{"id":1416,"date":"2011-04-04T20:53:48","date_gmt":"2011-04-04T20:53:48","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=1416"},"modified":"2011-05-18T21:17:01","modified_gmt":"2011-05-18T21:17:01","slug":"formally-smooth","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=1416","title":{"rendered":"Formally smooth"},"content":{"rendered":"<p>Just today I finally managed to fix the proof of &#8220;formally smooth + locally of finite presentation &lt;=&gt; smooth&#8221; for morphisms of algebraic spaces, see <a href=\"http:\/\/math.columbia.edu\/algebraic_geometry\/stacks-git\/locate.php?tag=04AM\">Lemma Tag 04AM<\/a>. In fact, the implication &#8220;=&gt;&#8221; isn&#8217;t hard, and is the result that is used in practice. In the current implementation, the proof of &#8220;&lt;=&#8221; uses infinitesimal deformation of maps, and in particular a topos theoretic description of first order thickenings of algebraic spaces which we alluded to in <a href=\"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=332\">this post<\/a>, see <a href=\"http:\/\/math.columbia.edu\/algebraic_geometry\/stacks-git\/locate.php?tag=05ZN\">Lemma Tag 05ZN<\/a> and <a href=\"http:\/\/math.columbia.edu\/algebraic_geometry\/stacks-git\/locate.php?tag=05ZN\">Lemma Tag 05ZN<\/a>.<\/p>\n<p>Here is a related fact:<\/p>\n<blockquote><p>Suppose that X &#8212;&gt; Y &#8212;&gt; Z are morphisms of algebraic spaces or schemes, that X &#8212;&gt; Y is etale and that X &#8212;&gt; Z is formally smooth. Then Y &#8212;&gt; Z is formally smooth too.&nbsp;<\/p><\/blockquote>\n<p>In other words, being formally smooth is etale local on the source and target. See <a href=\"http:\/\/math.columbia.edu\/algebraic_geometry\/stacks-git\/locate.php?tag=061K\">Lemma Tag 061K<\/a> for a more precise statement.<\/p>\n<p>If X, Y, Z are schemes, then one can prove this by reducing to the affine case, using that formal smoothness is equivalent to the cotangent complex being a projective module in degree 0 [Edit 5\/18\/2011: Wrong! See <a href=\"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=1595\">here<\/a>.], and using the distinguished triangle of cotangent complexes associated to a pair of compose-able ring maps.<\/p>\n<p>If X, Y, Z are algebraic spaces, then one has to do a bit more work (I think). The proof of the reference above uses the material mentioned in the first paragraph and that \u03a9_{X\/Z} is a locally projective, quasi-coherent O_X-module (see <a href=\"http:\/\/math.columbia.edu\/algebraic_geometry\/stacks-git\/locate.php?tag=061I\">Lemma Tag 061I<\/a>), which is fun in and of itself.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Just today I finally managed to fix the proof of &#8220;formally smooth + locally of finite presentation &lt;=&gt; smooth&#8221; for morphisms of algebraic spaces, see Lemma Tag 04AM. In fact, the implication &#8220;=&gt;&#8221; isn&#8217;t hard, and is the result that &hellip; <a href=\"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=1416\">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-1416","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\/1416","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=1416"}],"version-history":[{"count":12,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1416\/revisions"}],"predecessor-version":[{"id":1601,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1416\/revisions\/1601"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1416"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1416"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1416"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}