{"id":2090,"date":"2011-12-07T14:25:44","date_gmt":"2011-12-07T14:25:44","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=2090"},"modified":"2011-12-07T14:25:44","modified_gmt":"2011-12-07T14:25:44","slug":"update-18","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=2090","title":{"rendered":"Update"},"content":{"rendered":"

Since the last update<\/a>on October 12 we have added the following material<\/p>\n

    \n
  1. Gabber’s argument that categories of quasi-coherent modules form a Grothendieck abelian category (for schemes, spaces, and algebraic stacks),<\/li>\n
  2. an example of an fpqc space which is not an algebraic space,<\/li>\n
  3. an example of a quasi-compact non-quasi-separated morphism of schemes such that pushforward does not preserve quasi-coherency,<\/li>\n
  4. some material related to my course on commutative algebra: exercise, lemmas, shorten proof of ZMT, etc<\/li>\n
  5. introduced lisse-etale (and flat-fppf) sites,<\/li>\n
  6. functoriality of lisse-etale topos for smooth morphisms (and flat-fppf for flat morphisms),<\/li>\n
  7. material on Grothendieck abelian categories, incuding existence of injectives and existence of enough K-injective complexes (following Spaltenstein and Serp\\’e),<\/li>\n
  8. cohomology of unbounded complexes and adjointness of Lf^* and Rf_*,<\/li>\n
  9. a lot of material on D_{QCoh}(X) for an algebraic stack X, including Rf_* (on bounded below for quasi-compact and quasi-separated morphisms) and Lf^* (unbounded for general f).<\/li>\n<\/ol>\n

    In particular my suggestion in this post<\/a> worked out exactly as advertised. The existence of Rf_* is straightforward. It turns out that once you prove that the category D_{QCoh}(X) as defined in the blog post is equivalent to the version of D_{QCoh}(X) in L-MB or Martin Olsson’s paper (i.e. defined using the lisse-etale site), then you immediately obtain the existence of Lf^*. Namely, the existence of the lisse-etale site is used to prove that the Verdier quotient used to define D_{QCoh}(X) is a Bousfield colocalization (technically it is easier to use the flat-fppf site to do this, because we use the fppf topology as our default topology, but one can use either).<\/p>\n

    A bit of care is needed when working with the lisse-etale site and the lisse-etale topos. As discussed elsewhere, one reason is that the lisse-etale topos isn’t functorial for morphisms of algebraic stacks. Here is a another. There is a comparison morphism of topoi<\/p>\n

    g : Sh(X_{lisse,etale}) —-> Sh(X_{etale})<\/p><\/blockquote>\n

    The functor g^{-1} has a left adjoint denoted g_! (on sheaves of sets) and we have g^{-1}g_! = g^{-1}g_* = id. This means that Sh(X_{lisse,etale}) is an essential subtopos<\/em> of Sh(X_{etale}), see SGA 4, IV, 7.6 and 9.1.1. Let K be a sheaf of sets on X_{lisse,etale}. Let I be an injective abelian sheaf on X_{etale}. Question: H^p(K, g^{-1}I) = 0? In other words, is the pullback by g of an injective abelian sheaf limp<\/a>? If true this would be a convenient way to compare cohomology of sheaves on X_{etale} with cohomology of sheaves on the lisse-etale site. Unfortunately, we think this isn’t true (Bhargav made what is likely a counter example — but we haven’t fully written out all the details).<\/p>\n","protected":false},"excerpt":{"rendered":"

    Since the last updateon October 12 we have added the following material Gabber’s argument that categories of quasi-coherent modules form a Grothendieck abelian category (for schemes, spaces, and algebraic stacks), an example of an fpqc space which is not an … 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-2090","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\/2090","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=2090"}],"version-history":[{"count":12,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/2090\/revisions"}],"predecessor-version":[{"id":2102,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/2090\/revisions\/2102"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2090"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2090"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2090"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}