{"id":1127,"date":"2011-01-03T20:49:18","date_gmt":"2011-01-03T20:49:18","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=1127"},"modified":"2011-01-03T20:49:18","modified_gmt":"2011-01-03T20:49:18","slug":"a-sheaf","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=1127","title":{"rendered":"A sheaf"},"content":{"rendered":"

Max Lieblich asked if one could find an abelian sheaf G on the category of schemes in the \u00e9tale topology such that<\/p>\n

    \n
  1. G(X) = G(X_{red}),<\/li>\n
  2. G(X) = 0 when X has only one point,<\/li>\n
  3. G is not zero, and<\/li>\n
  4. G is limit preserving.<\/li>\n<\/ol>\n

    I’ll tell you why he asked in a minute, but first let me tell you an example: Let A be an abelian group. Let F be the presheaf on the category of schemes which associates to a scheme X the group of constructible functions a : |X| —> A modulo locally constant functions. Let G be the sheafification of F in the \u00e9tale topology. Then G works. (For more details, look at the the section entitled “Sheaves and constructible functions” in the chapter “Examples” of the stacks project.)<\/p>\n

    Why did this come up? Consider the stack [Spec(Z)\/G] classifying étale G-torsors. Then the morphism f : Spec(Z) —> [Spec(Z)\/G] is an equivalence of categories of sections over the spectrum of any field, f is formally étale, and the stack [Spec(Z)\/G] is limit preserving, but f is not an equivalence (as G is not zero). This answers a question posed by Dan Abramovich.<\/p>\n","protected":false},"excerpt":{"rendered":"

    Max Lieblich asked if one could find an abelian sheaf G on the category of schemes in the \u00e9tale topology such that G(X) = G(X_{red}), G(X) = 0 when X has only one point, G is not zero, and G … Continue reading →<\/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-1127","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\/1127","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=1127"}],"version-history":[{"count":11,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1127\/revisions"}],"predecessor-version":[{"id":1138,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1127\/revisions\/1138"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1127"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1127"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1127"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}