{"id":1358,"date":"2011-03-25T13:51:28","date_gmt":"2011-03-25T13:51:28","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=1358"},"modified":"2011-03-25T13:51:28","modified_gmt":"2011-03-25T13:51:28","slug":"characterizing-closed-immersions","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=1358","title":{"rendered":"Characterizing closed immersions"},"content":{"rendered":"<p><em>A universally closed, universally injective, and unramified morphism is a closed immersion.<\/em><\/p>\n<p>Here are some references. The result itself is here<\/p>\n<p>SCHEMES: <a href=\"http:\/\/math.columbia.edu\/algebraic_geometry\/stacks-git\/locate.php?tag=04XV\">Lemma Tag 04XV<\/a><br \/>\nSPACES: <a href=\"http:\/\/math.columbia.edu\/algebraic_geometry\/stacks-git\/locate.php?tag=05W8\">Lemma Tag 05W8<\/a><\/p>\n<p>The definition of an unramified morphisms is here<\/p>\n<p>RINGS: <a href=\"http:\/\/math.columbia.edu\/algebraic_geometry\/stacks-git\/locate.php?tag=00UT\">Definition Tag 00UT<\/a><br \/>\nSCHEMES: <a href=\"http:\/\/math.columbia.edu\/algebraic_geometry\/stacks-git\/locate.php?tag=02G4\">Definition Tag 02G4<\/a><br \/>\nSPACES: <a href=\"http:\/\/math.columbia.edu\/algebraic_geometry\/stacks-git\/locate.php?tag=03ZH\">Definition Tag 03ZH<\/a><\/p>\n<p>Formally unramified morphisms are defined here<\/p>\n<p>RINGS: <a href=\"http:\/\/math.columbia.edu\/algebraic_geometry\/stacks-git\/locate.php?tag=00UN\">Definition Tag 00UN<\/a><br \/>\nSCHEMES: <a href=\"http:\/\/math.columbia.edu\/algebraic_geometry\/stacks-git\/locate.php?tag=02H8\">Definition Tag 02H8<\/a><br \/>\nSPACES: <a href=\"http:\/\/math.columbia.edu\/algebraic_geometry\/stacks-git\/locate.php?tag=04G7\">Definition Tag 04G7<\/a><\/p>\n<p>and a morphism which is formally unramified and locally of finite type is unramified, see here<\/p>\n<p>SCHEMES: <a href=\"http:\/\/math.columbia.edu\/algebraic_geometry\/stacks-git\/locate.php?tag=02HE\">Lemma Tag 02HE<\/a><br \/>\nSPACES: <a href=\"http:\/\/math.columbia.edu\/algebraic_geometry\/stacks-git\/locate.php?tag=04GA\">Lemma Tag 04GA<\/a><\/p>\n<p>Enjoy!<\/p>\n","protected":false},"excerpt":{"rendered":"<p>A universally closed, universally injective, and unramified morphism is a closed immersion. Here are some references. The result itself is here SCHEMES: Lemma Tag 04XV SPACES: Lemma Tag 05W8 The definition of an unramified morphisms is here RINGS: Definition Tag &hellip; <a href=\"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=1358\">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-1358","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\/1358","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=1358"}],"version-history":[{"count":8,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1358\/revisions"}],"predecessor-version":[{"id":1366,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1358\/revisions\/1366"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1358"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1358"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1358"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}