Let be an -ring and let be an -module. is flat if
is a flat module over in the classical sense (i.e. is an exact functor);
For each , the induced map
is an isomorphism.
A map of -rings is flat if is flat when regarded as an -module.
The same definitions work for some other contexts of derived local algebra, e.g. dg-algebras.
A morphism of derived schemes is flat if for all affine subsets and the induced map on global sections is flat as a map of -rings.