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 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.

Let A’ \to A be a surjection of rings over some base ring Λ whose kernel is an ideal I having square zero.

1. If Y is a smooth proper algebraic space over A, then
   1. an obstruction to lifting Y to a smooth proper space over A’ lies in H^2(Y, T_{Y/A} ⊗ I),
   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} ⊗ I),
   3. the infinitesimal automorphism group of Y’ over Y is H^0(Y, T_{Y/A} ⊗ I)

   You can work this example out by yourself using just Cech cohomology methods.

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

   Again a Cech cohomology computation will show you why this is true.

3. 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
   1. an obstruction to lifting f to f’ : Y’ —> X’ with Y’ flat over A’ lies in Ext^2(L_{Y/X}, O_Y ⊗ I)
   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 ⊗ I)
   3. the infinitesimal automorphism group of f’ over f is Ext^0(L_{Y/X}, O_Y ⊗ I)

   For this one I recommend looking in Illusie.

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’ ⊗ A. Denote g : Y —> X’ the composition of f and the closed immersion X —> X’. Let C ∈ D(Y) be the cone of the map g^*L_{X’/A’} —> L_{Y/A}. Then
   1. an obstruction to lifting f to f’ : Y’ —> X’ with Y’ flat over A’ lies in Ext^2(C, O_Y ⊗ I)
   2. if f has a lift f’ then the set of isomorphism classes of lifts is principal homogeneous under Ext^1(C, O_Y ⊗ I)
   3. the infinitesimal automorphism group of f’ over f is Ext^0(C, O_Y ⊗ I)

   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 ⊗ 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

   Ext^1_{O_Y}(g^*L_{X’/A’}, O_Y ⊗ I) =
   Ext^1_{g^{-1}O_{X’}}(g^{-1}L_{X’/A’}, O_Y ⊗ I) =
   Ext^1_{g^{-1}O_{X’}}(L_{g^{-1}O_{X’}/A’}, O_Y ⊗ I) =
   Exal_{A’}(g^{-1}O_{X’}, O_Y ⊗ I)

   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 ⊗ I). A similar story goes for the other groups.

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 ⊗ ω^*_{Y/A}) where ω^*_{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!]

Working dually. 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.

Let X be a category fibred in groupoids on (Sch/Λ). Let us define a dual naive obstruction theory as being given by the following data

1. for every object x of X over a Λ-algebra A we get K_x* ∈ D(A),
2. for any surjection A’ —> A with square zero kernel I and x over A an element ξ ∈ H^2(K_x^* ⊗ I),
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^* ⊗ I) on the set of isomorphism classes of lifts,
4. for any surjection A’ —> A with square zero kernel I and x’ over A’, an identification of H^0(K_x^* ⊗ I) with the infinitesimal automorphisms of x’ over x.

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^* ⊗_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

A_1 —> A_2 —> A_3

which induce a short exact sequence 0 —> I_1 —> I_2 —> I_3 —> 0. Denote

∂ : H^n(K_x^* ⊗ I_3) —> H^{n + 1}(K_x^* ⊗ I_1)

the boundary operator on cohomology. Then, we require (using the functoriality axiom to identify some of the groups):

1. given lifts x_3 and x_3′ over A_3 differing by θ ∈ H^1(K_x^* ⊗ I_3) the obstructions to lifting x_3 and x_3′ to A_1 differ by ∂(θ) in H^2(K_x^* ⊗ I_1),
2. given a lift x_2 over A_2 and an infinitesimal automorphism θ ∈ H^0(K_x^* ⊗ I_3) of x_2|_{Spec(A_3)}, the obstruction to lifting θ to an infinitesimal automorphism of x_2 is ∂(θ) in H^2(K_x^* ⊗ I_1).

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

ξ(A, x) ∈ H^1(K_x^* ⊗ NL_{A/Λ})

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/Λ} and obtain a naive obstruction theory as in the previous post.

Let S be a scheme. Let X be a category fibred in groupoids over (Sch/S). In Artin’s work on algebraic stacks there is a notion of an obstruction theory for X. Artin splits the discussion into infinitesimal deformations and obstructions. Ideally we’d like to handle both at the same time. Sometimes the naive cotangent complex can be used to handle this.

Recall that the naive cotangent complex NL_{A/R} is the truncation τ_{≥ -1}L_{A/R} which is very weasy to work with, see Definition Tag 07BN. We can extend the definition of NL to schemes, algebraic spaces, and algebraic stacks (either by truncating the cotangent complex or by a direct construction we’ll come back to in the future).

Let’s define a naive obstruction theory for X over S as a rule which associated to every pair (T, x) where T is an affine scheme over S and x an object of X over T a map ξ : E —> NL_{T/S} in D(T) with the following properties:

1. the construction (E, ξ) is functorial in (T, x),
2. given a first order thickening T’ of T we have x lifts to x’ over T’ ⇔ the image of ξ in Hom(E, NL_{T’/T}) is zero,
3. the set of lifts x’ is principal homogeneous under Hom(E, NL_{T/T’}[-1]),
4. given two sections a,b : T’ —> T the lifts a^*x and b^*x differ by the element δ o ξ where δ = a – b : NL_{T/S} —-> NL_{T’/T}[-1] (see below), and
5. given a lift x’ then Inf(x’/x) = Hom(E, NL_{T/T’}[-2])

where Inf(x’/x) is the group of infinitesimal automorphisms of x’ over x. Note that NL_{T/T’} = I[1] where I is the ideal sheaf of T in T’ so the groups above are just Ext^{-1}(E, I), Hom(E, I), Ext^1(E, I). The map δ = a – b in 3 is just the composition NL_{T/S} —> Ω_{T/S} —> I associated to the difference between the ring maps a, b : O_T —> O_{T’}.

The motivation for this definition is the nonsensical formula “E = x^*NL_{X/S}”. It is nonsensical since we didn’t assume anything on X beyond being a category fibred in groupoids (b/c we’d like to use a naive obstruction theory to prove X is an algebraic stack). Thus a naive obstruction theory is an additional part of data. Of course, even a given algebraic stack X can have many different (naive) obstruction theories.

Example: If X is the stack whose category of sections over a scheme T is the category of families of smooth proper algebraic spaces of relative dimension d over T and x = (f : P —> T) then we can take E = Rf_*(ω_{P/T} ⊗ Ω^1_{P/T})[d – 1] and E —> NL_{T/S} the Kodaira spencer map.

Observations: (1) You do really have to take Rf_* because if P = P^1_T then in order for 3 to work you need E to be a rank 3 sheaf sitting in degree 1. (2) In order to define the Kodaira-Spencer map you use the triangle f^*NL_{T/S} —> NL_{P/S} —> NL_{P/T} and relative duality for f. (3) Using a bit of cohomology and base change, you can set E = dual perfect complex to Rf_*(T_{P/T}) and construct ξ whilst avoiding relative duality.

Versality. Now suppose that S is locally Noetherian and T of finite type over S. Let t be a closed point of T. Then we can ask if x is versal at t as defined in the chapter on Artin’s Axioms. If X has a naive obstruction theory, then (I haven’t checked all the details) x is versal at t if and only if

1. H^0(E ⊗ κ) —> H^0(NL_{T/S} ⊗ κ) is injective, and
2. H^{-1}(E ⊗ κ) —> H^{-1}(NL_{T/S} ⊗ κ) is surjective

where κ = κ(t).

Openness of versality. We’d like to show that if conditions i and ii hold, then the same is true in an open neighbourhood of t. Let C be the cone on the map ξ : E —> NL_{T/S}. Then conditions i and ii are equivalent to H^{-1}(C ⊗ κ) = 0. Provided that C has finite type cohomology modules, this condition then holds on an open neighbourhood of t, see Lemma Tag 068U as desired.

This is as it should be!

Let k be a field. Let G be a group scheme over k which is locally of finite type. Then X = [Spec(k)/G] is an algebraic stack over k (see for example Lemma Tag 06PL).

Let x_0 be the obvious 1-morphism Spec(k) —> X. Let’s look at the associated deformation problem, which in the stacks project is a category cofibred in groupoids

F = F_{X, k, x_0} —> C_k = (Artinian local k-algebras with residue field k)

see Section Tag 07T2. OK, so I was thinking about tangent spaces earlier today and it occurred to me that it is already somewhat fun to consider the example above. Namely, what is the tangent space TF in the situation above?

Your initial reaction might be “it is zero”. If you are a characteristic zero person, then you would be right, but before you read on: can you prove it?

Yeah, so the answer is that it is zero if G is a smooth group scheme over k (which is always the case in characteristic zero, see Lemma Tag 047N). Triviality of TF means that for every pair (T, t_0) where T is a G-torsor over Spec(k[ε]) and t_0 ∈ T(k), there exists a t ∈ T(k[ε]) which reduces to t_0 modulo ε. A torsor for a smooth group scheme is smooth. Hence the infinitesimal lifting criterion of smoothness implies that TF = 0.

But what if G isn’t smooth? In that case TF is always nontrivial. Namely, if TF = 0, then Spec(k) —> X is smooth (argument omitted) which isn’t true because Spec(k) x_X Spec(k) = G. I think that in general

dim(TF) + dim(G) = embedding dimension of G

but I haven’t tried to prove it or look it up. As usual, I welcome suggestions, comments, references, etc.