The Stacks project

98.3 Predeformation categories

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

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 29.16.1). 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 90.6.2. The construction will resemble the construction of Formal Deformation Theory, Remark 90.6.4.

First, by Morphisms, Lemma 29.16.1 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 90.3.1. 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

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 (98.3.0.1), 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

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 $\mathop{\mathrm{Spec}}(k) \to \mathop{\mathrm{Spec}}(A)$.

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

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 $\mathcal{F}_{\mathcal{Y}, k, y_0}(A)$.