{"id":3427,"date":"2013-07-28T12:28:22","date_gmt":"2013-07-28T12:28:22","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=3427"},"modified":"2013-07-28T12:28:44","modified_gmt":"2013-07-28T12:28:44","slug":"lemma-of-the-day-18","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=3427","title":{"rendered":"Lemma of the day"},"content":{"rendered":"<p>Let C be a site. Let O&#8217; &#8212;> O be a surjection of sheaves of rings whose kernel I is an ideal of square zero. Let F&#8217; be an O&#8217;-module and set F = F&#8217;\/I F&#8217;. The following are equivalent<\/p>\n<ol>\n<li> F&#8217; is a flat O&#8217;-module, and<\/li>\n<li>F is a flat O-module and I &otimes;<sub>O<\/sub> F &#8212;> F&#8217; is injective.<\/li>\n<\/ol>\n<p>See <a href=\"http:\/\/stacks.math.columbia.edu\/tag\/08M4\">Tag 08M4<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Let C be a site. Let O&#8217; &#8212;> O be a surjection of sheaves of rings whose kernel I is an ideal of square zero. Let F&#8217; be an O&#8217;-module and set F = F&#8217;\/I F&#8217;. The following are equivalent &hellip; <a href=\"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=3427\">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-3427","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\/3427","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=3427"}],"version-history":[{"count":11,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/3427\/revisions"}],"predecessor-version":[{"id":3437,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/3427\/revisions\/3437"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=3427"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=3427"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=3427"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}