The Stacks project

Lemma 106.16.4. Let $B \to C$ be a ring map. If

  1. the coprojections $C \to C \otimes _ B C$ are flat and

  2. $B \to C$ is universally injective,

then $B \to C$ is faithfully flat.

Proof. The map $\mathop{\mathrm{Spec}}(C) \to \mathop{\mathrm{Spec}}(B)$ is surjective as $B \to C$ is universally injective. Thus it suffices to show that $B \to C$ is flat which follows from Descent, Theorem 35.4.25. $\square$

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