{"id":1738,"date":"2011-08-10T18:41:45","date_gmt":"2011-08-10T18:41:45","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=1738"},"modified":"2011-08-10T18:41:45","modified_gmt":"2011-08-10T18:41:45","slug":"fppf-extensions","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=1738","title":{"rendered":"Fppf Extensions"},"content":{"rendered":"

Let X = Spec(A) be an affine scheme. Let M, N be A-modules. Let F, G be the sheaves of O_{big}-modules on the big fppf site of X associated to M and N, e.g., F(Spec(B)) = B ⊗_A M and similarly for G. As a by-product of the material on adequate modules<\/a> I proved the following formula<\/p>\n

Ext^i_{O_{big}}(F, G) = Pext^i_A(M, N)<\/p><\/blockquote>\n

The ext group on the left is the ext group in the category of all O_{big}-modules. The ext group on the right is the ith pure extension group<\/em> of M by N over A. This group is computed by taking a universally exact resolution 0 -> N -> I^0 -> I^1 -> … with each I^j pure injective and taking the ith cohomology group of Hom_A(M, I^*). An A-module I is pure injective<\/em> if for any universally injective map M_0 -> M_1 the map Hom_A(M_1, I) -> Hom_A(M_0, I) is surjective.<\/p>\n

There seems to be a lot of papers on pure modules, pure injectivity, etc. Gruson and Jensen characterized pure injective modules as those modules such that the functor – ⊗_A M : mod-A —> Ab is injective in the functor category (mod-A, Ab). Here mod-A is the category of finitely presented A-modules. It follows that<\/p>\n

Ext^i_{(mod-A, Ab)}(- ⊗_A M, – ⊗_A N) = Pext^i_A(M, N)<\/p><\/blockquote>\n

Our formula above is about O_{big}-modules, which in terms of functors means functors F : Alg_A —> Ab such that F(B) has the structure of a B-module for every A-algebra B and such that B —> B’ gives a B-linear map F(B) —> F(B’). These are called module-valued functors<\/em> (terminology due to Jaffe). Then we can rewrite the first equality above as<\/p>\n

Ext^i_{module-valued functors}(F, G) = Pext^i_A(M, N)<\/p><\/blockquote>\n

where F(B) = B ⊗_A M and G(B) = B ⊗_A N. In this formula you can let Alg_A be any sufficiently large category of A-algebras, e.g., the category of finitely presented A-algebras.<\/p>\n

The two results seem related. But there is a big difference between the functor categories (mod-A, Ab) and (Alg_A, Ab). Namely, if we look at Ext^i_{(Alg_A, Ab)}(F, G) then we get a completely different animal. For example suppose that G_a(B) = B for all A-algebras B and suppose that A is an F_p algebra. Then we see that Hom_{(Alg_A, Ab)}(G_a, G_a) contains the frobenius map frob : G_a —> G_a which on values over B raises every element to the pth power. In fact, the work of Breen<\/a> on ext groups of abelian sheaves on the fppf-site (warning: this is not exactly what he studies there) implies some of the higher ext groups Ext^i_{(Alg_A, Ab)}(G_a, G_a) are nonzero also (lowest case seems to be i = 2p)!<\/p>\n

Conclusion: module-valued functors over Alg_A and abelian group valued functors on mod-A somehow ends up giving the same ext groups for the functors associated to A-modules described above.<\/p>\n","protected":false},"excerpt":{"rendered":"

Let X = Spec(A) be an affine scheme. Let M, N be A-modules. Let F, G be the sheaves of O_{big}-modules on the big fppf site of X associated to M and N, e.g., F(Spec(B)) = B ⊗_A M and … 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-1738","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\/1738","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=1738"}],"version-history":[{"count":16,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1738\/revisions"}],"predecessor-version":[{"id":1754,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/1738\/revisions\/1754"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1738"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1738"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1738"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}