98.22 Obstruction theories
In this section we describe what an obstruction theory is. Contrary to the spaces of infinitesimal deformations and infinitesimal automorphisms, an obstruction theory is an additional piece of data. The formulation is motivated by the results of Lemma 98.21.2 and Remark 98.21.3.
Definition 98.22.1. Let $S$ be a locally Noetherian base. Let $\mathcal{X}$ be a category fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. An obstruction theory is given by the following data
for every $S$-algebra $A$ such that $\mathop{\mathrm{Spec}}(A) \to S$ maps into an affine open and every object $x$ of $\mathcal{X}$ over $\mathop{\mathrm{Spec}}(A)$ an $A$-linear functor
\[ \mathcal{O}_ x : \text{Mod}_ A \to \text{Mod}_ A \]
of obstruction modules,
for $(x, A)$ as in (1), a ring map $A \to B$, $M \in \text{Mod}_ A$, $N \in \text{Mod}_ B$, and an $A$-linear map $M \to N$ an induced $A$-linear map $\mathcal{O}_ x(M) \to \mathcal{O}_ y(N)$ where $y = x|_{\mathop{\mathrm{Spec}}(B)}$, and
for every deformation situation $(x, A' \to A)$ an obstruction element $o_ x(A') \in \mathcal{O}_ x(I)$ where $I = \mathop{\mathrm{Ker}}(A' \to A)$.
These data are subject to the following conditions
the functoriality maps turn the obstruction modules into a functor from the category of triples $(x, A, M)$ to sets,
for every morphism of deformation situations $(y, B' \to B) \to (x, A' \to A)$ the element $o_ x(A')$ maps to $o_ y(B')$, and
we have
\[ \text{Lift}(x, A') \not= \emptyset \Leftrightarrow o_ x(A') = 0 \]
for every deformation situation $(x, A' \to A)$.
This last condition explains the terminology. The module $\mathcal{O}_ x(A')$ is called the obstruction module. The element $o_ x(A')$ is the obstruction. Most obstruction theories have additional properties, and in order to make them useful additional conditions are needed. Moreover, this is just a sample definition, for example in the definition we could consider only deformation situations of finite type over $S$.
One of the main reasons for introducing obstruction theories is to check openness of versality. An example of this type of result is Lemma 98.22.2 below. The initial idea to do this is due to Artin, see the papers of Artin mentioned in the introduction. It has been taken up for example in the work by Flenner [Flenner], Hall [Hall-coherent], Hall and Rydh [rydh_axioms], Olsson [olsson_deformation], Olsson and Starr [olsson-starr], and Lieblich [lieblich-complexes] (random order of references). Moreover, for particular categories fibred in groupoids, often authors develop a little bit of theory adapted to the problem at hand. We will develop this theory later (insert future reference here).
reference
Lemma 98.22.2. Let $S$ be a locally Noetherian scheme. Let $\mathcal{X}$ be a category fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Assume
$\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ is representable by algebraic spaces,
$\mathcal{X}$ has (RS*),
$\mathcal{X}$ is limit preserving,
there exists an obstruction theory1,
for an object $x$ of $\mathcal{X}$ over $\mathop{\mathrm{Spec}}(A)$ and $A$-modules $M_ n$, $n \geq 1$ we have
$T_ x(\prod M_ n) = \prod T_ x(M_ n)$,
$\mathcal{O}_ x(\prod M_ n) \to \prod \mathcal{O}_ x(M_ n)$ is injective.
Then $\mathcal{X}$ satisfies openness of versality.
Proof.
We prove this by verifying condition (4) of Lemma 98.20.3. Let $(\xi _ n)$ and $(R_ n)$ be as in Remark 98.20.2 such that $\mathop{\mathrm{Ker}}(R_ m \to R_ n)$ is an ideal of square zero for all $m \geq n$. Set $A = R_1$ and $x = \xi _1$. Denote $M_ n = \mathop{\mathrm{Ker}}(R_ n \to R_1)$. Then $M_ n$ is an $A$-module. Set $R = \mathop{\mathrm{lim}}\nolimits R_ n$. Let
\[ \tilde R = \{ (r_1, r_2, r_3 \ldots ) \in \prod R_ n \text{ such that all have the same image in }A\} \]
Then $\tilde R \to A$ is surjective with kernel $M = \prod M_ n$. There is a map $R \to \tilde R$ and a map $\tilde R \to A[M]$, $(r_1, r_2, r_3, \ldots ) \mapsto (r_1, r_2 - r_1, r_3 - r_2, \ldots )$. Together these give a short exact sequence
\[ (x, R \to A) \to (x, \tilde R \to A) \to (x, A[M]) \]
of deformation situations, see Remark 98.21.5. The associated sequence of kernels $0 \to \mathop{\mathrm{lim}}\nolimits M_ n \to M \to M \to 0$ is the canonical sequence computing the limit of the system of modules $(M_ n)$.
Let $o_ x(\tilde R) \in \mathcal{O}_ x(M)$ be the obstruction element. Since we have the lifts $\xi _ n$ we see that $o_ x(\tilde R)$ maps to zero in $\mathcal{O}_ x(M_ n)$. By assumption (5)(b) we see that $o_ x(\tilde R) = 0$. Choose a lift $\tilde\xi $ of $x$ to $\mathop{\mathrm{Spec}}(\tilde R)$. Let $\tilde\xi _ n$ be the restriction of $\tilde\xi $ to $\mathop{\mathrm{Spec}}(R_ n)$. There exists elements $t_ n \in T_ x(M_ n)$ such that $t_ n \cdot \tilde\xi _ n = \xi _ n$ by Lemma 98.21.2 part (2)(b). By assumption (5)(a) we can find $t \in T_ x(M)$ mapping to $t_ n$ in $T_ x(M_ n)$. After replacing $\tilde\xi $ by $t \cdot \tilde\xi $ we find that $\tilde\xi $ restricts to $\xi _ n$ over $\mathop{\mathrm{Spec}}(R_ n)$ for all $n$. In particular, since $\xi _{n + 1}$ restricts to $\xi _ n$ over $\mathop{\mathrm{Spec}}(R_ n)$, the restriction $\overline{\xi }$ of $\tilde\xi $ to $\mathop{\mathrm{Spec}}(A[M])$ has the property that it restricts to the trivial deformation over $\mathop{\mathrm{Spec}}(A[M_ n])$ for all $n$. Hence by assumption (5)(a) we find that $\overline{\xi }$ is the trivial deformation of $x$. By axiom (RS*) applied to $R = \tilde R \times _{A[M]} A$ this implies that $\tilde\xi $ is the pullback of a deformation $\xi $ of $x$ over $R$. This finishes the proof.
$\square$
Example 98.22.3. Let $S = \mathop{\mathrm{Spec}}(\Lambda )$ for some Noetherian ring $\Lambda $. Let $W \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ W$-module flat over $S$. Consider the functor
\[ F : (\mathit{Sch}/S)_{fppf}^{opp} \longrightarrow \textit{Sets}, \quad T/S \longrightarrow H^0(W_ T, \mathcal{F}_ T) \]
where $W_ T = T \times _ S W$ is the base change and $\mathcal{F}_ T$ is the pullback of $\mathcal{F}$ to $W_ T$. If $T = \mathop{\mathrm{Spec}}(A)$ we will write $W_ T = W_ A$, etc. Let $\mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ be the category fibred in groupoids associated to $F$. Then $\mathcal{X}$ has an obstruction theory. Namely,
given $A$ over $\Lambda $ and $x \in H^0(W_ A, \mathcal{F}_ A)$ we set $\mathcal{O}_ x(M) = H^1(W_ A, \mathcal{F}_ A \otimes _ A M)$,
given a deformation situation $(x, A' \to A)$ we let $o_ x(A') \in \mathcal{O}_ x(A)$ be the image of $x$ under the boundary map
\[ H^0(W_ A, \mathcal{F}_ A) \longrightarrow H^1(W_ A, \mathcal{F}_ A \otimes _ A I) \]
coming from the short exact sequence of modules
\[ 0 \to \mathcal{F}_ A \otimes _ A I \to \mathcal{F}_{A'} \to \mathcal{F}_ A \to 0. \]
We have omitted some details, in particular the construction of the short exact sequence above (it uses that $W_ A$ and $W_{A'}$ have the same underlying topological space) and the explanation for why flatness of $\mathcal{F}$ over $S$ implies that the sequence above is short exact.
Example 98.22.4 (Key example). Let $S = \mathop{\mathrm{Spec}}(\Lambda )$ for some Noetherian ring $\Lambda $. Say $\mathcal{X} = (\mathit{Sch}/X)_{fppf}$ with $X = \mathop{\mathrm{Spec}}(R)$ and $R = \Lambda [x_1, \ldots , x_ n]/J$. The naive cotangent complex $\mathop{N\! L}\nolimits _{R/\Lambda }$ is (canonically) homotopy equivalent to
\[ J/J^2 \longrightarrow \bigoplus \nolimits _{i = 1, \ldots , n} R\text{d}x_ i, \]
see Algebra, Lemma 10.134.2. Consider a deformation situation $(x, A' \to A)$. Denote $I$ the kernel of $A' \to A$. The object $x$ corresponds to $(a_1, \ldots , a_ n)$ with $a_ i \in A$ such that $f(a_1, \ldots , a_ n) = 0$ in $A$ for all $f \in J$. Set
\begin{align*} \mathcal{O}_ x(A') & = \mathop{\mathrm{Hom}}\nolimits _ R(J/J^2, I)/\mathop{\mathrm{Hom}}\nolimits _ R(R^{\oplus n}, I) \\ & = \mathop{\mathrm{Ext}}\nolimits ^1_ R(\mathop{N\! L}\nolimits _{R/\Lambda }, I) \\ & = \mathop{\mathrm{Ext}}\nolimits ^1_ A(\mathop{N\! L}\nolimits _{R/\Lambda } \otimes _ R A, I). \end{align*}
Choose lifts $a_ i' \in A'$ of $a_ i$ in $A$. Then $o_ x(A')$ is the class of the map $J/J^2 \to I$ defined by sending $f \in J$ to $f(a_1', \ldots , a'_ n) \in I$. We omit the verification that $o_ x(A')$ is independent of choices. It is clear that if $o_ x(A') = 0$ then the map lifts. Finally, functoriality is straightforward. Thus we obtain an obstruction theory. We observe that $o_ x(A')$ can be described a bit more canonically as the composition
\[ \mathop{N\! L}\nolimits _{R/\Lambda } \to \mathop{N\! L}\nolimits _{A/\Lambda } \to \mathop{N\! L}\nolimits _{A/A'} = I[1] \]
in $D(A)$, see Algebra, Lemma 10.134.6 for the last identification.
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