Commutative algebra and differential geometry

[dx]

In Miles Reid's book on commutative algebra, he says that, given a ring of functions on a space X, the space X can be recovered from the maximal or prime ideals of that ring. How does this work?​

 

[fresh_42]

This is the subject of algebraic geometry. The basic principle is that zeros of multivariate polynomials are considered as the topological spaces and the ideals generated by those polynomials are used as their algebraic correspondence. Thus we can investigate geometric objects by algebraic methods.

A better explanation can be found on Wikipedia:
https://en.wikipedia.org/wiki/Algebraic_geometry
https://en.wikipedia.org/wiki/Affine_variety
and the correct explanation on:
http://www.math.lsa.umich.edu/~idolga/631.pdf

 

[Infrared]

For $p\in X$, let $I(p)$ be the ideal of functions vanishing at $p.$ These are exactly the maximal ideals of your ring of functions on $X,$ so there is a bijection between points of $X$ and maximal ideals in the ring of functions on $X.$

 

[mathwonk]

in other words, given a maximal ideal, look at all points where all functions in that ideal vanish. if the base field is algebraically closed, there will be exactly one such point. hence a maximal ideal recovers a point of the variety. if the base field is not algebraically closed, there will be more maximal ideals than points. e.g. over the real numbers R, if m is a maximal ideal of R[X], then R[X]/M will be isomorophic either to R or to C, the complex numbers. The ones corresponding to single points of R are the ones where the quotient field is R, and those maximal ideals where the quotient field is isomorphic to C, correspond to pairs of conjugate complex points. I.e. maximal ideals of R[X] are generated by irreducible polynomials over R, and these are either linear (corresponding to a single real point), or quadratic (corresponding to a pair of conjugate complex points).