Homework #2

Here is the second homework assignment. It is due February 25th.

1. Prove that for a finitely generated module M over a Noetherian ring R, that Min Supp(M) = Min Ass(M)

2. If R is Noetherian, and M and N are finitely generated R-modules, show that Ass(Hom(M,N)) = Supp M \cap Ass N

3. Let f : M --> N be a surjective map of R-modules, and L a proper submodule of N. Show that L is primary if and only if f^(-1)(L) is primary, and that in this case, they are primary for the same prime P of R.

4. Our goal will be to prove Krull's Intersection Theorem: Let R be Noetherian, I a proper ideal, and M a finitely generated R-module. Then there is an s \in I such that (1-s)\bigcap_{n=1}^\infty I^nM = (0).

a) Let N = \bigcap_{n=1}^\infty I^nM. Show that IN = N. To do this, consider a primary decomposition of IN = Q_1 \cap ... \cap Q_l, and show that N \subseteq Q_i for each i.

b) Deduce the existence of s \in I that satisfies the theorem.

c) Deduce that if R is a local Noetherian ring, and M a finitely generated R-module, then \bigcap_{n=1}^\infty I^nM = (0).

5. Let R be a ring. Prove that if R is a UFD, then R is normal.

6. Let R be an integral domain, and K its field of fractions. We say that x \in K is almost integral over R if there exists an element r \in R nonzero such that rx^n \in R for all n > 0. Show that if x is integral over R, then it is almost integral, and if R is Noetherian, the converse holds.

7. Let R be a subring of a ring S, with S integral over R. Show that dim R = dim S.