{"id":50,"date":"2010-01-31T18:09:37","date_gmt":"2010-01-31T18:09:37","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=50"},"modified":"2010-01-31T18:09:37","modified_gmt":"2010-01-31T18:09:37","slug":"grothendiecks-lemma","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=50","title":{"rendered":"Grothendieck&#8217;s lemma"},"content":{"rendered":"<p>Googling for <em>Grothendieck&#8217;s lemma<\/em> turns up a whole slew of different lemmas. For some reason I started thinking of Grothedieck&#8217;s lemma as the following result, of which there are two versions:<\/p>\n<ul>\n<li> If A &#8211;&gt; B is a flat local ring map of Noetherian local rings and f in B is a nonzero divisor in the fiber ring B\/m_AB, then B\/fB is flat over A.<\/li>\n<li>If A &#8211;&gt; B is a flat local ring map of local rings, B is essentially of finite presentation over A and f in B is a nonzero divisor in the fiber ring B\/m_AB, then B\/fB is flat over A.<\/li>\n<\/ul>\n<p>Leave a comment if you have an opinion about how to refer to this lemma. This result is (very) related to the <em>local criterion for flatness<\/em> which says instead<\/p>\n<ul>\n<li> If A &#8211;&gt; B is a local ring map of Noetherian local rings and I is an ideal of A such that B\/IB is flat over A\/I and moreover Tor_1^A(B, A\/I) is zero, then B is flat over A.<\/li>\n<li> If A &#8211;&gt; B is a local ring map of local rings, B is essentially if finite presentation over A and I is an ideal of A such that B\/IB is flat over A\/I and moreover Tor_1^A(B, A\/I) is zero, then B is flat over A.<\/li>\n<\/ul>\n<p>This is particularly useful when I = m_A because B\/m_A B is automatically flat over the field A\/m_A. In the Algebra chapter of the stacks project we prove both of these independently although it might have been better\/quicker to deduce the first from the second. Finally, there is another very related result which I think is usually called the <em>crit\u00e8re de platitude par fibre<\/em> which says roughly<\/p>\n<ul>\n<li> If X &#8211;&gt; Y is a morphism of locally Noetherian schemes flat over a locally Noetherian base S and if f induces flat morphisms between fibers, then f is flat.<\/li>\n<li> If X &#8211;&gt; Y is a morphism of schemes of flat and locally of finite presentation over a base S and if f induces flat morphisms between fibers, then f is flat.<\/li>\n<\/ul>\n<p>You can in fact weaken the assumptions a bit. Of course this is a completely algebraic fact which can be reformulated in terms of maps of local rings as above. There are also versions for modules which are potentially much more useful; for these and some other results see<a href=\"http:\/\/math.columbia.edu\/algebraic_geometry\/stacks-git\/locate.php?tag=00MD\"> Algebra, Section Tag 00MD<\/a>, <a href=\"http:\/\/math.columbia.edu\/algebraic_geometry\/stacks-git\/locate.php?tag=00R3\">Algebra, Section Tag 00R3<\/a>, and <a href=\"http:\/\/math.columbia.edu\/algebraic_geometry\/stacks-git\/locate.php?tag=039A\">More on Morphisms, Section Tag 039A<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Googling for Grothendieck&#8217;s lemma turns up a whole slew of different lemmas. For some reason I started thinking of Grothedieck&#8217;s lemma as the following result, of which there are two versions: If A &#8211;&gt; B is a flat local ring &hellip; <a href=\"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=50\">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-50","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\/50","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=50"}],"version-history":[{"count":13,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/50\/revisions"}],"predecessor-version":[{"id":63,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/50\/revisions\/63"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=50"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=50"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=50"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}