(faithfully flat)
An \(A\)-module \(N\) is faithfully flat if for every complex of \(A\)-modules \[ M' \to M \to M'' \] the complex is exact if and only if \[ M' \tensor_A N \to M \tensor_A N \to M'' \tensor_A N \] is exact. An \(A\)-algebra \(B\) is faithfully flat if it is so as an \(A\)-module.
A flat module satisfies the implication from exactness of the first complex to exactness of the second. Equivalently, \(N\) is faithfully flat if and only if \(N\) is flat and \(M \tensor_A N = 0\) implies \(M = 0\); for an algebra \(A \to B\), faithful flatness is equivalent to flatness together with surjectivity of \(\Spec B \to \Spec A\).
\(\QQ\) is flat over \(\ZZ\) but not faithfully flat, since \(\ZZ/p\tensor_\ZZ\QQ=0\). More generally, for a multiplicative subset \(S\subseteq A\), \(A\to S^{-1}A\) is flat, and it is faithfully flat only if \(\Spec S^{-1}A\to\Spec A\) is surjective, which fails when some \(s\in S\) lies in a prime ideal of \(A\). If \(A\to B\) is faithfully flat, then an \(A\)-module \(M\) is zero, finitely generated, or flat if and only if \(M\tensor_AB\) is.
Dependencies
D-DEFFLATFlat modulesuses
Backlinks
T-FFDESCENTFaithfully flat descent and the Amitsur complexusesFE-SCHFFLATFlatness of the affine line over the integers, and faithful flatness as faithfulnessuses