The Stacks project

Lemma 86.14.12. Let $A$ be an adic Noetherian topological ring. Let $\mathfrak p \subset A$ be a rig-closed prime. For any $n \geq 1$ the ring map

\[ A/\mathfrak p \longrightarrow A\{ x_1, \ldots , x_ n\} \otimes _ A A/\mathfrak p = A/\mathfrak p\{ x_1, \ldots , x_ n\} \]

is regular. In particular, the algebra $A\{ x_1, \ldots , x_ n\} \otimes _ A \kappa (\mathfrak p)$ is geometrically regular over $\kappa (\mathfrak p)$.

Proof. We will use some fact on regular ring maps the reader can find in More on Algebra, Section 15.41. Since $A/\mathfrak p$ is a complete local Noetherian ring it is excellent (More on Algebra, Proposition 15.52.3). Hence $A/\mathfrak p[x_1, \ldots , x_ n]$ is excellent (by the same reference). Hence $A/\mathfrak p[x_1, \ldots , x_ n] \to A/\mathfrak p\{ x_1, \ldots , x_ n\} $ is a regular ring homomorphism by More on Algebra, Lemma 15.50.14. Of course $A/\mathfrak p \to A/\mathfrak p[x_1, \ldots , x_ n]$ is smooth and hence regular. Since the composition of regular ring maps is regular the proof is complete. $\square$

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