Remark 15.90.6. While $R \to R_ f$ is always flat, $R \to R^\wedge $ is typically not flat unless $R$ is Noetherian (see Algebra, Lemma 10.97.2 and the discussion in Examples, Section 109.12). Consequently, we cannot in general apply faithfully flat descent as discussed in Descent, Section 35.3 to the morphism $R \to R^\wedge \oplus R_ f$. Moreover, even in the Noetherian case, the usual definition of a descent datum for this morphism refers to the ring $R^\wedge \otimes _ R R^\wedge $, which we will avoid considering in this section.

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