# Problem Set 8

HW 8 (due March 31)

1. Give an example (with proof) of a rank one locally free module over a Dedekind domain that is not free.
2. Give an example (with proof) of a Noetherian domain of Krull dimension one which is not a Dedekind domain.
3. Let $M$ be a finitely generated module over a Dedekind domain $A$. Show that $M$ has a projective resolution of length two. Conclude that Tor ${ }_{i}^{A}(M,-)$ and $\operatorname{Ext}_{A}^{i}(M,-)$ equal zero for $i>1$.
4. Let $M, N$ be a finitely generated modules over a Dedekind domain $A$. Show that $\operatorname{Tor}_{1}^{A}(M, N)$ is a torsion $A$-module. Can you identify its support?
5. Give an example of a Dedekind domain with uncountable Picard group.
6. Let $k$ be a field. Show that the Picard group of $k[t]$ is trivial, i.e. any rank one locally free sheaf over $k[t]$ is free.
7. Give an example of a domain with a maximal non-zero ideal $\mathfrak{m}$ such that $\mathfrak{m}^{2}=\mathfrak{m}$.