Local systems on a punctured line

Question

How to describe the category of local systems on $\mathbb{G}_{m, \mathbb{Z}}$ in different topologies? I think that in etale topology there is a version of Riemann-Hilbert correspondence which says that the category of etale local systems is equivalent to the category of representations of $\hat{\mathbb{Z}}$. Is it true? And what happens with Zariski, flat topologies? Any references are greatly appreciated!

Answer 1

Here's a partial answer to the part of your question I think is most reasonable/interesting.

The basically 'best'(=most general) version of the etale site currently available, is the pro-etale site of Bhatt and Scholze.

Definition (Bhatt--Scholze): Let $X$ be a locally topologically Noetherian scheme. Then, a morphism $Y\to X$ is weakly etale if both $Y\to X$ and the diagonal map $Y\to Y\times_X Y$ are flat.

We then define the pro-etale site of $X$, denoted $X_\mathrm{proet}$, to have as objects weakly etale morphisms $Y\to X$ and as covers fpqc covers (see Tag 03NW). Let us define $\mathrm{Loc}(X_\mathrm{proet})$ to be the category sheaves of sets $\mathcal{F}$ on $X_\mathrm{proet}$ which are locally trivial (i.e. for which there is an fpqc cover $\{Y_i\to X\}$ in $\mathrm{X}_{\mathrm{proet}}$ such that $\mathcal{F}_{Y_i}$ is constant for all $i$).

We then have the following theorem.

Theorem/Definition (Bhatt--Scholze): A morphism $X'\to X$ is called a geometric covering if it's partially proper (i.e. satisfies the valuative criterion for properness--see Tag 0BX5) and etale. Denote by $\mathrm{Cov}_X$ the category of geometric coverings of $X$. The functor

$$\mathrm{Cov}_X\to \mathrm{Loc(X_\mathrm{proet})},\qquad X'\mapsto h_{X'}:=\text{Hom}_X(-,X')$$

is an equivalence of categories.

In other words, all pro-etale local systems are representable, and representable precisely by geometric coverings.

There is also a group theoretic classification of pro-etale local systems. Bhatt and Scholze show that $\mathrm{Cov}_X$ is a so-called tame infinite Galois category (which is the analogue of classical Galois category theory where you are allowed to consider topological groups which aren't pro-finite). This means, by general theory, that there is a (Noohi) topological group $\mathrm{pi}_1^\mathrm{proet}(X,\overline{x})$ and a natural isomorphism

$$F_{\overline{x}}\colon \mathrm{Cov}_X\xrightarrow{\approx}\pi_1^\mathrm{proet}(X,\overline{x})\text{-}\mathbf{Set},$$

where this latter category is the category of discrete sets with a continuous action of $\pi_1^\mathrm{proet}(X,\overline{x})$. Here $F_{\overline{x}}(Y)$ is, as one would expect, the 'fiber' of $X'\to X$ over $\overline{x}$ (i.e. the underlying set of $X'\times_X \overline{x}$).

In particular, combining these results one sees that there are equivalence of categories

$$\mathrm{Loc}(X_\mathrm{proet})\overset{\approx}{\leftarrow}\mathrm{Cov}_X\overset{\approx}{\rightarrow}\pi_1^\mathrm{proet}(X,\overline{x})\text{-}\mathbf{Set}.$$

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So, what does this all mean for $X=\mathbb{G}_{m,k}$? Bhatt and Scholze prove that if $X$ is geometrically unibranch (e.g. normal) then

$$\mathrm{Cov}_X=\mathrm{UFEt}_X:=\left\{\bigsqcup_i X_i\to X: X_i\to X\text{ finite etale for all }i\right\},$$

and consequently

$$\pi_1^\mathrm{proet}(X,\overline{x})\cong\pi_1^\mathrm{fet}(X,\overline{x}).$$

For $X=\mathbb{G}_{m,k}$, if $k$ is algebraically closed (which I assume it is), every finite etale cover is of the form $\mathbb{G}_{m,k}\xrightarrow{x\mapsto x^n}\mathbb{G}_{m,k}$, and so $\pi_1^\mathrm{fet}(\mathbf{G}_{m,k},\overline{x})\cong\widehat{\mathbb{Z}}$. And, thus putting everything together we see that the pro-etale local systems $\mathcal{F}$ on $\mathbf{G}_{m,k}$ are precisely the sheaves of the form

$$\mathcal{F}=h_{X'},\qquad X'=\bigsqcup_{i\in I}X'_{n_i},\qquad (X'_{n_i}\to \mathbb{G}_{m,k})=(\mathbb{G}_{m,k}\xrightarrow{x\mapsto x^{n_i}}\mathbb{G}_{m,k}),$$

and that such data is classified precisely by discrete sets with a continuous action of $\widehat{\mathbb{Z}}$.

Exercise: decide which of these local systems is actually etale locally/Zariski locally trivial.

EDIT: Apparently you really did want $\mathbb{G}_{m,\mathbb{Z}}$. Since this scheme is also normal, we need only calculate its finite Galois covers.

Claim: Every connected finite cover $X\to\mathbb{G}_{m,\mathbb{Z}}$ is trivial.

Proof: I claim first that$ X_{\overline{\mathbb{Z}}}$ satisfies this is a disjoint union of copies of $\mathbb{G}_{m,\overline{\mathbb{Z}}}$, where $\overline{\mathbb{Z}}$ is the integral closure of $\mathbb{Z}$ in $\overline{\mathbb{Q}}$.

To see this note that working connected component by connected component, it suffices to assume that $X_{\overline{\mathbb{Z}}}$ is connected. Note that since $\overline{\mathbb{Z}}$ is normal, we have that $X$ is the normalization of $\mathbb{G}_{m,\overline{\mathbb{Z}}}$ in the finite etale cover $X_{\overline{\mathbb{Q}}}\to \mathbb{G}_{m,\overline{\mathbb{Q}}}$. By what we said above $X_{\overline{\mathbb{Q}}}\to\mathbb{G}_{m,\overline{\mathbb{Q}}}$ is isomorphic to $\mathbb{G}_{m,\overline{\mathbb{Q}}}\xrightarrow{x\mapsto x^n}\mathbb{G}_{m,\overline{\mathbb{Q}}}$ for some $n$. I leave it to you to verify that the normalization of $\mathbb{G}_{m,\overline{\mathbb{Z}}}$ in this is the map $\mathbb{G}_{m,\overline{\mathbb{Z}}}\xrightarrow{x\mapsto x^n}\mathbb{G}_{m,\overline{\mathbb{Z}}}$ for some $n\geqslant 1$. Since this map is assumed to be etale, thinking about what happens mod $p$ we deduce that $p\nmid n$ (since $d(x^n)=nx^{n-1}=0\mod p$ if $p\mid n$). Since $p$ was arbitrary we deduce that $n=1$ as desired.

So then, consider our $X\to\mathrm{Spec}(\mathbb{G}_{m,\mathbb{Z}})$. Again by loc. cit. this is the normalization of $\mathbb{G}_{m,\mathbb{Z}}$ of $X_\mathbb{Q}$. Choose a geometric point $\overline{x}$ of $\mathbb{G}_{m,\mathbb{Q}}$. Note that since $X_{\overline{\mathbb{Q}}}$ is trivial, since this is $X_{\overline{\mathbb{Z}}}$ base changed to $\overline{\mathbb{Q}}$, we have that the subgroup $\pi_1^\mathrm{et}(\mathbb{G}_{m,\overline{\mathbb{Q}}},\overline{x})$ acts trivially on $F_{\overline{x}}(X_{\mathbb{Q}})$. This means by Tag 0BTXthat $\pi_1(\mathbb{G}_{m,\mathbb{Q}},\overline{x})$ acts on $F_{\overline{x}}$ through $\pi_1(\mathrm{Spec}(\mathbb{Q}),\overline{x})$. This determines a finite etale algebra $E=L_1\times\cdots\times L_n$ over $\mathbb{Q}$ and it's easy to see that the $\pi_1(\mathbb{G}_{m,\mathbb{Q}},\overline{x})$-sets $\mathbb{G}_{m,E}$ and $X_\mathbb{Q}$ must be equal which, by the fundamental theorem of Galois categories, implies that $X_{\mathbb{Q}}$ is isomorphic to $\mathbb{G}_{m,E}$. Since $X_\mathbb{Q}$ is connected $E=L$ is a finite extension of $\mathbb{Q}$. Then $X$, which is the normalization of $\mathbb{G}_{m,\mathbb{Z}}$ in $X_\mathbb{Q}$, is easily seen to be $\mathbb{G}_{m,\mathcal{O}_L}$. But, by pulling back along $x=1$ this implies that $\mathrm{Spec}(\mathcal{O}_L)\to\mathrm{Spec}(\mathbb{Z})$ is etale, which implies that $L/\mathbb{Q}$ is nowhere ramified, which implies that $L=\mathbb{Q}$ (see the Hermite--Minkowski theorem), and thus $X=\mathbb{G}_{m,\mathbb{Z}}$ as desired. $\blacksquare$

From this, you see that every pro-etale local system on $\mathbb{G}_{m,\mathbb{Z}}$ is actually constant.