{"id":990,"date":"2010-10-24T01:40:11","date_gmt":"2010-10-24T01:40:11","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=990"},"modified":"2012-07-20T13:53:33","modified_gmt":"2012-07-20T13:53:33","slug":"formal-glueing","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=990","title":{"rendered":"Formal glueing"},"content":{"rendered":"<p>Thanks to Bhargav and some editing by yours truly we now have a section on formal glueing in the stacks project. In fact it is in a new chapter entitled &#8220;More Algebra&#8221;. The main results are <a href=\"http:\/\/stacks.math.columbia.edu\/tag\/05ER\">Proposition Tag 05ER<\/a>, <a href=\"http:\/\/stacks.math.columbia.edu\/tag\/05ES\">Theorem Tag 05ES<\/a>, and <a href=\"http:\/\/stacks.math.columbia.edu\/tag\/05ET\">Proposition Tag 05ET<\/a> (look up tags <a href=\"http:\/\/stacks.math.columbia.edu\/tag\">here<\/a>). The original more self-contained version can be found on <a href=\"http:\/\/www-personal.umich.edu\/~bhattb\/\">Bhargav&#8217;s home page<\/a>.<\/p>\n<p>What can you do with this? Well, the simplest application is perhaps the following. Suppose that you have a curve C over a field k and a closed point p \u2208 C. Denote D the spectrum of the completion of the local ring of C at p, and denote D* the punctured spectrum. Then there exists an equivalence of categories between quasi-coherent sheaves on C and triples (F_U, F_D, &phi;) where F_U is a quasi-coherent sheaf on on U = C &#8211; {p} and F_D is a quasi-coherent sheaf on D and &phi; : F_U|_{D*} &#8212;> F_D|_{D*} is an isomorphism of quasi-coherent sheaves on D*.<\/p>\n<p>An interesting special case occurs when considering vector bundles with trivial determinant, i.e., finite locally free sheaves with trivial determinant. Namely, in this case the sheaves F_U and F_D are automatically free(!) and we can think of &phi; as an invertible matrix with coefficients in O(D^*). In other words, the set of isomorphism classes of vector bundles of rank n with trivial determinant on C is given by the double coset space<\/p>\n<p>SL_n(O(U)) \\ SL_n(O(D*)) \/ SL_n(O(D))<\/p>\n<p>Another interesting application concerns the study of &#8220;models&#8221; of schemes over C. Namely, instead of considering quasi-coherent sheaves we could consider triples (X_U, X_D, &phi;) where X_U is a scheme over U, and so on. In this generality it is probably not the case that such triples correspond to schemes over C (counter example anybody?). But if X_U, resp. X_D is affine over U, resp. D or if they are endowed with compatible (via &phi;) relatively ample invertible sheaves, then the result above implies in a straightforward manner that the triple (X_U, X_D, &phi;) arises from a scheme X over C.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Thanks to Bhargav and some editing by yours truly we now have a section on formal glueing in the stacks project. In fact it is in a new chapter entitled &#8220;More Algebra&#8221;. The main results are Proposition Tag 05ER, Theorem &hellip; <a href=\"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=990\">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-990","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\/990","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=990"}],"version-history":[{"count":14,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/990\/revisions"}],"predecessor-version":[{"id":2654,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/990\/revisions\/2654"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=990"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=990"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=990"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}