The Stacks project

Lemma 10.97.2. Let $I$ be a ideal of a Noetherian ring $R$. Denote ${}^\wedge $ completion with respect to $I$.

  1. The ring map $R \to R^\wedge $ is flat.

  2. The functor $M \mapsto M^\wedge $ is exact on the category of finitely generated $R$-modules.

Proof. Consider $J \otimes _ R R^\wedge \to R \otimes _ R R^\wedge = R^\wedge $ where $J$ is an arbitrary ideal of $R$. According to Lemma 10.97.1 this is identified with $J^\wedge \to R^\wedge $ and $J^\wedge \to R^\wedge $ is injective. Part (1) follows from Lemma 10.39.5. Part (2) is a reformulation of Lemma 10.97.1 part (2). $\square$

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