(2) Various mathematicians have used a different method to construct scalar extensions with prescribed residue field, which seems 'more hopeful' to be functorial. In [b] (pp. 776-777) Kunz calls a special case of this construction the constant field extension. A version of this construction in equicharacteristic appears in [a] (pp. 18-19, 6.3). A more detailed description of this method can be found in [c] (pp. 4-7) and in [d] (pp. 36-38). I describe it in the equicharacteristic case: Given a local ring $(A,\mathfrak{m},k)$ and a field extension $K$ of $k$, take a coefficient field $k\hookrightarrow\hat{A}$ and complete $\hat{A}\otimes_kK$ with respect to the ideal $\mathfrak{m}(\hat{A}\otimes_kK)$. This is your $F_K(A)$. It is easy to see that this $F_K(A)$ is an scalar extension of $A$ with residue field $K$. ($F_K(A)$ depends on the choice of a coefficient field of $\hat{A}$, but is unique up to isomorphism).

Question B. Is the $F_K(\:\cdot\:)$ that was just described a functor from $\mathscr{C}_K$ to $\mathscr{C}_K$? To clarify the question, if $A_1\stackrel{\psi}{\longrightarrow} A_2$ is a local homomorphism of Noetherian local rings in $\mathscr{C}_K$, then does $\psi$ extend to a local homomorphism $F_K(A_1)\rightarrow F_K(A_2)$?

(2) Various mathematicians have used a different method to construct scalar extensions with prescribed residue field, which seems 'more hopeful' to be functorial. In [b] (pp. 776-777) Kunz calls a special case of this construction the constant field extension. A version of this construction in the equicharacteristic case appears in [a] (pp. 18-19, 6.3). A more detailed description of this method can be found in [c] (pp. 4-7) and in [d] (pp. 36-38). I describe it in the equicharacteristic case: Given a local ring $(A,\mathfrak{m},k)$ and a field extension $K$ of $k$, take a coefficient field $k\hookrightarrow\hat{A}$ and complete $\hat{A}\otimes_kK$ with respect to the ideal $\mathfrak{m}(\hat{A}\otimes_kK)$. This is your $F_K(A)$. It is easy to see that this $F_K(A)$ is an scalar extension of $A$ with residue field $K$. ($F_K(A)$ depends on the choice of a coefficient field of $\hat{A}$, but is unique up to isomorphism).

Question B. Is the $F_K(\:\cdot\:)$ that was just described a functor from $\mathscr{C}_K$ to $\mathscr{C}_K$? To clarify the question, if $A_1\stackrel{\psi}{\longrightarrow} A_2$ is a local homomorphism of Noetherian local rings in $\mathscr{C}_K$, then does $\psi$ extend to a local homomorphism $B_1:=F_K(A_1)\rightarrow B_2:=F_K(A_2)$?

A local homomorphism of local rings $(A,\mathfrak{m})\stackrel{\varphi}{\longrightarrow}(B,\mathfrak{n})$ is called a scalar extension (terminology due to Hans Schoutens) if:

Theorem. (Grothendieck, EGA III, Proposition 10.3.1, page 20). Let $(A,\mathfrak{m})$ be Noetherian local ring with residue field $k$, and let $K$ be a field extension of $k$. Then there exists a scalar extension $$(A,\mathfrak{m})\stackrel{\varphi}{\longrightarrow}(B,\mathfrak{n})$$ from $A$ to a Noetherian local ring $B$, with the property that $B/\mathfrak{n}$ is $k$-isomorphic to $K$.

Note. Other constructions of scalar extensions (not necessarily with this name) have appeared in

• Hochster and Huneke, $F$-regularity, test elements, and smooth base change, Trans. Amer. Math. Soc., 346 (1994) (see pages 18-19), and

• Schoutens, Classifying singularities up to analytic extensions of scalars, Ann. of Pure and Applied Logic, 162, (2011) (also available on the Arxiv, see pages 5-8).

Question. Is scalar extension of local rings a functor? To be more precise, suppose $A_1\stackrel{\psi}{\longrightarrow} A_2$ is a local homomorphism of Noetherian local rings with residue fields $k_1$ and $k_2$, and let $K$ be a common field extension of $k_1$ and $k_2$. Let $A_1\longrightarrow B_1$ and $A_2\longrightarrow B_2$ be two corresponding scalar extensions. Does $\psi$ extend to a local homomorphism $B_1\longrightarrow B_2$?

If you know of any reference where this question is discussed, please let me know.

EDIT: $B_1$ and $B_2$ are not any two arbitrary scalar extensions in my question. They are the scalar extensions obtained by the method described in the second reference by Schoutens.

Note. I have edited my question to make it more transparent, following some very good comments that I received. I am sorry if it is a bit long.

A local homomorphism of local rings $(A,\mathfrak{m})\stackrel{\varphi}{\longrightarrow}(B,\mathfrak{n})$ is called a scalar extension (terminology due to Hans Schoutens) if:

Let's fix a field $K$ (algebraically closed, if you wish) and let $\mathscr{C}_K$ be the category of Noetherian local rings whose residue field is a subfield of $K$, with morphisms being local homomorphisms.

Question A. Is there a functorial way of producing scalar extensions with a prescribed residue field? More precisely, is it possible to define a functor $F_K:\mathscr{C}_K\rightarrow\mathscr{C}_K$ in such a way that for every $A\in\mathscr{C}_K$ the local ring $F_K(A)$ is a scalar extension of $A$ with residue field $K$?

Here are some things that I know about this question:

(1) Grothendieck proved that scalar extensions with prescribed residue field always exist:

Theorem. (EGA III, Proposition 10.3.1, page 20). Let $(A,\mathfrak{m})$ be Noetherian local ring with residue field $k$, and let $K$ be a field extension of $k$. Then there exists a scalar extension $(A,\mathfrak{m})\stackrel{\varphi}{\longrightarrow}(B,\mathfrak{n})$ from $A$ to a Noetherian local ring $B$, with the property that $B/\mathfrak{n}$ is $k$-isomorphic to $K$.

Grothendieck's construction of the desired scalar extension depends on various 'choices' that he makes in his proof, and hence, does not produce a unique answer. For this reason I think it is hopeless to get a functor there.

(2) Various mathematicians have used a different method to construct scalar extensions with prescribed residue field, which seems 'more hopeful' to be functorial. In [b] (pp. 776-777) Kunz calls a special case of this construction the constant field extension. A version of this construction in equicharacteristic appears in [a] (pp. 18-19, 6.3). A more detailed description of this method can be found in [c] (pp. 4-7) and in [d] (pp. 36-38). I describe it in the equicharacteristic case: Given a local ring $(A,\mathfrak{m},k)$ and a field extension $K$ of $k$, take a coefficient field $k\hookrightarrow\hat{A}$ and complete $\hat{A}\otimes_kK$ with respect to the ideal $\mathfrak{m}(\hat{A}\otimes_kK)$. This is your $F_K(A)$. It is easy to see that this $F_K(A)$ is an scalar extension of $A$ with residue field $K$. ($F_K(A)$ depends on the choice of a coefficient field of $\hat{A}$, but is unique up to isomorphism).

Question B. Is the $F_K(\:\cdot\:)$ that was just described a functor from $\mathscr{C}_K$ to $\mathscr{C}_K$? To clarify the question, if $A_1\stackrel{\psi}{\longrightarrow} A_2$ is a local homomorphism of Noetherian local rings in $\mathscr{C}_K$, then does $\psi$ extend to a local homomorphism $F_K(A_1)\rightarrow F_K(A_2)$?

I can see how the method described in [c] provides an affirmative answer in equicharacteristic $0$ to Question B (it comes down to the fact that in equicharacteristic $0$ every maximal subfield of a complete local ring is a coefficient field) but I don't see how the method of [c] would still work in equicharacteristic $p>0$. I haven't checked the mixed characteristic case, yet, because I thought the equicharacteristic case is easier and if it cannot be settled positively, then there is even less hope for the mixed characteristic.

References.

a. M. Hochster and C. Huneke, $F$-regularity, test elements, and smooth base change, Trans. Amer. Math. Soc., 346 (1994).

b. E. Kunz, Characterizations of regular local rings of characteristic $p$, Amer. Jour. of Math., 41 (1969).

c. H. Schoutens, Classifying singularities up to analytic extensions of scalars, Ann. of Pure and Applied Logic, 162, (2011) (also available on the Arxiv, here).

d. H. Schoutens, The use of ultraproducts in commutative algebra, Lecture Notes in Mathematics, 1999, Speringer (2010).

A local homomorphism of local rings $(A,\mathfrak{m})\stackrel{\varphi}{\longrightarrow}(B,\mathfrak{n})$ is called a scalar extension (terminology due to Hans Schoutens) if:

• $\varphi(\mathfrak{m})B=\mathfrak{n}$, and
• $\varphi$ is a flat extension.

Theorem. (Grothendieck, EGA III, Proposition 10.3.1, page 20). Let $(A,\mathfrak{m})$ be Noetherian local ring with residue field $k$, and let $K$ be a field extension of $k$. Then there exists a scalar extension $$(A,\mathfrak{m})\stackrel{\varphi}{\longrightarrow}(B,\mathfrak{n})$$ from $A$ to a Noetherian local ring $B$, with the property that $B/\mathfrak{n}$ is $k$-isomorphic to $K$.

Note. Other constructions of scalar extensions (not necessarily with this name) have appeared in

• Hochster and Huneke, $F$-regularity, test elements, and smooth base change, Trans. Amer. Math. Soc., 346 (1994) (see pages 18-19), and

• Schoutens, Classifying singularities up to analytic extensions of scalars, Ann. of Pure and Applied Logic, 162, (2011) (also available on the Arxiv, see pages 5-8).

Question. Is scalar extension of local rings a functor? To be more precise, suppose $A_1\stackrel{\psi}{\longrightarrow} A_2$ is a local homomorphism of Noetherian local rings with residue fields $k_1$ and $k_2$, and let $K$ be a common field extension of $k_1$ and $k_2$. Let $A_1\longrightarrow B_1$ and $A_2\longrightarrow B_2$ be two corresponding scalar extensions. Does $\psi$ extend to a local homomorphism $B_1\longrightarrow B_2$?

If you know of any reference where this question is discussed, please let me know.

A local homomorphism of local rings $(A,\mathfrak{m})\stackrel{\varphi}{\longrightarrow}(B,\mathfrak{n})$ is called a scalar extension (terminology due to Hans Schoutens) if:

• $\varphi(\mathfrak{m})B=\mathfrak{n}$, and
• $\varphi$ is a flat extension.

Theorem. (Grothendieck, EGA III, Proposition 10.3.1, page 20). Let $(A,\mathfrak{m})$ be Noetherian local ring with residue field $k$, and let $K$ be a field extension of $k$. Then there exists a scalar extension $$(A,\mathfrak{m})\stackrel{\varphi}{\longrightarrow}(B,\mathfrak{n})$$ from $A$ to a Noetherian local ring $B$, with the property that $B/\mathfrak{n}$ is $k$-isomorphic to $K$.

Note. Other constructions of scalar extensions (not necessarily with this name) have appeared in

• Hochster and Huneke, $F$-regularity, test elements, and smooth base change, Trans. Amer. Math. Soc., 346 (1994) (see pages 18-19), and

• Schoutens, Classifying singularities up to analytic extensions of scalars, Ann. of Pure and Applied Logic, 162, (2011) (also available on the Arxiv, see pages 5-8).

Question. Is scalar extension of local rings a functor? To be more precise, suppose $A_1\stackrel{\psi}{\longrightarrow} A_2$ is a local homomorphism of Noetherian local rings with residue fields $k_1$ and $k_2$, and let $K$ be a common field extension of $k_1$ and $k_2$. Let $A_1\longrightarrow B_1$ and $A_2\longrightarrow B_2$ be two corresponding scalar extensions. Does $\psi$ extend to a local homomorphism $B_1\longrightarrow B_2$?

If you know of any reference where this question is discussed, please let me know.

EDIT: $B_1$ and $B_2$ are not any two arbitrary scalar extensions in my question. They are the scalar extensions obtained by the method described in the second reference by Schoutens.