## Thursday, February 25, 2010

### Homework #3

1. Let R be a Dedekind domain with fraction field K. Let L be a finite field extension of K, and let S be the integral closure of R in L. Prove that S is a Dedekind domain.

2. Let Q be a prime ideal of R, and let Q^(n) be the n^th symbolic power of Q, i.e., the Q-primary component of the ideal Q^n (we saw in class that powers of prime ideals are not necessarily primary). Since Q is minimal over Q^n, we know that the Q-primary component of Q^n is unique.

a) Prove that Q^(n) := {r \in R | sf \in Q^n for some s \in R, s \not\in Q}
b) Prove that if \phi : R --> R_Q is the localization map, then Q^(n) = \phi^(-1)(Q_Q^n)
c) Prove that (Q^(n))_Q = (Q_Q)^n.
d) Prove that elements outside Q are nonzerodivisors mod Q^(n).

3. Complete the proof of Krull's Principal ideal theorem.

4. Prove that any ideal in a Dedekind domain can be generated by at most two elements.

5. Prove that a finitely generated torsion-free module over a Dedekind domain is projective, and show it is isomorphic to a direct sum of ideals. Note that this direct sum decomposition is not unique (but there is a uniqueness here - it is formulated in terms of the ideal class group of R).

Hint for #5: Suppose R is a domain. For a f.g. torsion free module M of R, define the rank of M to be the dimension over Q(R) of M \otimes_R Q(R). Show that if M is a torsion free module over a Noetherian domain of rank n, then R^n embeds in M.
Next show that rank 1 torsion free modules are isomorphic to ideals over a Dedekind domain, and then argue by induction on the rank of M to get the general case.