Talk:Sheaf on an algebraic stack

Title -- contents mismatch

I think the article is in general quite nice already, but I am a bit puzzled why l-adic formalism should appear in an article on quasi-coherent sheaves? I see two options:

• remove the l-adic section

• rename the article to something like sheaves on stacks etc.?

Any ideas about this? Jakob.scholbach (talk) 20:15, 28 April 2019 (UTC)

Basically I had Behrend's paper in mind which generalizes the formalism to the stacky context. I think this is an important instance of a (quasi-coherent) sheaf on a stack: in particular, at least morally, $\mathbb {Q} _{\ell }$ is a sheaf on a stack, as a coefficient sheaf in cohomology of a stack. It is unrealistic to give a technically complete account but some sketch of use and construction is perfectly adequate for Wikipedia. -- Taku (talk) 00:55, 30 April 2019 (UTC)

You mean Q_l is a quasi-coherent sheaf? It is not, right (even on a scheme)? Jakob.scholbach (talk) 07:21, 30 April 2019 (UTC)

No, of course not. Now I think I see why "having quasi-coherent" is wrong or misleading (as you said and thank you for pointing it out). But, as I said above, I think it's still important to discuss an instance of a sheaf as a coefficient sheaf on cohomology of a stack. (I know H(-; Q_l) really means H(-; Z_l) tensored with Q_l.) At least removing "quasi-coherent" from the article title is a good idea for now. -- Taku (talk) 23:07, 1 May 2019 (UTC)

In fact, a more fundamental issue is that "as far as I can tell", l-adic sheaf, roughly an inverse system of Z/l-modules (right?), is not covered in Wikipedia at all! I will just go ahead and start l-adic sheaf; I still think the article is a better place fro the l-adic formalism in the stacky context. -- Taku (talk) 23:31, 2 May 2019 (UTC)