Co ordinate Rings

[@lachenmann]

Your explanation is really good professor, thank you very much for this material. I'm looking forward for your next videos. Are you going to organize them in playlists maybe? All the best from Chile

[@janakrani5698]

Can you suggest some introductory book for algebraic geometry?

[@jacopolanzoni5098]

Algebraic Geometry by Hartshorne is probably the best.

[@kapoioBCS]

@@jacopolanzoni5098 Introductory algebraic geometry and Hartshorne’s Book are two very different things 😂. I think you should start from a book in commutative algebra like Atiya’s since there are online solutions for all of its problems. And you can also try the Miles Reid’s books :)

[@abouland9404]

If you already have learned commutative algebra (which you should do before going into algebraic geometry), the golden standard nowadays, is Vakil's The Rising Sea. Hartshorne is better used as a reference textbook once you already learned the subject.

[@ejohn977]

Dr Chan, maybe this is a dumb question coming from a math PhD student, but why do we call it a "coordinate ring"? When I think of coordinates, I think of coordinate charts on a manifold. Are these two notions of "coordinates" related in some sense? My department here in the U.S. is small and we're a little deprived of algebraic geometry. Great introductory videos by the way!

[@dnlchn]

The coordinates like x,y in E.g.1, can be thought of as functions on the variety. The coordinate ring is thus just the k-algebra generated by these coordinate functions.

[@ejohn977]

That answer is way less dramatic than I was hoping for, but I see what you mean. Thank you very much for replying!

[@douglasholman6300]

How can one begin to understand pre-requisite concepts such as quotient rings and ideals? Those are gaps in my current understanding but I am eager to learn more

[@jacopolanzoni5098]

I don't think there is a way to understand algebraic geometry without first learning commutative algebra, I'm afraid.

[@kapoioBCS]

Quotient rings and ideals can be understood from a typical course in Abstract Algebra imo

[@jamesthorn9992]

A book of abstract algebra by Charles Pinter is a great place to start