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 $\overline{L}$.

  • 4
    $\begingroup$ $O_L \otimes_{O_K} P_K^n/P_K^{n+1} \cong O_L P_K^n/O_L P_K^{n+1}=P_L^n/P_L^{n+1}$. $\endgroup$
    – reuns
    Commented Dec 28, 2020 at 18:49
  • $\begingroup$ Thank you, this is absolutely true, but then what Serre writes Is wrong, am I right? $\endgroup$
    – rime
    Commented 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) $\endgroup$
    – reuns
    Commented 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). $\endgroup$
    – LSpice
    Commented Dec 28, 2020 at 21:30
  • 1
    $\begingroup$ The original french version is correct. $\endgroup$
    – A.G
    Commented Jan 12, 2021 at 11:56


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.