{"id":2514,"date":"2012-06-26T18:57:27","date_gmt":"2012-06-26T18:57:27","guid":{"rendered":"http:\/\/math.columbia.edu\/~dejong\/wordpress\/?p=2514"},"modified":"2012-06-28T21:52:51","modified_gmt":"2012-06-28T21:52:51","slug":"obstruction-theory","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/?p=2514","title":{"rendered":"Obstruction theory"},"content":{"rendered":"

This post continues the discussion started here<\/a>.<\/p>\n

Traditionally, an obstruction theory for a moduli problem is a way of computing infinitesimal automorphism groups, infinitesimal deformation spaces, and an obstruction space for a given moduli problem using cohomology. Moreover, in all cases where this can be done (as far as I know) these groups are computed as consecutive cohomology groups of a particular sheaf, or complex of sheaves, or sometimes consecutive ext groups. Let me give some examples.<\/p>\n

Let A’ \\to A be a surjection of rings over some base ring \u039b whose kernel is an ideal I having square zero.<\/p>\n

    \n
  1. If Y is a smooth proper algebraic space over A, then\n
      \n
    1. an obstruction to lifting Y to a smooth proper space over A’ lies in H^2(Y, T_{Y\/A} \u2297 I),<\/li>\n
    2. if Y has a lift Y’ then the set of isomorphism classes of lifts is principal homogeneous under H^1(Y, T_{Y\/A} \u2297 I),<\/li>\n
    3. the infinitesimal automorphism group of Y’ over Y is H^0(Y, T_{Y\/A} \u2297 I)<\/li>\n<\/ol>\n

      You can work this example out by yourself using just Cech cohomology methods.<\/li>\n

    4. If Y’ is a flat proper algebraic space over A’ and F is a finite locally free O_Y-module where Y = Y’ \u2297 A, then\n
        \n
      1. an obstruction to lifting F to a locally free module over Y’ lies in H^2(Y, End(F) \u2297 I)<\/li>\n
      2. if F has a lift F’ then the set of isomorphism classes of lifts is principal homogeneous under H^1(Y, End(F) \u2297 I)<\/li>\n
      3. the infinitesimal automorphism group of F’ over F is H^0(Y, End(F) \u2297 I)<\/li>\n<\/ol>\n

        Again a Cech cohomology computation will show you why this is true.<\/li>\n

      4. If X’ is an algebraic space flat over A’ and f : Y —> X is a morphism of algebraic spaces with Y flat and proper over A, then\n
          \n
        1. an obstruction to lifting f to f’ : Y’ —> X’ with Y’ flat over A’ lies in Ext^2(L_{Y\/X}, O_Y \u2297 I)<\/li>\n
        2. if f has a lift f’ then the set of isomorphism classes of lifts is principal homogeneous under Ext^1(L_{Y\/X}, O_Y \u2297 I)<\/li>\n
        3. the infinitesimal automorphism group of f’ over f is Ext^0(L_{Y\/X}, O_Y \u2297 I)<\/li>\n<\/ol>\n

          For this one I recommend looking in Illusie.<\/li>\n

        4. If X’ is an algebraic space over A’ (not necessarily flat) and f : Y —> X is a morphism of algebraic spaces with Y flat and proper over A and X = X’ \u2297 A. Denote g : Y —> X’ the composition of f and the closed immersion X —> X’. Let C \u2208 D(Y) be the cone of the map g^*L_{X’\/A’} —> L_{Y\/A}. Then\n
            \n
          1. an obstruction to lifting f to f’ : Y’ —> X’ with Y’ flat over A’ lies in Ext^2(C, O_Y \u2297 I)<\/li>\n
          2. if f has a lift f’ then the set of isomorphism classes of lifts is principal homogeneous under Ext^1(C, O_Y \u2297 I)<\/li>\n
          3. the infinitesimal automorphism group of f’ over f is Ext^0(C, O_Y \u2297 I)<\/li>\n<\/ol>\n

            For this one, I haven’t written out all the details. Note that the obstruction space maps to Ext^2(L_{Y\/A}, O_Y \u2297 I) and the obstruction in A maps to the obstruction to lifting Y to a flat space over A’. Once we have chosen a Y’ the obstruction of A is lifted to an element of<\/p>\n

            Ext^1_{O_Y}(g^*L_{X’\/A’}, O_Y \u2297 I) =
            \nExt^1_{g^{-1}O_{X’}}(g^{-1}L_{X’\/A’}, O_Y \u2297 I) =
            \nExt^1_{g^{-1}O_{X’}}(L_{g^{-1}O_{X’}\/A’}, O_Y \u2297 I) =
            \nExal_{A’}(g^{-1}O_{X’}, O_Y \u2297 I)<\/p><\/blockquote>\n

            which measures the obstruction to lifting f^# to a map g^{-1}O_{X’} —> O_{Y’}, i.e., measures the obstruction to lifting f to a morphism Y’ —> X’. Changing the choice of Y’ alters this obstruction by the corresponding element of Ext^1(L_{Y\/A}, O_Y \u2297 I). A similar story goes for the other groups.<\/li>\n<\/ol>\n

            In each of the cases above I think we can get a naive obstruction theory (as defined in the previous post). Essentially, each time the groups look like Ext^i(C, I), i = 0, 1, 2 for some object C of the derived category of some Y endowed with a proper flat morphism p : Y —> Spec(A). and you can take E = Rp_*(C \u2297 \u03c9^*_{Y\/A}) where \u03c9^*_{Y\/A} is the relative dualizing complex. [Edit June 28, 2012: This doesn’t work for case 4 because as Daivd Rydh points out below, the cone C may depend on A’. Thus you would have to allow for E to depend on the thickening… Ugh!]<\/p>\n

            Working dually.<\/strong> Folklore says that as soon as you can write down such a sequence of cohomology groups, then a naive obstruction theory should exist. The idea for the rest of this post is that you can try to axiomatize this. As stated here it only applies to cases 1 and 2 above; with some modifications it works in case 3 if you assume Y projective over A.<\/p>\n

            Let X be a category fibred in groupoids on (Sch\/\u039b). Let us define a dual naive obstruction theory<\/em> as being given by the following data<\/p>\n

              \n
            1. for every object x of X over a \u039b-algebra A we get K_x* \u2208 D(A),<\/li>\n
            2. for any surjection A’ —> A with square zero kernel I and x over A an element \u03be \u2208 H^2(K_x^* \u2297 I),<\/li>\n
            3. for any surjection A’ —> A with square zero kernel I and liftable x over A, a free transitive action of H^1(K_x^* \u2297 I) on the set of isomorphism classes of lifts,<\/li>\n
            4. for any surjection A’ —> A with square zero kernel I and x’ over A’, an identification of H^0(K_x^* \u2297 I) with the infinitesimal automorphisms of x’ over x.<\/li>\n<\/ol>\n

              We impose some axioms on these data; we refrain from listing them all here. An important axiom is functoriality: if we have A —> B and x over A with base change y to B, then K_x^* \u2297_A B = K_y^*. We will describe two other key axioms. Suppose that we have a pair (A, x) and three surjections A_i —> A, i = 1, 2, 3 with square zero kernels I_i. Moreover, suppose we have maps<\/p>\n

              A_1 —> A_2 —> A_3<\/p><\/blockquote>\n

              which induce a short exact sequence 0 —> I_1 —> I_2 —> I_3 —> 0. Denote<\/p>\n

              \u2202 : H^n(K_x^* \u2297 I_3) —> H^{n + 1}(K_x^* \u2297 I_1)<\/p><\/blockquote>\n

              the boundary operator on cohomology. Then, we require (using the functoriality axiom to identify some of the groups):<\/p>\n

                \n
              1. given lifts x_3 and x_3′ over A_3 differing by \u03b8 \u2208 H^1(K_x^* \u2297 I_3) the obstructions to lifting x_3 and x_3′ to A_1 differ by \u2202(\u03b8) in H^2(K_x^* \u2297 I_1),<\/li>\n
              2. given a lift x_2 over A_2 and an infinitesimal automorphism \u03b8 \u2208 H^0(K_x^* \u2297 I_3) of x_2|_{Spec(A_3)}, the obstruction to lifting \u03b8 to an infinitesimal automorphism of x_2 is \u2202(\u03b8) in H^2(K_x^* \u2297 I_1).<\/li>\n<\/ol>\n

                Now, I believe (I worked it out on the blackboard here yesterday but it got erased) that given such a theory one can construct a (somewhat canonical) element<\/p>\n

                \u03be(A, x) \u2208 H^1(K_x^* \u2297 NL_{A\/\u039b})<\/p><\/blockquote>\n

                which describes all the categories of lifts Lift(x, A’) for all surjections A’ —> A as above. Moreover, if K_x^* is a perfect complex, then we can set E = RHom_A(K_x^*, A) and use evaluation to get E —> NL_{A\/\u039b} and obtain a naive obstruction theory as in the previous post.<\/p>\n","protected":false},"excerpt":{"rendered":"

                This post continues the discussion started here. Traditionally, an obstruction theory for a moduli problem is a way of computing infinitesimal automorphism groups, infinitesimal deformation spaces, and an obstruction space for a given moduli problem using cohomology. Moreover, in all … 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-2514","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\/2514","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=2514"}],"version-history":[{"count":34,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/2514\/revisions"}],"predecessor-version":[{"id":2549,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/2514\/revisions\/2549"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2514"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2514"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~dejong\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2514"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}