Exercise 6, § 16

Let L be a finite Galois extension of the commutative field K, with Galois group G. Let V and $V'$ be K-vector spaces and $w$ (resp. $w'$) a tensor of type $(p, q)$ on V (resp. $V'$). An L-isomorphism from $(V, w)$ to $(V', w')$ is an L-linear isomorphism $u: V_{(L)}\rightarrow V'_{(L)}$ such that $u^{\otimes p}\otimes (^tu^{-1})^{\otimes q}$ sends the image of $w$ to that of $w'$. We denote by Aut$_L(V, w)$ the group of L-automorphisms of $(V, w)$, on which G acts by $^{\sigma}\varphi = (1_V\otimes \sigma )\circ \varphi \circ (1_V\otimes \sigma )^{-1}$.

a) Let $u$ be an L-isomorphism from $(V, w)$ to $(V', w')$ and $\sigma$ an element of G; the mapping $^{\sigma}u= (1_{V'}\otimes \sigma )\circ u\circ (1_V\otimes \sigma )^{-1}$ is an L-isomorphism from $(V, w)$ to $(V', w')$, so that $c(\sigma ) =u^{-1}\circ^{\sigma}u$ is an element of Aut$_L(V, w)$. Prove that the mapping $c: G\rightarrow$ Aut$_L(V, w)$ belongs to $Z^1(G$, Aut$_L(V, w))$ and that its class $\theta (V', w')$ in $H^1(G$, Aut$_L(V, w))$ does not depend on the choice of $u$.

b) Prove that the mapping $(V', t')\mapsto \theta (V', t')$ defines a bijection from the set of K-isomorphism classes of pairs $(V', w')$ L-isomorphic to $(V, w)$ to the set $H^1(G$, Aut$_L(V, w))$. (Given an element $c$ of $Z^1(G$, Aut$_L(V, w)$), deduce from Exercise 5, d) an L-automorphism $f$ of $V_{(L)}$ such that $c(\sigma ) =f^{-1}\circ^{\sigma}f$. Prove that the image $w'$ of $w$ by the automorphism deduced from $f$ is rational over K, and consider the pair $(V, w').)$

$*$c) Examples: The set $H^1(G,\mathbf{S}\mathbf{p}_{2n}(L))$ is reduced to one element. If Q is a quadratic form on K, then the set $H^1(G,\mathbf{O}(Q_{(L)}))$ can be identified with the set of equivalence classes of quadratic forms on K equivalent to Q over $L.*$

d) Let A be a K-algebra and M the automorphism group of the L-algebra $A_{(L)}$, endowed with the action of G defined above. The construction in b) defines a bijection from the set of isomorphism classes of pairs $(B, u)$, where B is a K-algebra and $u$ is an isomorphism from $B_{(L)}$ to $A_{(L)}$, to the set $Z^1(G,M)$. The inverse bijection sends an element $c$ of $Z^1(G,M)$ to the K-subalgebra of $A_{(L)}$ consisting of the elements $a$ satisfying $c(\sigma )(\sigma (a)) =a$ for every $\sigma \in G$.