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94.7 Change of field

This section is the analogue of Formal Deformation Theory, Section 86.29. As pointed out there, to discuss what happens under change of field we need to write $\mathcal{C}_{\Lambda , k}$ instead of $\mathcal{C}_\Lambda $. In the following lemma we use the notation $\mathcal{F}_{l/k}$ introduced in Formal Deformation Theory, Situation 86.29.1.

Lemma 94.7.1. 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 $l/k$ be a finite extension. Let $x_0$ be an object of $\mathcal{F}$ lying over $\mathop{\mathrm{Spec}}(k)$. Denote $x_{l, 0}$ the restriction of $x_0$ to $\mathop{\mathrm{Spec}}(l)$. Then there is a canonical functor

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

of categories cofibred in groupoids over $\mathcal{C}_{\Lambda , l}$. If $\mathcal{X}$ satisfies (RS), then this functor is an equivalence.

Proof. Consider a factorization

\[ \mathop{\mathrm{Spec}}(l) \to \mathop{\mathrm{Spec}}(B) \to S \]

as in (94.3.0.1). By definition we have

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

see Formal Deformation Theory, Situation 86.29.1. Thus an object of this is a morphism $x_0 \to x$ of $\mathcal{X}$ lying over the morphism $\mathop{\mathrm{Spec}}(k) \to \mathop{\mathrm{Spec}}(B \times _ l k)$. Choosing pullback functor for $\mathcal{X}$ we can associate to $x_0 \to x$ the morphism $x_{l, 0} \to x_ B$ where $x_ B$ is the restriction of $x$ to $\mathop{\mathrm{Spec}}(B)$ (via the morphism $\mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(B \times _ l k)$ coming from $B \times _ l k \subset B$). This construction is functorial in $B$ and compatible with morphisms.

Next, assume $\mathcal{X}$ satisfies (RS). Consider the diagrams

\[ \vcenter { \xymatrix{ l & B \ar[l] \\ k \ar[u] & B \times _ l k \ar[l] \ar[u] } } \quad \text{and}\quad \vcenter { \xymatrix{ \mathop{\mathrm{Spec}}(l) \ar[d] \ar[r] & \mathop{\mathrm{Spec}}(B) \ar[d] \\ \mathop{\mathrm{Spec}}(k) \ar[r] & \mathop{\mathrm{Spec}}(B \times _ l k) } } \]

The diagram on the left is a fibre product of rings. The diagram on the right is a pushout in the category of schemes, see More on Morphisms, Lemma 36.14.3. These schemes are all of finite type over $S$ (see remarks following Definition 94.5.1). Hence (RS) kicks in to give an equivalence of fibre categories

\[ \mathcal{X}_{\mathop{\mathrm{Spec}}(B \times _ l k)} \longrightarrow \mathcal{X}_{\mathop{\mathrm{Spec}}(k)} \times _{\mathcal{X}_{\mathop{\mathrm{Spec}}(l)}} \mathcal{X}_{\mathop{\mathrm{Spec}}(B)} \]

This implies that the functor defined above gives an equivalence of fibre categories. Hence the functor is an equivalence on categories cofibred in groupoids by (the dual of) Categories, Lemma 4.34.8. $\square$


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