{"id":748,"date":"2010-08-20T14:26:16","date_gmt":"2010-08-20T14:26:16","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=748"},"modified":"2010-08-20T14:26:16","modified_gmt":"2010-08-20T14:26:16","slug":"finite-flat-modules","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=748","title":{"rendered":"Finite flat modules"},"content":{"rendered":"<p>In my thesis, in the chapter on finite flat groupschemes, I made the mistake of thinking that a finite flat group scheme is the same thing as a finite locally free group scheme. In other words, I made the classic mistake of thinking that a finite flat module over a ring is finite locally free (or equivalently finitely presented). A counter example is given in the stacks project, see <a href=\"http:\/\/math.columbia.edu\/algebraic_geometry\/stacks-git\/examples.pdf\">examples.pdf<\/a>. Luckily I discovered this error (or maybe somebody else did and pointed it out to me) and the published version of my thesis does not have this mistake.<\/p>\n<p>Why I made this mistake I am not sure, maybe because I read Matsumura&#8217;s Commutative Algebra, where you can find the result that a finite flat module over a local ring is finite free.<\/p>\n<p>I have since learned that this is not as bad a mistake as one may think. Namely, it turns out that whether or not every finite flat R-module is finite locally free, is a property of R which depends only on the topology of X = Spec(R). The result is that every finite flat R-module is finite locally free if and only if every Z &sub; X which is closed and closed under generalizations is also open. A similar result holds for schemes. (I found this in some paper a while back, but now I cannot remember which paper.)<\/p>\n<p>I just added this to the stacks project this morning, see <a href=\"http:\/\/math.columbia.edu\/algebraic_geometry\/stacks-git\/locate.php?tag=052U\">Algebra, Lemma Tag 052U<\/a> and <a href=\"http:\/\/math.columbia.edu\/algebraic_geometry\/stacks-git\/locate.php?tag=053N\">Morphisms, Lemma Tag 053N<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>In my thesis, in the chapter on finite flat groupschemes, I made the mistake of thinking that a finite flat group scheme is the same thing as a finite locally free group scheme. In other words, I made the classic &hellip; <a href=\"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=748\">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-748","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\/748","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=748"}],"version-history":[{"count":4,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/748\/revisions"}],"predecessor-version":[{"id":752,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/748\/revisions\/752"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=748"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=748"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=748"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}