# Problem Set 5

HW 5 (due Feb 17): AM Chapter 2, exercises 24, 25, 26

1. Let $M$ be an $A$-module and let $F_{\bullet}: \cdots \rightarrow F_{2} \rightarrow F_{1} \rightarrow F_{0}$ be a flat resolution of $M$, i.e. a chain complex with each $F_{i}$ flat, and such that $H_{i}\left(F_{\bullet}\right)=0$ for $i>0$ and $H_{0}\left(F_{\bullet}\right)=M$. Show that for any $A$-module $N, H_{i}\left(F_{\bullet} \otimes_{A} N\right)=\operatorname{Tor}_{i}^{A}(M, N)$.
2. Let $M$ be a finitely-generated flat module over a Noetherian local ring $A$. Show that $M$ is free.
3. Carefully check that Ext is well-defined, i.e. independent of the choice of injective resolution in the definition.
4. Compute $\operatorname{Ext}_{\mathbb{Z}}^{i}(\mathbb{Z} / p \mathbb{Z}, \mathbb{Z})$ for each prime $p$.