{"id":64,"date":"2010-02-03T21:12:43","date_gmt":"2010-02-03T21:12:43","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=64"},"modified":"2010-02-03T21:12:43","modified_gmt":"2010-02-03T21:12:43","slug":"artins-trick-revisited","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=64","title":{"rendered":"Artin’s trick revisited"},"content":{"rendered":"

Well, my claims in this post<\/a> were a little premature and may not be correct after all. It turns out that an argument in the lemma on etale localization of groupoids was wrong and has to be fixed. It doesn’t matter much for the overall picture since I can certainly use the arguments in the paper of Keel and Mori to fix it, but I wanted to prove it using only “etale localization” which may be hard.<\/p>\n

The problem is the following: Suppose we have a groupoid scheme (U, R, s, t, c) with s, t flat, locally of finite presentation, and locally quasi-finite. Then we want to find many etale morphisms U’ –> U such that the restriction R’ of R to U’ is a disjoint union R’ = P \\coprod Rest, with P a subgroupoid which is finite<\/strong> over U’ under both s’ and t’. The technique that I used (wrongly) in the current version of proof of the lemma mentioned above does<\/em> prove that when we take U’ to be the spectrum of the henselization of the local ring of U at a point of U. But this doesn’t give you an etale morphism U’ —> U! What I am trying to see is if there is a kind of limit argument to descend P from the henselization to a smaller ring…<\/p>\n

Let me just state here for the record that I think this means we can use this version of the lemma (with the henselization I mean) to define the (strictly) henselian local ring of the coarse moduli space without knowing that the coarse moduli space exists! Namely, since s, t : P —> Spec(O_{U,u}^h) are finite, flat and locally of finite presentation, we obtain that P is affine, and the “usual” arguments show that O_{U,u}^h is integral over the subring C of P-invariant elements of O_{U, u}^h. Presumably C (or its strict henselization) is what we are looking for. I haven’t thought this through completely, however.<\/p>\n

I’ll post more here when I figure out how to repair the lemma.<\/p>\n","protected":false},"excerpt":{"rendered":"

Well, my claims in this post were a little premature and may not be correct after all. It turns out that an argument in the lemma on etale localization of groupoids was wrong and has to be fixed. It doesn’t … 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-64","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\/64","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=64"}],"version-history":[{"count":2,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/64\/revisions"}],"predecessor-version":[{"id":66,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/64\/revisions\/66"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=64"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=64"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=64"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}