{"id":2389,"date":"2012-05-11T22:40:26","date_gmt":"2012-05-11T22:40:26","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=2389"},"modified":"2012-05-11T22:40:26","modified_gmt":"2012-05-11T22:40:26","slug":"flat-is-not-enough","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=2389","title":{"rendered":"Flat is not enough"},"content":{"rendered":"<p>The title of this blog post is the opposite of <a href=\"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=1584\">this post<\/a>. But don&#8217;t click through yet, because it may be more fun to read this one first.<\/p>\n<p>I claim there exists a functor F on the category of schemes such that<\/p>\n<ol>\n<li>F is a sheaf for the etale topology,<\/li>\n<li>the diagonal of F is representable by schemes, and<\/li>\n<li>there exists a scheme U and a surjective, finitely presented, flat morphism U &#8212;> F<\/li>\n<\/ol>\n<p>but F is not an algebraic space. Namely, let k be a field of characteristic p &gt; 0 and let k &sub; k&#8217; be a nontrivial finite purely inseparable extension. Define<\/p>\n<blockquote><p>F(S) = {f : S &#8212;> Spec(k), f factors through Spec(k&#8217;) etale locally on S}<\/p><\/blockquote>\n<p>It is easy to see that F satisfies (1). It satisfies (2) as F &#8212;> Spec(k) is a monomorphism. It satisfies (3) because U = Spec(k&#8217;) &#8212;> F works. But F is not an algebraic space, because if it were, then F would be isomorphic to Spec(k) by <a href=\"http:\/\/math.columbia.edu\/algebraic_geometry\/stacks-git\/locate.php?tag=06MG\">Lemma Tag 06MG<\/a>.<\/p>\n<p>Ok, now go back and read the other blog post I linked to above. Conclusion: to get Artin&#8217;s result as stated in that blog post you definitively need to work with the fppf topology.<\/p>\n<p>(Thanks to Bhargav for a discussion.)<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The title of this blog post is the opposite of this post. But don&#8217;t click through yet, because it may be more fun to read this one first. I claim there exists a functor F on the category of schemes &hellip; <a href=\"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=2389\">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-2389","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\/2389","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=2389"}],"version-history":[{"count":8,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/2389\/revisions"}],"predecessor-version":[{"id":2397,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/2389\/revisions\/2397"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2389"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2389"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2389"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}