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GhaS Shee (Sep 02 2020 at 10:53):This might be a basic and an easy question of Algebraic Geometry.
As some projective space can be seen as half of a sphere,
I wonder why sheaves without any torsion element have been named flat modules.
I might expect Pure Ring Theory might have developed prior to Algebraic Geometry.
However, this really confused me a lot of times in learning Algebraic Geometry.
I would like to know the histroy or the truth.
Thank you.
GhaS Shee (Sep 02 2020 at 11:04):https://math.stackexchange.com/questions/778277/what-is-the-intuition-behind-the-name-flat-modules
Oh, I have been found some answer here.
And is there some thoughts of renaming it ?
GhaS Shee (Sep 02 2020 at 15:21):GhaS Shee said:
As some projective space can be seen as half of a sphere,
I wonder why sheaves without any torsion element have been named flat modules.
This part was wrong. I mistook the projective space as the sheaves on it.
So, the term "flat" makes sense at all.
Now, flat sounds like null homotopic.
John Baez (Sep 03 2020 at 23:02):I don't think anyone plans to rename flat modules; it's a very pleasant term and it's been generalized in various ways: for example, category theorists now study flat functors.
John Baez (Sep 03 2020 at 23:03):Tensoring with a flat module is an example of a flat functor.
GhaS Shee (Sep 04 2020 at 05:28):@John Baez
Thank you for clarifying!
I could agree the description in that link thanks to the viewpoint of Cisinski's Higher Categories!
I have just had a similar thinking since yesterday. For me, now the presheaf sSet has enough embedded projective objects from simplex.
I did not notice this yoneda world of such presheaves was, classically, the dual space of the projection from graded rings.