I'm trying to prove the Lazard theorem:Let M be an R-module. Then M is flat if and only if it is the colimit of a directed system of free finite R-modules.
I got this hint of one direction:
Proof. A colimit of a directed system of flat modules is flat, as taking directed colimits is exact and commutes with tensor product. Hence if M is the colimit of a directed system of free finite modules then M is flat.
I have some questions:
why do they say flat and not free in this sentence:
"directed system of flat"
Moreover I do not understand what is exact and how commuting with tensor product
helps to prove that the colimit is flat.