Lemma 23.8.2. Let $R$ be a regular ring. Let $\mathfrak p \subset R$ be a prime. Let $f_1, \ldots , f_ r \in \mathfrak p$ be a regular sequence. Then the completion of

$A = (R/(f_1, \ldots , f_ r))_\mathfrak p = R_\mathfrak p/(f_1, \ldots , f_ r)R_\mathfrak p$

is a complete intersection in the sense defined above.

Proof. The completion of $A$ is equal to $A^\wedge = R_\mathfrak p^\wedge /(f_1, \ldots , f_ r)R_\mathfrak p^\wedge$ because completion for finite modules over the Noetherian ring $R_\mathfrak p$ is exact (Algebra, Lemma 10.97.1). The image of the sequence $f_1, \ldots , f_ r$ in $R_\mathfrak p$ is a regular sequence by Algebra, Lemmas 10.97.2 and 10.68.5. Moreover, $R_\mathfrak p^\wedge$ is a regular local ring by More on Algebra, Lemma 15.43.4. Hence the result holds by our definition of complete intersection for complete local rings. $\square$

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