{"id":812,"date":"2010-09-06T15:05:00","date_gmt":"2010-09-06T15:05:00","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=812"},"modified":"2010-09-06T15:39:41","modified_gmt":"2010-09-06T15:39:41","slug":"gerbes","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=812","title":{"rendered":"Gerbes"},"content":{"rendered":"<p>Brian Conrad just emailed me to ask about gerbes in the stacks project. Unfortunately this is not yet in the stacks project (feel free to send your own write-up to <a href=\"mailto:stacks.project@gmail.com\">stacks.project@gmail.com<\/a> if you have one).<\/p>\n<p>This is how I would define them:<\/p>\n<p>(1) A gerbe over a site C is a stack in groupoids p : S &#8212;&gt; C with the following properties:<\/p>\n<ul>\n<li> for every object U of C there exists a covering {U_i &#8212;&gt; U} such that each fibre category S_{U_i} is nonempty, and<\/li>\n<li> for every object U of C and objects x, y in S_U there exists a covering {U_i &#8212;&gt; U} such that for every i in I we have that x|_{U_i} is isomorphic to y|_{U_i} in the fibre category S_{U_i}.<\/li>\n<\/ul>\n<p>Once this has been defined there should be a brief discussion of the &#8220;band&#8221; of a gerbe. In the case where the band is commutative it should be explained carefully that you get a sheaf of groups over the site. Actually, another important case is the case where you are given a sheaf of groups G on C and you consider gerbes whose band &#8220;is&#8221; G (this should be precisely defined). Also, it should be defined what is a &#8220;trivial&#8221; gerbe. I suggest we try to avoid cocycles as much as possible. For an informal discission for gerbes over topological spaces, see <a href=\"http:\/\/arxiv.org\/abs\/math\/0611317\">Lawrence Breen&#8217;s notes<\/a> or <a href=\"http:\/\/arxiv.org\/abs\/math\/0212266\">Ieke Moerdijk&#8217;s notes<\/a>. Another, less informal, reference would be the book by Giraud entitled &#8220;Cohomologie non abelienne&#8221;.<\/p>\n<p>(2) Let \\cX &#8212;&gt; X be a morphism from an algebraic stack \\cX to an algebraic space X. Then we say that \\cX is a gerbe over X if and only \\cX viewed as a stack in groupoids on (Sch\/X)_{fppf} is a gerbe as defined above. Moreover, all the notions defined in the abstract setting can be used in this setting also.<\/p>\n<p>This may not always correspond to the geometric picture of a gerbe, especially if the band (i.e., the automorphism group of an object) isn&#8217;t flat! But is it really always the case that gerbes in algebraic geometry have flat automorphism groups?<\/p>\n<p>As usual comments are welcome.<\/p>\n<p>[Edit: Brian adds that we could for instance prove (via erasable delta-functors) that gerbes describe H^2 with commutative coefficients (including its group structure *and* functoriality in both group and base space). And similarly give a torsor\/gerbe description of a 7-term exact sequence of pointed sets associated to a central extension of group sheaves. Most of this can be done without using cocycles. On pp. 144-145 of Milne&#8217;s book on etale cohomology he gives a nice little summary of the highlights on this aspect of Giraud&#8217;s book.<\/p>\n<p>Another great suggestion is: Explain as an example why Artin&#8217;s work shows that for an fppf group scheme (or algebraic space group) A, the A-gerbes are Artin stacks.]<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Brian Conrad just emailed me to ask about gerbes in the stacks project. Unfortunately this is not yet in the stacks project (feel free to send your own write-up to stacks.project@gmail.com if you have one). This is how I would &hellip; <a href=\"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=812\">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-812","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\/812","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=812"}],"version-history":[{"count":10,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/812\/revisions"}],"predecessor-version":[{"id":822,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/812\/revisions\/822"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=812"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=812"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=812"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}