Flat Module - Homological Algebra

Homological Algebra

Flatness may also be expressed using the Tor functors, the left derived functors of the tensor product. A left R-module M is flat if and only if Tor_nR(–, M) = 0 for all (i.e., if and only if Tor_nR(X, M) = 0 for all and all right R-modules X). Similarly, a right R-module M is flat if and only if Tor_nR(M, X) = 0 for all and all left R-modules X. Using the Tor functor's long exact sequences, one can then easily prove facts about a short exact sequence

• If A and C are flat, then so is B
• If B and C are flat, then so is A

If A and B are flat, C need not be flat in general. However, it can be shown that

• If A is pure in B and B is flat, then A and C are flat.