By the way, for descent conditions needed for infinite Galois extensions, at least in the case of quasi-projective varieties, see Corollary 16.25 here: http://www.jmilne.org/math/CourseNotes/AG16.pdf

This gets used all the time in, say, Shimura varieties, but is still semi-unsatisfying since it only works for quasi-projectives.

]]>For anyone reading Zhen’s comment, and not particularly enamored by the words ‘simplicial kernel’, let me just state another way of seeing it (which I assume both Zhen and Qiaochu know well). Namely, fpqc descent along involves, in particular, . Now, is fpqc for any a field extension. What makes the finite case different is that, there, $\displaystyle L\otimes_k L$ is just with the map being .

Thus, fpqc descent works for any extension, but explicitly being able to relate it to the Galois group is a failed endeavor for non-finite Galois extensions. This also suggestions how one might replace this notion by studying stacks on the subcategory of consisting of Galois objects, for other .

Again, so Zhen doesn’t yell at me, this is the exact same thing he wrote (I assume) in different language. š

]]>I’ve been seeing this for a few days now, and it’s the reason I stopped posting.

]]>I tried, but WordPress’s LaTeX is freaking out on me right now…

]]>(If you could fix the LaTeX there that would be much appreciated.)

]]>In a bit more detail: let be a field extension and consider the simplicial kernel of . By fpqc descent, it has an effective quotient (in the category of fpqc sheaves), namely the representable sheaf on , and moreover, sends this colimit diagram to a bilimit diagram. Since we are dealing with objects in a 2-category, we can truncate these (co)simplicial diagrams above degree 2. When is a finite Galois extension, the objects in the simplicial kernel are just disjoint unions of copies of . It is instructive to work out what the face operators are in terms of this identification: at the bottom, one is the diagonal embedding , and the other is , where is some enumeration of the Galois group.

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