Let $X$ be an algebraic variety (say, projective, irreducible and smooth), defined over a field $K$, and let $L$ be a Galois extension. I am interested in algebraic varieties $Y$, defined over $K$, such that there exists an isomorphism $\psi\colon X\_L\to Y\_L$ defined over $L$ (but not over $K$ in general).


The map $\psi$ induces a 1-co cycle from $Gal(L/K)$ to the group $Aut(X\_L) $ of $L$-automorphisms, and it is easy to check that the image in $H^1 (Gal, Aut(X\_L))$ only depends on the class of $Y$, up to $K$-isomorphisms. My question is the following: is every element of $H^1$ obtained? In other words, we have a map from the set of $L$-forms of $X$ to $H^1 (..)$, the map is injective but is it surjective?


If no, could it be true in some natural cases, if yes I would be happy to see a reference.



Forms of algebraic varieties
CW answer, copied from the comments of Timo Keller and abx:


See J.P. Serre's “Galois Cohomology”, Chapter III, 1.3, Proposition 5.


