Flat modules · Qual Corpus

(flat)

An $A$-module $N$ is flat if the functor $\wait \tensor_A N$ is exact. An $A$-algebra $B$ is flat if it is flat as an $A$-module.

Since $\wait \tensor_A N$ is right exact, $N$ is flat if and only if for every injection $M' \injects M$, the induced map $M' \tensor_A N \to M \tensor_A N$ is injective.

Free $\implies$ projective $\implies$ flat, and neither implication reverses in general. Over $A=\ZZ/6$, the module $\ZZ/2$ is projective, being a direct summand of $A\cong\ZZ/2\times\ZZ/3$, and not free. $\QQ$ is a flat $\ZZ$-module that is not projective; over a Noetherian local ring the three notions agree for finitely generated modules.

Over a domain, flat implies torsion-free. The ideal $(x,y)$ of $k[x,y]$ is torsion-free but not flat: its localization at $(x,y)$ is finitely generated over a Noetherian local ring and not free, since it has rank $1$ and needs two generators. Over a PID, flat and torsion-free coincide. Localisation $A \to \inverseof{S} A$ is flat; so on $\Spec A$, the functor $M\mapsto M_f=\Gamma(D(f),\widetilde M)$ is exact.

What is a flat module?

Give a module that is torsion-free but not flat.

How do free, projective, and flat compare?

Dependencies

D-DEFEXACTExactness, and left, right, and exact functorsuses

Backlinks

D-DEFFFLATFaithfully flat modules, and descent of exactnessuses
D-MORNORMREGNormal and regular morphismsuses
D-DEFTORThe $\Tor$ functors, and flatness as $\Tor$-vanishinguses
T-FLATCRITThe local and infinitesimal criteria for flatnessuses