Projective Modules, Flat Modules

Flat Module

Assume R is commutative. An R module M is flat if every short exact sequence tensored with M gives another short exact sequence.

Tensoring with R doesn't change a thing, hence R is a flat R module.

Direct Sum

Let M be the direct sum of modules M_i, and tensor M with 0 → A → B → C → 0. In an earlier section we showed that BM is the direct sum over BM_i, and the homomorphism from BM into CM is the direct sum of homomorphisms from BM_i into CM_i. The composite homomorphism from BM to CM is surjective iff each component homomorphism from BM_i to CM_i is surjective. (You can verify this yourself.) Similarly, the composite homomorphism from AM into BM is injective iff each component homomorphism from AM_i into BM_i is injective. Therefore, a direct sum of modules is flat iff each module is flat.

Using this, and the fact that R is flat, any free module is flat. For instance, R[x], the polynomials over R, is a flat R module.

Being a summand of a free module, every projective module is flat.

There are flat modules that are not projective, as demonstrated by Q, which is a flat Z module.

The Tensor Product of Two Flat Modules

If U and V are flat R modules then so is their tensor product. Tensor with U and get an exact sequence, then tensor with V and get an exact sequence. This is equivalent to tensoring with UV.

Flat S Module Becomes a Flat R Module

Let a ring homomorphism take R into S, where S is a flat R module. Let M be a flat S module. Note that M is also an R module, courtesy of the homomorphism from R into S. Tensor an exact sequence with S, giving an exact sequence of R modules. This is also an exact sequence of S modules. Tensor with M and find another exact sequence of S modules, which is an exact sequence of R modules. Thus, S×M is a flat R module.

The action of R on M already vectors through S, so M and S×M are isomorphic as S modules, and as R modules. thus M is a flat R module. Every flat S module becomes a flat R module.

Flat Homomorphism

A ring homomorphism h from R into S is flat if S is a flat R module.

As we saw above, a flat homomorphism from R onto S implies flat S modules are flat R modules.