Finite Galois descent for the Hopf algebras of multiplicative type

Every vector of $B$ lies in a finite-dimensional stable $L$-subspace: take the span of its finite $\mathrm{\Gamma }$-orbit. The same holds for any finite set. Apply F1 to these spaces to see $L\otimes B^{\mathrm{\Gamma }}\to B$ is surjective. Any tensor in its kernel involves finitely many invariant vectors, which lie in one finite-dimensional stable space; injectivity in F1 kills that tensor. Thus this map is an isomorphism even when $B$ is infinite-dimensional. The fixed algebra $A$ is closed under multiplication and contains $1$, so the isomorphism respects algebras. For canonical scalar extensions, each tensor involves a finite-dimensional $k$-space; F1 on that space gives $(L\otimes A_0{)}^{\mathrm{\Gamma }}=A_0$.

Tensoring the isomorphism gives $B{\otimes }_{L}B\cong L{\otimes }_k(A{\otimes }_{k}A)$ with canonical semilinear action. Step 1.1 gives $(B{\otimes }_{L}B{)}^{\mathrm{\Gamma }}=A\otimes A$. Equivariance of $\mathrm{\Delta },ϵ,S$ thus restricts them to $A$, with counit valued in $k$ by F3. Their identities hold because they hold after extension and tensoring with a field is faithful. These maps make $A$ a Hopf algebra. An equivariant map sends fixed elements to fixed elements and is determined by them after scalar extension, proving existence and uniqueness of descended maps. Conversely scalar extension of every Hopf map is equivariant.

Let $b_1,\dots ,b_t$ be algebra generators of $B$. Write each as a finite $L$-linear combination of elements of $A$ using step 1.1, and let $A^{\mathrm{\prime }}\subset A$ be the $k$-algebra generated by those finitely many elements. Then $L\otimes A^{\mathrm{\prime }}\to B=L\otimes A$ is onto; consequently $(A\mathrm{/}{A}^{\mathrm{\prime }})\otimes L=0$ as a vector-space quotient. Faithfulness gives $A^{\mathrm{\prime }}=A$. Thus $A$ is finitely generated, and F2 gives the descended finite-type affine group.