . Extending this to a morphism , we obtain a finite map with , and thus is affine.

*Remark 15:* There is almost certainly an easier way to do the above. For example, if one uses this your life becomes much easier. In the above I don’t really use this proposition (save I use part of it in the proof of the non-smooth case of Lemma 12). The reason I opted to not use it is that while I think this approach is more elementary, it’s somehow trickier than what I did above. A lot of the techniques used above are incredibly common and worth internalizing (albeit maybe overkill).

We are now ready to start classifying one-dimensional connected geometrically reduced group varieties over a field . We will break our discussion in to its two separated, and distinct cases: the affine case and the projective case.

In short, the projective one-dimensional geometrically reduced group varieties are just elliptic curves. This is what we endeavor to prove. But, before we state this precisely let us remind ourselves what an elliptic curve is.

*Definition 16:* Let be a field. Then, an *elliptic curve* over is a pair where is variety of the form is a smooth curve where has homogenous degree and .

Of course, we know that elliptic curves over have a group law. More explicitly, let be distinct. There exists a unique line (i.e. a closed subscheme of the form where is a degree one homogenous polynomial) such that passes through and . If we set to be the unique line in passing through and which is tangent to at (i.e. is the line where ).

Note then that by Bezout’s theorem that for any (possibly with ) we have that (by which we rigorously mean the fiber product ) is a finite -scheme with -dimensional global sections. If then it’s easy to see that is the disjoint union of three -points which are and a third point . If then is the disjoint union of (where for a closed subscheme of we denote by the ideal of induced by ) and an -point . In either case we see that we obtain a third point from .

Note then the exact same ideas yield associated to a line and a third point of intersection of the line and . We denote this third -point of by .

We then have the following:

Theorem 17:Let be a field and an elliptic curve over .

- The operation given by described above endows with the structure of a group.
- There exists a unique group variety structure on such that the group structure on agrees with that from 1.

*Proof:* The proof of 1. is [Sil, Proposition 2.2]. The unicity of the group scheme structure in 2. is clear. Indeed, let be a multiplication map. Since is a dense subset of , and all schemes in consideration are separated, the map is determined by its value on . The existence of such (i.e. the algebraicity of the group operation from 1. and the fact that it’s defined over ) follows from [Sil, Group Law Algorithm 2.3].

We are now able to state our desired broad classification in the projective case:

Proposition 18:Let be a field. Every elliptic cruve is a one-dimensional geometrically connected proper group variety over . Conversely, if is a one-dimensional geometrically connected projective group variety over then is isomorphic (as a group variety) to an elliptic curve.

*Proof:* Let us first observe that elliptic curves are certainly one-dimensional, smooth, and projective (by definition). Thus, it remains to show why they are connected. There a multitude ways to prove this (Bezout’s lemma, [Vak, Exercise 11.3.F],…) but we list here a cohomological one. It suffices to show that is a one-dimensional space since this then forces the ring to be and thus to have no non-trivial idempotents. But, note the ideal sheaf of is evidently and, in particular, is a line bundle. We have the short exact sequence of sheaves

where is the tautological closed embedding. We then get a long exact sequence on cohomology group that contains the portion

But, evidently (e.g. see [Vak, Theorem 18.1.3]) and equally evident is and thus we see that

as desired.

Conversely, suppose that is a one-dimensional connected geometrically reduced group variety. By the Rigidity Lemma (which immediately succeeds this proof) it suffices to show that there exists an isomorphism of pointed varieties where is an elliptic curve and and are the respective identity elements. Of course, it suffices to show that there exists an isomorphism of varieties since we could postcompose with translation by an appropriate element of to guarantee that mapsto .

We begin by noting that necessarily has genus . There are several ways to see this, but we proceed with the one that is simplest. Namely, it’s not hard to show that any group variety is parallelizable (i.e. has cotangent bundle which is free) by using the theory of invariant differentials (e.g. see [BLR, §4.2 Corollary 3]). In particular, we see that and thus . But, (e.g. see [Har, Example 1.3.3]). Thus, as desired.

We now need to show that is isomorphic to an actual variety of the form where is a smooth cubic function. To do this let us begin by noting that since we have by standard theory (e.g. see [Vak, Conclusion 19.2.11]) that is very ample. Let us note also that, by Riemann-Roch, we have that

but since , which has negative degree, the left-hand side of the above simplifies to . So, . Thus, we see determines a closed embedding . So, is isomorphic to a closed subscheme of . Since we know that we have that corresponds to a height one homogenous prime in which is automatically principal. Thus, (or more precisely the image of ) is actually a hypersurface, say .

Thus, it remains to show that . But, let us note that if then the ideal sheaf of is . Using the long exact sequence in cohomology for

(as we’ve already discussed in this proof) we see that (again using [Vak, Theorem 18.3.1]) that

But, note that since is a closed embedding we know from standard theory (e.g. [Har, Exercise 8.2]) that

and the right-hand side is one dimensional since has genus . Thus, in conclusion we see that . But, (e.g. see again [Vak, THeorem 18.3.1]). Thus, as desired.

Lemma 19(The Rigidity Lemma):Let be a field. Let and be geometrically integral schemes of finite type over and a separated -scheme. Let be a morphismof -schemes, and assume further that

- is proper.
- For some algebraically closed extension there exists some such that the restriction to is a constant map to some .
Then is independent of ; i.e. there exists a unique morphism of -schemes such that .

In particular, if are smooth geometrically integral proper -varieties with identity elements then any morphism of pointed -varieties is a morphism of group varieties.

*Proof:* See [Con, Theorem 1.7.1] for the first statement. The second statement follows from the first by noting that the map given on points by is constant on and thus is trivial. This implies that is a group morphism.

This was taken care of in the previous post. In particular, we have the following:

Theorem 20:Let be a field and let be an affine one-dimensional connected geometrically reduced group variety over . Then, or . If then the assumption that is geometrically reduced is unnecessary.

Combining our two cases we arrive that the following:

Theorem 21:Let be a field and let be a one-dimensional connected geometrically reduced group variety over . Then, one of the following holds:

- is an elliptic curve.
If then the assumption that is geometrically reduced is unnecessary.

Theorem 21 is lacking in two orthogonal ways. The proper case gives us a nice classification over but it’s pretty inexplicit (how do we explicitly parameterize elliptic curves up to isomorphism?). The affine case is very explicit, but only works over . Our goal now is to remedy both of these issues in (essentially) full generality. In particular, the main assumption we will often make is that is perfect.

We begin by trying to understand how to explicitly parameterize elliptic curves over . We begin with a well-known and simple first case:

Proposition 22:Let be an algebraically closed field. Then, associated to every elliptic curve over is an element . This integer only depends on the isomorphism class of and induces a bijection

The element from Proposition 22 is the so-called *-invariant* of . Its definition can be given purely in terms of a defining equation for as in [Sil, §3.1]. The proof of Proposition 22 is then the contents of [Sil, Proposition III.1.4.(b)]. If we want to explicitly construct an inverse for the bijection in Proposition one associates to the elliptic curve with the polynomial

at least if . If one can take the elliptic curve with equation

and if one can take the elliptic curve with equation

which covers all cases. We denote these explicit elliptic curves as for any

In fact, from the simple observation that has a model over (that with the same equation) we actually deduce the following strengthening of Proposition 22:

Proposition 23:Let be a field. Then, the mapis a surjection with fibers the sets

Thus it really remains to explicitly understand the sets . These sets are exactly the sets of ‘twists’ of (e.g. see the discussion of twists in the notes of this post). For our sanity, we now assume that is perfect so that is Galois. We then have the following:

Lemma 24:Let be a perfect field. Then, there is a natural bijection

*Proof:* For a down-to-earth proof see [Sil, §X.5]. The high level proof is as follows. Let be the category fibered in groupoids over the small etale site for . This is a stack (e.g. see [Ols, Theorem 13.1.2]). Thus, by Theorem 5.1 of the notes from this post the claim essentially follows. Namely, the only other reduction is the observation that by standard theory one has that

from where the conclusion follows.

*Remark 25:* I don’t truthfully know what happens over non-perfect fields. I suspect that as long as the characteristic is not or then the fact that is smooth should imply that all isomorphisms which occur over actually occur over in which case Lemma 24 is still valid. Even if this is true, I don’t know what can happen in the case of characteristic or . Please feel free to enlighten me if you know the answer.

So, all that remains to do to completely classify elliptic curves over , at least in the case when is perfect, is to calculate the groups for all . This turns out to be quite simple in the case when the characteristic of is not or . Indeed, we have the following:

Lemma 26:Let be a perfect field of characteristic not or . Then,where this is isomorphisms as group schemes over .

*Proof:* This is precisely [Sil, III.10.2].

Now, the Galois cohomology of is very simple in general. Namely, by Hilbert’s theorem 90 we know that in we have a canonical bijection and . In particular, we see that we have bijections

Thus, we can enhance Proposition 22 to the following:

Theorem 27:Let be a perfect field of characteristic not nor . Then, there is an explicit bijectionwhere if and where

The explictness in the statement of Theorem 27 means that there is an explicit inverse (since the map itself just sends to where is the element of that comes from our discussion). For an explicit description see [Sil, Corollary 5.4.3]. We mention though, since it’s the most common and most simple case, that if then as soon as one writes where is the homogenization of a polynomial of the form (which is always possible since we’re in characteristic different from or ) then the element that corresponds to is the elliptic curve whose affine equation is –this is the so-called *quadratic twist* of .

The affine case was covered in the previous post, so we just summarize the results here. Namely, we have the following:

Theorem 28:Let be a perfect field of characteristic not . Then, a one-dimensional connected affine geometrically reduced group variety over is either or a torus. Moreover, there is a bijectionsuch that that identity element of corresponds to and non-identity corresponds to the torus

Summarizing everything above we get the following:

Theorem 29:Let be a field. Then every one-dimensional connected geometrically reduced group variety over is either affine or an elliptic curve. If then the geometrically reduced assumption is automatically satisfied.Moreover, we have the following parameterization of these two families:

- If is perfect and of characteristic not or then there is an explicit bijection
where if and where

- If is perfect and of characteristic not then an affine is either or a one-dimensional torus. Moreover, there is a bijection
such that that identity element of corresponds to and non-identity corresponds to the torus

*Remark 30:* One can almost certainly remove the perfectness hypotheses from 1. in Theorem 29. Indeed, let and be elliptic curves over . We need only show that if then . But, note that is an fppf torsor for . But, is for by Lemma 26. Its simple to show that any fppf torsor for is representable and is smooth since is. So, is a representable by a smooth (in fact etale) -scheme over . So, it evidently has a point from where the conclusion follows.

[BLR] Bosch, S., Lütkebohmert, W. and Raynaud, M., 2012. *Néron models* (Vol. 21). Springer Science & Business Media.

[Con] Conrad, B., 2015. *Abelian varieties*. https://stuff.mit.edu/people/fengt/249C.pdf

[Har] Hartshorne, R., 2013. *Algebraic geometry* (Vol. 52). Springer Science & Business Media.

[Mil] Milne, J.S., 2017. *Algebraic groups: The theory of group schemes of finite type over a field* (Vol. 170). Cambridge University Press.

[Ols] Olsson, M., 2016. *Algebraic spaces and stacks* (Vol. 62). American Mathematical Soc.

[Qin] Liu, Q., 2002. *Algebraic geometry and arithmetic curves* (Vol. 6). Oxford University Press on Demand.

[Sil] Silverman, J.H., 2009. *The arithmetic of elliptic curves* (Vol. 106). Springer Science & Business Media.

[Vak] Vakil, R., 2017. *The Rising Sea: Foundations of Algebraic Geometry* (Ver. Nov. 18 2017). http://math.stanford.edu/~vakil/216blog/FOAGnov1817public.pdf

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The goal of the notes was to understand some of the representation theory surrounding Scholze’s paper on the cohomology of the Lubin–Tate tower. In particular I, Koji Shimizu, DongGyu Lim, and Sander Mack-Crane were/are interested in understanding whether there is a function field analogue of this paper.

In particular, it has an eye towards modular (i.e. mod ) representations of -adic groups. So, it discusses some of the classical theory of representations of -adic groups from a categorical perspective which serves one better in the modular representation setting. It also discusses the fascinating theorem of Kazhdan relating Hecke algebras for and where and are soemthing like the tilts of and .

Enjoy!

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Please feel free to leave any constructive comments!

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I will be continuing to edit this post with the most recent version of the exercises.

Please feel free to point out any errors and/or suggest any good problems!

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While the explanation of my work was the original goal of the notes, they have since evolved into a motivation for the Langlands program in terms of the cohomology of Shimura varieties, as well as explaining some directions that the relationships between Shimura varieties and Langlands has taken in the last few decades (including my own work).

I hope that it’s useful to any reader out there. Part I was mostly written with me, four years ago, in mind. So, in a perfect world someone out there will be in the same headspace as I was, in which case it will (hopefully) be enlightening.

In case you’re wondering the intended level for the reader is probably: 1-3 year graduate student with interest in number theory and/or arithmetic geometry. In particular, for Part I there is an assumption that the reader has some basic knowledge about: Lie groups, algebraic geometry, number theory (e.g. be comfortable with what a Galois representation is), algebraic group theory, and etale cohomology (although this can be black-boxed in the standard way–e.g. all one needs to know is the contents of Section 3 of this set of notes). Part II is mostly written as an introduction to a research topic, and so requires more background.

Enjoy!

PS, feel encouraged to point out any mistakes/improvements that you think are worth mentioning.

The Langlands conjecture and the cohomology of Shimura varieties

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