{"id":1954,"date":"2011-10-16T15:54:41","date_gmt":"2011-10-16T15:54:41","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=1954"},"modified":"2011-10-16T15:54:41","modified_gmt":"2011-10-16T15:54:41","slug":"derived-pullback","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=1954","title":{"rendered":"Derived pullback"},"content":{"rendered":"

This post is a follow-up on the post on adequate modules<\/a>. There I described a construction of the higher direct images of a quasi-coherent sheaf in terms of the morphism of big fppf sites associated to a quasi-compact and quasi-separated morphism of schemes. As I mentioned in my last<\/a> post, this is now implemented (in the stacks project) for quasi-compact and quasi-separated morphisms of algebraic stacks, with the slight modification that we work with locally quasi-coherent modules with the flat base change property. (In this post, ‘module’ means fppf O-module.)<\/p>\n

Given an algebraic stack X let’s denote M_X either the abelian category of locally quasi-coherent modules with the flat base change property, or the abelian category of adequate modules. Since M_X is a weak Serre subcategory of Mod(O_X) we have the derived category D_M(X) := D_{M_X}(O_X) of complexes of O_X-modules whose cohomology sheaves are objects of M_X. The category of parasitic objects in M_X is a Serre subcategory. We define<\/p>\n

D_{QCoh}(X) = D_M(X) \/ complexes with parasitic cohomology sheaves<\/p><\/blockquote>\n

If X is a scheme, then this definition recovers the usual notion (see chapter on adequate modules for the adequate case). So now let’s think about the derived pullback of a quasi-coherent module along a morphism f : X —> Y of algebraic stacks. It is clear that we have an induced functor<\/p>\n

f^* : D_M(Y) —> D_M(X)<\/p><\/blockquote>\n

In fact f^* : Mod(O_Y) —> Mod(O_X) is exact (big sites!) and transforms objects of M_Y into objects of M_X. But f^* does not preserve parasitic modules if f isn’t a flat morphism of algebraic stacks. We define<\/em> Lf^* : D_{QCoh}(Y) —> D_{QCoh}(X) as the left derived functor (in the sense of Deligne, see Definition Tag 05S9<\/a>) of the displayed functor f^* above! What could be more natural?<\/p>\n

Thus the question isn’t “What is derived pullback?” but it is “When is derived pullback everywhere defined?”<\/p>\n

However, a better question is: “Does there exist a functor L^* : D_{QCoh}(Y) —> D_M(Y) which is left adjoint to the quotient functor q_Y : D_M(Y) —> D_{QCoh}(Y)?” If it exists, then Lf^* = q_X o f^* o L^* where q_X : D_M(X) —> D_{QCoh}(X) is the quotient functor for X, so derived pullback exists for any morphism with target Y. The existence of L^* is equivalent to asking the quotient map q_Y to be a Bousfield colocalization<\/em>: for every E in D_M(Y) there should be a distinguished triangle<\/p>\n

E’ —> E —> C —> E'[1]<\/p><\/blockquote>\n

in D_M(Y) where C is parasitic and Hom(E’, C’) = 0 for every parasitic object C’ of D_M(Y). Formulated in this way, there is lots of general theory we can rely on to (dis)prove the existence.<\/p>\n

For example if the subcategory of parasitic objects of D_M(Y) has products and they agree with products in D_M(Y) we are through I think; this isn’t as crazy as it sounds, e.g., the category of quasi-coherent sheaves on a scheme has products, see Akhil Mathew’s post<\/a>. (In any case being parasitic is preserved under products.) Hmm? I’ll think more.<\/p>\n","protected":false},"excerpt":{"rendered":"

This post is a follow-up on the post on adequate modules. There I described a construction of the higher direct images of a quasi-coherent sheaf in terms of the morphism of big fppf sites associated to a quasi-compact and quasi-separated … 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-1954","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\/1954","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=1954"}],"version-history":[{"count":27,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1954\/revisions"}],"predecessor-version":[{"id":1981,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1954\/revisions\/1981"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1954"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1954"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1954"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}