§ 4. Flat modules and "Tor" functors

4. Flat modules and "tor" functors

For the benefit of readers conversant with Homological Algebra (*), we shall indicate quickly how the theory of flat modules is related to that of Tor functors.

Proposition 1

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.

Proposition 2

Let R, S be two rings, $\rho : \mathbf{R} \to \mathbf{S}$ a homomorphism and F a left R-module. The following two properties are equivalent:* (a) $\operatorname{Tor}_1^R(\rho_*(E), F) = 0$ for every right S-module E. (b) *The left S-module $\varphi^*(F) = F_{(S)} = S \otimes_R F$ is flat and $\operatorname{Tor}_1^R(\rho_*(S_d), F) = 0$ .

Suppose that (a) holds. Taking $E = S_d$ , we see that $\operatorname{Tor}_1^R(\rho_*(S_d), F) = 0$ . We show also that $F_{(S)}$ is a flat S-module. For that, we note that, if E is a right S-module, the additive group $E \otimes_S F_{,,}$ is identified with $\rho_*(E) \otimes_R F$ . Then if there is an exact sequence of right S-modules

0 \to E' \to E \to E'' \to 0

we obtain, using (a), an exact sequence

0 \to \rho_*(E') \otimes_R F \to \rho_*(E) \otimes_R F \to \rho_*(E'') \otimes_R F \to 0

or also

0 \to E' \otimes_S F_{(S)} \to E \otimes_S F_{(S)} \to E'' \otimes_S F_{(S)} \to 0

which proves that $F_{(S)}$ is flat.

Conversely, if (b) holds, we have first of all, for every free right S-module $L = S_d^{(I)}$ , $\operatorname{Tor}_1^R(\rho_*(L), F) = (\operatorname{Tor}_1^R(\rho_*(S_d), F))^{(I)} = 0$ . Every right S-module E can be written in the form $E = L/H$ for a suitable free S-module L; then we have the exact sequence

(2) $0 = \operatorname{Tor}_1^R(\rho_*(L), F) \to \operatorname{Tor}_1^R(\rho_*(E), F) \to \rho_*(H) \otimes_R F \to \rho_*(L) \otimes_R F.$

But as $F_{,,}$ is flat, the homomorphism $H \otimes_S F_{(S)} \to L \otimes_S F_{(S)}$ is injective and is identified with the homomorphism

\rho_*(H) \otimes_R F \to \rho_*(L) \otimes_R F.

Then it follows from (2) that $\operatorname{Tor}_1^R(\rho_*(E), F) = 0$ .

Remark

Proposition 2 also follows from the existence of the exact sequence

E \otimes_S \operatorname{Tor}_1^R(\rho_*(S_d), F) \to \operatorname{Tor}_1^R(\rho_*(E), F) \to \operatorname{Tor}_1^S(E, S_d \otimes_R F) \to 0

arising from the spectral sequence of the "associativity" of the Tor functors.

Exercises

See the exercises for § 4.