Flatness and base change

For modules over a commutative unital ring R, tensoring is right exact: M′→M→M″→0 remains exact after tensoring with any N. N is flat when every short exact sequence remains short exact, equivalently when tensoring preserves all injections. Free modules are flat because tensoring with a direct sum of copies of R gives a direct sum of the original sequence. Projective modules are flat as direct summands of free modules. Localization is flat by its denominator-clearing exactness argument, while a general quotient module is not.

The simplest failure is integer multiplication by three. It is injective on Z, but after tensoring with Z/3Z it becomes the zero map on a nonzero module. The missing kernel is not a failure of right exactness; it is exactly the left-side information flatness would preserve. Higher derived tensor functors package such failures as Tor groups, but this elementary injection test already proves nonflatness. Over a principal ideal domain, torsion-free modules are flat; that useful criterion does not generalize to arbitrary domains without further hypotheses.

A ring map R→S is flat when S is flat as an R-module. If T is another R-algebra, then S⊗_R T is flat over T whenever S is flat over R. Tensor associativity turns an exact T-module sequence into the same scalar-extension test. Likewise, extending a flat R-module to T gives a flat T-module. A finitely presented flat module is projective, and over a local ring it is finite free; retaining finite presentation is important when stating this useful local-freeness conclusion.

A flat map can still annihilate nonzero modules. Rationalization Z→Q is flat because it is localization, but Z/nZ tensors to zero for every nonzero integer n. Thus observing an isomorphism only after rationalization can miss torsion in its kernel or cokernel. Specialization to a residue field is more drastic and is often nonflat: evaluating t=0 can turn multiplication by t from an injection into zero. One must distinguish a flat extension from a chosen fiber and distinguish both from a faithful extension.

A flat R-algebra S is faithfully flat when N⊗_R S=0 implies N=0 for every R-module N. Equivalently, its spectrum map is surjective, and tensoring preserves and reflects exactness. To understand reflection, flatness identifies the homology of a tensored complex with the tensor product of its original homology; faithfulness then says a vanished homology module was already zero. This is a theorem about all modules, not merely the finite ones in a convenient example. Faithful flatness is also preserved under base change.

Polynomial extension R→R[t] is faithfully flat: as an R-module the target is free, and N⊗R[t] is a polynomial list of coefficients from N, with its constant coefficient retaining any nonzero element. A field extension is similarly faithful and flat. The Noetherian local completion from the previous lesson is another important example. These extensions support descent of exactness and many finite-presentation properties under their proper theorems, but they do not justify saying every basis or every chosen equation literally descends without compatibility data. Faithful detection is precise information, not a universal reversal of every construction.