In "stack project", there is a lemma on finite locally free morphisms, saying that a finite locally free morphism of schemes is equivalent to a morphism which is finite, flat, and locally of finite presentation.


For the proof, they refer to the commutative algebra fact that a module is finite locally free iff it is flat and finitely presented. 


In order to complete the proof, I have the following gap:


Let $A \to B$ be a morphism of rings. Then $B$ can either be viewed as an $A$-algbra or an $A$-module. Is it true that the following two statements are equivalent?


1. $B$ is a flat $A$-module and is finitely presented as an $A$-module.
2. $B$ is a finitely generated flat $A$-module, and is finitely presented as an $A$-algebra.



Does 'finite + finitely presented as an algebra' equal 'finitely presented as a module'?
EGA IV$\_1$, 1.4.7.


