Consider the topology τ on the category of schemes where a covering is a finite family of proper morphisms which are jointly surjective. (Dear reader: does this topology have a name?) For the purpose of this post proper hypercoverings will be τ-hypercoverings as defined in the chapter on hypercoverings. Proper hypercoverings are discussed specifically in Brian Conrad’s write up. In this post I wanted to explain an example which I was recently discussing with Bhargav on email. I’d love to hear about other “explicit” examples that you know about; please leave a comment.

The example is an example of proper hypercovering for curves. Namely, consider a separable degree 2 map X —> Y of projective nonsingular curves over an algebraically closed field and let y be a ramification point. The simplicial scheme X_* with X_i = normalization of (i + 1)st fibre product of X over Y is NOT a proper hypercovering of Y. Namely, consider the fibre above y (recall that the base change of a proper hypercovering is a proper hypercovering). Then we see that X_0 has one point above y, X_1 has 2 points above y, and X_2 has 4 points above y. But if X_2 is supposed to surject onto the degree 2 part of cosk_1(X_1 => X_0) then the fibre of X_2 over y has to have at least 8 points!!!!

Namely cosk_1(S —> *) where S is a set and * is a singleton set is the simplicial set with S^3 in degree 2, S^6 in degree 3, etc because an n-simplex should exist for any collection of (n + 1 choose 2) 1-simplices since each of the 1-simplices bounds the unique 0-simplex on both sides, see for example Remark 0189. So I think that to construct the proper hypercovering we have to throw in some extra points in simplicial degree 2 which sort of glue the two components of X_1.

Now, as X_* does work over the complement of the ramification locus in Y, I think you can argue that it really does suffice to add finite sets of points to X_* (over ramification points) to get a proper hypercovering!

PS: Proper hypercoverings are interesting since they can be used to express the cohomology of a (singular) variety in terms of cohomologies of smooth varieties. But that’s for another post.

Re your PS, also proper hypercoverings come up in Joe Ross’s work (as I am sure you are already aware).

Does the pretopology tau you describe generate the same topology as the pretopology where a cover is just a proper map? Or in other words, is tau a superextensive pretopology http://ncatlab.org/nlab/show/superextensive+site ?

Yeah, I think so, but you are missing the word “surjective” in your sentence.

Oh, yes, that’s right: surjective proper maps.

It is perhaps worth pointing out that, if X –> Y is proper and surjective, then the simplicial scheme X_* with X_i = seminormalization of the (i + 1)st fiber product of X over Y, is a proper hypercovering of Y. (This is true before seminormalizing X_i, and the seminormalization doesn’t change the topology.)

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