# On Serre's "Local fields"

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 $$\overline{L}$$.

• $O_L \otimes_{O_K} P_K^n/P_K^{n+1} \cong O_L P_K^n/O_L P_K^{n+1}=P_L^n/P_L^{n+1}$. Commented Dec 28, 2020 at 18:49
• Thank you, this is absolutely true, but then what Serre writes Is wrong, am I right?
– rime
Commented Dec 28, 2020 at 20:27
• $O_L \otimes_{O_K} P_K^n/P_K^{n+1}\cong O_L/P_K O_L \otimes_{O_K/P_K} P_K^n/P_K^{n+1}=\overline{L}\otimes_{\overline{K}} P_K^n/P_K^{n+1}$ (nobody uses this notation for the residue field) Commented Dec 28, 2020 at 20:33
• 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). Commented Dec 28, 2020 at 21:30
• The original french version is correct.
– A.G
Commented Jan 12, 2021 at 11:56