{"id":903,"date":"2010-09-26T01:22:30","date_gmt":"2010-09-26T01:22:30","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=903"},"modified":"2010-11-10T15:12:08","modified_gmt":"2010-11-10T15:12:08","slug":"faithfully-flat-descent-of-projectivity","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=903","title":{"rendered":"Faithfully flat descent of projectivity"},"content":{"rendered":"<p>Alex Perry wrote a paper about faithfully flat descent of projectivity for modules, and submitted it to the stacks project. His writeup largely follows the exposition in Raynaud and Gruson&#8217;s paper [RG]. But, as Brian Conrad mentioned <a href=\"http:\/\/mathoverflow.net\/questions\/6719\/is-projectiveness-a-zariski-local-property-of-modules-answered-yes\/16448#16448\">here<\/a>, there is an error in [RG]. Alex avoids this error by proving that the Mittag-Leffler condition satisfies flat descent in the presence of flatness.<\/p>\n<p>The result of the title of this blog post is Theorem <a href=\"http:\/\/math.columbia.edu\/algebraic_geometry\/stacks-git\/locate.php?tag=05A9\">Tag 05A9<\/a> of the stacks project. To get an (almost) self contained proof of the theorem start reading the introductory Section <a href=\"http:\/\/math.columbia.edu\/algebraic_geometry\/stacks-git\/locate.php?tag=058B\">Tag 058B<\/a> entitled &#8220;Faithfully flat descent for projectivity of modules&#8221;.<\/p>\n<p>Edit: You can now find Alex&#8217;s write-up at <a href=\"http:\/\/arxiv.org\/abs\/1011.0038v1\">arXiv:1011.0038v1<\/a> [math.AC].<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Alex Perry wrote a paper about faithfully flat descent of projectivity for modules, and submitted it to the stacks project. His writeup largely follows the exposition in Raynaud and Gruson&#8217;s paper [RG]. But, as Brian Conrad mentioned here, there is &hellip; <a href=\"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=903\">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-903","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\/903","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=903"}],"version-history":[{"count":8,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/903\/revisions"}],"predecessor-version":[{"id":1008,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/903\/revisions\/1008"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=903"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=903"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=903"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}