{"id":697,"date":"2010-07-31T18:49:38","date_gmt":"2010-07-31T18:49:38","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=697"},"modified":"2010-07-31T18:49:38","modified_gmt":"2010-07-31T18:49:38","slug":"more-flattening","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=697","title":{"rendered":"More flattening"},"content":{"rendered":"

This is a continuation of previous post<\/a> on flattening stratifications. The experts reading this blog could probably tell that I hadn’t really understood what is going on at all. I still haven’t mastered the subject but I think I know a little bit more now.<\/p>\n

Let f : X —> S be a morphism of schemes. Consider the functor F : (Sch)^{opp} —> (Sets) which to a scheme T associates the set of morphisms T —> S such that the base change X_T is flat over T. Clearly the map F —> S is a monomorphism. We propose to introduce the following<\/p>\n

Definition: We say the flattening stratification of f exists<\/em> if F is an algebraic space.<\/p>\n

What I added to the stacks project last Friday is the following: Assume S is the spectrum of a Noetherian complete local ring and f is of finite type. Then there exists a biggest closed subscheme Z of S such that X_Z —> Z is flat at all the points of the closed fibre. Moreover, Z satisfies a universal property which is formulated in terms of local morphisms of local schemes and flatness at points of the special fibre. If in addition X —> S is closed, then it follows that X_Z —> Z is flat as the set of points where X_Z —> Z is flat is an open set.<\/p>\n

Assume S Noetherian and f of finite type and proper. In terms of Artin’s axioms for F the result in the previous paragraph takes care of the existence of a formal versal deformation. I think there is a straightforward little argument which takes care of openness of versality (but I did not write this out completely). Since f is of finite presentation, it follows that F is of finite presentation by the usual arguments on limits and flatness. Relative representability is OK too. Hence, if S is excellent then F is an algebraic space by Artin’s theorem. But of course we can descend X —> S to a situation of finite type over Z and hence we get the result in general (with same hypotheses). In fact, using limit arguments we may be able to prove the same thing when S is arbitrary and f proper and of finite presentation.<\/p>\n

Still, my answer to Jason’s question here<\/a> was a bit premature. Some of the above may work exactly as stated in the generality of Jason’s question. But I was trying to prove flattening stratifications exist without using Artin’s theorem. In particular, it should be possible to avoid using general N\\’eron desingularization.<\/p>\n

The reason I started looking at flattening stratifications was to construct Quot and Hilbert schemes\/spaces\/stacks. And the reason to discuss those was that Artin’s trick uses Hilbert spaces. However, it only uses the Hilbert space parametrizing closed subschemes of length n on a space. Of course I could take the easy way out and just use one of the explicit constructions of Hilb^n. But once I started looking at the problem of constructing flattening stratifications (which is related to descent of flat modules) I just couldn’t stop myself.<\/p>\n","protected":false},"excerpt":{"rendered":"

This is a continuation of previous post on flattening stratifications. The experts reading this blog could probably tell that I hadn’t really understood what is going on at all. I still haven’t mastered the subject but I think I know … 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-697","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\/697","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=697"}],"version-history":[{"count":2,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/697\/revisions"}],"predecessor-version":[{"id":699,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/697\/revisions\/699"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=697"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=697"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=697"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}