u/Desvl

>Vector bundles are to the geometer what representations or modules are to the algebraist. In fact the modern algebraic geometer hardly distinguishes between the two.
-- Sir Michael Atiyah, ICM 1962

>... vector bundles are obtained from projective modules just as smooth manifolds are obtained from smooth algebras.
-- Jet Nestruev

>Think geometrically, prove algebraically. —John Tate.

Projective module is not a rather advanced concept so it's possible to find it in many (introductory) books on algebra. However the introduction of projective modules always goes like this: there are three or four or five equivalent definitions, and as an exercise prove that these conditions are equivalent, and then we will see some applications if any.

Proving the equivalence of these conditions is indeed a reasonable exercise in commutative algebra but such an exercise gives the student very little motivation and intuition. Nevertheless, I think the text authors, who are experts in the field, should all be aware of the analogy and the strong connexion between projective modules and vector bundles, which can be derived from Serre-Swan theorem.

Speaking of vector bundles, the example of Möbius band would make a difference already, even without rigorous definition. There are two line bundles of the unit circle S^1, the cylinder and the Möbius band. The cylinder is a trivial bundle but the Möbius band is not - it is instead locally trivial (if you cut down a piece it's just a rectangle, which is a line bundle of a whole segment). The Möbius band itself is a direct summand of a trivial vector bundle, i.e. the open donut (which is homeomorphic to SL(2,R) by the way).

We can replace "trivial" by "free" and see what is going on in the world of projective module. A projective module is not necessarily free but always locally free. A projective module is always a direct summand of a free module (corresponding to the sense trivial bundle).

I believe experts can come up with better exposition on the analogy than that what I have written in a few minutes, but the introductory texts that mention the connexion between projective modules and vector bundles are rare (I search on Google "projective module" for lecture notes and I don't see any geometrical interpretation), if we do not take K-theory books into account of course. I think this is a shame. Like, what prevented the authors from mentioning that? Are we obliged to include the whole definition of vector bundles, or are we afraid of adding more confusion by an informal discussion? In my opinion a 10 lines long remark on the connection can already make the first exposition to projective modules much better. And it's not only about projective modules of course. In general, we should not hesitate to deliver arithmetical/geometrical motivation in the study of algebra.

Let me know about your thoughts!

To be clear, this is not a ranting post. I have never published a book, but recently I have been wondering why are authors writing maths books that is extremely long, say, 600-1600 pages, and the inclusion of exercices makes the question more complicated.

Indeed, if a maths book does not have any exercise, then we can somewhat suppose that it serves as a reference book and the book won't play a big role as a textbook. For example nobody complains the lack of exercise on EGA.

But in my opinion, if a maths book includes exercises, it automatically qualifies as a textbook. So I wonder, when the book is extremely long, what are the author expecting? Finishing a book of < 500 pages within one or two semesters can be feasible, but for a book, rather advanced and extraordinarily long, like Hatcher's Algebraic Topology, Bump's Automorphic Forms, Evan's PDE, Görtz-Wedhorn's Algebraic Geometry (1600 pages!) and many other books that I can't name, the reading of the book can already be extremely time-consuming. In this case, what were the authors expecting when they are writing the book? I have a few guesses:

I have a folder of lecture notes and exercise sheets in my hand, so it's a good idea to compile them into a book, if it happens to be a long book, so be it.
I had no idea about the length of the book before submitting the draft to the editor.
The priority is the completeness of the book and ideally it will work as a repository of lecture resource.
I kinda imagine that the reader finish all of them. Other than that I don't care. If someone cites my book for an important result appeared in an exercise of my book, that's cool as well.

So if you are a textbook author, would you like to rectify my guesses or share your opinion?

We should have talked about the (geometrical) motivation/analogy of projective modules in introductory texts.