Nonabelian $H^2$ and Galois descent

Question

I would like to know whether the following metatheorem on nonabelian $H^2$ has been ever stated and/or proved.

Let $k$ be a perfect field and $k^s$ its fixed separable closure. Let $X^s$ be a variety with additional structure over $k^s$ (I don't want to specify what I mean by additional structure). By a $k$-model of $X^s$ I mean a variety with additional structure $X$ over $k$ together with a $k^s$-isomorphism $$ X\times_k k^s\overset{\sim}{\to} X^s.$$

Metatheorem. Let $k$ be a perfect field and $k^s$ its fixed separable closure. Let $X^s$ be a variety with additional structure over $k^s$. Write $A^s=\mathrm{Aut}(X^s)$, and assume that $A^s$ "is" an algebraic group over $k^s$. Assume that for any $\sigma\in\mathrm{Gal}(k^s/k)$ there exists a $k^s$-isomorphism $$\lambda_\sigma\colon \sigma X^s\to X^s,$$ where $\sigma X^s$ is the variety obtained from $X^s$ by transport of structure. Assume also that $X^s$ admits a $k_1$-model over a finite Galois extension $k_1/k$ contained in $k^s$. Then these data define a $k$-kernel $$\kappa\colon\mathrm{Gal}(k^s/k)\to \mathrm{Out}(A^s)$$ and a cohomology class $\eta\in H^2(k,A^s,\kappa)$. If $\eta$ is not neutral, then $X^s$ has no $k$-model. If $\eta$ is neutral and the variety $X^s$ is quasi-projective, then $X^s$ admits a $k$-model $X$. Moreover, set $A=\mathrm{Aut}(X)$, then there is a canonical bijection between $H^1(k,A)$ and the set of isomorphism classes of $k$-models of $X^s$.

Example of application of the metatheorem: If $k=\mathbb{R}$, $k^s=\mathbb{C}$, $A^s$ is a finite abelian group of odd order, then $H^2(\mathbb{R},A)=1$ and $H^1(\mathbb{R},A)=1$ (because $\mathrm{Gal}(\mathbb{C}/\mathbb{R})$ is of order 2), hence $X^s$ has a unique model over $\mathbb{R}$.

I would be also glad to have references where this metatheorem was proved in special cases.

I know that it was proved in the case when $X^s$ is a principal homogeneous space of $G^s$ dominating $Y^s$, where $Y$ is a given homogeneous space (not necessarily principal) of an algebraic group $G$ defined over $k$, see Springer, Nonabelian $H^2$ in Galois cohomology. In: Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), pages 164--182. Amer. Math. Soc., Providence, R.I., 1966. Borovoi, Abelianization of the second nonabelian Galois cohomology, Duke Math. J. 72(1), 217--239, 1993. Flicker, Scheiderer, Sujatha, Grothendieck's theorem on nonabelian $H^2$ and local-global principles. J. Amer. Math. Soc. 11(3), 731--750, 1998.

Answer 1

Let me elaborate more on the remark above. Let $k$ be a perfect field. Let $\mathrm{Field}_k$ denote the category of finite extensions of $k$, i.e., the objects of $\mathrm{Field}_k$ are fields $k'$ equipped with an embedding $k \to k'$ such that $k'$ is finite dimensional over $k$. The morphisms are the maps of fields $k' \to k''$ which respect the embedding (here we do not think of all the $k'$ as subfield of a fixed seperable closure). Suppose we have a functor $F:\mathrm{Field}_k \to \mathrm{Grpd}$ to the category of small groupoids. For example, $F$ may be the functor which sends $k'$ to the groupoid whose objects are varieties with a certain structure defined over $k'$ and whose morphisms are structure preserving $k'$-isomorphisms between them. For $k'$ in $\mathrm{Fields}_k$ the inclusion $\iota:k \to k'$ can be considered as a morphism in $\mathrm{Fields}_k$, and we consequently have an associated functor $F(\iota):F(k) \to F(k')$ which we can call the base change functor. If $k'$ is also a Galois extension of $k$ with (finite) Galois group $G$, then the automorphism group of $k'$ in the category $\mathrm{Fields}_k$ is exactly $G$. In particular, $G$ now acts on the groupoid $F(k')$ (via functors). Given an object $X \in F(k')$ let us denote by $X^{\sigma}$ the image of $X$ under the action of $\sigma \in G$ on $F(k')$. Now whenever we have a group $G$ acting on a groupoid $Z$, we have an associated notion of a $G$-equivariant object of $Z$. This is an object $X \in Z$ equipped with a compatible collection of (iso)morphisms $f_{\sigma}: X \to X^{\sigma}$. We may also call this a twisted action of $G$ on $X$. Let us denote by $Z^{hG}$ the groupoid of $G$-equivariant objects in $Z$ (where the notation echoes the fact that we think of $G$-equivariant objects as homotopy fixed points). Now the mere fact that $F$ is a functor implies that if $X$ is an object of $F(k)$ then the object $F(\iota)(X) \in F(k')$ carries a natural twisted action of $G$. We hence obtain a functor $$ T_{k'/k}:F(k) \to F(k')^{hG} .$$ We can now say that $F$ satisfies Galois descent if $T_{k'/k}$ is an equivalence of groupoids for every finite Galois extension $k'/k$.

Now the $H^2$ and $H^1$ business is something that has only to do with computing groupoids of equivariant objects, and has nothing to do with, say, algebraic varieties. Let $Z$ be a groupoid equipped with an action of a group $G$. Let $\pi_0(Z)$ denote the set of isomorphism classes of $Z$, so that we have an induced action of $G$ on $\pi_0(Z)$. If $x \in \pi_0(Z)$ is an isomorphism class fixed by $G$, then we have an induced action of $G$ on the connected component $Z_x \subseteq Z$ corresponding to $x$. Let $X \in Z_x$ be any object and let $A = Aut(X)$ be its automorphism group. Since $Z_x$ is a connected groupoid, the group of connected components of self-homotopy equivalences of $Z_x$ is naturally isomorphic to $Out(A)$. We hence obtain a natural map $G \to Out(A)$, i.e., a pseudo-action of $G$ on $A$. Classical obstruction theory now associates to $X$ an obstruction element $o_X \in H^2(G,A)$, which is neutral if and only if $X$ admits a $G$-equivariant structure (i.e., a twisted action of $G$). The object $o_X$ is the one associated with a certain group extension $$ 1 \to A \to G_X \to G \to 1 $$ where $G_X$ is the group whose elements are pairs $(f,\sigma)$ where $\sigma$ is an element of $G$ and $f:X \to X^{\sigma}$ is a morphism (composition of elements is defined in a natural way). If the obstruction element $o_X$ is neutral then we can choose a section $G \to G_X$. Each such section determines a twisted action of $G$ on $X$. Furthermore, two such twisted actions result in isomorphic $G$-equivariant objects if and only if the two sections are conjugate by an element of $A$. This data is now classified by the cohomology group $H^1(G,A)$, and we obtain a bijection between $H^1(G,A)$ and the set of isomorphism classes of $G$-equivariant objects in the component $Z_x$. This is the way to compute groupoids of $G$-equivariant objects.

Edit:

If $Z,W$ are two groupoids then the functor category ${\rm Fun}(Z,W)$ is a groupoid as well. Two functors $f,g: Z \to W$ are homotopic if they are isomorphic in ${\rm Fun}(Z,W)$, and a functor $f: Z \to W$ is a homotopy equivalence if it has an inverse up to homotopy. For a groupoid $Z$ we have the full subgroupoid ${\rm Equiv}(Z,Z) \subseteq {\rm Fun}(Z,Z)$ spanned by the homotopy equivalences. Then $\pi_0{\rm Equiv}(Z,Z)$ (i.e., the set of isomorphism classes of the groupoid ${\rm Equiv}(Z,Z)$) is naturally a group by composition. This is the "group of connected components of self homotopy equivalences" alluded to in the answer.