My coauthors and I are writing a paper based on MO questions and answers: Friedrich Knop's answer, my answer 1 and my answer 2. For a linear algebraic group $G$ over a perfect field $k$, I consider a cohomology class $\xi\in H^1(k,G)$. For a finite extension $K/k$, I consider the image of $\xi$ in $H^1(K,G)$, the restriction of $\xi$ to the smaller Galois group.

Question. Is there a standard notation for the image in $H^1(K,G)$ of $\xi\in H^1(k,G)$ ? Is it ${\rm Res}_k^K(\xi)$ or ${\rm Res}_K^k(\xi)$ ?

If there is no standard notation, I would be grateful for suggestions of reasonable notation.

  • $\begingroup$ How about $\xi_K$? More compact, and similar to the common notation $X_K$ for scalar extension of a $k$-scheme $X$ through a ring map $k \to K$. $\endgroup$ – nfdc23 Nov 15 '17 at 1:06
  • $\begingroup$ @nfdc23: I thought about that, but I need the formula ${\rm Res}^L_K({\rm Res}^K_k(\xi))={\rm Res}^L_k(\xi)$, which is difficult to write with your compact notation. $\endgroup$ – Mikhail Borovoi Nov 15 '17 at 6:02
  • $\begingroup$ You could write it as "$(\xi_K)_L = \xi_L$" much as we do with restriction of a function $f:X \to Y$ to a subset $U \subset X$ and then to a subset $V \subset U$, namely "$(f|_U)|_V = f|_V$". But to each their own. $\endgroup$ – nfdc23 Nov 15 '17 at 8:35
  • $\begingroup$ @nfdc23: Thank you, maybe I will do that. $\endgroup$ – Mikhail Borovoi Nov 15 '17 at 8:38

I've seen both $\text{Res}_k^K$ and $\text{Res}_{K/k}$, and similarly for the inflation map if $K/k$ is Galois. More generally, if $\Gamma$ is a group acting on a group $G$ and if $\Lambda\subseteq\Gamma$ is a subgroup, the restriction map $H^1(\Gamma,G)\to H^1(\Lambda,G)$ is commonly written as $\text{Res}^\Gamma_\Lambda$.


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