{"id":753,"date":"2010-08-20T14:46:26","date_gmt":"2010-08-20T14:46:26","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=753"},"modified":"2010-08-23T16:07:16","modified_gmt":"2010-08-23T16:07:16","slug":"flat-finite-type-ring-extensions","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=753","title":{"rendered":"Flat finite type ring extensions"},"content":{"rendered":"<p>Encouraged by the success in studying finite flat modules, see the <a href=\"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=748\">preceding post<\/a>, let&#8217;s think a bit about flat, finite type ring extensions.<\/p>\n<p>Question: For which rings R is every finite type flat ring map R &#8212;&gt; S of finite presentation?<\/p>\n<p>A Noetherian ring satisfies this property. In the paper by Raynaud and Gruson they prove that this holds if R is a domain. I recently added this result to the stacks project (with a purely algebraic proof), see <a href=\"http:\/\/math.columbia.edu\/algebraic_geometry\/stacks-git\/locate.php?tag=053G\"> Algebra, Proposition Tag 053G<\/a>. If R is a local ring whose maximal ideal is nilpotent then the result is true as well. But I don&#8217;t know what happens if the maximal ideal is only assumed to be locally nilpotent, i.e., every element of the maximal ideal is nilpotent, i.e., the maximal ideal is \u221a(0). Do you?<\/p>\n<p>By the way, I still want more ideas about the question I posted <a href=\"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=744\">here<\/a>! [Edit: this question has now been answered.]<\/p>\n<p>[Edit on August 23, 2010: As David Rydh points out in a comment below any ring which has finitely many associated primes satisfies the condition. This follows trivially from Raynaud-Gruson Theorem 3.4.6. Don&#8217;t know why I did not see this! Anyway, so a local ring whose maximal ideal is locally nilpotent is an example too.]<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Encouraged by the success in studying finite flat modules, see the preceding post, let&#8217;s think a bit about flat, finite type ring extensions. Question: For which rings R is every finite type flat ring map R &#8212;&gt; S of finite &hellip; <a href=\"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=753\">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-753","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\/753","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=753"}],"version-history":[{"count":13,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/753\/revisions"}],"predecessor-version":[{"id":776,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/753\/revisions\/776"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=753"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=753"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=753"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}