What is the need for torsion in the definition of lisse sheaves?

Question

I am studying the basics of constructible and lisse sheaves, and am trying to understand SGA 4, IX. As Grothendieck himself observes at the beginning of the chapter, one is forced to work with torsion sheaves, and there are several "moral" explanations for this: Serre's example of an elliptic curve with quaternionic multiplication which cannot provide a $\mathbb{R}$-valued representation of the quaternions, the fact that constructible/lisse sheaves should correspond to continuous representations of $\pi_1^\mathrm{et}$ and these are of pro-finite image, and so on. But I am trying to understand a more technical detail, basically around Lemma 2.1 of SGA 4, Chap. IX.

Grothendieck fixes a topos $T$ and goes on to find sufficient conditions for a subsheaf $\mathcal{F}$ of a constant sheaf $S_T$ to be in turn (locally) constant: this depends a bit on the kind of object $S$ is. If $S$ is an abelian group, and we look at $S_T$ and $\mathcal{F}$ as being $\textbf{Ab}$-valued, then we need $S$ to be finite. If we have fixed a (noetherian) ring $A$, $S$ is an $A$-module and $\mathcal{F}$ to be $\textbf{Mod-A}$-valued, then we need $S$ to be finitely generated. These conditions turn out to be precisely those needed to make the category of locally constant, and hence of constructible, sheaves abelian. As a consequence (see Remark 2.3.1 ibid.) a constructible sheaf of $\mathbb{Z}$-modules is not the same thing as a constructible sheaf of abelian groups. I have tried to come up with examples to see that these finiteness assumptions are really needed, but did not succeed.

1. Is there an example of a subsheaf of a locally constant sheaf with stalks (say) $\mathbb{Z}^n$, which is not locally constant as $\textbf{Ab}$-valued étale sheaf?
2. Being a noetherian object in the category of $\mathbb{Z}$-modules or in that of abelian groups is the same thing: what is the reason for imposing that constructible $\mathbf{Ab}$-sheaves be valued in finite groups (which, in passing, kills injective objects)?

Let me add that I'd prefer to see examples in an algebraic setting, so working with either étale or Zariski topology: I was able to cook up an example of an infinitely-generated locally constant sheaf and a non-locally constant subsheaf on the one-point compactification of $\mathbb{Z}$ (with discrete topology) but this is not the kind of topological space which look natural to me.

Edit: In a first version, I asked also about $\mathbb{Z}_\ell$-sheaves and Nicolás adressed this in his very correct answer; but I am more interested in the true need for the finiteness conditions mentioned above. Here is the rest of the old question.

In the literature, a "constructible $\mathbb{Z}_\ell$-sheaf" is usually defined as a projective system of constructible $\mathbb{Z}/\ell^n$-sheaves (plus some conditions) whereas a definition was a priori already available. What is an example of a $\mathbb{Z}_\ell$-constructible sheaf $\mathcal{F}=(\mathcal{F}_n)$ defined as a projective system which does not come from a "naive" constructible sheaf over the ring $A=\mathbb{Z}_\ell$? Why is the former the "right" definition?

Answer 1

I think you are misinterpreting things slightly. Lemma 2.1 says nothing about abelian groups. Lemma 2.1(i) is about sets, and Lemma 2.1(ii) is about A-modules. For $A = \mathbb Z$, $\mathbb Z$-modules are equivalent to abelian groups and so Lemma 2.1(ii) applies.

It is only 2.1(i) that is used in Remark 2.3(i). That is used to show that a sheaf of groups that is constructible as a sheaf of sets is also constructible as a sheaf of groups - in this case we take $f$ to be the multiplication or inversion map. The last line, that $\mathbb Z_X$ is constructible as a $\mathbb Z$-module, but not as a sheaf of groups, follows immediately from the Definition 2.3, as of course $\mathbb Z$ is constant and of finite presentation, but isn't finite. Nothing about the category of sheaves of $\mathbb Z$-modules or the category of sheaves of groups forces you to do that, because they are equivalent categories.

So why did Grothendieck define constructible sheaves of abelian groups to be finite? I don't know, but I believe it's exactly for all the other reasons one must work with torsion coefficients. Remember that if Grothendieck wants to talk about constructible sheaves of groups without the torsion condition, he can just work with constructible sheaves of $\mathbb Z$-modules.

Answer 2

There are two different issues here which I think you are confusing: 1) that the right coefficients are $l$-adic instead of archimedean (these are the "moral" reasons you mention), and 2) the issue about building the theory first with torsion coefficients and then passing to the limit to get to characteristic $0$ (this is where the issue arises that the characteristic $0$ sheaves are defined as "limits" of torsion sheaves, both in the real and derived world).

Regarding 2), I believe this was mostly for technical reasons and nowadays the definition can be actually done in one step (see e.g. Bhatt's and Scholze's pro-etale topology https://arxiv.org/pdf/1309.1198.pdf).