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.