Mathematics 506        Spring 2004

Homework

• Due Wednesday, April 7:
  • Section 15.1: 4, 5, 6, 7, 8 (recall theorem 1 in section 12.1), 19 (assume that f is nonconstant), 25
• Due Wednesday, April 14:
  • Section 15.2: 4, 8, 19, 23 (you may use the results from problem 2 in any of these).
  • The following is sometimes called the Nullstellensatz: Let k be a field. If F is a field extension of k and F is finitely generated as a k-algebra, then F is a finite extension of k.
    Show that this is equivalent to the weak Nullstellensatz.
• Due Wednesday, April 21:
  • Section 15.2: 45. (There are a few typos: the ideal A should be defined to be the radical of (F). In part (b), instead of letting I_n be the ideal of all functions vanishing for x < n, it should perhaps be those which vanish when x < 1/n. Finally, part (f) should say: prove that there exists a prime ideal Q which contains (F) and is properly contained in P.)
  • Section 15.3: 15, 16 (you may use problem 18 from Section 15.2 without proof).
  • A problem from the 1997 prelim: find all of the maximal ideals of R[x,y] (where R is the real numbers). For any maximal ideal m in R[x,y], find polynomials f and g so that m=(f,g). (There are several ways to do this: you can imitate the proof from class of the weak form of the Nullstellensatz, or you can use the equivalent formulation of the Nullstellensatz that I gave on last week's homework.)
• Due Wednesday, April 28:
  • Section 15.4: 10, 19 (you may use any earlier problems, like 18, or 26 in section 15.1), 21, 22, 23, 24
• Due Wednesday, May 5:
  • Section 17.1: 4, 9, 11 (you may use problem 10 if you want; note that the "canonical projection maps" should have codomain I/I_i, not I_i), 19, 21.
• Due Wednesday, May 12:
  • Section 17.1: 26
  • Find a ring R and a finitely generated R-module that is not finitely presented.
  • Let k be a field, m > 1 an integer, and R = k[x]/(x^m). Since k is isomorphic to R/(x), k has an R-module structure (on which x acts trivially). Compute Ext_R^s (k, k) for all non-negative s.
  • With the same setup as in the previous problem, fix an integer i with 0 < i < m+1 and compute Ext_R^s (k, R/(xⁱ)) for all non-negative s.
  • (Similarly, it is not hard to compute Ext_R^s (R/(xⁱ), R/(x^j)) for all i and j. One can use the classification of finitely generated modules over a PID to show that in this situation, every finitely generated R-module is a direct sum of modules of the form R/(xⁱ), and thus you could use problem 10 in Section 17.1 to compute Ext_R^s (M, N) for all finitely generated M and N and all s.)
• Due Wednesday, May 19:
  • Section 17.2: 8, 18, 19
  • Let k be a field, let p be a prime, and let G be a cyclic group of order p. Show that the group algebra kG is isomorphic to k[x] / (x^p - 1).
  • With the same notation, show that if the characteristic of k is p, then kG is isomorphic to k[y] / (y^p), and use last week's homework to compute H^*(G, k), where k is the trivial module.
  • With the same notation, show that if the characteristic of k is not p, then the trivial module k is projective as a kG-module, and use this to compute H^*(G, k).
• Due Wednesday, May 26:
  • Section 18.1: 1, 2, 3, 15, 16, 17, 20 (13 and 14 are also important)
• No homework due Wednesday, June 2.

       John Palmieri    Padelford C-538    (206) 543-1785    email    web page