I know that for a finitely presented $A$-module $M$ ($A$ a commutative ring), TFAE:


1. $M$ is *projective*;
2. $M$ is *max-locally free*, meaning that $M\_{\mathfrak m}$ is free for every maximal ideal $\mathfrak m$;
3. $M$ is *prime-locally free*, meaning that $M\_{\mathfrak{p}}$ is free for every prime ideal $\mathfrak p$;
4. $M$ is *Zariski-locally free*, meaning that there are some $f\_1,\ldots,f\_n$ generating the unit ideal in $A$ such that each $M\_{f\_i}$ is free.


(Reference: Eisenbud commutative algebra, p. 136 / end of chapter 4).


I know that (1) implies (2) without finite presentation: see [Kaplansky (1958): *Projective Modules*](https://www.jstor.org/stable/1970252), p. 374. (He doesn't even assume $A$ is commutative, and uses an awesome lemma that any projective module is a direct sum of countably-generated submodules.) Finite presentation is used to prove (3) implies (4), as is often the case when passing from stalks of a sheaf to actual open sets.


So now I'm wondering in particular if you need finite presentation to prove (4) implies (1), and more generally,



> 
> If $M$ is *Zariski-locally projective* (meaning there are some $f\_1,\ldots,f\_n$ generating the unit ideal in $A$ such that each $M\_{f\_i}$ is projective), is it *projective*?
> 
> 
> 


If so, how can I see this directly / commutative-algebraically?




---


**Follow up:** I checked out Bhargav's reference, [Raynaud and Gruson: *Critères de platitude et de projectivité*](https://doi.org/10.1007/BF01390094). It turns out (on p. 81) they actually use the same technique as Kaplansky in the paper I linked above, of writing a module as a transfinite union with countably generated successive quotients, which they call a "Kaplansky division" when these quotients are direct summands. The conclusion that projectiveness is Zariski-local is stated as Example 3.1.4(3) on the bottom of page 82.


Tricky stuff!



Is projectiveness a Zariski-local property of modules? (Answered: Yes!)
Being projective is indeed a local property for the Zariski topology. In fact, it is even local for the fpqc topology --- this is a famous theorem of Raynaud and Gruson (see MR0308104).


