{"id":3392,"date":"2013-07-21T13:40:57","date_gmt":"2013-07-21T13:40:57","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=3392"},"modified":"2013-07-21T13:40:57","modified_gmt":"2013-07-21T13:40:57","slug":"lemma-of-the-day-15","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=3392","title":{"rendered":"Lemma of the day"},"content":{"rendered":"<p>Let P be a property of morphisms of algebraic spaces. Assume<\/p>\n<ol>\n<li>P is smooth local on the source,<\/li>\n<li>P is smooth local on the target, and<\/li>\n<li>P is stable under postcomposing with smooth morphisms: if f : X &#8212;> Y has P and Y &#8212;> Z is a smooth morphism then X &#8212;> Z has P.<\/li>\n<\/ol>\n<p>Then P is smooth local on the source-and-target. See <a href=\"http:\/\/stacks.math.columbia.edu\/tag\/06FB\">Tag 06FB<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Let P be a property of morphisms of algebraic spaces. Assume P is smooth local on the source, P is smooth local on the target, and P is stable under postcomposing with smooth morphisms: if f : X &#8212;> Y &hellip; <a href=\"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=3392\">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-3392","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\/3392","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=3392"}],"version-history":[{"count":2,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/3392\/revisions"}],"predecessor-version":[{"id":3394,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/3392\/revisions\/3394"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=3392"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=3392"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=3392"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}