{"id":1595,"date":"2011-05-18T21:14:33","date_gmt":"2011-05-18T21:14:33","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=1595"},"modified":"2011-05-18T21:14:33","modified_gmt":"2011-05-18T21:14:33","slug":"cotangent-complex-formal-smoothness","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=1595","title":{"rendered":"Cotangent complex &#038; formal smoothness"},"content":{"rendered":"<p>In <a href=\"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=1416\">this post<\/a> I claimed that a formally smooth ring map has a cotangent complex which is quasi-isomorphic to a projective module sitting in degree 0. I thought this was in Illusie&#8217;s thesis. But when Wansu Kim asked me for a reference, and when I tried to find it today, I couldn&#8217;t find it.<\/p>\n<p>Now I think it is simply wrong! I constructed what I think is a counter example and put it in the <a href=\" http:\/\/math.columbia.edu\/algebraic_geometry\/stacks-git\/examples.pdf\">chapter on examples<\/a> (search for cotangent complex). Let me know if I made a mistake&#8230; again.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>In this post I claimed that a formally smooth ring map has a cotangent complex which is quasi-isomorphic to a projective module sitting in degree 0. I thought this was in Illusie&#8217;s thesis. But when Wansu Kim asked me for &hellip; <a href=\"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=1595\">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-1595","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\/1595","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=1595"}],"version-history":[{"count":5,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1595\/revisions"}],"predecessor-version":[{"id":1600,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1595\/revisions\/1600"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1595"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1595"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1595"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}