{"id":3163,"date":"2013-02-21T22:22:50","date_gmt":"2013-02-21T22:22:50","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=3163"},"modified":"2013-02-21T22:22:50","modified_gmt":"2013-02-21T22:22:50","slug":"derived-lower-shriek","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=3163","title":{"rendered":"Derived lower shriek"},"content":{"rendered":"

Let (X, OX<\/sub>) be a ringed space. Let π : C<\/i> —> X be a stack over X where we use the topology on X to view X as a site. Endow C<\/i> with the topology inherited from X (see Definition 06NV<\/a>). This (roughly) means that the fibre categories C<\/i>U<\/sub><\/i> where U ⊂ X is open are endowed with the chaotic topology. Denote B = π -1<\/sup>OX<\/sub> and think of C<\/i> as a ringed site and π as a morphism of ringed sites<\/p>\n

π : (C<\/i>, B) —-> (X, OX<\/sub>)<\/p><\/blockquote>\n

The functor π*<\/sup> = π -1<\/sup> : Mod(OX<\/sub>) —> Mod(B) commutes with all limits and colimits on modules and hence has a left adjoint π!<\/sub> : Mod(B) —> Mod(OX<\/sub>). In fact, if F is a sheaf of B-modules on C<\/i>, then we can describe π!<\/sub>F as the sheaf associated to the presheaf<\/p>\n

U |—> colimξ in opposite of C<\/i>U<\/sub><\/sub> F(ξ)<\/p><\/blockquote>\n

on the topological space X. (Colimit taken in category OX<\/sub>(U)-modules.) Actually, it turns out that the situation above is a special case of this section of the Stacks project<\/a> and we obtain a left derived extension Lπ!<\/sub> : D(B) —> D(OX<\/sub>) for free (note there are no boundedness assumptions).<\/p>\n

In fact, the construction shows a little bit more. Namely, let ξ be an object of C<\/i> lying over the open U ⊂ X. Then we can consider the localization morphism jξ<\/sub> : C<\/i>\/ξ —> C<\/i> and the sheaf Oξ<\/sub> = jξ, !<\/sub>B|ξ<\/sub>. Any B-module is a quotient of a direct sum of these Oξ<\/sub> and we have<\/p>\n

Lπ!<\/sub> Oξ<\/sub> = π!<\/sub> Oξ<\/sub><\/p><\/blockquote>\n

Cool, so this gives us a bit of control in trying to compute Lπ!<\/sub>.<\/p>\n

Let x be a point of X. Let C<\/i>x<\/sub> denote the category<\/p>\n

colimx ∈ U ⊂ X<\/sub> C<\/i>U<\/sub><\/i><\/p><\/blockquote>\n

This makes sense as C<\/i> is a stack over X so we can think of it as a sheaf of categories. If F is a sheaf of B-modules on C<\/i>, then the stalk of π!<\/sub>F is just the colimit of the “values” of F over C<\/i>x<\/sub>. Since taking stalks is exact, I think this should mean that we can compute the stalk of Lπ!<\/sub>F at x by taking the corresponding construction over the category C<\/i>x<\/sub> with its chaotic topology.<\/p>\n

Another tool to compute Lπ!<\/sub> should be that if C<\/i> is given as the stackification of a category C<\/i>‘ fibred over X, then it should be sufficient to compute with C<\/i>‘. Going back to the discussion and especially the example in this post<\/a> we have to replace our choice of C<\/i> there. We should start with the fibred category C<\/i>‘ of immersions φ : U —> A<\/strong>n<\/sup>B<\/sub> (not necessarily closed) and commutative diagrams over B. Then C<\/i> should be the stackification of that. Then with all of the above you’d get the cotangent complex of X\/B by doing the same construction as in the affine case<\/a>. The key is that affine locally C<\/i>‘ has a good co-simplicial object computing the derived lower shriek functor. You use the localization of sheaves of algebras construction to provide C<\/i> with a sheaf of rings surjecting onto the pullback of the structure sheaf of X (and not to change the underlying category).<\/p>\n

A similar procedure is going to define the base change C<\/i>S<\/sub> given a morphism of schemes S —> B, i.e., as underlying fibred category start with some category of diagrams of schemes and use the localization of sheaves of algebras construction to endow this with a structure sheaf.<\/p>\n

I think this will just work and in fact it simplifies the original idea I had for the stacks C<\/i> and C<\/i>S<\/sub>. We’ll see.<\/p>\n","protected":false},"excerpt":{"rendered":"

Let (X, OX) be a ringed space. Let π : C —> X be a stack over X where we use the topology on X to view X as a site. Endow C with the topology inherited from X (see … 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-3163","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\/3163","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=3163"}],"version-history":[{"count":29,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/3163\/revisions"}],"predecessor-version":[{"id":3192,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/3163\/revisions\/3192"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=3163"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=3163"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=3163"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}