Proposition 1

Tag 044W, permanent. Cite it as [Bourbaki, Tag 044W].

Commutative Algebra, chapter I, § 4. Flat modules and "Tor" functors

Let E be a right A-module. The following four properties are equivalent:

(a) E is flat. (b) For every left A-module F and every integer n $\geq 1$ , $\mathrm{Tor}_n^A(E, F) = 0$ . (c) For every left A-module F, $\mathrm{Tor}_1^A(E, F) = 0$ . (d) For every finitely generated left ideal $a$ of A,

\mathrm{Tor}_1^A(E, A_s/a) = 0.

We show that (a) implies (b). Let

\cdots \to L_n \to L_{n-1} \to \cdots \to L_0 \to F \to 0

be a free resolution of F. As E is flat, the sequence

(1) \quad \cdots \to E \otimes L_n \to E \otimes L_{n-1} \to \cdots \to E \otimes L_0 \to E \otimes F \to 0

is exact. As the

\mathrm{Tor}_n^A(E, F)

are isomorphic to homology groups of the complex (1), they are zero for

n \geq 1

. It is trivial that (b) implies (c) and (c) implies (d). We show finally that (d) implies (a). The exact sequence

0 \to a \to A, \to A_s/a \to 0

gives the exact sequence

\mathrm{Tor}_1^A(E, A_s/a) \to E \otimes_A a \to E \otimes_A A.

As (d) holds, the canonical homomorphism

E \otimes_A a \to E \otimes_A A = E

is injective, which means that E is flat (§ 2, no. 3, Proposition 1).

Proposition 1 provides a characterization of flat modules which is often useful in applications. We shall restrict ourselves, by way of an example, to giving a

(*) See the part of this Treatise devoted to categories and, in particular, Abelian categories (in preparation). Until this is published, the reader can consult H. CARTAN-S. EILENBERG, Homological Algebra, Princeton, 1956, or R. GODEMENT, Théorie des Faisceaux, Paris (Hermann), 1958.

\operatorname{Tor}_1^A(E', F) \to \operatorname{Tor}_1^A(E, F) \to \operatorname{Tor}_1^A(E'', F)

shows that $\operatorname{Tor}_1^A(E, F) = 0$ for every left A-module F, hence E is flat. If E and E'' are flat, the exact sequence

\operatorname{Tor}_2^A(E'', F) \to \operatorname{Tor}_1^A(E', F) \to \operatorname{Tor}_1^A(E, F)

shows that $\operatorname{Tor}_1^A(E', F) = 0$ , hence E' is flat.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

§ 2, no. 3, Proposition 1