**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:

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

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 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 $B_1:=F_K(A_1)\rightarrow B_2:=F_K(A_2)$?

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).

completion of$A_1$along$K$. Regarding the functor question: which category should be the domain? $\endgroup$15more comments