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Let $k$ be a perfect field of characteristic $p$. I read that $W_n(k)$ is injective as a $W_n(k)$-module. I did not find a direct reference for this, but I assume it has to do with this question.

My question is, is this still the case for the infinite Witt vectors $W(k)$? If not in general, are there conditions under which this might be true?

rollover
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  • @KReiser would you mind giving a reference? As far as I know, $W(k)$ is a DVR, no? See https://ncatlab.org/nlab/show/ring+of+Witt+vectors. In particular the $p$-adic integers are not a field. – rollover Nov 03 '18 at 12:49
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    @KReiser A DVR is an integral domain. According to Wikipedia, "no integral domain that is not a field is self-injective". – rollover Nov 03 '18 at 12:53
  • Ugh, I misremembered and was careless. Yeah, you're correct, you always get the ring of integers of some extension of $\Bbb Z_p$, so it's always a DVR and thus never injective over itself. Yeesh. – KReiser Nov 03 '18 at 18:08

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