While I was reading J.P. Serre's book "Local Fields" I found something strange in Chapter V. When Serre discusses properties of norm for unramified extensions, he says it is possible to identify the quotient $\mathfrak{p}_L^{n}/\mathfrak{p}_L^{n+1}$ with the tensor product $\overline{L}\otimes_K\mathfrak{p}_K^n/\mathfrak{p}_K^{n+1}$. Clearly the setting is the typical one, K is a field complete wrt a discrete valuation, $\mathfrak{p}_K$ is its unique prime and $L$ is a finite unramified extension of K. $\overline{L}$ is its residue field. Actually, I cannot see how the tensor product is defined as there are no maps from K to $\overline{L}$ at least when the characteristic of K is different from the one of $\overline{L}$.
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4$\begingroup$ $O_L \otimes_{O_K} P_K^n/P_K^{n+1} \cong O_L P_K^n/O_L P_K^{n+1}=P_L^n/P_L^{n+1}$. $\endgroup$– reunsCommented Dec 28, 2020 at 18:49

$\begingroup$ Thank you, this is absolutely true, but then what Serre writes Is wrong, am I right? $\endgroup$– rimeCommented Dec 28, 2020 at 20:27

2$\begingroup$ $O_L \otimes_{O_K} P_K^n/P_K^{n+1}\cong O_L/P_K O_L \otimes_{O_K/P_K} P_K^n/P_K^{n+1}=\overline{L}\otimes_{\overline{K}} P_K^n/P_K^{n+1}$ (nobody uses this notation for the residue field) $\endgroup$– reunsCommented Dec 28, 2020 at 20:33

$\begingroup$ So, just to have it written: yes, what Serre writes is wrong, or, more likely, it is a misprint. For what it's worth, Tits also uses $\overline K$ for the residue field (at least in the Corvallis article). $\endgroup$– LSpiceCommented Dec 28, 2020 at 21:30

1$\begingroup$ The original french version is correct. $\endgroup$– A.GCommented Jan 12, 2021 at 11:56
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