Reading: Class notes. Eisenbud: Sections 11.3, 11.4 (skip Theorem 11.8), 13.1 (through Corollary 13.4, then Proposition 13.10, Theorem 13.9, and Corollary 13.6), 8.2.1 (Theorem A1), 13.2, 13.3 (Corollary 13.13), 1.9 (review), 15.1, and 15.2.
Exercises: Eisenbud: 10. 4, 10.6, 11.7, 13.2 (use the hint), 13.9 (Feel free to prove an analogue of what we called the "weak Noether normalization" in class; the right statement is that k[[x1, ..., xn]] is a Noether normalization of a complete Noetherian local ring R containing a field, where k is the residue class field of R, if and only if x1, ..., xn is a system of parameters.)
Exercise F: Theorem 13.3 (well, it is basically Lemma 13.2c) guarantees that over an infinite field k, you can always find a "linear" Noether normalization. Produce a counterexample to this over a finite field as follows. Let k = ℤ/2 and let S = k[x,y]. Find a polynomial f ∈ k[x,y] such that the ring R := S/(f) does not admit a linear Noether normalization.
Last modified: (2020-03-24 02:26:51 CDT)