Lemma 98.23.1. Let $A \to k$ be a ring map with $k$ a field. Let $E \in D^-(A)$. Then $\mathop{\mathrm{Ext}}\nolimits ^ i_ A(E, k) = \mathop{\mathrm{Hom}}\nolimits _ k(H^{-i}(E \otimes ^\mathbf {L} k), k)$.
98.23 Naive obstruction theories
The title of this section refers to the fact that we will use the naive cotangent complex in this section. Let $(x, A' \to A)$ be a deformation situation for a given category fibred in groupoids over a locally Noetherian scheme $S$. The key Example 98.22.4 suggests that any obstruction theory should be closely related to maps in $D(A)$ with target the naive cotangent complex of $A$. Working this out we find a criterion for versality in Lemma 98.23.3 which leads to a criterion for openness of versality in Lemma 98.23.4. We introduce a notion of a naive obstruction theory in Definition 98.23.5 to try to formalize the notion a bit further.
In the following we will use the naive cotangent complex as defined in Algebra, Section 10.134. In particular, if $A' \to A$ is a surjection of $\Lambda $-algebras with square zero kernel $I$, then there are maps
\[ \mathop{N\! L}\nolimits _{A'/\Lambda } \to \mathop{N\! L}\nolimits _{A/\Lambda } \to \mathop{N\! L}\nolimits _{A/A'} \]
whose composition is homotopy equivalent to zero (see Algebra, Remark 10.134.5). This doesn't form a distinguished triangle in general as we are using the naive cotangent complex and not the full one. There is a homotopy equivalence $\mathop{N\! L}\nolimits _{A/A'} \to I[1]$ (the complex consisting of $I$ placed in degree $-1$, see Algebra, Lemma 10.134.6). Finally, note that there is a canonical map $\mathop{N\! L}\nolimits _{A/\Lambda } \to \Omega _{A/\Lambda }$.
Proof. Omitted. Hint: Replace $E$ by a bounded above complex of free $A$-modules and compute both sides. $\square$
Lemma 98.23.2. Let $\Lambda \to A \to k$ be finite type ring maps of Noetherian rings with $k = \kappa (\mathfrak p)$ for some prime $\mathfrak p$ of $A$. Let $\xi : E \to \mathop{N\! L}\nolimits _{A/\Lambda }$ be morphism of $D^{-}(A)$ such that $H^{-1}(\xi \otimes ^{\mathbf{L}} k)$ is not surjective. Then there exists a surjection $A' \to A$ of $\Lambda $-algebras such that
$I = \mathop{\mathrm{Ker}}(A' \to A)$ has square zero and is isomorphic to $k$ as an $A$-module,
$\Omega _{A'/\Lambda } \otimes k = \Omega _{A/\Lambda } \otimes k$, and
$E \to \mathop{N\! L}\nolimits _{A/A'}$ is zero.
Proof. Let $f \in A$, $f \not\in \mathfrak p$. Suppose that $A'' \to A_ f$ satisfies (a), (b), (c) for the induced map $E \otimes _ A A_ f \to \mathop{N\! L}\nolimits _{A_ f/\Lambda }$, see Algebra, Lemma 10.134.13. Then we can set $A' = A'' \times _{A_ f} A$ and get a solution. Namely, it is clear that $A' \to A$ satisfies (a) because $\mathop{\mathrm{Ker}}(A' \to A) = \mathop{\mathrm{Ker}}(A'' \to A) = I$. Pick $f'' \in A''$ lifting $f$. Then the localization of $A'$ at $(f'', f)$ is isomorphic to $A''$ (for example by More on Algebra, Lemma 15.5.3). Thus (b) and (c) are clear for $A'$ too. In this way we see that we may replace $A$ by the localization $A_ f$ (finitely many times). In particular (after such a replacement) we may assume that $\mathfrak p$ is a maximal ideal of $A$, see Morphisms, Lemma 29.16.1.
Choose a presentation $A = \Lambda [x_1, \ldots , x_ n]/J$. Then $\mathop{N\! L}\nolimits _{A/\Lambda }$ is (canonically) homotopy equivalent to
\[ J/J^2 \longrightarrow \bigoplus \nolimits _{i = 1, \ldots , n} A\text{d}x_ i, \]
see Algebra, Lemma 10.134.2. After localizing if necessary (using Nakayama's lemma) we can choose generators $f_1, \ldots , f_ m$ of $J$ such that $f_ j \otimes 1$ form a basis for $J/J^2 \otimes _ A k$. Moreover, after renumbering, we can assume that the images of $\text{d}f_1, \ldots , \text{d}f_ r$ form a basis for the image of $J/J^2 \otimes k \to \bigoplus k\text{d}x_ i$ and that $\text{d}f_{r + 1}, \ldots , \text{d}f_ m$ map to zero in $\bigoplus k\text{d}x_ i$. With these choices the space
\[ H^{-1}(\mathop{N\! L}\nolimits _{A/\Lambda } \otimes ^{\mathbf{L}}_ A k) = H^{-1}(\mathop{N\! L}\nolimits _{A/\Lambda } \otimes _ A k) \]
has basis $f_{r + 1} \otimes 1, \ldots , f_ m \otimes 1$. Changing basis once again we may assume that the image of $H^{-1}(\xi \otimes ^{\mathbf{L}} k)$ is contained in the $k$-span of $f_{r + 1} \otimes 1, \ldots , f_{m - 1} \otimes 1$. Set
\[ A' = \Lambda [x_1, \ldots , x_ n]/(f_1, \ldots , f_{m - 1}, \mathfrak pf_ m) \]
By construction $A' \to A$ satisfies (a). Since $\text{d}f_ m$ maps to zero in $\bigoplus k\text{d}x_ i$ we see that (b) holds. Finally, by construction the induced map $E \to \mathop{N\! L}\nolimits _{A/A'} = I[1]$ induces the zero map $H^{-1}(E \otimes _ A^\mathbf {L} k) \to I \otimes _ A k$. By Lemma 98.23.1 we see that the composition is zero. $\square$
The following lemma is our key technical result.
Lemma 98.23.3. Let $S$ be a locally Noetherian scheme. Let $\mathcal{X}$ be a category fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$ satisfying (RS*). Let $U = \mathop{\mathrm{Spec}}(A)$ be an affine scheme of finite type over $S$ which maps into an affine open $\mathop{\mathrm{Spec}}(\Lambda )$. Let $x$ be an object of $\mathcal{X}$ over $U$. Let $\xi : E \to \mathop{N\! L}\nolimits _{A/\Lambda }$ be a morphism of $D^{-}(A)$. Assume
for every deformation situation $(x, A' \to A)$ we have: $x$ lifts to $\mathop{\mathrm{Spec}}(A')$ if and only if $E \to \mathop{N\! L}\nolimits _{A/\Lambda } \to \mathop{N\! L}\nolimits _{A/A'}$ is zero, and
there is an isomorphism of functors $T_ x(-) \to \mathop{\mathrm{Ext}}\nolimits ^0_ A(E, -)$ such that $E \to \mathop{N\! L}\nolimits _{A/\Lambda } \to \Omega ^1_{A/\Lambda }$ corresponds to the canonical element (see Remark 98.21.8).
Let $u_0 \in U$ be a finite type point with residue field $k = \kappa (u_0)$. Consider the following statements
$x$ is versal at $u_0$, and
$\xi : E \to \mathop{N\! L}\nolimits _{A/\Lambda }$ induces a surjection $H^{-1}(E \otimes _ A^{\mathbf{L}} k) \to H^{-1}(\mathop{N\! L}\nolimits _{A/\Lambda } \otimes _ A^{\mathbf{L}} k)$ and an injection $H^0(E \otimes _ A^{\mathbf{L}} k) \to H^0(\mathop{N\! L}\nolimits _{A/\Lambda } \otimes _ A^{\mathbf{L}} k)$.
Then we always have (2) $\Rightarrow $ (1) and we have (1) $\Rightarrow $ (2) if $u_0$ is a closed point.
Proof. Let $\mathfrak p = \mathop{\mathrm{Ker}}(A \to k)$ be the prime corresponding to $u_0$.
Assume that $x$ versal at $u_0$ and that $u_0$ is a closed point of $U$. If $H^{-1}(\xi \otimes _ A^{\mathbf{L}} k)$ is not surjective, then let $A' \to A$ be an extension with kernel $I$ as in Lemma 98.23.2. Because $u_0$ is a closed point, we see that $I$ is a finite $A$-module, hence that $A'$ is a finite type $\Lambda $-algebra (this fails if $u_0$ is not closed). In particular $A'$ is Noetherian. By property (c) for $A'$ and (i) for $\xi $ we see that $x$ lifts to an object $x'$ over $A'$. Let $\mathfrak p' \subset A'$ be kernel of the surjective map to $k$. By Artin-Rees (Algebra, Lemma 10.51.2) there exists an $n > 1$ such that $(\mathfrak p')^ n \cap I = 0$. Then we see that
\[ B' = A'/(\mathfrak p')^ n \longrightarrow A/\mathfrak p^ n = B \]
is a small, essential extension of local Artinian rings, see Formal Deformation Theory, Lemma 90.3.12. On the other hand, as $x$ is versal at $u_0$ and as $x'|_{\mathop{\mathrm{Spec}}(B')}$ is a lift of $x|_{\mathop{\mathrm{Spec}}(B)}$, there exists an integer $m \geq n$ and a map $q : A/\mathfrak p^ m \to B'$ such that the composition $A/\mathfrak p^ m \to B' \to B$ is the quotient map. Since the maximal ideal of $B'$ has $n$th power equal to zero, this $q$ factors through $B$ which contradicts the fact that $B' \to B$ is an essential surjection. This contradiction shows that $H^{-1}(\xi \otimes _ A^{\mathbf{L}} k)$ is surjective.
Assume that $x$ versal at $u_0$. By Lemma 98.23.1 the map $H^0(\xi \otimes _ A^{\mathbf{L}} k)$ is dual to the map $\mathop{\mathrm{Ext}}\nolimits ^0_ A(\mathop{N\! L}\nolimits _{A/\Lambda }, k) \to \text{Ext}^0_ A(E, k)$. Note that
\[ \mathop{\mathrm{Ext}}\nolimits ^0_ A(\mathop{N\! L}\nolimits _{A/\Lambda }, k) = \text{Der}_\Lambda (A, k) \quad \text{and}\quad T_ x(k) = \mathop{\mathrm{Ext}}\nolimits ^0_ A(E, k) \]
Condition (ii) assures us the map $\mathop{\mathrm{Ext}}\nolimits ^0_ A(\mathop{N\! L}\nolimits _{A/\Lambda }, k) \to \text{Ext}^0_ A(E, k)$ sends a tangent vector $\theta $ to $U$ at $u_0$ to the corresponding infinitesimal deformation of $x_0$, see Remark 98.21.8. Hence if $x$ is versal, then this map is surjective, see Formal Deformation Theory, Lemma 90.13.2. Hence $H^0(\xi \otimes _ A^{\mathbf{L}} k)$ is injective. This finishes the proof of (1) $\Rightarrow $ (2) in case $u_0$ is a closed point.
For the rest of the proof assume $H^{-1}(E \otimes _ A^\mathbf {L} k) \to H^{-1}(\mathop{N\! L}\nolimits _{A/\Lambda } \otimes _ A^\mathbf {L} k)$ is surjective and $H^0(E \otimes _ A^\mathbf {L} k) \to H^0(\mathop{N\! L}\nolimits _{A/\Lambda } \otimes _ A^\mathbf {L} k)$ injective. Set $R = A_\mathfrak p^\wedge $ and let $\eta $ be the formal object over $R$ associated to $x|_{\mathop{\mathrm{Spec}}(R)}$. The map $d\underline{\eta }$ on tangent spaces is surjective because it is identified with the dual of the injective map $H^0(E \otimes _ A^{\mathbf{L}} k) \to H^0(\mathop{N\! L}\nolimits _{A/\Lambda } \otimes _ A^{\mathbf{L}} k)$ (see previous paragraph). According to Formal Deformation Theory, Lemma 90.13.2 it suffices to prove the following: Let $C' \to C$ be a small extension of finite type Artinian local $\Lambda $-algebras with residue field $k$. Let $R \to C$ be a $\Lambda $-algebra map compatible with identifications of residue fields. Let $y = x|_{\mathop{\mathrm{Spec}}(C)}$ and let $y'$ be a lift of $y$ to $C'$. To show: we can lift the $\Lambda $-algebra map $R \to C$ to $R \to C'$.
Observe that it suffices to lift the $\Lambda $-algebra map $A \to C$. Let $I = \mathop{\mathrm{Ker}}(C' \to C)$. Note that $I$ is a $1$-dimensional $k$-vector space. The obstruction $ob$ to lifting $A \to C$ is an element of $\mathop{\mathrm{Ext}}\nolimits ^1_ A(\mathop{N\! L}\nolimits _{A/\Lambda }, I)$, see Example 98.22.4. By Lemma 98.23.1 and our assumption the map $\xi $ induces an injection
\[ \mathop{\mathrm{Ext}}\nolimits ^1_ A(\mathop{N\! L}\nolimits _{A/\Lambda }, I) \longrightarrow \mathop{\mathrm{Ext}}\nolimits ^1_ A(E, I) \]
By the construction of $ob$ and (i) the image of $ob$ in $\mathop{\mathrm{Ext}}\nolimits ^1_ A(E, I)$ is the obstruction to lifting $x$ to $A \times _ C C'$. By (RS*) the fact that $y/C$ lifts to $y'/C'$ implies that $x$ lifts to $A \times _ C C'$. Hence $ob = 0$ and we are done. $\square$
The key lemma above allows us to conclude that we have openness of versality in some cases.
Lemma 98.23.4. Let $S$ be a locally Noetherian scheme. Let $\mathcal{X}$ be a category fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$ satisfying (RS*). Let $U = \mathop{\mathrm{Spec}}(A)$ be an affine scheme of finite type over $S$ which maps into an affine open $\mathop{\mathrm{Spec}}(\Lambda )$. Let $x$ be an object of $\mathcal{X}$ over $U$. Let $\xi : E \to \mathop{N\! L}\nolimits _{A/\Lambda }$ be a morphism of $D^{-}(A)$. Assume
for every deformation situation $(x, A' \to A)$ we have: $x$ lifts to $\mathop{\mathrm{Spec}}(A')$ if and only if $E \to \mathop{N\! L}\nolimits _{A/\Lambda } \to \mathop{N\! L}\nolimits _{A/A'}$ is zero,
there is an isomorphism of functors $T_ x(-) \to \mathop{\mathrm{Ext}}\nolimits ^0_ A(E, -)$ such that $E \to \mathop{N\! L}\nolimits _{A/\Lambda } \to \Omega ^1_{A/\Lambda }$ corresponds to the canonical element (see Remark 98.21.8),
the cohomology groups of $E$ are finite $A$-modules.
If $x$ is versal at a closed point $u_0 \in U$, then there exists an open neighbourhood $u_0 \in U' \subset U$ such that $x$ is versal at every finite type point of $U'$.
Proof. Let $C$ be the cone of $\xi $ so that we have a distinguished triangle
\[ E \to \mathop{N\! L}\nolimits _{A/\Lambda } \to C \to E[1] \]
in $D^{-}(A)$. By Lemma 98.23.3 the assumption that $x$ is versal at $u_0$ implies that $H^{-1}(C \otimes ^\mathbf {L} k) = 0$. By More on Algebra, Lemma 15.76.4 there exists an $f \in A$ not contained in the prime corresponding to $u_0$ such that $H^{-1}(C \otimes ^\mathbf {L}_ A M) = 0$ for any $A_ f$-module $M$. Using Lemma 98.23.3 again we see that we have versality for all finite type points of the open $D(f) \subset U$. $\square$
The technical lemmas above suggest the following definition.
Definition 98.23.5. Let $S$ be a locally Noetherian base. Let $\mathcal{X}$ be a category fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Assume that $\mathcal{X}$ satisfies (RS*). A naive 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 $\mathop{\mathrm{Spec}}(\Lambda ) \subset S$ and every object $x$ of $\mathcal{X}$ over $\mathop{\mathrm{Spec}}(A)$ we are given an object $E_ x \in D^-(A)$ and a map $\xi _ x : E \to \mathop{N\! L}\nolimits _{A/\Lambda }$,
given $(x, A)$ as in (1) there are transformations of functors
for $(x, A)$ as in (1) and a ring map $A \to B$ setting $y = x|_{\mathop{\mathrm{Spec}}(B)}$ there is a functoriality map $E_ x \to E_ y$ in $D(A)$.
\[ \text{Inf}_ x( - ) \to \mathop{\mathrm{Ext}}\nolimits ^{-1}_ A(E_ x, -) \quad \text{and}\quad T_ x(-) \to \mathop{\mathrm{Ext}}\nolimits ^0_ A(E_ x, -) \]
These data are subject to the following conditions
in the situation of (3) the diagram
\[ \xymatrix{ E_ y \ar[r]_{\xi _ y} & \mathop{N\! L}\nolimits _{B/\Lambda } \\ E_ x \ar[u] \ar[r]^{\xi _ x} & \mathop{N\! L}\nolimits _{A/\Lambda } \ar[u] } \]is commutative in $D(A)$,
given $(x, A)$ as in (1) and $A \to B \to C$ setting $y = x|_{\mathop{\mathrm{Spec}}(B)}$ and $z = x|_{\mathop{\mathrm{Spec}}(C)}$ the composition of the functoriality maps $E_ x \to E_ y$ and $E_ y \to E_ z$ is the functoriality map $E_ x \to E_ z$,
the maps of (2) are isomorphisms compatible with the functoriality maps and the maps of Remark 98.21.3,
the composition $E_ x \to \mathop{N\! L}\nolimits _{A/\Lambda } \to \Omega _{A/\Lambda }$ corresponds to the canonical element of $T_ x(\Omega _{A/\Lambda }) = \mathop{\mathrm{Ext}}\nolimits ^0(E_ x, \Omega _{A/\Lambda })$, see Remark 98.21.8,
given a deformation situation $(x, A' \to A)$ with $I = \mathop{\mathrm{Ker}}(A' \to A)$ the composition $E_ x \to \mathop{N\! L}\nolimits _{A/\Lambda } \to \mathop{N\! L}\nolimits _{A/A'}$ is zero in
\[ \mathop{\mathrm{Hom}}\nolimits _ A(E_ x, \mathop{N\! L}\nolimits _{A/\Lambda }) = \mathop{\mathrm{Ext}}\nolimits ^0_ A(E_ x, \mathop{N\! L}\nolimits _{A/A'}) = \mathop{\mathrm{Ext}}\nolimits ^1_ A(E_ x, I) \]if and only if $x$ lifts to $A'$.
Thus we see in particular that we obtain an obstruction theory as in Section 98.22 by setting $\mathcal{O}_ x( - ) = \mathop{\mathrm{Ext}}\nolimits ^1_ A(E_ x, -)$.
Lemma 98.23.6. Let $S$ and $\mathcal{X}$ be as in Definition 98.23.5 and let $\mathcal{X}$ be endowed with a naive obstruction theory. Let $A \to B$ and $y \to x$ be as in (3). Let $k$ be a $B$-algebra which is a field. Then the functoriality map $E_ x \to E_ y$ induces bijections for $i = 0, 1$.
\[ H^ i(E_ x \otimes _ A^{\mathbf{L}} k) \to H^ i(E_ y \otimes _ B^{\mathbf{L}} k) \]
Proof. Let $z = x|_{\mathop{\mathrm{Spec}}(k)}$. Then (RS*) implies that
\[ \textit{Lift}(x, A[k]) = \textit{Lift}(z, k[k]) \quad \text{and}\quad \textit{Lift}(y, B[k]) = \textit{Lift}(z, k[k]) \]
because $A[k] = A \times _ k k[k]$ and $B[k] = B \times _ k k[k]$. Hence the properties of a naive obstruction theory imply that the functoriality map $E_ x \to E_ y$ induces bijections $\mathop{\mathrm{Ext}}\nolimits ^ i_ A(E_ x, k) \to \text{Ext}^ i_ B(E_ y, k)$ for $i = -1, 0$. By Lemma 98.23.1 our maps $H^ i(E_ x \otimes _ A^{\mathbf{L}} k) \to H^ i(E_ y \otimes _ B^{\mathbf{L}} k)$, $i = 0, 1$ induce isomorphisms on dual vector spaces hence are isomorphisms. $\square$
Lemma 98.23.7. Let $S$ be a locally Noetherian scheme. Let $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}^{opp}$ be a category fibred in groupoids. Assume that $\mathcal{X}$ satisfies (RS*) and that $\mathcal{X}$ has a naive obstruction theory. Then openness of versality holds for $\mathcal{X}$ provided the complexes $E_ x$ of Definition 98.23.5 have finitely generated cohomology groups for pairs $(A, x)$ where $A$ is of finite type over $S$.
Proof. Let $U$ be a scheme locally of finite type over $S$, let $x$ be an object of $\mathcal{X}$ over $U$, and let $u_0$ be a finite type point of $U$ such that $x$ is versal at $u_0$. We may first shrink $U$ to an affine scheme such that $u_0$ is a closed point and such that $U \to S$ maps into an affine open $\mathop{\mathrm{Spec}}(\Lambda )$. Say $U = \mathop{\mathrm{Spec}}(A)$. Let $\xi _ x : E_ x \to \mathop{N\! L}\nolimits _{A/\Lambda }$ be the obstruction map. At this point we may apply Lemma 98.23.4 to conclude. $\square$
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