Naive obstruction theories

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!

3 thoughts on “Naive obstruction theories

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