{"id":4742,"date":"2022-02-13T02:11:10","date_gmt":"2022-02-13T02:11:10","guid":{"rendered":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=4742"},"modified":"2022-02-13T02:11:10","modified_gmt":"2022-02-13T02:11:10","slug":"supports-of-flat-modules-part-b","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=4742","title":{"rendered":"Supports of flat modules, part B"},"content":{"rendered":"<p>Part A is <a href=\"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=4727\">this post<\/a>. Let me prove the opposite of what the exercise in part A wrongly claimed (sigh).<\/p>\n<p><strong>Lemma:<\/strong> Let Z &#8212;> Y be a finite morphism of affine schemes. Then there exists a closed immersion Z &#8212;> Z&#8217; of schemes over Y such that Z&#8217; is finite syntomic over Y.<\/p>\n<p><strong>Remark:<\/strong> If we embed Z&#8217; into a smooth scheme X over Y, then F = O_{Z&#8217;} is a coherent O_X-module flat over Y such that the generic points of Z are associated points of the restriction of F to their fibres.<\/p>\n<p><strong>Proof:<\/strong> Write Y = Spec(A) and Z = Spec(B). Choose generators b_1, &#8230;, b_r of B as an algebra over A. As B is finite over A, each b_i is the root of a monic polynomial P_i with coefficients in A. Then B&#8217; = A[x_1, &#8230;, x_r]\/(P_1(x_1), &#8230;, P_r(x_r)) is finite syntomic over A and Z&#8217; = Spec(B&#8217;) works. EndProof.<\/p>\n<p>The point I want to make in this post is that we have some equidimensionality result for associated points of flat modules, namely EGA IV, Proposition 12.1.1.5 (see <a href=\"https:\/\/stacks.math.columbia.edu\/tag\/0GSF\">Tag 0GSF<\/a>). It implies the following: suppose that f : X &#8212;> Y is smooth with Y Noetherian and irreducible. Suppose that F is coherent on X and flat over Y. Let x be a point of the generic fibre of f which is an associated point of F. Then the zariski closure Z &subset; X of the singleton {x} has the property that Z &#8212;> Y is equidimensional!<\/p>\n<p>So for example, there is no finite module M over k[x, y, z] which is flat over k[x, y] such that (x &#8211; yt) is an associated prime of M. Presumably, you can see this directly? Is it easy? I didn&#8217;t try.<\/p>\n<p>Enjoy!<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Part A is this post. Let me prove the opposite of what the exercise in part A wrongly claimed (sigh). Lemma: Let Z &#8212;> Y be a finite morphism of affine schemes. Then there exists a closed immersion Z &#8212;> &hellip; <a href=\"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=4742\">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-4742","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\/4742","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=4742"}],"version-history":[{"count":12,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/4742\/revisions"}],"predecessor-version":[{"id":4754,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/4742\/revisions\/4754"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=4742"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=4742"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=4742"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}