Redpill me on Arithmetic Schemes Veeky Forums

Redpill me on Arithmetic Schemes Veeky Forums

Come on Veeky Forums. You gotta know something.

What do you want to know? It's not clear from your first post

Why is SpecZ there? I don't know much about arithmetic schemes because I consider C-algebras and go from there, but I would imagine that an arithmetic scheme has a cover by affine opens the same way all schemes do, so why does the picture suggest you can only take the curve specZ? I think you're just looking at an example there OP. I'll tell you think, I bet they're complicated as fuck. They would always have the frobenius automorphism over F_p...

>Why is SpecZ there?
because typically by definition that's what an arithmetic scheme is, a scheme over Spec Z (arithmetic meaning looking at Z instead of some field k)

>e, but I would imagine that an arithmetic scheme has a cover by affine opens
They do. But the affine schemes are always SpecZ.

You'd think they'd be interesting.
But they're not.


Complex algebraic geometry best algebraic geometry.

But like SpecZ is by definition the prime ideals of Z. And the prime ideals of Z are the principal ideals generated by prime numbers. There must be something interesting going on here.

just learn stone duality. do topology

>doing algebraic geometry over fields of characteristic 0
It's like you don't want to say Frobenius four times per sentence.

i am convinced algebraic geometry is some alien shit and that grothendieck was from the future

i have been struggling with this stuff for fucking ever and i thought i had made progress but i don't even know what the shit an arithmetic scheme is

just fucking kill me

"The Geometry of Schemes" by Eisenbud&Harris is probably the most "intuitive" introduction to Schemes.

how old are the people who understand this
i feel like i'm probably years behind everyone

I'm 18.

How comfortable are you with the classical notions of alegraic geometry?

vague understanding at 23

Op here. 19, basic understanding.

About 50 pages into this book.

god damnit
i'm 23 and i'm still having trouble with some prerequisite ring theory stuff
i'm about ready to just give up

Studying the Geometry has actually helped me understand some Ring theory stuff better.
Specifically, localization of rings. Made a lot more sense why we would want to do that when studying the Zariski Tangent Space.

I think it helps to think of localizations as rational functions without poles. This viewpoint especially makes the name meaningful.

Yeah I realized that when studying the geometry. Locally ringed spaces and the stalks of their structure sheaves.

schemes are something you learn in a high-level graduate course. Think 600-700 level. They require a ton of prerequisites including category theory, topology, algebraic topology, basic algebraic geometry, commutative algebra, linear algebra, ....

>2016
>not learning schemes in preschool

Yes, I think that's one of the many things in commutative algebra you really need to get the geometric intuition for. That said, it's amazing how many purely algebraic things boil down to "prove this is true for a local ring and then you're done."

>be taking undergraduate algebra
>not too bad
>get through groups, rings and polynomials just fine
>get to fields, finite fields and Galois theory
>suddenly struggling

Am I just an idiot?

You need to spend more time on it. I distinctly remember working through a problem set in my course on those topics and suddenly all of the ideas behind Galois theory simultaneously clicked.

I tend to disagree. You don't need much category theory, topology or algebraic topology to get started with schemes. The very basic notions of category, functor and topological space should be enough.

kek'd

knowledge accretes faster as you learn more.

I don't really have time to learn AG bc its just a hobby

But after things barely making sense when I was ~23, at 25 after using category theory and sheaves a lot more I could pick up Hartshorne and read the first few chapters without too much struggle