## 90.17 Lifts of objects

The content of this section is that the tangent space has a principal homogeneous action on the set of lifts along a small extension in the case of a deformation category.

Definition 90.17.1. Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal{C}_\Lambda$. Let $f: A' \to A$ be a map in $\mathcal{C}_\Lambda$. Let $x \in \mathcal{F}(A)$. The category $\textit{Lift}(x, f)$ of lifts of $x$ along $f$ is the category with the following objects and morphisms.

1. Objects: A lift of $x$ along $f$ is a morphism $x' \to x$ lying over $f$.

2. Morphisms: A morphism of lifts from $a_1 : x'_1 \to x$ to $a_2 : x'_2 \to x$ is a morphism $b : x'_1 \to x'_2$ in $\mathcal{F}(A')$ such that $a_2 = a_1 \circ b$.

The set $\text{Lift}(x, f)$ of lifts of $x$ along $f$ is the set of isomorphism classes of $\textit{Lift}(x, f)$.

Remark 90.17.2. When the map $f: A' \to A$ is clear from the context, we may write $\textit{Lift}(x, A')$ and $\text{Lift}(x, A')$ in place of $\textit{Lift}(x, f)$ and $\text{Lift}(x, f)$.

Remark 90.17.3. Let $\mathcal{F}$ be a category cofibred in groupoids over $\mathcal{C}_\Lambda$. Let $x_0 \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{F}(k))$. Let $V$ be a finite dimensional vector space. Then $\text{Lift}(x_0, k[V])$ is the set of isomorphism classes of $\mathcal{F}_{x_0}(k[V])$ where $\mathcal{F}_{x_0}$ is the predeformation category of objects in $\mathcal{F}$ lying over $x_0$, see Remark 90.6.4. Hence if $\mathcal{F}$ satisfies (S2), then so does $\mathcal{F}_{x_0}$ (see Lemma 90.10.6) and by Lemma 90.12.2 we see that

$\text{Lift}(x_0, k[V]) = T\mathcal{F}_{x_0} \otimes _ k V$

as $k$-vector spaces.

Remark 90.17.4. Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal C_\Lambda$ satisfying (RS). Let

$\xymatrix{ A_1 \times _ A A_2 \ar[r] \ar[d] & A_2 \ar[d] \\ A_1 \ar[r] & A }$

be a fibre square in $\mathcal{C}_\Lambda$ such that either $A_1 \to A$ or $A_2 \to A$ is surjective. Let $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{F}(A))$. Given lifts $x_1 \to x$ and $x_2 \to x$ of $x$ to $A_1$ and $A_2$, we get by (RS) a lift $x_1 \times _ x x_2 \to x$ of $x$ to $A_1 \times _ A A_2$. Conversely, by Lemma 90.16.2 any lift of $x$ to $A_1 \times _ A A_2$ is of this form. Hence a bijection

$\text{Lift}(x, A_1) \times \text{Lift}(x, A_2) \longrightarrow \text{Lift}(x, A_1 \times _ A A_2).$

Similarly, if $x_1 \to x$ is a fixed lifting of $x$ to $A_1$, then there is a bijection

$\text{Lift}(x_1, A_1 \times _ A A_2) \longrightarrow \text{Lift}(x, A_2).$

Now let

$\xymatrix{ A_1' \times _ A A_2 \ar[r] \ar[d] & A_1 \times _ A A_2 \ar[r] \ar[d] & A_2 \ar[d] \\ A_1' \ar[r] & A_1 \ar[r] & A }$

be a composition of fibre squares in $\mathcal{C}_\Lambda$ with both $A'_1 \to A_1$ and $A_1 \to A$ surjective. Let $x_1 \to x$ be a morphism lying over $A_1 \to A$. Then by the above we have bijections

\begin{align*} \text{Lift}(x_1, A_1' \times _ A A_2) & = \text{Lift}(x_1, A_1') \times \text{Lift}(x_1, A_1 \times _ A A_2) \\ & = \text{Lift}(x_1, A_1') \times \text{Lift}(x, A_2). \end{align*}

Lemma 90.17.5. Let $\mathcal{F}$ be a deformation category. Let $A' \to A$ be a surjective ring map in $\mathcal{C}_\Lambda$ whose kernel $I$ is annihilated by $\mathfrak m_{A'}$. Let $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{F}(A))$. If $\text{Lift}(x, A')$ is nonempty, then there is a free and transitive action of $T\mathcal{F} \otimes _ k I$ on $\text{Lift}(x, A')$.

Proof. Consider the ring map $g : A' \times _ A A' \to k[I]$ defined by the rule $g(a_1, a_2) = \overline{a_1} \oplus a_2 - a_1$ (compare with Lemma 90.10.8). There is an isomorphism

$A' \times _ A A' \xrightarrow {\sim } A' \times _ k k[I]$

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 of $A' \times _ A A'$ and $A' \times _ k k[I]$ to $A$. Thus there is a bijection

90.17.5.1
$$\label{formal-defos-equation-one} \text{Lift}(x, A' \times _ A A') \longrightarrow \text{Lift}(x, A' \times _ k k[I])$$

By Remark 90.17.4 there is a bijection

90.17.5.2
$$\label{formal-defos-equation-two} \text{Lift}(x, A') \times \text{Lift}(x, A') \longrightarrow \text{Lift}(x, A' \times _ A A')$$

There is a commutative diagram

$\xymatrix{ A' \times _ k k[I] \ar[r] \ar[d] & A \times _ k k[I] \ar[r] \ar[d] & k[I] \ar[d] \\ A' \ar[r] & A \ar[r] & k. }$

Thus if we choose a pushforward $x \to x_0$ of $x$ along $A \to k$, we obtain by the end of Remark 90.17.4 a bijection

90.17.5.3
$$\label{formal-defos-equation-three} \text{Lift}(x, A' \times _ k k[I]) \longrightarrow \text{Lift}(x, A') \times \text{Lift}(x_0, k[I])$$

Composing (90.17.5.2), (90.17.5.1), and (90.17.5.3) we get a bijection

$\Phi : \text{Lift}(x, A') \times \text{Lift}(x, A') \longrightarrow \text{Lift}(x, A') \times \text{Lift}(x_0, k[I]).$

This bijection commutes with the projections on the first factors. By Remark 90.17.3 we see that $\text{Lift}(x_0, k[I]) = T\mathcal{F} \otimes _ k I$. If $\text{pr}_2$ is the second projection of $\text{Lift}(x, A') \times \text{Lift}(x, A')$, then we get a map

$a = \text{pr}_2 \circ \Phi ^{-1} : \text{Lift}(x, A') \times (T\mathcal{F} \otimes _ k I) \longrightarrow \text{Lift}(x, A').$

Unwinding all the above we see that $a(x' \to x, \theta )$ is the unique lift $x'' \to x$ such that $g_*(x', x'') = \theta$ in $\text{Lift}(x_0, k[I]) = T\mathcal{F} \otimes _ k I$. To see this is an action of $T\mathcal{F} \otimes _ k I$ on $\text{Lift}(x, A')$ we have to show the following: if $x', x'', x'''$ are lifts of $x$ and $g_*(x', x'') = \theta$, $g_*(x'', x''') = \theta '$, then $g_*(x', x''') = \theta + \theta '$. This follows from the commutative diagram

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

The action is free and transitive because $\Phi$ is bijective. $\square$

Remark 90.17.6. The action of Lemma 90.17.5 is functorial. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism of deformation categories. Let $A' \to A$ be a surjective ring map whose kernel $I$ is annihilated by $\mathfrak m_{A'}$. Let $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{F}(A))$. In this situation $\varphi$ induces the vertical arrows in the following commutative diagram

$\xymatrix{ \text{Lift}(x, A') \times (T\mathcal{F} \otimes _ k I) \ar[d]_{(\varphi , d\varphi \otimes \text{id}_ I)} \ar[r] & \text{Lift}(x, A') \ar[d]^\varphi \\ \text{Lift}(\varphi (x), A') \times (T\mathcal{G} \otimes _ k I) \ar[r] & \text{Lift}(\varphi (x), A') }$

The commutativity follows as each of the maps (90.17.5.2), (90.17.5.1), and (90.17.5.3) of the proof of Lemma 90.17.5 gives rise to a similar commutative diagram.

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