# Problem Set 7

HW 7 (due Thursday, March 17, after Spring Break):

1. Let $M$ be an $A$-module, and $f \in A$. Construct an isomorphism between $M_{f}$ and $\underset{\longrightarrow}{\rightarrow} M \stackrel{\cdot f}{\rightarrow} M \stackrel{\cdot f}{\rightarrow} M \stackrel{\cdot f}{\rightarrow} \cdots$.
2. Construct a module $M$ over a ring $A$ such that for each prime ideal $\mathfrak{p}$ of $A, M_{\mathfrak{p}}$ is finitely generated, but $M$ is not finitely-generated.
3. Let $M$ be a finitely-generated module over a Noetherian ring. Show that $M=0$ if and only if the support of $M$ is empty.
4. Suppose $\operatorname{Spec}(A)=V_{1} \sqcup V_{2}$, where $V_{1}, V_{2}$ are clopen disjoint subsets. Show that there exists a direct sum decomposition $A=A_{1} \oplus A_{2}$ such that the natural quotient maps $A \rightarrow A_{i}$ induce isomorphisms $\operatorname{Spec}\left(A_{i}\right) \rightarrow V_{i}$ for $i=1,2$.
5. Show that exactness of a long exact sequence is a local property.
6. Let $A$ be a Noetherian local domain with residue field $k$ and fraction field $K$, and $M$ a finitely-generated $A$-module. Show that the following are equivalent:
1. $M$ is free
2. $\operatorname{dim}_{k} M \otimes_{A} k=\operatorname{dim}_{K} M \otimes_{A} K$.
7. Let $A$ be a Noetherian ring and $M$ finitely-generated. Show that the following are equivalent:
1. $M$ is locally free
2. $M$ is projective
3. $M$ is flat.
8. Show that any Artinian ring is Noetherian
9. Show that if $A$ is a Noetherian ring such that $\operatorname{Spec}(A)$ is Hausdorff, then $A$ is Artinian.