Reading: Class notes. Eisenbud: Sections 10.3, 11.1 (read on your own), 11.2, 11.3 (through Theorem 11.6).
Exercises: Eisenbud: 10.3, 10.8, 11.1, 11.3. Exercise 11.3 should be reworded, though: Exercise 11.3: Show that a valuation ring is Noetherian iff it is a DVR or a field. By the way, there is another error in the textbook in the very definition of a valuation ring. Here is a correct definition: A valuation ring is a domain R with a valuation ν: K(R)* → G such that R = { x ∈ K(R)* | ν(x) ≥ 0} ∪ {0}.
Exercise D: Find an example of a local ring (R, 𝔪) such that 𝔪 has four generators, but dim R = 2. Find an explicit system of parameters {f1, f2} for R and prove that this is a system of parameters. Is it also a regular sequence?
Exercise E: Consider the ring k[x, y, z]/(xz, yz), localized at the maximal ideal (x,y,z).
Last modified: (2020-03-22 14:10:20 CDT)