{"id":1229,"date":"2011-02-16T20:33:21","date_gmt":"2011-02-16T20:33:21","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=1229"},"modified":"2011-02-16T20:33:21","modified_gmt":"2011-02-16T20:33:21","slug":"a-challenge","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=1229","title":{"rendered":"A challenge"},"content":{"rendered":"<p>Here is a challenge to an commutative algebraist out there. Give a direct algebraic proof of the following statement (see <a href=\"http:\/\/math.columbia.edu\/algebraic_geometry\/stacks-git\/locate.php?tag=05U9\">Lemma Tag 05U9<\/a>):<\/p>\n<blockquote><p>Let A &#8212;> B be a local ring homomorphism which is essentially of finite type. Let N be a finite type B-module. Let M be a flat A-module. Let u : N &#8212;> N be an A-module map such that N\/m_AN &#8212;> M\/m_AM is injective. Then u is A-universally injective, N is a B-module of finite presentation, and N is flat as an A-module.<\/p><\/blockquote>\n<p>To my mind it is at least conceivable that there is a direct proof of this (not using the currently used technology). It wouldn&#8217;t directly imply all the wonderful things proved by Raynaud and Gruson but it would go a long way towards verifying some of them. In particular, it would give an independent proof of the following result (see <a href=\"http:\/\/math.columbia.edu\/algebraic_geometry\/stacks-git\/locate.php?tag=05UA\">Theorem Tag 05UA<\/a>):<\/p>\n<blockquote><p>Let f : X &#8212;> S be a finite type morphism of schemes. Let x &isin; X with s = f(x) &isin; S. Suppose that X is flat over S at all points x&#8217; &isin; Ass(X_s) which specialize to x. Then X is flat over S at x.<\/p><\/blockquote>\n<p>This result is used in an essential way in the main result on universal flattening which I will explain in the next blog post.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Here is a challenge to an commutative algebraist out there. Give a direct algebraic proof of the following statement (see Lemma Tag 05U9): Let A &#8212;> B be a local ring homomorphism which is essentially of finite type. Let N &hellip; <a href=\"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=1229\">Continue reading <span class=\"meta-nav\">&rarr;<\/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-1229","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\/1229","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=1229"}],"version-history":[{"count":5,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1229\/revisions"}],"predecessor-version":[{"id":1234,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1229\/revisions\/1234"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1229"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1229"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1229"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}