# Commutative algebra and differential geometry

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In summary, algebraic geometry is a branch of mathematics that uses algebraic methods to study geometric objects. Miles Reid's book on commutative algebra explains how, given a ring of functions on a space X, the space X can be recovered from the maximal or prime ideals of that ring. This is done by considering the zeros of multivariate polynomials as topological spaces and using the corresponding ideals as their algebraic correspondence. This method can be further explored by looking at Wikipedia and the provided resources. However, when the base field is not algebraically closed, there may be more maximal ideals than points, and they correspond to either single points or pairs of conjugate complex points.
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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?​

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

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.##

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).

Infrared

## 1. What is the difference between commutative algebra and differential geometry?

Commutative algebra is a branch of mathematics that studies commutative rings, which are algebraic structures that satisfy the commutative property of multiplication. Differential geometry, on the other hand, is a branch of mathematics that studies smooth manifolds and the geometric properties of curves and surfaces. While commutative algebra focuses on algebraic structures, differential geometry focuses on geometric objects.

## 2. How are commutative algebra and differential geometry related?

Commutative algebra and differential geometry are closely related because many concepts in differential geometry, such as tangent spaces and differential forms, can be described using commutative algebra. In fact, commutative algebra provides the algebraic foundations for many of the mathematical tools used in differential geometry.

## 3. What are some applications of commutative algebra and differential geometry?

Commutative algebra and differential geometry have many practical applications, particularly in physics and engineering. For example, commutative algebra is used in cryptography to study codes and ciphers, while differential geometry is used in computer graphics to model and render 3D objects.

## 4. What are some important theorems in commutative algebra and differential geometry?

In commutative algebra, one of the most important theorems is the Nullstellensatz, which states that there is a one-to-one correspondence between the prime ideals of a polynomial ring and the algebraic varieties defined by those polynomials. In differential geometry, the Gauss-Bonnet theorem is a fundamental result that relates the curvature of a smooth surface to its topology.

There are many resources available for learning about commutative algebra and differential geometry, including textbooks, online courses, and academic journals. It is recommended to start with a basic understanding of algebra and calculus before delving into these subjects. Additionally, seeking guidance from a mathematician or taking a course at a university can provide a deeper understanding and help with any questions that may arise.

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