The Stacks project

88.29 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 88.3.1. 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 88.29.1. 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

88.29.1.1
\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 (88.29.1.1) 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 88.3.4) 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 (88.29.1.1) 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 88.6.4. Here are some elementary observations.

Lemma 88.29.2. With notation and assumptions as in Situation 88.29.1.

  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 88.6.2.

Part (3) follows as the functor (88.29.1.1) commutes with fibre products and transforms surjective maps into surjective maps, see Definition 88.10.1.

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 88.10.1. Applying the functor (88.29.1.1) 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 88.10.4 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 88.16.4 part (2) and the fact that (88.29.1.1) commutes with fibre products. $\square$

The following lemma applies in particular when $\mathcal{F}$ satisfies (S2) and is a predeformation category, see Lemma 88.10.5.

Lemma 88.29.3. With notation and assumptions as in Situation 88.29.1. 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 88.29.2 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 88.12.2. 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 88.12.2. 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 88.11.4 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 88.29.4. With notation and assumptions as in Situation 88.29.1. 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 88.19.4. 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 88.29.3 and the fact that $\mathit{Aut}(x_0)$ satisfies (RS) by Lemma 88.19.6 (whence we have (S2) by Lemma 88.16.6). $\square$

Lemma 88.29.5. With notation and assumptions as in Situation 88.29.1. 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 (88.29.1.1) 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 88.29.6. With notation and assumptions as in Situation 88.29.1. 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 88.9.5.)

Since $\xi $ is versal, the transformation $\underline{\xi } : \underline{R}|_{\mathcal{C}_{\Lambda , k}} \to \mathcal{F}$ is smooth. By Lemma 88.29.5 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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