Recall that every extension of scalars functor for modules is cocontinuous and a fortiori admits a right adjoint. Hence this definition is in an agreement with the more general notion of a flat functor in category and topos theory.
Flat modules: A right -module is flat if the functor from the category of left -modules to the category Ab of abelian groups is exact.
A morphism of schemes is faithfully flat if it is flat and epi.
flat precisely if exhibits as a flat module over ;
faithfully flat if exhibits as a faithfully flat module over .
A flat morphism is faithfully flat if it is an epimorphism.
(e.g Milne, footnote 18)
These are the Grothendieck topologies in which covers consist of flat morphisms which set-theoretically cover the target scheme with some additional finiteness conditions. The usual choices are the fppf (fr. fidèlement plat de presentation finie, ‘faithfully flat and of finite presentation’) and fpqc (fr. fidèlement plat et quasicompact, ‘faithfully flat and quasicompact’) topologies. Cf. wikipedia:flat topology