The Stacks project

95.8 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 87.12 we have defined the tangent space

95.8.0.1
\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 87.19 we have defined

95.8.0.2
\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 87.12.2 (assumptions hold by Lemma 95.6.1 and Remark 95.6.2). Moreover, Formal Deformation Theory, Lemma 87.19.9 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 98.17).

Lemma 95.8.1. 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 95.3 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 69.16.2 we can find a finite extension $l \supset k$ 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 95.7.1 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 87.29.3 and 87.29.4 (these are applicable by Lemmas 95.5.2 and 95.6.1 and Remark 95.6.2). 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 95.3.3 and Formal Deformation Theory, Lemma 87.26.2 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 91.14.2. Also, $\mathcal{R}$ is representable by an algebraic space $R$ smooth over $U$ (via either projection, see Algebraic Stacks, Lemma 91.16.2). 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 (95.3.1.1)

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

is smooth by Lemma 95.3.2. 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 87.12.4) and surjective (as smooth maps of predeformation categories induce surjective maps on tangent spaces by Formal Deformation Theory, Lemma 87.8.8). 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 95.8.2. 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 95.5.3 we see that $\mathcal{W}$ satisfies (RS) and hence the lemma makes sense. To see the lemma is true, apply Lemmas 95.3.3 and 95.6.1 and Formal Deformation Theory, Lemma 87.20.1. $\square$


Comments (1)

Comment #1588 by Ariyan on

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


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 07WY. Beware of the difference between the letter 'O' and the digit '0'.