The Stacks project

89.29 Change of residue field

In this section we quickly discuss what happens if we replace the residue field $k$ by a finite extension. Let $\Lambda $ be a Noetherian ring and let $\Lambda \to k$ be a finite ring map where $k$ is a field. Throughout this whole chapter we have used $\mathcal{C}_\Lambda $ to denote the category of Artinian local $\Lambda $-algebras whose residue field is identified with $k$, see Definition 89.3.1. However, since in this section we will discuss what happen when we change $k$ we will instead use the notation $\mathcal{C}_{\Lambda , k}$ to indicate the dependence on $k$.

Situation 89.29.1. Let $\Lambda $ be a Noetherian ring and let $\Lambda \to k \to l$ be a finite ring maps where $k$ and $l$ are fields. Thus $l/k$ is a finite extensions of fields. A typical object of $\mathcal{C}_{\Lambda , l}$ will be denoted $B$ and a typical object of $\mathcal{C}_{\Lambda , k}$ will be denoted $A$. We define
\begin{equation} \label{formal-defos-equation-comparison} \mathcal{C}_{\Lambda , l} \longrightarrow \mathcal{C}_{\Lambda , k}, \quad B \longmapsto B \times _ l k \end{equation}

Given a category cofibred in groupoids $p : \mathcal{F} \to \mathcal{C}_{\Lambda , k}$ we obtain an associated category cofibred in groupoids

\[ p_{l/k} : \mathcal{F}_{l/k} \longrightarrow \mathcal{C}_{\Lambda , l} \]

by setting $\mathcal{F}_{l/k}(B) = \mathcal{F}(B \times _ l k)$.

The functor ( makes sense: because $B \times _ l k \subset B$ we have

\begin{align*} [k : k']\ \text{length}_{B \times _ l k}(B \times _ l k) & = \text{length}_\Lambda (B \times _ l k) \\ & \leq \text{length}_\Lambda (B) \\ & = [l : k']\ \text{length}_ B(B) < \infty \end{align*}

(see Lemma 89.3.4) hence $B \times _ l k$ is Artinian (see Algebra, Lemma 10.53.6). Thus $B \times _ l k$ is an Artinian local ring with residue field $k$. Note that ( commutes with fibre products

\[ (B_1 \times _ B B_2) \times _ l k = (B_1 \times _ l k) \times _{(B \times _ l k)} (B_2 \times _ l k) \]

and transforms surjective ring maps into surjective ring maps. We use the “expensive” notation $\mathcal{F}_{l/k}$ to prevent confusion with the construction of Remark 89.6.4. Here are some elementary observations.

Lemma 89.29.2. With notation and assumptions as in Situation 89.29.1.

  1. We have $\overline{\mathcal{F}_{l/k}} = (\overline{\mathcal{F}})_{l/k}$.

  2. If $\mathcal{F}$ is a predeformation category, then $\mathcal{F}_{l/k}$ is a predeformation category.

  3. If $\mathcal{F}$ satisfies (S1), then $\mathcal{F}_{l/k}$ satisfies (S1).

  4. If $\mathcal{F}$ satisfies (S2), then $\mathcal{F}_{l/k}$ satisfies (S2).

  5. If $\mathcal{F}$ satisfies (RS), then $\mathcal{F}_{l/k}$ satisfies (RS).

Proof. Part (1) is immediate from the definitions.

Since $\mathcal{F}_{l/k}(l) = \mathcal{F}(k)$ part (2) follows from the definition, see Definition 89.6.2.

Part (3) follows as the functor ( commutes with fibre products and transforms surjective maps into surjective maps, see Definition 89.10.1.

Part (4). To see this consider a diagram

\[ \xymatrix{ & l[\epsilon ] \ar[d] \\ B \ar[r] & l } \]

in $\mathcal{C}_{\Lambda , l}$ as in Definition 89.10.1. Applying the functor ( we obtain

\[ \xymatrix{ & k[l\epsilon ] \ar[d] \\ B \times _ l k \ar[r] & k } \]

where $l\epsilon $ denotes the finite dimensional $k$-vector space $l\epsilon \subset l[\epsilon ]$. According to Lemma 89.10.4 the condition of (S2) for $\mathcal{F}$ also holds for this diagram. Hence (S2) holds for $\mathcal{F}_{l/k}$.

Part (5) follows from the characterization of (RS) in Lemma 89.16.4 part (2) and the fact that ( commutes with fibre products. $\square$

The following lemma applies in particular when $\mathcal{F}$ satisfies (S2) and is a predeformation category, see Lemma 89.10.5.

Lemma 89.29.3. With notation and assumptions as in Situation 89.29.1. Assume $\mathcal{F}$ is a predeformation category and $\overline{\mathcal{F}}$ satisfies (S2). Then there is a canonical $l$-vector space isomorphism

\[ T\mathcal{F} \otimes _ k l \longrightarrow T\mathcal{F}_{l/k} \]

of tangent spaces.

Proof. By Lemma 89.29.2 we may replace $\mathcal{F}$ by $\overline{\mathcal{F}}$. Moreover we see that $T\mathcal{F}$, resp. $T\mathcal{F}_{l/k}$ has a canonical $k$-vector space structure, resp. $l$-vector space structure, see Lemma 89.12.2. Then

\[ T\mathcal{F}_{l/k} = \mathcal{F}_{l/k}(l[\epsilon ]) = \mathcal{F}(k[l\epsilon ]) = T\mathcal{F} \otimes _ k l \]

the last equality by Lemma 89.12.2. More generally, given a finite dimensional $l$-vector space $V$ we have

\[ \mathcal{F}_{l/k}(l[V]) = \mathcal{F}(k[V_ k]) = T\mathcal{F} \otimes _ k V_ k \]

where $V_ k$ denotes $V$ seen as a $k$-vector space. We conclude that the functors $V \mapsto \mathcal{F}_{l/k}(l[V])$ and $V \mapsto T\mathcal{F} \otimes _ k V_ k$ are canonically identified as functors to the category of sets. By Lemma 89.11.4 we see there is at most one way to turn either functor into an $l$-linear functor. Hence the isomorphisms are compatible with the $l$-vector space structures and we win. $\square$

Lemma 89.29.4. With notation and assumptions as in Situation 89.29.1. Assume $\mathcal{F}$ is a deformation category. Then there is a canonical $l$-vector space isomorphism

\[ \text{Inf}(\mathcal{F}) \otimes _ k l \longrightarrow \text{Inf}(\mathcal{F}_{l/k}) \]

of infinitesimal automorphism spaces.

Proof. Let $x_0 \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{F}(k))$ and denote $x_{l, 0}$ the corresponding object of $\mathcal{F}_{l/k}$ over $l$. Recall that $\text{Inf}(\mathcal{F}) = \text{Inf}_{x_0}(\mathcal{F})$ and $\text{Inf}(\mathcal{F}_{l/k}) = \text{Inf}_{x_{l, 0}}(\mathcal{F}_{l/k})$, see Remark 89.19.4. Recall that the vector space structure on $\text{Inf}_{x_0}(\mathcal{F})$ comes from identifying it with the tangent space of the functor $\mathit{Aut}(x_0)$ which is defined on the category $\mathcal{C}_{k, k}$ of Artinian local $k$-algebras with residue field $k$. Similarly, $\text{Inf}_{x_{l, 0}}(\mathcal{F}_{l/k})$ is the tangent space of $\mathit{Aut}(x_{l, 0})$ which is defined on the category $\mathcal{C}_{l, l}$ of Artinian local $l$-algebras with residue field $l$. Unwinding the definitions we see that $\mathit{Aut}(x_{l, 0})$ is the restriction of $\mathit{Aut}(x_0)_{l/k}$ (which lives on $\mathcal{C}_{k, l}$) to $\mathcal{C}_{l, l}$. Since there is no difference between the tangent space of $\mathit{Aut}(x_0)_{l/k}$ seen as a functor on $\mathcal{C}_{k, l}$ or $\mathcal{C}_{l, l}$, the lemma follows from Lemma 89.29.3 and the fact that $\mathit{Aut}(x_0)$ satisfies (RS) by Lemma 89.19.6 (whence we have (S2) by Lemma 89.16.6). $\square$

Lemma 89.29.5. With notation and assumptions as in Situation 89.29.1. If $\mathcal{F} \to \mathcal{G}$ is a smooth morphism of categories cofibred in groupoids over $\mathcal{C}_{\Lambda , k}$, then $\mathcal{F}_{l/k} \to \mathcal{G}_{l/k}$ is a smooth morphism of categories cofibred in groupoids over $\mathcal{C}_{\Lambda , l}$.

Proof. This follows immediately from the definitions and the fact that ( preserves surjections. $\square$

There are many more things you can say about the relationship between $\mathcal{F}$ and $\mathcal{F}_{l/k}$ (in particular about the relationship between versal deformations) and we will add these here as needed.

Lemma 89.29.6. With notation and assumptions as in Situation 89.29.1. Let $\xi $ be a versal formal object for $\mathcal{F}$ lying over $R \in \mathop{\mathrm{Ob}}\nolimits (\widehat{\mathcal{C}}_{\Lambda , k})$. Then there exist

  1. an $S \in \mathop{\mathrm{Ob}}\nolimits (\widehat{\mathcal{C}}_{\Lambda , l})$ and a local $\Lambda $-algebra homomorphism $R \to S$ which is formally smooth in the $\mathfrak m_ S$-adic topology and induces the given field extension $l/k$ on residue fieds, and

  2. a versal formal object of $\mathcal{F}_{l/k}$ lying over $S$.

Proof. Construction of $S$. Choose a surjection $R[x_1, \ldots , x_ n] \to l$ of $R$-algebras. The kernel is a maximal ideal $\mathfrak m$. Set $S$ equal to the $\mathfrak m$-adic completion of the Noetherian ring $R[x_1, \ldots , x_ n]$. Then $S$ is in $\widehat{\mathcal{C}}_{\Lambda , l}$ by Algebra, Lemma 10.97.6. The map $R \to S$ is formally smooth in the $\mathfrak m_ S$-adic topology by More on Algebra, Lemmas 15.37.2 and 15.37.4 and the fact that $R \to R[x_1, \ldots , x_ n]$ is formally smooth. (Compare with the proof Lemma 89.9.5.)

Since $\xi $ is versal, the transformation $\underline{\xi } : \underline{R}|_{\mathcal{C}_{\Lambda , k}} \to \mathcal{F}$ is smooth. By Lemma 89.29.5 the induced map

\[ (\underline{R}|_{\mathcal{C}_{\Lambda , k}})_{l/k} \longrightarrow \mathcal{F}_{l/k} \]

is smooth. Thus it suffices to construct a smooth morphism $\underline{S}|_{\mathcal{C}_{\Lambda , l}} \to (\underline{R}|_{\mathcal{C}_{\Lambda , k}})_{l/k}$. To give such a map means for every object $B$ of $\mathcal{C}_{\Lambda , l}$ a map of sets

\[ \mathop{\mathrm{Mor}}\nolimits _{\widehat{\mathcal{C}}_{\Lambda , l}}(S, B) \longrightarrow \mathop{\mathrm{Mor}}\nolimits _{\widehat{\mathcal{C}}_{\Lambda , k}}(R, B \times _ l k) \]

functorial in $B$. Given an element $\varphi : S \to B$ on the left hand side we send it to the composition $R \to S \to B$ whose image is contained in the sub $\Lambda $-algebra $B \times _ l k$. Smoothness of the map means that given a surjection $B' \to B$ and a commutative diagram

\[ \xymatrix{ S \ar[r] & B \ar@{=}[r] & B \\ R \ar[u] \ar[r] & B' \times _ l k \ar[u] \ar[r] & B' \ar[u] } \]

we have to find a ring map $S \to B'$ fitting into the outer rectangle. The existence of this map is guaranteed as we chose $R \to S$ to be formally smooth in the $\mathfrak m_ S$-adic topology, see More on Algebra, Lemma 15.37.5. $\square$

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