{"id":669,"date":"2010-07-20T14:48:10","date_gmt":"2010-07-20T14:48:10","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=669"},"modified":"2010-07-20T14:48:10","modified_gmt":"2010-07-20T14:48:10","slug":"stratification-into-gerbs","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=669","title":{"rendered":"Stratification into gerbs"},"content":{"rendered":"<p>Here is a fun example. Take U = Spec(k[x_0, x_1, x_2, &#8230;]) and let G_m act by t(x_0, x_1, x_2, &#8230;) = (tx_0, t^px_1, t^{p^2}x_2, &#8230;) where p is a prime number. Let X = [U\/G_m]. This is an algebraic stack. There is a stratification of X by strata<\/p>\n<ul>\n<li>X_0 is where x_0 is not zero,<\/li>\n<li>X_1 is where x_0 is zero but x_1 is not zero,<\/li>\n<li>X_2 is where x_0, x_1 are zero, but x_2 is not zero,<\/li>\n<li>and so on&#8230;<\/li>\n<li>X_{infty} is where all the x_i are zero<\/li>\n<\/ul>\n<p>Each stratum is a gerb over a scheme with group \\mu_{p^i} for X_i and G_m for X_{infty}. The strata are reduced locally closed substacks. There is no coarser stratification with the same properties.<\/p>\n<p>So clearly, in order to prove a very general result as in the title of this post then we need to allow infinite stratifications&#8230;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Here is a fun example. Take U = Spec(k[x_0, x_1, x_2, &#8230;]) and let G_m act by t(x_0, x_1, x_2, &#8230;) = (tx_0, t^px_1, t^{p^2}x_2, &#8230;) where p is a prime number. Let X = [U\/G_m]. This is an algebraic &hellip; <a href=\"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=669\">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-669","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\/669","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=669"}],"version-history":[{"count":2,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/669\/revisions"}],"predecessor-version":[{"id":671,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/669\/revisions\/671"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=669"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=669"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=669"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}