Talk:ℓ-adic sheaf

motivation

I added a paragraph on why lisse sheaves are necessary in place of local systems. I read this from an article, which is cited. In the article it uses a specific scheme (a nodal curve) for demonstrating the counterexample but I found it works for general schemes (I might be wrong . Please tell me if so.) Please review this edit. Sorry in advance for this is my first edit on Wikipedia so there might be guidelines or requirements that I'm not properly following. Thanks. --Fourier-Deligne Transgirl (talk) 02:51, 26 March 2022 (UTC)

Unfortunately you are wrong about it working for general schemes, which I describe below, but I also think there are some more inaccuracies here:

The example you give isn't motivation for introducing lisse sheaves, it's a demonstration that the profinite fundamental group doesn't classify the correct notion of lisse Q_l local systems! The reason for defining lisse sheaves is much simpler: one notices that etale cohomology doesn't behave well with non-torsion coefficients so one builds Z_l sheaves as inverse limits of torsion sheaves. For 'naive' Z_l locally constant sheaves we'd have instead (for X normal) e.g.: H^1(X,Z_l)=Hom_cont(pi_1(X),Z_l), but where the latter group has the discrete topology! So this always vanishes. The reason is that in the etale topos we don't really have a good way to take the topology on Z_l into account. This is fixed by the pro-etale topos.

Now lisse Z_l sheaves are in fact classified by the etale fundamental group! But as you describe, lisse Q_l sheaves can't in general be glued together from local data. Your example doesn't give us a lisse Q_l sheaf by the classical definition (which is just a lisse Z_l sheaf, tensored with Q), but that's actually undesirable as we didn't do anything unnatural. Note also that your example relies on X being not normal, as otherwise multiplication by l cannot be a descent datum, being of infinite order. This is true in general: For normal X, the profinite fundamental group classifies the correct notion of Q_l local systems (which can be glued), but for non-normal X it fails. Some of these 'anomalous' local systems get recovered when considering the SGA3 enlarged fundamental group, but not all: There is an example in the Bhatt-Scholze paper due to Deligne for which this doesn't work. However, your argument fails when applied to the SGA3 fundamental group as it relies on the group being profinite, but the enlarged group is only prodiscrete. The good news is that everything works out fine with the proetale fundamental group.

I looked briefly at your source, and it seems confused in similar ways, which is perhaps not totally surprising as it's a Masters thesis haha ;) Sloth sisyphos (talk) 22:23, 21 April 2023 (UTC)

@Sloth sisyphos: Thank you so much! This is actually why I left this message here as then I felt a bit unconfident of that argument in that source... Do you care to contribute to the page itself? Oh and I guess it's my turn now so: Welcome back to Wikipedia! Please rollback my edit, until we put Sloth sisyphos' wonderful insight into the page. --Fourier-Deligne Transgirl (talk) 16:10, 4 May 2023 (UTC)

I have no idea what's going on here or why, specifically, you need an administrator to undo an edit you can undo yourself, but I undid it anyway because the English was subpar.--Bbb23 (talk) 17:36, 4 May 2023 (UTC)

@Bbb23: Thank you. (though what do you mean by "the English is subpar"? --Fourier-Deligne Transgirl (talk) 19:21, 4 May 2023 (UTC)