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.}

Answer 2

Maybe there are more obvious connections (But the appearance of projective modules was contemporaneaous with that of this type of geometry, ~50's)

This connection resides in the universal property of the $n$ dimensional projective space $\mathbb{P}^{n}$. Indeed, provided a scheme $X$, $X$-points of $\mathbb{P}^{n}$ (that is morphisms $X \to \mathbb{P}^{n}$) are classified by pairs of the form $(\mathcal{L},\psi)$ where $\mathcal{L}$ is an invertible sheaf over $X$ and $\psi : \mathcal{O}_{X}^{\oplus n+1} \to \mathcal{L}$ an epimorphism of sheaves (as in https://stacks.math.columbia.edu/tag/01NE).

An invertible sheaf over $X$ is a locally free $\mathcal{O}_{X}$-module of rank one (these are exactly the quasi-coherent sheaves that are invertible for $\otimes_{\mathcal{O}_{X}}$, hence the name). It represents a line bundle over $X$. Before explaining the correspondence between points and such pairs, let's understand why invertible sheaves do represent line bundles.

In algebraic geometry, locally free modules play the role of vector bundles. Provided such a module $\mathcal{F}$ (let's say it is finite), one may attach to it the total space $\mathbb{V}(\mathcal{F}) = \text{Spec}_{X}(\text{Sym}(\mathcal{F}^{\vee}))$ of the vector bundle it defines. Locally on zariski open $U \subset X$, $\mathcal{F}$ is free of finite rank $n$, from which follows that $\text{Sym}(\mathcal{F}_{|U}^{\vee}) \simeq \text{Sym}(\mathcal{O}_{U}^{\oplus n})\simeq \mathcal{O}_{U}[t_1,\dots,t_{n}]$ and $\mathbb{V}(U) \simeq \mathbb{A}_{X}^{n}$ is the $n$ dimensional affine space over $X$. Whence the condition of $\mathcal{F}$ being locally free is that of the vector bundle being locally trivial. (Conversely, provided an affine space $V$ over $X$ that is locally trivial $\simeq \mathbb{A}^{n}$, the sheaf of sections of $V \to X$ is a locally free $\mathcal{O}_{X}$-module.) In the affine and finite case, locally free modules are exactly the finite projective ones (see https://stacks.math.columbia.edu/tag/00NX).

As mentionned above, line bundles are objects of interest for the study of $\mathbb{P}^{n}$. Working over $\mathbb{Z}$ and given a ring $A$, $A$-points of $\mathbb{P}^{n}$ are thus related to projective rank one $A$-modules. There is a "tautological" line bundle on $\mathbb{P}^{n}$ for which the fiber of a point is simply the line in $\mathbb{A}^{n+1}$ it represents. The associated invertible sheaf is the twisted sheaf $\mathcal{O}_{\mathbb{P}^{n}}(1)$. This sheaf is endowed with $n+1$ "tautological" sections which yield an epimorphism of sheaves $\pi : \mathcal{O}_{\mathbb{P}^{n}}^{\oplus n+1} \to \mathcal{O}_{\mathbb{P}^{n}}(1)$ (taking the usual affine cover $U_{1},\dots,U_{n}$ of $\mathbb{P}^{n}$ trivializes $\mathcal{O}_{\mathbb{P}^{n}}(1)$ and gives $n+1$-sections locally generating $\mathcal{O}_{\mathbb{P}^{n}}(1)$). Provided a morphism $f:X \to \mathbb{P}^{n}$, taking the pulled back line bundle $\mathcal{L}=f^{*}\mathcal{O}_{\mathbb{P}^{n}}(1)$ and epimorphism $\psi = f^{*}\pi : \mathcal{O}_{X}^{\oplus n+1} \to \mathcal{L}$ establish the above mentioned correspondence.