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Flat Modules

 
  • Jan 18, 2024
$\DeclareMathOperator{\Tor}{Tor}
Definition 6.25. A module $F$ is flat if $-βŠ—_R F$ is an exact functor. Lemma 6.26. Let $B$ be a left $R$-module. The following are equivalent:
  1. $B$ is flat.
  2. $\Tor_n^R(A, B)=0$ for all $n \geq 1$ and all left $R$-modules $A$.
  3. $\Tor_1^R(A, B)=0$ for all left $R$-modules $A$.
Proof. Suppose that $B$ is flat. Let $F_β€’ β†’ A$ be a free resolution of $A$. Since $-βŠ—_R B$ is exact, the sequence \[ … β†’ F_2 βŠ—_R B β†’ F_1 βŠ—_R B β†’ F_0 βŠ—_R B β†’ A βŠ—_R B β†’ 0 \] is exact, so the homology of \[ … β†’ F_2 βŠ—_R B β†’ F_1 βŠ—_R B β†’ F_0 βŠ—_R B β†’ 0 \] vanishes in positive degree. Therefore, we have (1)⟹(2). The implication (2)⟹(3) is trivial. Finally, (3)⟹(1) follows from the long exact sequence of Tor, since for any short exact sequenceΒ $0 β†’ X β†’ Y β†’ Z β†’ 0$, we have that \[ 0=\operatorname{Tor}_1^R(Z, B) β†’ X βŠ— B β†’ Y βŠ— B β†’ Z βŠ— B β†’ 0 \] is exact.