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Every faithfully flat ring map is injective

Statement

Assume the Axiom of Choice.

Every faithfully flat homomorphism of commutative rings is injective.

Facts & Assumptions

Given: The Axiom of Choice and a faithfully flat homomorphism $f : R \to S$ .

[L1]

Faithful flatness of the ring map means that $S$ is a faithfully flat $R$ -module, hence in particular a flat $R$ -module (Flat and faithfully flat modules and ring homomorphisms).

[L2]

For a faithfully flat module $T$ , a nonzero module cannot tensor to zero: if $N \otimes_{R} T = 0$ , then $N = 0$ (For a flat module, faithful flatness is equivalent to detecting nonzero modules and residue fields).

Proof

technique · direct

1.1

Let $K = \ker f$ . Since $S$ is flat over $R$ by [L1], tensoring the exact sequence $0 \to K \to R \overset{f}{\to} S$ with $S$ remains exact: $0 \to K \otimes_{R} S \to R \otimes_{R} S \to S \otimes_{R} S .$

L1given

2.1

Under the canonical identification $R \otimes_{R} S ≅ S$ , the middle map in step 1.1 is $s \mapsto 1 \otimes s$ . Multiplication $μ : S \otimes_{R} S \to S$ , $μ (s \otimes t) = s t$ , is a left inverse to this map, so the middle map is injective. Therefore $K \otimes_{R} S = 0$ .

step 1.1algebra

3.1

Because $S$ is faithfully flat over $R$ by [L1], [L2] applied to $N = K$ shows that $K = 0$ .

L1L2step 2.1

4.1

Hence $f$ is injective.

algebra∎

Depends on

Used by

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Dependency tree · two levels

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Sources

J. S. Milne, A Primer of Commutative Algebra, Proposition 11.12 (standard reference, not scraped)