Tags: #todo #MOC #todo/learning

Commutative Algebra

References

https://www.math.ucla.edu/~merkurev/210c.1.20s/notes.pdf

Topics

#todo/questions

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,chain conditions, primary decomposition, length, Hilbert functions, dimension theory, completion, Dedekind domain.

• Operations
  • localization
    • See some actual computations of computing “at a prime” or “away from a prime”
    • fraction field
  • completion
• Types of rings
  • Complete ring
  • Noetherian ring,\
  • local ring
  • reduced ring,
  • normal ring,
  • Definition,
  • excellent ring
  • Cohen-Macauly
• Big Theorems / Results
  • Noether Normalization
  • Nakayama
• algebra over a ring
• regular sequence
• nilradical