{"id":1235,"date":"2011-02-16T22:20:57","date_gmt":"2011-02-16T22:20:57","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=1235"},"modified":"2011-02-16T22:20:57","modified_gmt":"2011-02-16T22:20:57","slug":"universal-flattening","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=1235","title":{"rendered":"Universal flattening"},"content":{"rendered":"

In this post<\/a> I talked a bit about flattening of morphisms. Meanwhile I have written some more about this in the stacks project which led to a change in definitions. Namely, I have formally introduced the following terminology:<\/p>\n

    \n
  1. Given a morphism of schemes X —> S we say there exists a universal flattening of X<\/em> if there exists a monomorphism of schemes S’ —> S such that the base change X_{S’} of X is flat over S’ and such that for any morphism of schemes T —> S we have that X_T is flat over T if and only if T —> S factors through S’.<\/li>\n
  2. Given a morphism of schemes X —> S we say there exists a flattening stratification of X<\/em> if there exists a universal flattening S’ —> S and moreover S’ is isomorphic as an S-scheme to the disjoint union of locally closed subschemes of S.<\/li>\n<\/ol>\n

    Of course the definition of “having a flattening stratification” this is a bit nonsensical, since we really want to know how to “enumerate” the locally closed subschemes so obtained. Please let me know if you think this terminology isn’t suitable.<\/p>\n

    Perhaps the simplest case where a universal flattening doesn’t exist is the immersion of A^1 – {0} into A^2. Currently the strongest existence result in the stacks project is (see Lemma Tag 05UH<\/a>):<\/p>\n

    If f : X —> S is of finite presentation and X is S-pure then a universal flattening S’ —> S of X exists.<\/p><\/blockquote>\n

    Note that the assumptions hold f is proper and of finite presentation. It is much easier to prove that a flattening stratification exists if f is projective and of finite presentation and I strongly urge the reader to always use the result on projective morphisms, and only use the result quoted above if absolutely necessary.<\/p>\n

    PS: I recently received a preprint by Andrew Kresch where, besides other results, he gives examples of cases where the universal flattening exists (he call this the “flatification”) but where there does not exist a flattening stratification.<\/p>\n","protected":false},"excerpt":{"rendered":"

    In this post I talked a bit about flattening of morphisms. Meanwhile I have written some more about this in the stacks project which led to a change in definitions. Namely, I have formally introduced the following terminology: Given a … 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-1235","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\/1235","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=1235"}],"version-history":[{"count":7,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1235\/revisions"}],"predecessor-version":[{"id":1242,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1235\/revisions\/1242"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1235"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1235"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1235"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}