The Stacks project

Lemma 15.44.3. If $A \to B$ is an ├ętale ring map and $\mathfrak q$ is a prime of $B$ lying over $\mathfrak p \subset A$, then $A_{\mathfrak p}$ is regular if and only if $B_{\mathfrak q}$ is regular.

Proof. By Lemma 15.44.1 we may assume both $A_\mathfrak p$ and $B_\mathfrak q$ are Noetherian in order to prove the equivalence. Let $x_1, \ldots , x_ t \in \mathfrak pA_\mathfrak p$ be a minimal set of generators. As $A_\mathfrak p \to B_\mathfrak q$ is faithfully flat we see that the images $y_1, \ldots , y_ t$ in $B_\mathfrak q$ form a minimal system of generators for $\mathfrak pB_\mathfrak q = \mathfrak q B_\mathfrak q$ (Algebra, Lemma 10.143.5). Regularity of $A_\mathfrak p$ by definition means $t = \dim (A_\mathfrak p)$ and similarly for $B_\mathfrak q$. Hence the lemma follows from the equality $\dim (A_\mathfrak p) = \dim (B_\mathfrak q)$ of Lemma 15.44.2. $\square$


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