# Punctured disks and punctured curves and cohomology, oh my! (Part I: the proper curve case)

This is the first in a series of posts whose goal is to compute the cohomology of the (several times) punctured closed $p$-adic disk $\mathbb{B}_{\mathbb{C}_p}-\{p_1,\ldots,p_m\}$ as well as the cohomology of smooth (and some non-smooth) algebraic curves.

# Motivation

The original goal for this post was to discuss the calculation of the etale cohomology for the adic space $\mathbb{B}_{\mathbb{C}_p}-\{p_1,\ldots,p_m\}$ where $p_i\in\mathbb{D}_{\mathbb{C}_p}(\mathbb{C}_p)$ are classical points. Moreover, I wanted to illustrate some interesting theory (the theory of cohomology with supports for adic spaces) that helps with such computations and some thorny details which make the computation different than one might expect if one copied verbatim the algebraic analogue.

Because of this last sentence, but also to give the (essentially correct) blueprint for how we would compute this cohomology I wanted to discuss computing the cohomology of smooth connected curves or, equivalently, the cohomology of smooth connected projective curves which are punctured (several times). So, this post has grown quite a bit longer than originally intended.

# A model: the cohomology of curves

Before we begin actually computing the cohomology of  $\mathbb{B}_{\mathbb{C}_p}-\{p_1,\ldots,p_m\}$ we discuss how to compute the cohomology of an integral smooth curve $U$ over an algebraically closed field $k$. This will be useful since, in some sense, it contains the exact blueprint necessary to compute the cohomology of the multiply punctured disk.

## The setup

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 $k$ be an algebraically closed field and $U$ a smooth connected curve over $k$. There exists a unique smooth projective connected curve $X$ over $k$ such that $U=X-\{p_1,\ldots,p_m\}$ where $p_i\in X(k)$.

Proof: See the discussion in the subsection titled Affineness or projectiveness in this post. $\blacksquare$

So, let us now fix a prime $\ell$ invertible in $k$. We then wish to compute the etale cohomology $H^1_\mathrm{et}(U,\mathbb{Z}/\ell^n\mathbb{Z})$ for all $n\geqslant 1$. Namely, we show the following:

Proposition 2: Let $k$ be an algebraically closed field and let $X$ be a smooth projective connected curve over $k$ of genus $g$. Let $p_1,\ldots,p_m\in X(k)$ and set $U:=X-\{p_1,\ldots,p_m\}$. Then, for every prime $\ell$ invertible in $k$ and $n\geqslant 1$ we have the following:

$H^i_\mathrm{et}(U,\mathbb{Z}/\ell^n\mathbb{Z})=\begin{cases}\mathbb{Z}/\ell^n\mathbb{Z} & \mbox{if}\quad i=0\\ (\mathbb{Z}/\ell^n\mathbb{Z})^{2g+N(m)} &\mbox{if}\quad i=1\\ (\mathbb{Z}/\ell^n\mathbb{Z})^{M(m)} & \mbox{if}\quad i=2\\ 0 & \mbox{if}\quad i>2\end{cases}$

Here $N(m)$ and $M(m)$ are simple integers depending on $m$:

$N(m)=\begin{cases}0 & \mbox{if}\quad m=0\\ m-1 & \mbox{if}\quad m>0\end{cases}$

and

$M(m)=\begin{cases}1 & \mbox{if}\quad m=0\\ 0 & \mbox{if}\quad m>0\end{cases}$

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 $m=0$) and the affine case on equal footing and computes them by essentially by the same method (e.g. see [Stacks, Tag03RR]). The method we elect to take here is to use machinery which, in general, helps one relate the cohomology of a scheme and an open subset of the scheme. I speak, of course, of the notion of cohomology with supports. I prefer this method since I think it’s a) more fun, and b) containing ideas that are more generally applicable.

## The proper case

We handle the proper case as an initial base case from which we will inductively remove points (and see how the cohomology changes) to reach the general case.

Remark 3: The approach we take here is, essentially, the canonical/consummate one. All presentations of the proof of this result are, as far as I know, essentially equivalent with small tweaks here or there depending on taste. If one would like more details than what I write here (including a nice introduction to the Brauer group of a scheme) I highly recommend Dan Litt’s excellently written notes [Lit].

Let us simplify Proposition 2 to the proper case:

Proposition 4: Let $k$ be an algebraically closed field and let $X$ be a smooth proper integral curve over $k$ of genus $g$. Then, for every prime $\ell$ invertible in $k$ and integer $n\geqslant 1$, we have the following:

$H^i_\mathrm{et}(X,\mathbb{Z}/\ell^n\mathbb{Z})\cong \begin{cases} \mathbb{Z}/\ell^n\mathbb{Z} & \mbox{if}\quad i=0\\ (\mathbb{Z}/\ell^n\mathbb{Z})^{2g} & \mbox{if}\quad i=1\\ \mathbb{Z}/\ell^n\mathbb{Z} & \mbox{if}\quad i=2\\ 0 & \mbox{if}\quad i>2\end{cases}$

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 $S$ be any scheme and let $N$ be a positive integer invertible in $\mathcal{O}_S(S)$. Then, there is exact sequence of sheaves on the (small or big) etale site of $S$ given by

$1\to \mu_{N,S}\to\mathbb{G}_{m,S}\xrightarrow{x\mapsto x^N}\mathbb{G}_{m,S}\to 1$

Proof: It’s evident that this sequence is exact save the fact that the map of etale sheaves $\mathbb{G}_{m,S}\xrightarrow{x\mapsto x^N}\mathbb{G}_{m,S}$ is surjective. To do this, let $T\to S$ be an etale morphism. Then, we need to show that for every $\alpha\in \mathcal{O}_T(T)^\times$ that there exists some etale cover $f:T'\to T$ such that $f^\ast\alpha$ is an $N^\text{th}$-power. But, such a covering is given by

$\underline{\mathrm{Spec}}(\mathcal{O}_T[x]/(x^N-\alpha))\to T$

where, here, $\mathcal{O}_T[x]/(x^N-\alpha)$ is the obvious quasi-coherent $\mathcal{O}_T$-algebra and $\underline{\mathrm{Spec}}$ stands for the relative spectrum. The reason that this map is etale is precisely because $N$ is invertible in $\mathcal{O}_S(S)$ and thus in $\mathcal{O}_T(T)$$\blacksquare$

As a corollary we get the following:

Corollary 6: Let $S$ be any scheme and let $N$ be a positive integer invertible in $\mathcal{O}_S(S)^\times$. Then, there is a long exact sequence of abelian groups

$\cdots\to H^i_\mathrm{et}(X,\mu_{N,S})\to H^i_\mathrm{et}(X,\mathbb{G}_{m,S})\xrightarrow{[N]}H^i_\mathrm{et}(X,\mathbb{G}_{m,S})\to\cdots$

where $[N]$ denotes the multiplication-by-$n$ map.

This seems somewhat irrelevant for our computation of $H^i_\mathrm{et}(X,\mathbb{Z}/\ell^n\mathbb{Z})$ since it seems to be useful for computing only the cohomology of  the sheaf $\mu_{\ell^n}$. Of course, this feeling only persists until one makes the following incredibly simple, and incredibly basic observation:

Lemma 7: Let $S$ be $\displaystyle \mathbb{Z}\left[\frac{1}{N},\zeta_N\right]$-scheme (where $\zeta_N$ is a primitive $N^\text{th}$-root of unity). Then, there is an isomorphism of sheaves $\mu_N\cong \underline{\mathbb{Z}/N\mathbb{Z}}_S$ on the (big or small) etale site of $S$.

Proof: One needs only check that the group schemes $\mu_{N,S}$ and the constant group scheme $\underline{\mathbb{Z}/N\mathbb{Z}}_S$ are isomorphic on the small big site of $S$. But, note that $\mu_{N,S}$ and $\underline{\mathbb{Z}/N\mathbb{Z}}_S$ are the pullbacks from the group schemes $\mu_{N,S_0}$ and $\underline{\mathbb{Z}/N\mathbb{Z}}_{S_0}$ where $S_0:=\mathrm{Spec}(\mathbb{Z}\left[\frac{1}{N},\zeta_N\right])$. 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). $\blacksquare$

Thus, we see that, in particular, if we have $S\to\mathrm{Spec}(\mathbb{Z}\left[\frac{1}{N},\zeta_N\right])$ one has that

$H^i_\mathrm{et}(S,\mathbb{Z}/\ell^n\mathbb{Z})\cong H^i_\mathrm{et}(S, \mu_{N,S})$

In particular, since $k$ is algebraically closed and $\ell$ is invertible on $k$ we know that $X$ is actually a $\mathbb{Z}\left[\frac{1}{N},\zeta_N\right]$-scheme and thus we can profitably use the Kummer sequence to attack Proposition 4 (at least hopefully).

In particular, we see that to compute $H^i_\mathrm{et}(X,\mathbb{Z}/\ell^n\mathbb{Z})$ we must compute $H^i_\mathrm{et}(X,\mathbb{G}_{m,X})$ for all $i\geqslant 0$. This, as it turns out, is not too bad since the function fields of curves are particularly simple.

Now, since $H^0(X,\mathbb{G}_{m,X})=k^\times$ we begin the computation of $H^i_\mathrm{et}(X,\mathbb{G}_{m,X})$ in earnest at degree $i=1$. We do this by citing a very general result:

Lemma 8 (Hilbert’s Theorem 90): Let $S$ be any scheme. Then, there is a canonical isomorphism of abelian groups

$H^1_\mathrm{fppf}(S,\mathbb{G}_{m,S})\cong H^1_\mathrm{et}(S,\mathbb{G}_{m,S})\cong H^1_\mathrm{Zar}(S,\mathbb{G}_{m,S})\cong \mathrm{Pic}(S)$

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 $H^1_\mathcal{C}(S,\mathbb{G}_{m,S})$ classifying for any site $\mathcal{C}$? The answer is $\mathbb{G}_{m,S}$-torsors. But, note that $\mathbb{G}_{m,S}\cong \mathrm{Aut}(\mathcal{O}_X)$ (these are automorphisms of $\mathcal{O}_X$-modules) and by the ‘theory of twists’ (e.g. see Theorem 5.1 of the notes from this post) this means that $H^1_\mathcal{C}(S,\mathbb{G}_{m,S})$ is classifying ‘$\mathcal{O}_X$-modules on $S$ locally free of rank $1$ in the $\mathcal{C}$-topology’. So, this theorem comes down to a statement that whether $\mathcal{O}_X$-module is locally free of rank $1$ in a topology $\mathcal{C}\in\{\mathrm{et},\mathrm{Zar},\mathrm{fppf}\}$ doesn’t actually depend on the choice. An even more general useful version of this is seen as [Ols, Proposition 4.3.8]. $\blacksquare$

In particular, we see that $H^1_\mathrm{et}(X,\mathbb{G}_{m,X})\cong \mathrm{Pic}(X)$. This leaves us to try and compute $H^i_\mathrm{et}(X,\mathbb{G}_{m,X})$ for $i>1$. This is where we will have to take advantage of the aforementioned niceness of the function field of $X$. We will do this by appealing to yet another useful exact sequence.

To do this, let $j:\eta_X\hookrightarrow X$ be the inclusion of the generic point $\eta_X:=\mathrm{Spec}(K(X))$ of $X$ in to $X$. We then have the following pivotal sequence:

Lemma 9(the divisor sequence): Let $S$ be any Noetherian, integral, separated, and locally factorial scheme. Let $\eta_S$ be the generic point of $S$ and let $j:\eta_S\hookrightarrow S$ be the natural inclusion. Then, there is a short exact sequence of sheaves on the (small) etale site of $S$:

$\displaystyle 0\to \mathbb{G}_{m,S}\to j_\ast\mathbb{G}_{m,\eta_S}\to \bigoplus_{x\in |X|^{(1)}}(i_x)_\ast \underline{\mathbb{Z}}\to 0$

where $|X|^{(1)}$ denotes the points of $X$ of codimension $1$ and $i_x:\{x\}\hookrightarrow X$ 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. $\blacksquare$

This lemma tells us that to compute the cohomology of $\mathbb{G}_{m,X}$ we really only have to compute the cohomology of $j_\ast \mathbb{G}_{m,\eta_X}$ and the cohomology of $\displaystyle \bigoplus_{x\in |X|^{(1)}}(i_x)_\ast\underline{\mathbb{Z}}$. One of these is decidedly easier than the other:

Lemma 10: Let $k$ be an algebraically closed field and let $X$ be a connected smooth projective curve over $k$. Then,

$\displaystyle H^i_\mathrm{et}\left(X,\bigoplus_{x\in |X|^{(1)}}(i_x)_\ast\underline{\mathbb{Z}}\right)=\begin{cases}\displaystyle \bigoplus_{x\in |X|^{(1)}}\mathbb{Z} & \mbox{if}\quad i=0\\ 0 & \mbox{if}\quad i>0\end{cases}$

Proof: Since cohomology commutes with direct sums (e.g. see [Stacks,Tag0F11 (2)]) and $i_x$ is acyclic (e.g. see [Fu, Proposition 5.7.4]) one reduces this computation to the claim that for all $x\in |X|^{(1)}$ one has that for all $i>0$ that $H^i_\mathrm{et}(\mathrm{Spec}(k(x)),\underline{\mathbb{Z}})=0$. But, note that since $x$ is a closed point, since $X$ is a curve, that $k(x)/k$ is finite and thus $k(x)=k$ is algebraically closed. Thus, the claim follows by standard theory (e.g. see [Fu, Proposition 5.7.8]). $\blacksquare$

To compute $H^i_\mathrm{et}(X,j_\ast\mathbb{G}_{m,\eta_X})$ we would like to somehow reduce this to computing $H^i_\mathrm{et}(\eta_X,\mathbb{G}_{m,\eta_X})$ 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) $H^i_\mathrm{et}(\eta_X,\mathbb{G}_{m,\eta_X})$:

Lemma 11: Let $K$ be an algebraically closed field and let $L/K$ be an extension of transcendence degree $1$. Then, $H^i_\mathrm{et}(\mathrm{Spec}(L),\mathbb{G}_{m,L})=0$ for $i>0$.

Proof: As usual, see [Lit] or [Stacks,Tag03RG] but also take a look at [Star] and [Poo, Chapter 1]. $\blacksquare$

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 $H^i_\mathrm{et}(X,j_\ast\mathbb{G}_{m,X})$ since, of course, there is no reason that $R^q j_\ast\mathbb{G}_{m,\eta_X}=0$. 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 $f:X\to Y$ be a map of schemes and $\mathcal{F}$ an abelian sheaf on the small etale site of $X$. Then, there is an $E_2$ spectral sequence:

$E_2^{p,q}=H^p_\mathrm{et}(Y,R^qf_\ast\mathcal{F})\implies H^{p+q}(X,\mathcal{F})$

Proof: See [Stacks, Tag03QB]. $\blacksquare$

Applying this for $j:\eta_X\hookrightarrow X$ we get the spectral sequence

$E_2^{p,q}=H^p_\mathrm{et}(X,R^q j_\ast \mathbb{G}_{m,\eta_X})\implies H_\mathrm{et}^{p+q}(\eta_X,\mathbb{G}_{m,\eta_X})$

So, if we can prove that $R^qj_\ast \mathbb{G}_{m,\eta_X}=0$ we’ll essentially be done since this spectral sequence will collapse showing that

$H^i_\mathrm{et}(X,j_\ast\mathbb{G}_{m,\eta_X})\cong H^i_\mathrm{et}(\eta_X,\mathbb{G}_{m,\eta_X})$

and the latter is zero (for $i>0$) by Lemma $11$ since $\mathrm{tr.deg}(K(x)/k)=1$.

So, how can we show that $R^qj_\ast\mathbb{G}_{m,\eta_X}=0$? Well, as per usual, it suffices to show that $(R^qj_\ast\mathbb{G}_{m,\eta_X})_{\overline{s}}$ for every geometric point $\overline{s}$ of $X$. But, (see [Stacks, Tag03Q9] and/or [Stacks, Tag03Q6]) one has the following

\begin{aligned}(R^q j_\ast\mathbb{G}_{m,X})_{\overline{s}} &=\varinjlim_{(U,\overline{u})} H^q_\mathrm{et}(\eta_X\times_X U,\mathbb{G}_{m,\eta_X\times_X U})\\ &= H^i_\mathrm{et}(\eta_X\times_X (\varprojlim_{(U,\overline{u})}U),\mathbb{G}_m)\\ &= H^i_\mathrm{et}(\eta_X \times_X \mathrm{Spec}(\mathcal{O}_{X,\overline{s}}),\mathbb{G}_m)\end{aligned}

where $\mathrm{Spec}(\mathcal{O}_{X,\overline{s}})$ is the local ring of $x$ at $\overline{s}$ in the etale topology. But, one can check (e.g. see [Stacks, Tag03RJ]) that $\eta_X \times_X \mathrm{Spec}(\mathcal{O}_{X,\overline{s}})$ is always the spectrum a field extension of $k$ of transcendence degree at most $1$. Thus, we deduce the vanishing of $(R^qj_\ast\mathbb{G}_{m,\eta_X})_{\overline{s}}$ also by Lemma 11.

So, we deduce the the following:

Lemma 14: Let $k$ be an algebraically closed field and let $X$ be a smooth connected projective curve over $k$. Then, $H^i_\mathrm{et}(X,j_\ast\mathbb{G}_{m,\eta_X})=0$ for $i>0$.

Thus, returning to the divisor sequence we see that we get a long exact sequence

$\displaystyle \cdots 0\to H^i_\mathrm{et}(X,\mathbb{G}_{m,X})\to H^i_\mathrm{et}(X,j_\ast\mathbb{G}_{m,\eta_X})\to H^i\left(X,\bigoplus_{x\in |X|^{(1)}}\underline{\mathbb{Z}}\right)\to \cdots$

which, putting together Lemma 10 and Lemma 14, easily gives us the following:

Lemma 15: Let $k$ be an algebraically closed field and let $X$ be a smooth connected projective curve over $k$. Then, the following holds:

$H^i_\mathrm{et}(X,\mathbb{G}_{m,X})=\begin{cases}k^\times & \mbox{if}\quad i=0\\ \mathrm{Pic}(X) & \mbox{if}\quad i=1\\ 0 & \mbox{if}\quad i>1\end{cases}$

Then, finally, returning to the Kummer sequence we deduce the following exact sequence

$0\to H^1_\mathrm{et}(X,\mathbb{Z}/\ell^n\mathbb{Z})\to \mathrm{Pic}(X)\xrightarrow{[\ell^n]}\mathrm{Pic}(X)\to H^2_\mathrm{et}(X,\mathbb{Z}/\ell^n\mathbb{Z})\to 0$

(where we have cut off prior terms in the long exact sequence since $k^\times\xrightarrow{[\ell^n]}k^\times$ is surjective and we cut off successive terms since $H^i_\mathrm{et}(X,\mathbb{G}_{m,X})$ vanishes). Thus, we deduce the following:

Lemma 16: Let $k$ be an algebraically closed field and let $X$ be a smooth connected projective curve over $k$. Then for all primes $\ell$ invertible on $X$ and integers $n\geqslant 1$ the following holds:

$H^i_\mathrm{et}(X,\mathbb{Z}/\ell^n\mathbb{Z})=\begin{cases}\mathbb{Z}/\ell^n\mathbb{Z} & \mbox{if}\quad i=0\\ \ker(\mathrm{Pic}(X)\xrightarrow{[\ell^n]}\mathrm{Pic}(X)) & \mbox{if}\quad i=1\\ \mathrm{coker}(\mathrm{Pic}(X)\xrightarrow{[\ell^n]}\mathrm{Pic}(X)) & \mbox{if}\quad i=2\\ 0 & \mbox{if}\quad i>2\end{cases}$

Thus, to prove Proposition 4 it suffices to understand the group $\mathrm{Pic}(X)$. Recall that since $X$ is a smooth connected proper curve there is a surjective group map $\mathrm{deg}:\mathrm{Pic}(X)\to \mathbb{Z}$ and we denote by $\mathrm{Pic}^0(X)$ the kernel of this map. Evidently there is a non-canonical splitting

$\mathrm{Pic}(X)\cong \mathrm{Pic}^0(X)\times\mathbb{Z}$

so it suffices to understand the structure of $\mathrm{Pic}^0(X)$. This is the so-called theory of the Jacobian. Namely, we have the following well-known proposition:

Lemma 17: Let $k$ be an algebraically closed field and let $X$ be a smooth connected projective curve over $k$ of genus $g$. Then, the group functor

$\mathrm{Pic}^0:\mathsf{Sch}/X\to \mathsf{Set}$

given by

$\displaystyle (f:T\to X)\mapsto \frac{\left\{\mathscr{L}\in\mathrm{Pic}(X\times_k T): \mathrm{deg}(\mathscr{L}_t)=0\text{ for all }t\in T\right\}}{f^\ast\mathrm{Pic}(T)}$

is representable by a $g$-dimensional abelian variety.

Proof: See [Poo, §5.7.1] and the references therein. $\blacksquare$

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 $k=\mathbb{C}$. In this case know by Serre’s GAGA theorem that $\mathrm{Pic}(X)=\mathrm{Pic}(X^\mathrm{an})$ and $\mathrm{Pic}^0(X)=\mathrm{Pic}^0(X^\mathrm{an})$ . But, note that we have a short exact sequence of sheaves on $X^\mathrm{an}$ (the so-called exponential sequence)

$0\to \underline{2\pi i \mathbb{Z}}\to \mathcal{O}_{X^\mathrm{an}}\xrightarrow{\exp} \mathcal{O}_{X^\mathrm{an}}^\times\to 0$

In particular, one gets an exact sequence which contains the terms

$H^1(X^\mathrm{an},\underline{2\pi i \mathbb{Z}})\to H^1(X^\mathrm{an},\mathcal{O}_X)\to H^1(X^\mathrm{an},\mathcal{O}_{X^\mathrm{an}}^\times)\to H^2(X^\mathrm{an},\underline{2\pi i\mathbb{Z}})$

Now, using the identifications

$H^1(X^\mathrm{an},\mathcal{O}_{X^\mathrm{an}}^\times)\cong\mathrm{Pic}(X^\mathrm{an}),\qquad H^2(X^\mathrm{an},\underline{2\pi i \mathbb{Z}})\cong H^2_\mathrm{sing}(X^\mathrm{an},\mathbb{Z})\cong \mathbb{Z}$

one can identify

$H^1(X^\mathrm{an},\mathcal{O}_{X^\mathrm{an}}^\times)\to H^2(X^\mathrm{an},\underline{2\pi i\mathbb{Z}})$

with

$\deg:\mathrm{Pic}(X^\mathrm{an})\to\mathbb{Z}$

and so we see that we can identify

$\mathrm{coker}(H^1(X^\mathrm{an},\underline{2\pi i \mathbb{Z}})\to H^1(X,\mathcal{O}_{X^\mathrm{an}}))\xrightarrow{\approx}\mathrm{Pic}^0(X^\mathrm{an})$

Now, we have identifications

$H^1(X^\mathrm{an},\underline{2\pi i \mathbb{Z}})\cong H^1_\mathrm{sing}(X^\mathrm{an},\mathbb{Z})\cong \mathbb{Z}^{2g},\qquad H^1(X,\mathbb{O}_X)\cong \mathbb{C}^g$

and Hodge theory shows that the image of

$H^1(X^\mathrm{an},\underline{2\pi i \mathbb{Z}})\to H^1(X,\mathcal{O}_{X^\mathrm{an}})$

is a full lattice so that

$\mathrm{coker}(H^1(X^\mathrm{an},\underline{2\pi i \mathbb{Z}})\to H^1(X,\mathcal{O}_{X^\mathrm{an}}))$

is a $g$-dimensional complex torus. Moreover, Hodge theory also provides a polarization on

$\mathrm{coker}(H^1(X^\mathrm{an},\underline{2\pi i \mathbb{Z}})\to H^1(X,\mathcal{O}_{X^\mathrm{an}}))$

and thus $\mathrm{Pic}^0(X^\mathrm{an})$ is really a $g$-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 $k$ be an algebraically closed field and $A$ a $g$-dimensional abelian variety over $k$. Then, for a prime $\ell$ invertible in $k$ one has that $[\ell^n]:A(k)\to A(k)$ is surjective and has kernel isomorphic to $(\mathbb{Z}/\ell^n\mathbb{Z})^{2g}$.

Proof: See [Con, Proposition 4.2.2]. $\blacksquare$

Remark 20: There are two improvements upon Proposition 4 that should be noted. First, one is able to replace the condition that $k$ is algebraically closed by the condition that $k$ 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 $k(t)$ is trivial if $k$ is algebraically closed. This is actually false if $k$ 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 $k$ is an arbitrary field by which we mean one would like to know what the action of $\mathrm{Gal}(k^\mathrm{sep}/k)$ on $H^i_\mathrm{et}(X_{k^\mathrm{sep}},\mathbb{Z}/\ell^n\mathbb{Z})$ is. For this one can see Theorem 7 of this post.

# References

[Con] Conrad, B., 2015. Abelian varietieshttps://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 Curveshttps://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