{"id":1898,"date":"2011-10-04T13:35:07","date_gmt":"2011-10-04T13:35:07","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=1898"},"modified":"2012-07-14T00:24:13","modified_gmt":"2012-07-14T00:24:13","slug":"tale-algebraic-stacks","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=1898","title":{"rendered":"&Eacute;tale algebraic stacks"},"content":{"rendered":"<p>Just yesterday, upon some prodding from Michael Thaddeus, I added a two short sections comparing algebraic spaces and algebraic stacks in the fppf and &eacute;tale topology (see the corresponding sections of chapters <a href=\"http:\/\/www.math.columbia.edu\/algebraic_geometry\/stacks-git\/bootstrap.pdf\">Bootstrap<\/a> and <a href=\"http:\/\/www.math.columbia.edu\/algebraic_geometry\/stacks-git\/criteria.pdf\">Criteria for Representability<\/a>). Let me just tell you what the statement is. For algebraic spaces the result is that<\/p>\n<blockquote><p>If F is a sheaf on (Sch) in the &eacute;tale topology whose diagonal is representable by schemes and which has an &eacute;tale covering by a scheme, then F is also a sheaf in the fppf topology hence an algebraic space (as defined in the stacks project).<\/p><\/blockquote>\n<p>For algebraic stacks the result is that<\/p>\n<blockquote><p>If X is a stack in groupoids over (Sch) with the &eacute;tale topology whose diagonal is representable by algebraic spaces and which has a smooth covering by a scheme, then X is also a stack for the fppf topology hence an algebraic stack (as defined in the stacks project).<\/p><\/blockquote>\n<p>Till yesterday I had filed away this material under the heading: &#8220;Things that have to go into the stacks project at some point but which are not as interesting as other material I am working on now.&#8221; However, I probably should have worked it out sooner as some related remarks in the stacks project were misleading (I have now removed these remarks, see the red text in this <a href=\"https:\/\/github.com\/stacks\/stacks-project\/commit\/a54d1da506cb3035ae237792d07fe1babeceb80d\">commit<\/a>).<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Just yesterday, upon some prodding from Michael Thaddeus, I added a two short sections comparing algebraic spaces and algebraic stacks in the fppf and &eacute;tale topology (see the corresponding sections of chapters Bootstrap and Criteria for Representability). Let me just &hellip; <a href=\"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=1898\">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-1898","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\/1898","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=1898"}],"version-history":[{"count":10,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1898\/revisions"}],"predecessor-version":[{"id":2625,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1898\/revisions\/2625"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1898"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1898"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1898"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}