{"id":1440,"date":"2011-04-06T14:27:17","date_gmt":"2011-04-06T14:27:17","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=1440"},"modified":"2011-04-06T14:27:17","modified_gmt":"2011-04-06T14:27:17","slug":"quasi-quasi-coherent-sheaves","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=1440","title":{"rendered":"Quasi quasi-coherent sheaves"},"content":{"rendered":"

On any ringed topos there is a notion of a quasi-coherent sheaf, see Definition Tag 03DL<\/a>. The pullback of a quasi-coherent module via any morphism of ringed topoi is quasi-coherent, see Lemma Tag 03DO<\/a>.<\/p>\n

Let (X, O_X) be a scheme. Let tau = fppf, syntomic, etale, smooth, or Zariski. The site (Sch\/X)_{tau} is a ringed site with sheaf of rings O. The category of quasi-coherent O_X-modules on X is equivalent to the category of quasi-coherent O-modules on(Sch\/X)_{tau}, see Proposition Tag 03DX<\/a>. This equivalence is compatible with pullback, but in general not with pushforward, see Proposition Tag 03LC<\/a>.<\/p>\n

Let me explain this last point a bit. Suppose\u00a0 f : X —> Y is a quasi-compact and quasi-separated morphism of schemes. Denote f_{big} the morphism of big tau sites. Let F be a quasi-coherent O_X-module on X. The corresponding quasi-coherent O-module F^a on (Sch\/X)_{tau} is given by the rule F^a(U) = Γ(U, φ^*F) if φ : U —> X is an object of (Sch\/X)_{tau}. In general, for a sheaf G on (Sch\/X)_{tau} we have f_{big, *}G(V) = G(V \\times_Y X). Hence we see that the restriction of f_{big, *}F^a to V_{Zar} is given by the (usual) pushforward via the projection V \\times_Y X —> V of the (usual) pullback of F to V \\times_Y X via the other projection. It follows from the description of quasi-coherent sheaves on (Sch\/Y)_{tau} as associated to usual quasi-coherent sheaves on Y that f_{big, *}F^a is quasi-coherent on (Sch\/Y)_{tau} if and only if formation of f_*F commutes with arbitrary base change. This is simply not the case, even for morphisms of varieties, etc.<\/p>\n

On the other hand, we know that f_*F commutes with any flat base change (still assuming f quasi-compact and quasi-separated). Hence f_{big, *}F^a is a sheaf H on (Sch\/Y)_{tau} such that H|_{V_{Zar}} and H|_{V_{etale}} are quasi-coherent. Moreover, the same argument shows that if G is any sheaf of O-modules on (Sch\/X)_{tau} such that G|_{U_{Zar}} or G|_{U_{etale}} is quasi-coherent for every U\/X then H = f_{big, *}G is a sheaf such that H|_{V_{Zar}} or H|_{V_{etale}} are quasi-coherent for any object V of (Sch\/Y)_{tau}. Moreover, this property is also preserved by f_{big}^* as this is just given by restriction.<\/p>\n

Thus a convenient class of O-modules on (Sch\/X)_{tau} appears to be the category of sheaves of O-modules F such that F|_{U_{etale}} is quasi-coherent for all U\/X. These “quasi quasi-coherent sheaves” are preserved under any pullback and pushforward along quasi-compact and quasi-separated morphisms. Via the approach I sketched here<\/a> they give a notion of quasi quasi-coherent sheaves on the tau site of any algebraic stack with arbitrary pullbacks and pushforward along quasi-compact and quasi-separated morphisms. An interesting example of a quasi quasi-coherent sheaf is the sheaf of differentials Ω on the etale site that I mentioned in here<\/a>.<\/p>\n

Can anybody suggest a better name for these sheaves?<\/p>\n","protected":false},"excerpt":{"rendered":"

On any ringed topos there is a notion of a quasi-coherent sheaf, see Definition Tag 03DL. The pullback of a quasi-coherent module via any morphism of ringed topoi is quasi-coherent, see Lemma Tag 03DO. Let (X, O_X) be a scheme. … 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-1440","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\/1440","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=1440"}],"version-history":[{"count":8,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1440\/revisions"}],"predecessor-version":[{"id":1448,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1440\/revisions\/1448"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1440"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1440"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1440"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}