{"id":623,"date":"2010-07-06T15:16:51","date_gmt":"2010-07-06T15:16:51","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=623"},"modified":"2010-07-07T13:02:40","modified_gmt":"2010-07-07T13:02:40","slug":"epimorphism-of-rings","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=623","title":{"rendered":"Epimorphism of rings"},"content":{"rendered":"<p>We added a section on epimorphisms of rings to the <a href=\"http:\/\/www.math.columbia.edu\/algebraic_geometry\/stacks-git\/algebra.pdf\">algebra chapter<\/a>. Everything is completely straightforward except for the following fact: If A &#8212;> B is an epimorphism of rings then |B| &le; |A|. Since an epimorphism of rings is not necessarily surjective this is not a triviality. We learned this from the <a href=\"http:\/\/www.numdam.org\/item?id=SAC_1967-1968__2__A2_0\">exposee by Mazet<\/a> in the Seminaire Samuel.<\/p>\n<p>You can use this to show that if X &#8212;>Y is a monomorphism of schemes then size(X) &le; size(Y), which is a technical condition on the cardinalities of some sets associated to X and Y. See the <a href=\"http:\/\/www.math.columbia.edu\/algebraic_geometry\/stacks-git\/sets.pdf\">chapter on sets<\/a>.<\/p>\n<p>Here is a consequence: Given a scheme Y there is a set worth of isomorphism classes of monomorphisms X &#8212;> Y. I don&#8217;t think this is formal since I vaguely remember reading somewhere about a category (maybe spaces up to homotopy?) where such a thing is not true. Leave a comment if you know the correct statement.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>We added a section on epimorphisms of rings to the algebra chapter. Everything is completely straightforward except for the following fact: If A &#8212;> B is an epimorphism of rings then |B| &le; |A|. Since an epimorphism of rings is &hellip; <a href=\"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=623\">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-623","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\/623","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=623"}],"version-history":[{"count":11,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/623\/revisions"}],"predecessor-version":[{"id":634,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/623\/revisions\/634"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=623"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=623"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=623"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}