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### Presentation of Witt vectors of polynomial rings

The main trick I know of for getting your hands on Witt vectors (of $\mathbb F_p$-algebras) is to reduce to the case of perfect $\mathbb F_p$-algebras, where we can appeal to the theorem that $W(-)$ ...
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### Questions about the Cayley-Hamilton theorem for modules

I think I'm a bit late! Either way: Yes, we require $\varphi(M) \subseteq IM$ to guarantee that $p_j \in I^j$ (this is particularly useful for commutative algebra when $I$ is prime). Note that $I = R$...
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### Geometric interpretation of Minimal Prime Ideals

First, note that when one is looking for minimal primes over an ideal $I$, one can work with the radical $\sqrt{I}$ instead: $\sqrt{I}=\bigcap_{P\supset I, \text{ P prime}} P$ (ref), so if $P$ is a ...
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I think I've found an answer to my question. We construct a sequence $(x_i\in M)$ by: $x_0=0$ and $\pi x_i=x_{-1}$. Now define a map $K\rightarrow M$ by $\pi^{-i}a\mapsto a.x_i$, then the kernel is ...