Playlist - Rings and modules (Borcherds)

This is the introductory lecture, where we recall some basic definitions and examples, and describe the analogy between groups and rings.

We decribe some examples of rings constructed from groups and monoids, such as group rings and rings of Dirichlet polynomials.

We discuss a few assorted examples of rings. The Burnside ring of a group is a ring constructed form the permutation representations. The ring of differentail operators is a ring whose modules are related to differential equations.

We discuss unique factorization in rings, showing the implications (Integers) implies (Euclidean domain) implies (Principal ideal domain) implies (Unique factorization domain). We give a few examples to illustrate these implications.

We give some examples to illustrate unique factorization. We use the fact that the Gaussian integers have unique factorization to prove Fermat's theorem about primes that are sums of two squares. Then we discuss a few other quadratic fields with or without unique factorization.

We discuss prime and maximal ideals of a (commutative) ring, use them to construct the spectrum of a ring, and give a few examples.

We discuss the operation of inverting the elements of a subset S of a ring R, called localization. We describe the localization in detail for commutative rings, and briefly discuss the non-commutative case.

We mainly discuss the problem of whether free modules over a ring have a well defined rank, generalizing the dimension of a vector space. We show that they do over many rings, including all non-zero commutative rings, but give an example of a non-commutative ring where the rank of a free module is not defined.

We define projective modules, and give several examples of them, including the Moebius band, a non-principal ideal, and the tangent bundle of the sphere.

We define tensor products of abelian groups, and calculate them for many common examples using the fact that tensor products preserve colimits.

We define tensor products of modules over more general rings, and give some examples: coproducts of commutative rings, tensors in differential geometry, tensor products of group representations, and tensor products of fields.

We descibe some notions of duality for modules generalizing the dual of a vector space. We first discuss duality for free and projective modules, which is very siilar to the vector space case. Then we discuss duality for finite abelian groups, which is a sort of analog of Fourier analysis. As an application of duality we show the existence of many injective modules over a ring.

We discuss the problem of when a colimit of exact sequences is exact. We show that the colimit of exact sequences is at least right exact, but give an example to show that it is not always left exact. We find some conditions under which it is exact, such as direct sums, colimits over directed posets, and colimits over filtered categories.

We discuss when taking limits of modules preserves exactness. In particular we give the Mittag-Leffler condition that ensures that taking inverse limits of modules preserves exactness.

We review basic properties of polynomials over a field, and show that polynomials in any number of variables over a field or the integers have unique factorization.

We discuss the problem of factorising polynomials with integer coefficients, and in particular give some tests to see whether they are irreducible.

We define Noetherian rings, give several equivalent properties, and give some examples of rings that are or are not Noetherian.

This will be continued in the next lecture about Hilbert's finiteness theorems.

We prove Hilbert's theorem that poynomial rings over fields are Noetherian, and use this to prove Hilbert's theorem about finite generation of algebras of invariants, at least for finite groups over the complex numbers.

We show that symmetric polynomials are polynomials in the elementary symmetric functions. Then we prove Newton's identities relating sums of powers to the elementary symmetric functions, and briefly discuss their relations with Adams operations.

We describe the reultant of two polynomials and use it to calculate the discriminant of a polynomial.

We study rings of formal power series over a field in several variables. We first prove they are Noetherian, then use the Weierstrass preparation theorem to show they are unique factorization domains.

We continue the previous lecture on complete rings by discussing Hensel's lemma for finding roots of polynomials over p-adic rings or over power series rings. We sketch two proofs, by slowly improving a root one digit at a time, and the other using Newton's method.