{"id":3352,"date":"2013-07-11T12:16:24","date_gmt":"2013-07-11T12:16:24","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=3352"},"modified":"2013-07-11T12:16:24","modified_gmt":"2013-07-11T12:16:24","slug":"lemma-of-the-day-10","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=3352","title":{"rendered":"Lemma of the day"},"content":{"rendered":"<p>Let A be a valuation ring. Let A\u2192B be a ring map of finite type. Let M be a finite B-module.<\/p>\n<ol>\n<li>If B is flat over A, then B is a finitely presented A-algebra.<\/li>\n<li>If M is flat as an A-module, then M is finitely presented as a B-module.<\/li>\n<\/ol>\n<p>See <a href=\"http:\/\/stacks.math.columbia.edu\/tag\/053E\">Tag 053E<\/a>.<\/p>\n<p>PS: Much more is true, see the <a href=\"http:\/\/stacks.math.columbia.edu\/download\/flat.pdf\">this chapter<\/a> in the stacks project. The proof of the lemma above however is quite easy.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Let A be a valuation ring. Let A\u2192B be a ring map of finite type. Let M be a finite B-module. If B is flat over A, then B is a finitely presented A-algebra. If M is flat as an &hellip; <a href=\"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=3352\">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-3352","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\/3352","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=3352"}],"version-history":[{"count":2,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/3352\/revisions"}],"predecessor-version":[{"id":3354,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/3352\/revisions\/3354"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=3352"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=3352"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=3352"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}