The Stacks project

97.15 Axioms for functors

Let $S$ be a scheme. Let $F : (\mathit{Sch}/S)_{fppf}^{opp} \to \textit{Sets}$ be a functor. Denote $\mathcal{X} = \mathcal{S}_ F$ the category fibred in sets associated to $F$, see Algebraic Stacks, Section 93.7. In this section we provide a translation between the material above as it applies to $\mathcal{X}$, to statements about $F$.

Let $S$ be a locally Noetherian scheme. Let $F : (\mathit{Sch}/S)_{fppf}^{opp} \to \textit{Sets}$ be a functor. Let $k$ be a field of finite type over $S$. Let $x_0 \in F(\mathop{\mathrm{Spec}}(k))$. The associated predeformation category (97.3.0.2) corresponds to the functor

\[ F_{k, x_0} : \mathcal{C}_\Lambda \longrightarrow \textit{Sets}, \quad A \longmapsto \{ x \in F(\mathop{\mathrm{Spec}}(A)) \mid x|_{\mathop{\mathrm{Spec}}(k)} = x_0 \} . \]

Recall that we do not distinguish between categories cofibred in sets over $\mathcal{C}_\Lambda $ and functor $\mathcal{C}_\Lambda \to \textit{Sets}$, see Formal Deformation Theory, Remarks 89.5.2 (11). Given a transformation of functors $a : F \to G$, setting $y_0 = a(x_0)$ we obtain a morphism

\[ F_{k, x_0} \longrightarrow G_{k, y_0} \]

see (97.3.1.1). Lemma 97.3.2 tells us that if $a : F \to G$ is formally smooth (in the sense of More on Morphisms of Spaces, Definition 75.13.1), then $F_{k, x_0} \longrightarrow G_{k, y_0}$ is smooth as in Formal Deformation Theory, Remark 89.8.4.

Lemma 97.4.1 says that if $Y' = Y \amalg _ X X'$ in the category of schemes over $S$ where $X \to X'$ is a thickening and $X \to Y$ is affine, then the map

\[ F(Y \amalg _ X X') \to F(Y) \times _{F(X)} F(X') \]

is a bijection, provided that $F$ is an algebraic space. We say a general functor $F$ satisfies the Rim-Schlessinger condition or we say $F$ satisfies (RS) if given any pushout $Y' = Y \amalg _ X X'$ where $Y, X, X'$ are spectra of Artinian local rings of finite type over $S$, then

\[ F(Y \amalg _ X X') \to F(Y) \times _{F(X)} F(X') \]

is a bijection. Thus every algebraic space satisfies (RS).

Lemma 97.6.1 says that given a functor $F$ which satisfies (RS), then all $F_{k, x_0}$ are deformation functors as in Formal Deformation Theory, Definition 89.16.8, i.e., they satisfy (RS) as in Formal Deformation Theory, Remark 89.16.5. In particular the tangent space

\[ TF_{k, x_0} = \{ x \in F(\mathop{\mathrm{Spec}}(k[\epsilon ])) \mid x|_{\mathop{\mathrm{Spec}}(k)} = x_0\} \]

has the structure of a $k$-vector space by Formal Deformation Theory, Lemma 89.12.2.

Lemma 97.8.1 says that an algebraic space $F$ locally of finite type over $S$ gives rise to deformation functors $F_{k, x_0}$ with finite dimensional tangent spaces $TF_{k, x_0}$.

A formal object1 $\xi = (R, \xi _ n)$ of $F$ consists of a Noetherian complete local $S$-algebra $R$ whose residue field is of finite type over $S$, together with elements $\xi _ n \in F(\mathop{\mathrm{Spec}}(R/\mathfrak m^ n))$ such that $\xi _{n + 1}|_{\mathop{\mathrm{Spec}}(R/\mathfrak m^ n)} = \xi _ n$. A formal object $\xi $ defines a formal object $\xi $ of $F_{R/\mathfrak m, \xi _1}$. We say $\xi $ is versal if and only if it is versal in the sense of Formal Deformation Theory, Definition 89.8.9. A formal object $\xi = (R, \xi _ n)$ is called effective if there exists an $x \in F(\mathop{\mathrm{Spec}}(R))$ such that $\xi _ n = x|_{\mathop{\mathrm{Spec}}(R/\mathfrak m^ n)}$ for all $n \geq 1$. Lemma 97.9.5 says that if $F$ is an algebraic space, then every formal object is effective.

Let $U$ be a scheme locally of finite type over $S$ and let $x \in F(U)$. Let $u_0 \in U$ be a finite type point. We say that $x$ is versal at $u_0$ if and only if $\xi = (\mathcal{O}_{U, u_0}^\wedge , x|_{\mathop{\mathrm{Spec}}(\mathcal{O}_{U, u_0}/\mathfrak m_{u_0}^ n)})$ is a versal formal object in the sense described above.

Let $S$ be a locally Noetherian scheme. Let $F : (\mathit{Sch}/S)_{fppf}^{opp} \to \mathit{Sch}$ be a functor. Here are the axioms we will consider on $F$.

  1. a set theoretic condition2 to be ignored by readers who are not interested in set theoretical issues,

  2. $F$ is a sheaf for the ├ętale topology,

  3. $F$ is limit preserving,

  4. $F$ satisfies the Rim-Schlessinger condition (RS),

  5. every tangent space $TF_{k, x_0}$ is finite dimensional,

  6. every formal object is effective,

  7. $F$ satisfies openness of versality.

Here limit preserving is the notion defined in Limits of Spaces, Definition 69.3.1 and openness of versality means the following: Given a scheme $U$ locally of finite type over $S$, given $x \in F(U)$, and given a finite type point $u_0 \in U$ such that $x$ is versal at $u_0$, then there exists an open neighbourhood $u_0 \in U' \subset U$ such that $x$ is versal at every finite type point of $U'$.

[1] This is what Artin calls a formal deformation.
[2] The condition is the following: the supremum of all the cardinalities $|F(\mathop{\mathrm{Spec}}(k))|$ where $k$ runs over the finite type fields over $S$ is $\leq $ than the size of some object of $(\mathit{Sch}/S)_{fppf}$.

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