## 98.21 Infinitesimal deformations

In this section we discuss a generalization of the notion of the tangent space introduced in Section 98.8. To do this intelligently, we borrow some notation from Formal Deformation Theory, Sections 90.11, 90.17, and 90.19.

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 90.17.1 and Remark 90.17.2. 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 90.19.1 and Formal Deformation Theory, Remark 90.19.8.

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 98.21.1. 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

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

$x$ 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 98.21.2. 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 $\mathop{\mathrm{Spec}}(A)$.

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 $I = \mathop{\mathrm{Ker}}(A' \to A)$.

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

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

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 infinitesimal deformations of $x$ to $\mathop{\mathrm{Spec}}(A[M])$. We apply Formal Deformation Theory, Lemma 90.11.4 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.
$\square$

Lemma 98.21.6. 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 98.18.3 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 98.21.1.

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 90.11.5.

The rest of the proof is exactly the same as the proof of Formal Deformation Theory, Lemma 90.20.1.
$\square$

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