Math 502 (Rezk, 10:00 MWF, 141 Altgeld Hall).

Homework

• Friday, September 2. Matsumura 1.2, 1.3, 1.5, 2.1, 3.1, 3.2, 3.6. Here are solutions.
• Monday, September 19.
  • Matsumura 4.4, 4,5, 4.6, 4.7, 4.10.
  • Matsumura 9.1, 9.7.
  • Let k be a prime field (Q or F_p), and let K be an algebraically closed field having infinite transcendence degree over k. Let A=K[X₁, ..., X_r]. The following exercise gives a quick proof of the Nullstellensatz for such rings (e.g., the case K=the complex numbers).
    1. Let P be a prime ideal of A. Show that there exist a finite set of elements c₁,...,c_n of K such that, if F is any subfield of K containing them, and Q is the contraction of P to F[X₁,...,X_r], then P=QA.
    2. Let be F is a subfield as in part 1 which has finite transcendence degree over k. Show that the fraction field of F[X₁,...,X_r]/Q can be embedded in K, extending the inclusion of $F$ in $K$. (This uses the hypotheses on the field K.)
    3. Show that if f is an element of A-P, there is an element a=(a₁,...,a_r) in K^r such that a is in Z(P) but f(a)≠0. (Use a suitable embedding as in part 2 to produce a candidate a.)
    4. Conclude that every prime ideal P of A is an intersection of ideals of the form (X₁-a₁,...,X_r-a_r), with a in Z(P).
• Wednesday, October 5.
  • Let A be a ring which as an abelian group is finitely generated. Let P be a prime ideal of A. Show: (a) if P is maximal, then A/P is a finite field, and (b) if P is not maximal, then it is a minimal prime.
  • Matsumura 6.3, 6.4, 6.5, 6.7.
  • Show that if A is a Noetherian local ring with maximal ideal M, the M-primary ideals I are exactly those such that I contains Mⁿ for some n. Show that if A is a Noetherian ring, the P-primary ideals are exactly those which are contractions of PA_P-primary ideals of the localization A_P.
• Wednesday, November 2.
  • Matsumura 7.1, 7.3, 7.9.
  • Matsumura 8.1, 8.2, 8.5.
• Friday, November 18.
  • Matsumura 10.7.
  • Matsumura 11.2, 11.3, 11.8, 11.9, 11.10.
  • Matsumura 13.1, 13.5.
• Friday, December 9.
  • Matsumura 14.1.
  • Matsumura 15.1.
  • Suppose a homomorphism from A to B is an integral extension of Noetherian rings, and M a finite B-module. Show that dim_B (M) = dim_A (M).

Last modified 30 November 2004 by Charles Rezk. Email: