Are projective modules related to projective spaces?

Question

I am aware of the possible motivations for calling "projective" a projective module (such as, for example, these). However, I have been asked by a student if there is some connection between projective modules and projective spaces, since they share a common name. After a first moment in which I have been tempted to answer negatively, I realized that I actually don't know if this is the case or not. Does anybody have ever thought about this?

Answer 1

As far as I know, the naming is coincidental.*

I think projective spaces as understood in projective geometry get the "projective" notion from the idea of light and images. That is, one can understand 2-d perspective as light rays being collapsed into points ("projected onto") a canvas from different angles.

But I think the term for modules arises from the mapping property, that if $A\to B$ is a surjection, and $P\to B$ is any homomorphism, then $P$ "projects onto" $A$ as in, "there exists a homomorphism $g:P\to A$". (It does not necessarily have to be an onto mapping, which is why I have it in scare quotes.)

Besides, if you thought there was an analogy between projective spaces and projective modules, then wouldn't there be an analogy between injective spaces and injective modules? Maybe there is: personally I'd never heard of an injective space until I looked it up just now. The term is a real thing, apparently.

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_{*So being only an evaluation based on my experience, there is still a chance there is some deep connection I'm not aware of. Or some connections that establish a connection, and yet were not known historically when the two things were named.}