{"id":1982,"date":"2011-10-17T00:35:16","date_gmt":"2011-10-17T00:35:16","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=1982"},"modified":"2012-07-14T00:23:10","modified_gmt":"2012-07-14T00:23:10","slug":"quasi-coherent-sheaves","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=1982","title":{"rendered":"Quasi-coherent sheaves"},"content":{"rendered":"<p>This is a follow-up to <a href=\"http:\/\/amathew.wordpress.com\/2011\/07\/30\/quasi-coherent-sheaves-presentable-categories-and-a-result-of-gabber\/\">Akhil Mathew&#8217;s blog post<\/a> which explains that the category of quasi-coherent sheaves on a scheme X is a Grothendieck abelian category. The key is a result of Gabber: given a scheme X there exists a cardinal &kappa; such that every quasi-coherent sheaf is the directed colimit of its &kappa;-generated quasi-coherent subsheaves. It follows by a standard argument that the embedding QCoh(O_X) &#8212;> Mod(O_X) has a right adjoint, whence limits exist in QCoh(O_X) and QCoh(O_X) has enough injectives.<\/p>\n<p>Earlier today I wrote this up for the stacks project (it is in the <a href=\"http:\/\/math.columbia.edu\/algebraic_geometry\/stacks-git\/properties.pdf\">chapter on properties of schemes<\/a>) and it occurred to me that the exact same results hold for algebraic stacks, with the exact same proof. (The proof is one of these &#8220;randomly pick elements and see what happens&#8221; arguments, kinda like <a href=\"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=1766\">this post<\/a>.) I&#8217;ll check the details and write out the proof some time later this week; keep watching <a href=\"https:\/\/github.com\/stacks\/stacks-project\/commits\/master.atom\">this feed<\/a> to see it appear.<\/p>\n<p>Anyway, I guess it is just one of those general facts&#8230; easy to prove but hard to use.<\/p>\n<p>Edit 10\/17\/2011. Beware of the following facts on quasi-coherent modules:<\/p>\n<ul>\n<li>It isn&#8217;t true that a product of quasi-coherent modules is quasi-coherent.<\/li>\n<li>An injective object in QCoh(O_X) is not always injective O_X-module.<\/li>\n<li>Cohomology using resolutions in QCoh(O_X) does not agree with cohomology.<\/li>\n<li>There exists a ring A and an injective A-module I such that the quasi-coherent sheaf I~ associated to I isn&#8217;t flasque, I~ isn&#8217;t an injective O_X-module, and there exists an open U of Spec(A) such that I~|_U isn&#8217;t an injective object of QCoh(O_U).<\/li>\n<li>D^+_{QCoh}(O_X) isn&#8217;t equivalent to D^+(QCoh(O_X)) in general.<\/li>\n<li>The coherator Q : Mod(O_X) &#8212;> QCoh(O_X) isn&#8217;t exact in general.<\/li>\n<li>And so on.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>This is a follow-up to Akhil Mathew&#8217;s blog post which explains that the category of quasi-coherent sheaves on a scheme X is a Grothendieck abelian category. The key is a result of Gabber: given a scheme X there exists a &hellip; <a href=\"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=1982\">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-1982","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\/1982","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=1982"}],"version-history":[{"count":14,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1982\/revisions"}],"predecessor-version":[{"id":2624,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1982\/revisions\/2624"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1982"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1982"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1982"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}