The Stacks project

98.8 Tangent spaces

Let $S$ be a locally Noetherian scheme. Let $\mathcal{X}$ be a category fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Let $k$ be a field of finite type over $S$ and let $x_0$ be an object of $\mathcal{X}$ over $k$. In Formal Deformation Theory, Section 90.12 we have defined the tangent space
\begin{equation} \label{artin-equation-tangent-space} T\mathcal{F}_{\mathcal{X}, k, x_0} = \left\{ \begin{matrix} \text{isomorphism classes of morphisms} \\ x_0 \to x\text{ over }\mathop{\mathrm{Spec}}(k) \to \mathop{\mathrm{Spec}}(k[\epsilon ]) \end{matrix} \right\} \end{equation}

of the predeformation category $\mathcal{F}_{\mathcal{X}, k, x_0}$. In Formal Deformation Theory, Section 90.19 we have defined
\begin{equation} \label{artin-equation-infinitesimal-automorphisms} \text{Inf}(\mathcal{F}_{\mathcal{X}, k, x_0}) = \mathop{\mathrm{Ker}}\left( \text{Aut}_{\mathop{\mathrm{Spec}}(k[\epsilon ])}(x'_0) \to \text{Aut}_{\mathop{\mathrm{Spec}}(k)}(x_0) \right) \end{equation}

where $x_0'$ is the pullback of $x_0$ to $\mathop{\mathrm{Spec}}(k[\epsilon ])$. If $\mathcal{X}$ satisfies the Rim-Schlessinger condition (RS), then $T\mathcal{F}_{\mathcal{X}, k, x_0}$ comes equipped with a natural $k$-vector space structure by Formal Deformation Theory, Lemma 90.12.2 (assumptions hold by Lemma 98.6.1 and Remark 98.6.2). Moreover, Formal Deformation Theory, Lemma 90.19.9 shows that $\text{Inf}(\mathcal{F}_{\mathcal{X}, k, x_0})$ has a natural $k$-vector space structure such that addition agrees with composition of automorphisms. A natural condition is to ask these vector spaces to have finite dimension.

The following lemma tells us this is true if $\mathcal{X}$ is locally of finite type over $S$ (see Morphisms of Stacks, Section 101.17).

Lemma 98.8.1. Let $S$ be a locally Noetherian scheme. Assume

  1. $\mathcal{X}$ is an algebraic stack,

  2. $U$ is a scheme locally of finite type over $S$, and

  3. $(\mathit{Sch}/U)_{fppf} \to \mathcal{X}$ is a smooth surjective morphism.

Then, for any $\mathcal{F} = \mathcal{F}_{\mathcal{X}, k, x_0}$ as in Section 98.3 the tangent space $T\mathcal{F}$ and infinitesimal automorphism space $\text{Inf}(\mathcal{F})$ have finite dimension over $k$.

Proof. Let us write $\mathcal{U} = (\mathit{Sch}/U)_{fppf}$. By our definition of algebraic stacks the $1$-morphism $\mathcal{U} \to \mathcal{X}$ is representable by algebraic spaces. Hence in particular the 2-fibre product

\[ \mathcal{U}_{x_0} = (\mathit{Sch}/\mathop{\mathrm{Spec}}(k))_{fppf} \times _\mathcal {X} \mathcal{U} \]

is representable by an algebraic space $U_{x_0}$ over $\mathop{\mathrm{Spec}}(k)$. Then $U_{x_0} \to \mathop{\mathrm{Spec}}(k)$ is smooth and surjective (in particular $U_{x_0}$ is nonempty). By Spaces over Fields, Lemma 72.16.2 we can find a finite extension $l/k$ and a point $\mathop{\mathrm{Spec}}(l) \to U_{x_0}$ over $k$. We have

\[ (\mathcal{F}_{\mathcal{X}, k , x_0})_{l/k} = \mathcal{F}_{\mathcal{X}, l, x_{l, 0}} \]

by Lemma 98.7.1 and the fact that $\mathcal{X}$ satisfies (RS). Thus we see that

\[ T\mathcal{F} \otimes _ k l \cong T\mathcal{F}_{\mathcal{X}, l, x_{l, 0}} \quad \text{and}\quad \text{Inf}(\mathcal{F}) \otimes _ k l \cong \text{Inf}(\mathcal{F}_{\mathcal{X}, l, x_{l, 0}}) \]

by Formal Deformation Theory, Lemmas 90.29.3 and 90.29.4 (these are applicable by Lemmas 98.5.2 and 98.6.1 and Remark 98.6.2). Hence it suffices to prove that $T\mathcal{F}_{\mathcal{X}, l, x_{l, 0}}$ and $\text{Inf}(\mathcal{F}_{\mathcal{X}, l, x_{l, 0}})$ have finite dimension over $l$. Note that $x_{l, 0}$ comes from a point $u_0$ of $\mathcal{U}$ over $l$.

We interrupt the flow of the argument to show that the lemma for infinitesimal automorphisms follows from the lemma for tangent spaces. Namely, let $\mathcal{R} = \mathcal{U} \times _\mathcal {X} \mathcal{U}$. Let $r_0$ be the $l$-valued point $(u_0, u_0, \text{id}_{x_0})$ of $\mathcal{R}$. Combining Lemma 98.3.3 and Formal Deformation Theory, Lemma 90.26.2 we see that

\[ \text{Inf}(\mathcal{F}_{\mathcal{X}, l, x_{l, 0}}) \subset T\mathcal{F}_{\mathcal{R}, l, r_0} \]

Note that $\mathcal{R}$ is an algebraic stack, see Algebraic Stacks, Lemma 94.14.2. Also, $\mathcal{R}$ is representable by an algebraic space $R$ smooth over $U$ (via either projection, see Algebraic Stacks, Lemma 94.16.2). Hence, choose an scheme $U'$ and a surjective ├ętale morphism $U' \to R$ we see that $U'$ is smooth over $U$, hence locally of finite type over $S$. As $(\mathit{Sch}/U')_{fppf} \to \mathcal{R}$ is surjective and smooth, we have reduced the question to the case of tangent spaces.

The functor (

\[ \mathcal{F}_{\mathcal{U}, l, u_0} \longrightarrow \mathcal{F}_{\mathcal{X}, l, x_{l, 0}} \]

is smooth by Lemma 98.3.2. The induced map on tangent spaces

\[ T\mathcal{F}_{\mathcal{U}, l, u_0} \longrightarrow T\mathcal{F}_{\mathcal{X}, l, x_{l, 0}} \]

is $l$-linear (by Formal Deformation Theory, Lemma 90.12.4) and surjective (as smooth maps of predeformation categories induce surjective maps on tangent spaces by Formal Deformation Theory, Lemma 90.8.8). Hence it suffices to prove that the tangent space of the deformation space associated to the representable algebraic stack $\mathcal{U}$ at the point $u_0$ is finite dimensional. Let $\mathop{\mathrm{Spec}}(R) \subset U$ be an affine open such that $u_0 : \mathop{\mathrm{Spec}}(l) \to U$ factors through $\mathop{\mathrm{Spec}}(R)$ and such that $\mathop{\mathrm{Spec}}(R) \to S$ factors through $\mathop{\mathrm{Spec}}(\Lambda ) \subset S$. Let $\mathfrak m_ R \subset R$ be the kernel of the $\Lambda $-algebra map $\varphi _0 : R \to l$ corresponding to $u_0$. Note that $R$, being of finite type over the Noetherian ring $\Lambda $, is a Noetherian ring. Hence $\mathfrak m_ R = (f_1, \ldots , f_ n)$ is a finitely generated ideal. We have

\[ T\mathcal{F}_{\mathcal{U}, l, u_0} = \{ \varphi : R \to l[\epsilon ] \mid \varphi \text{ is a } \Lambda \text{-algebra map and } \varphi \bmod \epsilon = \varphi _0\} \]

An element of the right hand side is determined by its values on $f_1, \ldots , f_ n$ hence the dimension is at most $n$ and we win. Some details omitted. $\square$

Lemma 98.8.2. Let $S$ be a locally Noetherian 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 $k$ be a field of finite type over $S$ and let $w_0$ be an object of $\mathcal{W} = \mathcal{X} \times _\mathcal {Y} \mathcal{Z}$ over $k$. Denote $x_0, y_0, z_0$ the objects of $\mathcal{X}, \mathcal{Y}, \mathcal{Z}$ you get from $w_0$. Then there is a $6$-term exact sequence

\[ \xymatrix{ 0 \ar[r] & \text{Inf}(\mathcal{F}_{\mathcal{W}, k, w_0}) \ar[r] & \text{Inf}(\mathcal{F}_{\mathcal{X}, k, x_0}) \oplus \text{Inf}(\mathcal{F}_{\mathcal{Z}, k, z_0}) \ar[r] & \text{Inf}(\mathcal{F}_{\mathcal{Y}, k, y_0}) \ar[lld] \\ & T\mathcal{F}_{\mathcal{W}, k, w_0} \ar[r] & T\mathcal{F}_{\mathcal{X}, k, x_0} \oplus T\mathcal{F}_{\mathcal{Z}, k, z_0} \ar[r] & T\mathcal{F}_{\mathcal{Y}, k, y_0} } \]

of $k$-vector spaces.

Proof. By Lemma 98.5.3 we see that $\mathcal{W}$ satisfies (RS) and hence the lemma makes sense. To see the lemma is true, apply Lemmas 98.3.3 and 98.6.1 and Formal Deformation Theory, Lemma 90.20.1. $\square$

Comments (1)

Comment #1588 by Ariyan on

Typo: In the proof of Lemma 78.8.1 "representably" should be "representable".

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