{"id":1244,"date":"2011-02-21T19:00:41","date_gmt":"2011-02-21T19:00:41","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=1244"},"modified":"2012-07-14T00:28:20","modified_gmt":"2012-07-14T00:28:20","slug":"conditions-on-diagonal-not-needed","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=1244","title":{"rendered":"Conditions on diagonal not needed"},"content":{"rendered":"<p>In a <a href=\"https:\/\/github.com\/stacks\/stacks-project\/commit\/2ed2b578d4ba774091feb38563531fbaf266aa7e\">recent contribution<\/a> of Jonathan Wang to the stacks project we find the following criterion of algebraicity of stacks (see <a href=\"http:\/\/math.columbia.edu\/algebraic_geometry\/stacks-git\/locate.php?tag=05UL\">Lemma Tag 05UL<\/a>):<\/p>\n<blockquote><p>If X is a stack in groupoids over (Sch\/S)_{fppf} such that there exists an algebraic space U and a morphism u : U &#8212;&gt; X which is representable by algebraic spaces, surjective, and smooth, then X is an algebraic stack.<\/p><\/blockquote>\n<p>In other words, you do not need to check that the diagonal is representable by algebraic spaces. The analogue of this statement for algebraic spaces is <a href=\"http:\/\/math.columbia.edu\/algebraic_geometry\/stacks-git\/locate.php?tag=046K\">Lemma Tag 046K<\/a> (for etale maps) and <a href=\"http:\/\/math.columbia.edu\/algebraic_geometry\/stacks-git\/locate.php?tag=04S6\">Theorem Tag 04S6<\/a> (for smooth maps).<\/p>\n<p>The quoted result is closely related to the statement that the stack associated to a smooth groupoid in algebraic spaces is an algebraic stack (<a href=\"http:\/\/math.columbia.edu\/algebraic_geometry\/stacks-git\/locate.php?tag=04TK\">Theorem Tag 04TK<\/a>). Namely, given u : U &#8212;> X as above you can construct a groupoid by taking R = U x_X U and show that X is equivalent to [U\/R] as a stack. But somehow the statements have different flavors. Finally, the result as quoted above is often how one comes about it in moduli theory: Namely, given a moduli stack M we often already have a scheme U and a representable smooth surjective morphism u : U &#8212;> M. Please try this out on your favorite moduli problem!<\/p>\n","protected":false},"excerpt":{"rendered":"<p>In a recent contribution of Jonathan Wang to the stacks project we find the following criterion of algebraicity of stacks (see Lemma Tag 05UL): If X is a stack in groupoids over (Sch\/S)_{fppf} such that there exists an algebraic space &hellip; <a href=\"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=1244\">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-1244","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\/1244","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=1244"}],"version-history":[{"count":14,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1244\/revisions"}],"predecessor-version":[{"id":2629,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1244\/revisions\/2629"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1244"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1244"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1244"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}