Lemma 106.16.4 (0GRM)—The Stacks project

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Table of contents
Part 7: Algebraic Stacks
Chapter 106: More on Morphisms of Stacks
Section 106.16: Tensor functors
Lemma 106.16.4 (cite)

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

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