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added to étale topos some basics in the section Properties – Base change and sheaf cohomology
Added a section Sheaf condition and examples of etale sheaves
I don’t understand the scope of these constructions. In ’General Concrete’ we’re dealing with etale sites and so certainly schemes. In ’General Abstract’ it sounds like we’re in any differential cohesive $\infty$-topos. When ’etale’ turns up there, does this mean it’s not restricted to schemes?
Yes, that’s the idea. With étaleness formulated in terms of differential cohesive modalities, the notion of étale topos exists very generally.
But the entry is currently lacking detailed discussion of how the general construction reduces to that on schemes. I’ll try to add a clarifying remark now, to make more clear what’s going on.
OK good, that seemed to be where you were heading. But then how broadly should we take the associated concepts? I mean why restrict etale cohomology to sheaves on sites of schemes, why restrict Weil cohomology to projective varieties? Either you devise new abstract general names, or you redefine the concrete ones so that they’re now abstract general.
Also, something I keep encountering is that when you use ’cohesive’ and ’differential cohesive’ it’s often in an absolute context, i.e., relative to $\infty Grpd$. But presumably there are versions of the relevant statements for the relative case, where we have $\mathbf{H}$ cohesive over $\mathbf{J}$, and a differential extension $\mathbf{H}_{th}$
Hi David, if I understand correctly, then the answer to your first question is: because I haven’t yet found time and energy to do more.
It seems to me that this was the very point that Grothendieck kept pursuing while he was active: that whatever you encounter in algebraic geometry should be understood as a special case of a more general concept in topos theory. It is curious that while everyone will hail Grothendieck as the greatest such-and-such etc., the wide-spread tendency is to actively counteract this point of view.
A fun example is the Wikipedia page on Etale cohomology which has the following cute passage:
Grothendieck originally developed étale cohomology in an extremely general setting, working with concepts such as Grothendieck toposes and Grothendieck universes. With hindsight, much of this machinery proved unnecessary
Heh.
I hope you do find time to clarify the connection. At the moment it is very, very far from obvious that whether there is any real connection or whether it is just an analogy.
@David
The reason why Weil cohomology is defined for smooth projective varieties over an algebraically closed field (!) is because the axioms only hold in that case, and we neither want nor need them to hold in the more general setting. (For instance, it is totally unreasonable to expect finiteness or Poincaré duality for varieties that are not projective.) And étale cohomology is just a name for the sheaf cohomology of the étale site. (But this is not to be confused with $\ell$-adic cohomology, which is a subtly more complicated construction.)
I didn’t mean ’why haven’t you done it yet?’, just ’ought this to be done’? I can see it’s quite tricky when writing this material up to do so at an optimal level of generality, given existing naming conventions, etc., and tricky to decide whether to extend the use of terms to more general settings, or devise new terms. ’Etale cohomology’ sounds like it could stand being taken more generally. Perhaps when a name is involved, like Weil or Zariski, it’s harder to have it applied away from the context in which they were working.
It’s interesting to consider then when special properties kick in, or when we should expect analogues. E.g., will there be analogues for the Weil conjectures in other differential cohesive situations?
The reason why Weil cohomology is defined for smooth projective varieties over an algebraically closed field (!) is because the axioms only hold in that case, and we neither want nor need them to hold in the more general setting.
You don’t need any restriction on the base field, it’s the field of coefficients which you need to be of characteristic zero.
@Zhen Lin,
it’s really not mysterious. There is a coreflection of rings inside infinitesimally extended rings which induces a quadruple of adjoints on the (pre)sheaf categories. Etale morphisms are those which are modal with respect to the corresponding “infinitesimal shape modality”, which is just a slight paraphrase of what Kontsevich-Rosenberg observed in section 4.1 here.
That gives the étale toposes in the general sense as discussed in the entry. The usual étale site is found in there by taking the canonical site and restricting to a subcategory of finite and regular objects (the étale schemes over the base inside all formally etale maps of sheaves into the base).
But, yeah, it should be written out in more detail anyway.
@Urs
That’s cheating! It’s not really a special case if you still have to do something special to recover the original definition!
It’s just a size restriction on the fibers of the formally étale maps. Just a few days back we have been discussing that this is usefully relaxed also in the “original definition” to pro-etaleness.
Urs, on etale topos, what should \ref{FormallyEtaleInHTh} point to?
Thanks for catching this, this is a copy-and-paste problem, because that material originates in the page differential cohesion. There the definition in question is this here.
I have made the link come out properly now. (There might be more broken links like this in the text?)
Yes, I saw two more which I also fixed.
Oh, sorry. Thanks, then!
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