{"id":1170,"date":"2011-01-11T20:11:25","date_gmt":"2011-01-11T20:11:25","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=1170"},"modified":"2011-01-11T20:11:25","modified_gmt":"2011-01-11T20:11:25","slug":"isomorphism-colimit","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=1170","title":{"rendered":"Isomorphism colimit"},"content":{"rendered":"<p>Today I encountered the following lemma:<\/p>\n<p>Let A be a ring. Let u : M &#8212;> N be an A-module map. Let R = colim_i R_i be a directed colimit of A-algebras. Assume that M is a finite A-module, N is a finitely presented A-module, and u &otimes; 1 : M &otimes; R &#8212;> N &otimes; R is an isomorphism. Then there exists an i &isin; I such that u &otimes; 1 : M &otimes; R_i &#8212;> N &otimes; R_i is an isomorphism. (All tensor products over A.)<\/p>\n<p>What I like about this statement is that M only needs to be a finite A-module. This is similar to what happened in <a href=\"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=1070\">this post<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Today I encountered the following lemma: Let A be a ring. Let u : M &#8212;> N be an A-module map. Let R = colim_i R_i be a directed colimit of A-algebras. Assume that M is a finite A-module, N &hellip; <a href=\"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=1170\">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-1170","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\/1170","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=1170"}],"version-history":[{"count":5,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1170\/revisions"}],"predecessor-version":[{"id":1175,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1170\/revisions\/1175"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1170"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1170"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1170"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}