## 96.21 Infinitesimal deformations

In this section we discuss a generalization of the notion of the tangent space introduced in Section 96.8. To do this intelligently, we borrow some notation from Formal Deformation Theory, Sections 88.11, 88.17, and 88.19.

Let $S$ be a scheme. Let $\mathcal{X}$ be a category fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Given a homomorphism $A' \to A$ of $S$-algebras and an object $x$ of $\mathcal{X}$ over $\mathop{\mathrm{Spec}}(A)$ we write $\textit{Lift}(x, A')$ for the category of lifts of $x$ to $\mathop{\mathrm{Spec}}(A')$. An object of $\textit{Lift}(x, A')$ is a morphism $x \to x'$ of $\mathcal{X}$ lying over $\mathop{\mathrm{Spec}}(A) \to \mathop{\mathrm{Spec}}(A')$ and morphisms of $\textit{Lift}(x, A')$ are defined as commutative diagrams. The set of isomorphism classes of $\textit{Lift}(x, A')$ is denoted $\text{Lift}(x, A')$. See Formal Deformation Theory, Definition 88.17.1 and Remark 88.17.2. If $A' \to A$ is surjective with locally nilpotent kernel we call an element $x'$ of $\text{Lift}(x, A')$ a (infinitesimal) deformation of $x$. In this case the group of infinitesimal automorphisms of $x'$ over $x$ is the kernel

$\text{Inf}(x'/x) = \mathop{\mathrm{Ker}}\left( \text{Aut}_{\mathcal{X}_{\mathop{\mathrm{Spec}}(A')}}(x') \to \text{Aut}_{\mathcal{X}_{\mathop{\mathrm{Spec}}(A)}}(x)\right)$

Note that an element of $\text{Inf}(x'/x)$ is the same thing as a lift of $\text{id}_ x$ over $\mathop{\mathrm{Spec}}(A')$ for (the category fibred in sets associated to) $\mathit{Aut}_\mathcal {X}(x')$. Compare with Formal Deformation Theory, Definition 88.19.1 and Formal Deformation Theory, Remark 88.19.8.

If $M$ is an $A$-module we denote $A[M]$ the $A$-algebra whose underlying $A$-module is $A \oplus M$ and whose multiplication is given by $(a, m) \cdot (a', m') = (aa', am' + a'm)$. When $M = A$ this is the ring of dual numbers over $A$, which we denote $A[\epsilon ]$ as is customary. There is an $A$-algebra map $A[M] \to A$. The pullback of $x$ to $\mathop{\mathrm{Spec}}(A[M])$ is called the trivial deformation of $x$ to $\mathop{\mathrm{Spec}}(A[M])$.

Lemma 96.21.1. Let $S$ be a scheme. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Let

$\xymatrix{ B' \ar[r] & B \\ A' \ar[u] \ar[r] & A \ar[u] }$

be a commutative diagram of $S$-algebras. Let $x$ be an object of $\mathcal{X}$ over $\mathop{\mathrm{Spec}}(A)$, let $y$ be an object of $\mathcal{Y}$ over $\mathop{\mathrm{Spec}}(B)$, and let $\phi : f(x)|_{\mathop{\mathrm{Spec}}(B)} \to y$ be a morphism of $\mathcal{Y}$ over $\mathop{\mathrm{Spec}}(B)$. Then there is a canonical functor

$\textit{Lift}(x, A') \longrightarrow \textit{Lift}(y, B')$

of categories of lifts induced by $f$ and $\phi$. The construction is compatible with compositions of $1$-morphisms of categories fibred in groupoids in an obvious manner.

Proof. This lemma proves itself. $\square$

Let $S$ be a base scheme. Let $\mathcal{X}$ be a category fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. We define a category whose objects are pairs $(x, A' \to A)$ where

1. $A' \to A$ is a surjection of $S$-algebras whose kernel is an ideal of square zero,

2. $x$ is an object of $\mathcal{X}$ lying over $\mathop{\mathrm{Spec}}(A)$.

A morphism $(y, B' \to B) \to (x, A' \to A)$ is given by a commutative diagram

$\xymatrix{ B' \ar[r] & B \\ A' \ar[u] \ar[r] & A \ar[u] }$

of $S$-algebras together with a morphism $x|_{\mathop{\mathrm{Spec}}(B)} \to y$ over $\mathop{\mathrm{Spec}}(B)$. Let us call this the category of deformation situations.

Lemma 96.21.2. Let $S$ be a scheme. Let $\mathcal{X}$ be a category fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Assume $\mathcal{X}$ satisfies condition (RS*). Let $A$ be an $S$-algebra and let $x$ be an object of $\mathcal{X}$ over $\mathop{\mathrm{Spec}}(A)$.

1. There exists an $A$-linear functor $\text{Inf}_ x : \text{Mod}_ A \to \text{Mod}_ A$ such that given a deformation situation $(x, A' \to A)$ and a lift $x'$ there is an isomorphism $\text{Inf}_ x(I) \to \text{Inf}(x'/x)$ where $I = \mathop{\mathrm{Ker}}(A' \to A)$.

2. There exists an $A$-linear functor $T_ x : \text{Mod}_ A \to \text{Mod}_ A$ such that

1. given $M$ in $\text{Mod}_ A$ there is a bijection $T_ x(M) \to \text{Lift}(x, A[M])$,

2. given a deformation situation $(x, A' \to A)$ there is an action

$T_ x(I) \times \text{Lift}(x, A') \to \text{Lift}(x, A')$

where $I = \mathop{\mathrm{Ker}}(A' \to A)$. It is simply transitive if $\text{Lift}(x, A') \not= \emptyset$.

Proof. We define $\text{Inf}_ x$ as the functor

$\text{Mod}_ A \longrightarrow \textit{Sets},\quad M \longrightarrow \text{Inf}(x'_ M/x) = \text{Lift}(\text{id}_ x, A[M])$

mapping $M$ to the group of infinitesimal automorphisms of the trivial deformation $x'_ M$ of $x$ to $\mathop{\mathrm{Spec}}(A[M])$ or equivalently the group of lifts of $\text{id}_ x$ in $\mathit{Aut}_\mathcal {X}(x'_ M)$. We define $T_ x$ as the functor

$\text{Mod}_ A \longrightarrow \textit{Sets},\quad M \longrightarrow \text{Lift}(x, A[M])$

of isomorphism classes of infintesimal deformations of $x$ to $\mathop{\mathrm{Spec}}(A[M])$. We apply Formal Deformation Theory, Lemma 88.11.4 to $\text{Inf}_ x$ and $T_ x$. This lemma is applicable, since (RS*) tells us that

$\textit{Lift}(x, A[M \times N]) = \textit{Lift}(x, A[M]) \times \textit{Lift}(x, A[N])$

as categories (and trivial deformations match up too).

Let $(x, A' \to A)$ be a deformation situation. Consider the ring map $g : A' \times _ A A' \to A[I]$ defined by the rule $g(a_1, a_2) = \overline{a_1} \oplus a_2 - a_1$. There is an isomorphism

$A' \times _ A A' \longrightarrow A' \times _ A A[I]$

given by $(a_1, a_2) \mapsto (a_1, g(a_1, a_2))$. This isomorphism commutes with the projections to $A'$ on the first factor, and hence with the projections to $A$. Thus applying (RS*) twice we find equivalences of categories

\begin{align*} \textit{Lift}(x, A') \times \textit{Lift}(x, A') & = \textit{Lift}(x, A' \times _ A A') \\ & = \textit{Lift}(x, A' \times _ A A[I]) \\ & = \textit{Lift}(x, A') \times \textit{Lift}(x, A[I]) \end{align*}

Using these maps and projection onto the last factor of the last product we see that we obtain “difference maps”

$\text{Inf}(x'/x) \times \text{Inf}(x'/x) \longrightarrow \text{Inf}_ x(I) \quad \text{and}\quad \text{Lift}(x, A') \times \text{Lift}(x, A') \longrightarrow T_ x(I)$

These difference maps satisfy the transitivity rule “$(x'_1 - x'_2) + (x'_2 - x'_3) = x'_1 - x'_3$” because

$\xymatrix{ A' \times _ A A' \times _ A A' \ar[rrrrr]_-{(a_1, a_2, a_3) \mapsto (g(a_1, a_2), g(a_2, a_3))} \ar[rrrrrd]_{(a_1, a_2, a_3) \mapsto g(a_1, a_3)} & & & & & A[I] \times _ A A[I] = A[I \times I] \ar[d]^{+} \\ & & & & & A[I] }$

is commutative. Inverting the string of equivalences above we obtain an action which is free and transitive provided $\text{Inf}(x'/x)$, resp. $\text{Lift}(x, A')$ is nonempty. Note that $\text{Inf}(x'/x)$ is always nonempty as it is a group. $\square$

Remark 96.21.3 (Functoriality). Assumptions and notation as in Lemma 96.21.2. Suppose $A \to B$ is a ring map and $y = x|_{\mathop{\mathrm{Spec}}(B)}$. Let $M \in \text{Mod}_ A$, $N \in \text{Mod}_ B$ and let $M \to N$ an $A$-linear map. Then there are canonical maps $\text{Inf}_ x(M) \to \text{Inf}_ y(N)$ and $T_ x(M) \to T_ y(N)$ simply because there is a pullback functor

$\textit{Lift}(x, A[M]) \to \textit{Lift}(y, B[N])$

coming from the ring map $A[M] \to B[N]$. Similarly, given a morphism of deformation situations $(y, B' \to B) \to (x, A' \to A)$ we obtain a pullback functor $\textit{Lift}(x, A') \to \textit{Lift}(y, B')$. Since the construction of the action, the addition, and the scalar multiplication on $\text{Inf}_ x$ and $T_ x$ use only morphisms in the categories of lifts (see proof of Formal Deformation Theory, Lemma 88.11.4) we see that the constructions above are functorial. In other words we obtain $A$-linear maps

$\text{Inf}_ x(M) \to \text{Inf}_ y(N) \quad \text{and}\quad T_ x(M) \to T_ y(N)$

such that the diagrams

$\vcenter { \xymatrix{ \text{Inf}_ y(J) \ar[r] & \text{Inf}(y'/y) \\ \text{Inf}_ x(I) \ar[r] \ar[u] & \text{Inf}(x'/x) \ar[u] } } \quad \text{and}\quad \vcenter { \xymatrix{ T_ y(J) \times \text{Lift}(y, B') \ar[r] & \text{Lift}(y, B') \\ T_ x(I) \times \text{Lift}(x, A') \ar[r] \ar[u] & \text{Lift}(x, A') \ar[u] } }$

commute. Here $I = \mathop{\mathrm{Ker}}(A' \to A)$, $J = \mathop{\mathrm{Ker}}(B' \to B)$, $x'$ is a lift of $x$ to $A'$ (which may not always exist) and $y' = x'|_{\mathop{\mathrm{Spec}}(B')}$.

Remark 96.21.4 (Automorphisms). Assumptions and notation as in Lemma 96.21.2. Let $x', x''$ be lifts of $x$ to $A'$. Then we have a composition map

$\text{Inf}(x'/x) \times \mathop{Mor}\nolimits _{\textit{Lift}(x, A')}(x', x'') \times \text{Inf}(x''/x) \longrightarrow \mathop{Mor}\nolimits _{\textit{Lift}(x, A')}(x', x'').$

Since $\textit{Lift}(x, A')$ is a groupoid, if $\mathop{Mor}\nolimits _{\textit{Lift}(x, A')}(x', x'')$ is nonempty, then this defines a simply transitive left action of $\text{Inf}(x'/x)$ on $\mathop{Mor}\nolimits _{\textit{Lift}(x, A')}(x', x'')$ and a simply transitive right action by $\text{Inf}(x''/x)$. Now the lemma says that $\text{Inf}(x'/x) = \text{Inf}_ x(I) = \text{Inf}(x''/x)$. We claim that the two actions described above agree via these identifications. Namely, either $x' \not\cong x''$ in which the claim is clear, or $x' \cong x''$ and in that case we may assume that $x'' = x'$ in which case the result follows from the fact that $\text{Inf}(x'/x)$ is commutative. In particular, we obtain a well defined action

$\text{Inf}_ x(I) \times \mathop{Mor}\nolimits _{\textit{Lift}(x, A')}(x', x'') \longrightarrow \mathop{Mor}\nolimits _{\textit{Lift}(x, A')}(x', x'')$

which is simply transitive as soon as $\mathop{Mor}\nolimits _{\textit{Lift}(x, A')}(x', x'')$ is nonempty.

Remark 96.21.5. Let $S$ be a scheme. Let $\mathcal{X}$ be a category fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Let $A$ be an $S$-algebra. There is a notion of a short exact sequence

$(x, A_1' \to A) \to (x, A_2' \to A) \to (x, A_3' \to A)$

of deformation situations: we ask the corresponding maps between the kernels $I_ i = \mathop{\mathrm{Ker}}(A_ i' \to A)$ give a short exact sequence

$0 \to I_3 \to I_2 \to I_1 \to 0$

of $A$-modules. Note that in this case the map $A_3' \to A_1'$ factors through $A$, hence there is a canonical isomorphism $A_1' = A[I_1]$.

Lemma 96.21.6. Let $S$ be a scheme. Let $p : \mathcal{X} \to \mathcal{Y}$ and $q : \mathcal{Z} \to \mathcal{Y}$ be $1$-morphisms of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Assume $\mathcal{X}$, $\mathcal{Y}$, $\mathcal{Z}$ satisfy (RS*). Let $A$ be an $S$-algebra and let $w$ be an object of $\mathcal{W} = \mathcal{X} \times _\mathcal {Y} \mathcal{Z}$ over $A$. Denote $x, y, z$ the objects of $\mathcal{X}, \mathcal{Y}, \mathcal{Z}$ you get from $w$. For any $A$-module $M$ there is a $6$-term exact sequence

$\xymatrix{ 0 \ar[r] & \text{Inf}_ w(M) \ar[r] & \text{Inf}_ x(M) \oplus \text{Inf}_ z(M) \ar[r] & \text{Inf}_ y(M) \ar[lld] \\ & T_ w(M) \ar[r] & T_ x(M) \oplus T_ z(M) \ar[r] & T_ y(M) }$

of $A$-modules.

Proof. By Lemma 96.18.3 we see that $\mathcal{W}$ satisfies (RS*) and hence $T_ w(M)$ and $\text{Inf}_ w(M)$ are defined. The horizontal arrows are defined using the functoriality of Lemma 96.21.1.

Definition of the “boundary” map $\delta : \text{Inf}_ y(M) \to T_ w(M)$. Choose isomorphisms $p(x) \to y$ and $y \to q(z)$ such that $w = (x, z, p(x) \to y \to q(z))$ in the description of the $2$-fibre product of Categories, Lemma 4.35.7 and more precisely Categories, Lemma 4.32.3. Let $x', y', z', w'$ denote the trivial deformation of $x, y, z, w$ over $A[M]$. By pullback we get isomorphisms $y' \to p(x')$ and $q(z') \to y'$. An element $\alpha \in \text{Inf}_ y(M)$ is the same thing as an automorphism $\alpha : y' \to y'$ over $A[M]$ which restricts to the identity on $y$ over $A$. Thus setting

$\delta (\alpha ) = (x', z', p(x') \to y' \xrightarrow {\alpha } y' \to q(z'))$

we obtain an object of $T_ w(M)$. This is a map of $A$-modules by Formal Deformation Theory, Lemma 88.11.5.

The rest of the proof is exactly the same as the proof of Formal Deformation Theory, Lemma 88.20.1. $\square$

Remark 96.21.7 (Compatibility with previous tangent spaces). Let $S$ be a locally Noetherian scheme. Let $\mathcal{X}$ be a category fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Assume $\mathcal{X}$ has (RS*). Let $k$ be a field of finite type over $S$ and let $x_0$ be an object of $\mathcal{X}$ over $\mathop{\mathrm{Spec}}(k)$. Then we have equalities of $k$-vector spaces

$T\mathcal{F}_{\mathcal{X}, k, x_0} = T_{x_0}(k) \quad \text{and}\quad \text{Inf}(\mathcal{F}_{\mathcal{X}, k, x_0}) = \text{Inf}_{x_0}(k)$

where the spaces on the left hand side of the equality signs are given in (96.8.0.1) and (96.8.0.2) and the spaces on the right hand side are given by Lemma 96.21.2.

Remark 96.21.8 (Canonical element). Assumptions and notation as in Lemma 96.21.2. Choose an affine open $\mathop{\mathrm{Spec}}(\Lambda ) \subset S$ such that $\mathop{\mathrm{Spec}}(A) \to S$ corresponds to a ring map $\Lambda \to A$. Consider the ring map

$A \longrightarrow A[\Omega _{A/\Lambda }], \quad a \longmapsto (a, \text{d}_{A/\Lambda }(a))$

Pulling back $x$ along the corresponding morphism $\mathop{\mathrm{Spec}}(A[\Omega _{A/\Lambda }]) \to \mathop{\mathrm{Spec}}(A)$ we obtain a deformation $x_{can}$ of $x$ over $A[\Omega _{A/\Lambda }]$. We call this the canonical element

$x_{can} \in T_ x(\Omega _{A/\Lambda }) = \text{Lift}(x, A[\Omega _{A/\Lambda }]).$

Next, assume that $\Lambda$ is Noetherian and $\Lambda \to A$ is of finite type. Let $k = \kappa (\mathfrak p)$ be a residue field at a finite type point $u_0$ of $U = \mathop{\mathrm{Spec}}(A)$. Let $x_0 = x|_{u_0}$. By (RS*) and the fact that $A[k] = A \times _ k k[k]$ the space $T_ x(k)$ is the tangent space to the deformation functor $\mathcal{F}_{\mathcal{X}, k, x_0}$. Via

$T\mathcal{F}_{U, k, u_0} = \text{Der}_\Lambda (A, k) = \mathop{\mathrm{Hom}}\nolimits _ A(\Omega _{A/\Lambda }, k)$

(see Formal Deformation Theory, Example 88.11.11) and functoriality of $T_ x$ the canonical element produces the map on tangent spaces induced by the object $x$ over $U$. Namely, $\theta \in T\mathcal{F}_{U, k, u_0}$ maps to $T_ x(\theta )(x_{can})$ in $T_ x(k) = T\mathcal{F}_{\mathcal{X}, k, x_0}$.

Remark 96.21.9 (Canonical automorphism). Let $S$ be a locally Noetherian scheme. Let $\mathcal{X}$ be a category fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Assume $\mathcal{X}$ satisfies condition (RS*). Let $A$ be an $S$-algebra such that $\mathop{\mathrm{Spec}}(A) \to S$ maps into an affine open and let $x, y$ be objects of $\mathcal{X}$ over $\mathop{\mathrm{Spec}}(A)$. Further, let $A \to B$ be a ring map and let $\alpha : x|_{\mathop{\mathrm{Spec}}(B)} \to y|_{\mathop{\mathrm{Spec}}(B)}$ be a morphism of $\mathcal{X}$ over $\mathop{\mathrm{Spec}}(B)$. Consider the ring map

$B \longrightarrow B[\Omega _{B/A}], \quad b \longmapsto (b, \text{d}_{B/A}(b))$

Pulling back $\alpha$ along the corresponding morphism $\mathop{\mathrm{Spec}}(B[\Omega _{B/A}]) \to \mathop{\mathrm{Spec}}(B)$ we obtain a morphism $\alpha _{can}$ between the pullbacks of $x$ and $y$ over $B[\Omega _{B/A}]$. On the other hand, we can pullback $\alpha$ by the morphism $\mathop{\mathrm{Spec}}(B[\Omega _{B/A}]) \to \mathop{\mathrm{Spec}}(B)$ corresponding to the injection of $B$ into the first summand of $B[\Omega _{B/A}]$. By the discussion of Remark 96.21.4 we can take the difference

$\varphi (x, y, \alpha ) = \alpha _{can} - \alpha |_{\mathop{\mathrm{Spec}}(B[\Omega _{B/A}])} \in \text{Inf}_{x|_{\mathop{\mathrm{Spec}}(B)}}(\Omega _{B/A}).$

We will call this the canonical automorphism. It depends on all the ingredients $A$, $x$, $y$, $A \to B$ and $\alpha$.

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