010_Commutative Algebra

Commutative Algebra Background

Topics: Hilbert’s Basis Theorem and Nullstellensatz, ideals, spectra, localization, primary decomposition, Artin-Rees Lemma, flat families and Tor, completions of rings, Noether Normalization, systems of parameters, DVRs, dimension theory, Hilbert-Samuel polynomials, depth, Cohen-Macaulay and regular rings, homological methods.

Exactness, direct limits, tensor products, Cayley-Hamilton theorem, integral dependence, localization, Cohen-Seidenberg theory, Noether normalization, Nullstellensatz, chain conditions, primary decomposition, length, Hilbert functions, dimension theory, completion, Dedekind domains.

Definitions

  • What is \(k[x]\)? \(k(x)\)? \(k[[x]]\)? \(k((x))\)?
  • What does it mean to be finitely generated as a module? As a field extension?
  • What is the transcendence degree of a field extension?
  • What is the integral closure of an integral domain?
    • What does it mean for an integral domain to be integrally closed?
  • What is a homogeneous ideal?
  • What is a finite type module \(M\) over a commutative algebra \(A\)?
  • What is a primary ideal?
    • What is the primary decomposition of an ideal?
  • What is a Dedekind domain?
  • What is the Krull dimension of a ring?
  • What is the irrelevant ideal?
  • What is the Rees algebra of an ideal?
  • What is a reduced \(k{\hbox{-}}\)algebra?
  • What is the homogenization of a polynomial? The dehomogenization?
  • What is a DVR?
  • What is the Krull dimension?
  • What is the primary decomposition of an ideal?
  • What is the integral closure?
  • What is Serre’s criterion?
  • What is a Cohen-Macaulay ring?
  • What is a flat morphism of rings?
  • What is a regular ring?
  • What is the global dimension of a ring?
  • What is the depth of an ideal?
  • What is the codimension of an ideal?

Results

  • What is Nakayama’s lemma?
  • What is the Artin-Rees lemma?
  • What is the Artin-Rees lemma?
  • What is Nakayama’s lemma?
  • What is the Hilbert Basis Theorem?
  • What is Noether normalization?

Problems

  • When is a localization a subring of the fraction field?
  • Show that \(R\) Noetherian implies \(R[[x]]\) Noetherian.
  • Show that \(\sqrt{I}\) is an ideal if \(I\) is an ideal.
  • Show that an ideal \(I{~\trianglelefteq~}k[x_1, \cdots, x_{n}]\) is homogeneous iff it is graded, i.e. \(I = \bigoplus I_d\) where \(I_d \coloneqq I \cap k[x_1, \cdots, x_{n}]_d\), the homogeneous degree \(d\) part of the graded ring \(k[x_1, \cdots, x_{n}]\).
  • Let \(k[V]\) be the coordinate ring of a variety, and show that every maximal ideal \({\mathfrak{m}}\in \operatorname{mSpec}k[V]\) is of the form \({\mathfrak{m}}_p \coloneqq\left\{{f\in k[V] {~\mathrel{\Big\vert}~}f(p) = 0}\right\}\) for some point \(p\in V\).