***Notations and terminology***: Let $k$ be a field and $X$ be a $k$-scheme. Denote by $X\_L$ the scheme $X\times\_k\rm Spec(L)$. For a field extension $L/k$, *a $L/k$ form* is a $k$-scheme $Y$ such that there is a $L$ isomorphism $X\_L\cong Y\_L$. For a galois extension $L/k$ with the galois group $\rm Gal(L/k)$, we denote isomorphism classes of $L/k$ forms by $A(L/k)$. For $\sigma\in\rm Gal(L/k)$ we obtain an automorphism $\sigma f$ of $X\_L$ which is given by $\sigma f=(1\otimes\sigma)f(1\otimes\sigma)^{-1}$


Now there is injection $\phi: A(L/k)\to H^1(L/k,\rm Aut\_L(X))$; here $\phi$ is given as follows: Let $Y\in A(L/k)$ and fix $f:X\_L\to Y\_L$. For every $\sigma\in G$ we define 1-cocycle $a\_\sigma=f^{-1}(\sigma f)$. Here $H^1(L/k, \rm Aut\_L(X))$ is first non-abelian cohomology group.


It is easy to see that the map $\phi$ injective.


**Question:** 1. If we assume our $k$-scheme is affine of finite type then how does one show that the map $\phi$ is surjective? 2. What other schemes there are for which the map $\phi$ is surjective?


Note that for an affine $k$ scheme $X$, any $L/k$ form $Y$ must be affine $k$ scheme. So for Question 1 we require to find affine $k$ schems. I am not able to think a way to construct an affine $k$ scheme $Y\in A(L/k)$ which will be corresponds to given $a\_\sigma\in H^1(L/k, \rm Aut\_L(X))$.


Thanks!



$L/k$ forms for affine schemes of finite type
See [Serre, Galois cohomology], Chapter III, §1 Forms, and the references there ([Serre, Local Fields], X.§2, and [Giraud, Cohomologie Non Abélienne]).


