{"id":2190,"date":"2012-02-26T17:01:04","date_gmt":"2012-02-26T17:01:04","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=2190"},"modified":"2012-02-26T17:01:04","modified_gmt":"2012-02-26T17:01:04","slug":"proper-hypercovering","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=2190","title":{"rendered":"Proper hypercovering"},"content":{"rendered":"

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<\/a>. Proper hypercoverings are discussed specifically in Brian Conrad’s write up<\/a>. 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.<\/p>\n

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!!!!<\/p>\n

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<\/a>. 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.<\/p>\n

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!<\/p>\n

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.<\/p>\n","protected":false},"excerpt":{"rendered":"

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 … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-2190","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/2190","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=2190"}],"version-history":[{"count":8,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/2190\/revisions"}],"predecessor-version":[{"id":2198,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/2190\/revisions\/2198"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2190"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2190"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2190"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}