# Problem Set 4

:::{.problem title="?"}
Let $f: M \rightarrow N$ be a map of modules over a local ring $A$, with $N$ finitely generated. Show that if the induced map $M / \mathfrak{m} M \rightarrow N / \mathfrak{m} N$ is surjective, the same is true for $f$. Does a similar statement hold for injectivity?
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:::{.problem title="?"}
Let $M$ be a finitely-generated module over a ring $A$, and let $p$ be a prime ideal of $A$. Suppose $M_{p}=0$. Show that there exists a finite set $x_{1}, \cdots, x_{n}$ of elements of $A \backslash \mathfrak{p}$ such that the localization of $M$ at the multiplicative set generated by the $x_{i}$ is zero.

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:::{.problem title="?"}
Let $M$ be a finitely-generated module over a Noetherian ring $A$ (that means any submodule of a finitely-generated module is finitely generated), and let $p$ be a prime ideal of $A$. Suppose $M_{p}$ is free. Show that there exists a finite set $x_{1}, \cdots, x_{n}$ of elements of $A \backslash p$ such that the localization of $M$ at the multiplicative set generated by the $x_{i}$ is free.
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:::{.problem title="?"}
Give an example of a module $M$ over a ring $A$ such that $M_{p}$ is free for each prime ideal p of $A$, but $M$ itself is not free. Such modules are called locally free.
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:::{.problem title="?"}
Give an example of a flat module which is not projective.
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:::{.problem title="?"}
Write a careful proof of the Cayley-Hamilton theorem over an arbitrary field.
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