{"id":473,"date":"2010-06-07T17:23:19","date_gmt":"2010-06-07T17:23:19","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=473"},"modified":"2010-06-07T17:23:19","modified_gmt":"2010-06-07T17:23:19","slug":"local-on-source-and-target","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=473","title":{"rendered":"Local on source and target"},"content":{"rendered":"<p>What does it mean for a property P of morphisms of schemes to be etale local on the source and target? In Deligne-Mumford they use the following definition (page 100): for any family of commutative squares<br \/>\n<a href=\"http:\/\/math.columbia.edu\/~dejong\/wordpress\/wp-content\/uploads\/2010\/06\/DM.bmp\"><img decoding=\"async\" src=\"http:\/\/math.columbia.edu\/~dejong\/wordpress\/wp-content\/uploads\/2010\/06\/DM.bmp\" alt=\"commutative diagram\" title=\"DM\" class=\"alignnone size-full wp-image-475\" \/><\/a><br \/>\nwhere {h_i : X_i &#8212;> X}, {g_i : Y_i &#8212;> Y} are etale coverings we have P(f) <=> P(f_i) for all i. And of course this is exactly the minimum needed to be able to define what it means for a morphism of Deligne-Mumford stacks to have a certain property&#8230;<\/p>\n<p>However, here are some very confusing points<\/p>\n<ol>\n<li>the condition does NOT imply that P is preserved under post-composing with open immersions,<\/li>\n<li>if P is etale local on the source and P is etale local on the target, then P does not necessarily satisfy Deligne and Mumford&#8217;s condition.<\/li>\n<\/ol>\n<p>Now it turns out that this NEVER leads to any confusion, since if P is preserved under post-composing with open immersions, which is a condition always satisfied in practice, then all three conceivable notions agree. Moreover, in that case the property is preserved under post-composing with <em>etale<\/em> morphisms. To see all the gory details, see the section entitled &#8220;Properties of morphisms local on source-and-target&#8221; in <a href=\"http:\/\/math.columbia.edu\/algebraic_geometry\/stacks-git\/descent.pdf\">Descent.pdf<\/a>.<\/p>\n<p>PS: This may be good material to read if you are having trouble falling asleep.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>What does it mean for a property P of morphisms of schemes to be etale local on the source and target? In Deligne-Mumford they use the following definition (page 100): for any family of commutative squares where {h_i : X_i &hellip; <a href=\"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=473\">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-473","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\/473","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=473"}],"version-history":[{"count":13,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/473\/revisions"}],"predecessor-version":[{"id":487,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/473\/revisions\/487"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=473"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=473"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=473"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}