The Stacks project

95.3 Predeformation categories

Let $S$ be a locally Noetherian base scheme. Let

\[ p : \mathcal{X} \longrightarrow (\mathit{Sch}/S)_{fppf} \]

be a category fibred in groupoids. Let $k$ be a field and let $\mathop{\mathrm{Spec}}(k) \to S$ be a morphism of finite type (see Morphisms, Lemma 29.15.1). We will sometimes simply say that $k$ is a field of finite type over $S$. Let $x_0$ be an object of $\mathcal{X}$ lying over $\mathop{\mathrm{Spec}}(k)$. Given $S$, $\mathcal{X}$, $k$, and $x_0$ we will construct a predeformation category, as defined in Formal Deformation Theory, Definition 87.6.2. The construction will resemble the construction of Formal Deformation Theory, Remark 87.6.4.

First, by Morphisms, Lemma 29.15.1 we may pick an affine open $\mathop{\mathrm{Spec}}(\Lambda ) \subset S$ such that $\mathop{\mathrm{Spec}}(k) \to S$ factors through $\mathop{\mathrm{Spec}}(\Lambda )$ and the associated ring map $\Lambda \to k$ is finite. This provides us with the category $\mathcal{C}_\Lambda $, see Formal Deformation Theory, Definition 87.3.1. The category $\mathcal{C}_\Lambda $, up to canonical equivalence, does not depend on the choice of the affine open $\mathop{\mathrm{Spec}}(\Lambda )$ of $S$. Namely, $\mathcal{C}_\Lambda $ is equivalent to the opposite of the category of factorizations

95.3.0.1
\begin{equation} \label{artin-equation-factor} \mathop{\mathrm{Spec}}(k) \to \mathop{\mathrm{Spec}}(A) \to S \end{equation}

of the structure morphism such that $A$ is an Artinian local ring and such that $\mathop{\mathrm{Spec}}(k) \to \mathop{\mathrm{Spec}}(A)$ corresponds to a ring map $A \to k$ which identifies $k$ with the residue field of $A$.

We let $\mathcal{F} = \mathcal{F}_{\mathcal{X}, k, x_0}$ be the category whose

  1. objects are morphisms $x_0 \to x$ of $\mathcal{X}$ where $p(x) = \mathop{\mathrm{Spec}}(A)$ with $A$ an Artinian local ring and $p(x_0) \to p(x) \to S$ a factorization as in (95.3.0.1), and

  2. morphisms $(x_0 \to x) \to (x_0 \to x')$ are commutative diagrams

    \[ \xymatrix{ x & & x' \ar[ll] \\ & x_0 \ar[lu] \ar[ru] } \]

    in $\mathcal{X}$. (Note the reversal of arrows.)

If $x_0 \to x$ is an object of $\mathcal{F}$ then writing $p(x) = \mathop{\mathrm{Spec}}(A)$ we obtain an object $A$ of $\mathcal{C}_\Lambda $. We often say that $x_0 \to x$ or $x$ lies over $A$. A morphism of $\mathcal{F}$ between objects $x_0 \to x$ lying over $A$ and $x_0 \to x'$ lying over $A'$ corresponds to a morphism $x' \to x$ of $\mathcal{X}$, hence a morphism $p(x' \to x) : \mathop{\mathrm{Spec}}(A') \to \mathop{\mathrm{Spec}}(A)$ which in turn corresponds to a ring map $A \to A'$. As $\mathcal{X}$ is a category over the category of schemes over $S$ we see that $A \to A'$ is $\Lambda $-algebra homomorphism. Thus we obtain a functor

95.3.0.2
\begin{equation} \label{artin-equation-predeformation-category} p : \mathcal{F} = \mathcal{F}_{\mathcal{X}, k, x_0} \longrightarrow \mathcal{C}_\Lambda . \end{equation}

We will use the notation $\mathcal{F}(A)$ to denote the fibre category over an object $A$ of $\mathcal{C}_\Lambda $. An object of $\mathcal{F}(A)$ is simply a morphism $x_0 \to x$ of $\mathcal{X}$ such that $x$ lies over $\mathop{\mathrm{Spec}}(A)$ and $x_0 \to x$ lies over $\mathop{\mathrm{Spec}}(k) \to \mathop{\mathrm{Spec}}(A)$.

Lemma 95.3.1. The functor $p : \mathcal{F} \to \mathcal{C}_\Lambda $ defined above is a predeformation category.

Proof. We have to show that $\mathcal{F}$ is (a) cofibred in groupoids over $\mathcal{C}_\Lambda $ and (b) that $\mathcal{F}(k)$ is a category equivalent to a category with a single object and a single morphism.

Proof of (a). The fibre categories of $\mathcal{F}$ over $\mathcal{C}_\Lambda $ are groupoids as the fibre categories of $\mathcal{X}$ are groupoids. Let $A \to A'$ be a morphism of $\mathcal{C}_\Lambda $ and let $x_0 \to x$ be an object of $\mathcal{F}(A)$. Because $\mathcal{X}$ is fibred in groupoids, we can find a morphism $x' \to x$ lying over $\mathop{\mathrm{Spec}}(A') \to \mathop{\mathrm{Spec}}(A)$. Since the composition $A \to A' \to k$ is equal the given map $A \to k$ we see (by uniqueness of pullbacks up to isomorphism) that the pullback via $\mathop{\mathrm{Spec}}(k) \to \mathop{\mathrm{Spec}}(A')$ of $x'$ is $x_0$, i.e., that there exists a morphism $x_0 \to x'$ lying over $\mathop{\mathrm{Spec}}(k) \to \mathop{\mathrm{Spec}}(A')$ compatible with $x_0 \to x$ and $x' \to x$. This proves that $\mathcal{F}$ has pushforwards. We conclude by (the dual of) Categories, Lemma 4.34.2.

Proof of (b). If $A = k$, then $\mathop{\mathrm{Spec}}(k) = \mathop{\mathrm{Spec}}(A)$ and since $\mathcal{X}$ is fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$ we see that given any object $x_0 \to x$ in $\mathcal{F}(k)$ the morphism $x_0 \to x$ is an isomorphism. Hence every object of $\mathcal{F}(k)$ is isomorphic to $x_0 \to x_0$. Clearly the only self morphism of $x_0 \to x_0$ in $\mathcal{F}$ is the identity. $\square$

Let $S$ be a locally Noetherian base scheme. Let $F : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism between categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Let $k$ is a field of finite type over $S$. Let $x_0$ be an object of $\mathcal{X}$ lying over $\mathop{\mathrm{Spec}}(k)$. Set $y_0 = F(x_0)$ which is an object of $\mathcal{Y}$ lying over $\mathop{\mathrm{Spec}}(k)$. Then $F$ induces a functor

95.3.1.1
\begin{equation} \label{artin-equation-functoriality} F : \mathcal{F}_{\mathcal{X}, k, x_0} \longrightarrow \mathcal{F}_{\mathcal{Y}, k, y_0} \end{equation}

of categories cofibred over $\mathcal{C}_\Lambda $. Namely, to the object $x_0 \to x$ of $\mathcal{F}_{\mathcal{X}, k, x_0}(A)$ we associate the object $F(x_0) \to F(x)$ of $\mathcal{F}_{\mathcal{Y}, k, y_0}(A)$.

Lemma 95.3.2. Let $S$ be a locally Noetherian scheme. Let $F : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Assume either

  1. $F$ is formally smooth on objects (Criteria for Representability, Section 94.6),

  2. $F$ is representable by algebraic spaces and formally smooth, or

  3. $F$ is representable by algebraic spaces and smooth.

Then for every finite type field $k$ over $S$ and object $x_0$ of $\mathcal{X}$ over $k$ the functor (95.3.1.1) is smooth in the sense of Formal Deformation Theory, Definition 87.8.1.

Proof. Case (1) is a matter of unwinding the definitions. Assumption (2) implies (1) by Criteria for Representability, Lemma 94.6.3. Assumption (3) implies (2) by More on Morphisms of Spaces, Lemma 73.19.6 and the principle of Algebraic Stacks, Lemma 91.10.9. $\square$

Lemma 95.3.3. Let $S$ be a locally Noetherian scheme. Let

\[ \xymatrix{ \mathcal{W} \ar[d] \ar[r] & \mathcal{Z} \ar[d] \\ \mathcal{X} \ar[r] & \mathcal{Y} } \]

be a $2$-fibre product of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Let $k$ be a finite type field over $S$ and $w_0$ an object of $\mathcal{W}$ over $k$. Let $x_0, z_0, y_0$ be the images of $w_0$ under the morphisms in the diagram. Then

\[ \xymatrix{ \mathcal{F}_{\mathcal{W}, k, w_0} \ar[d] \ar[r] & \mathcal{F}_{\mathcal{Z}, k, z_0} \ar[d] \\ \mathcal{F}_{\mathcal{X}, k, x_0} \ar[r] & \mathcal{F}_{\mathcal{Y}, k, y_0} } \]

is a fibre product of predeformation categories.

Proof. This is a matter of unwinding the definitions. Details omitted. $\square$


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