Let $K$ be a field with $2\in K^\times$, let $A$ be a separable $K$-algebra (i.e. $A$ is finite-dimensional, semisimple and its center is an etale $K$-algebra), and let $\sigma:A\to A$ be a $K$-involution.

Consider $\mathbf{U}(A,\sigma)$, the unitary algebraic $K$-group of $(A,\sigma)$; it is the affine algebraic $K$-group with sections given by $$\mathbf{U}(A,\sigma)(L)=U(A_L,\sigma_L):=\{a\in A_L\,:\,a^\sigma a=1\},$$ where $A_L=A\otimes_K L$, $\sigma_L=\sigma\otimes_K\mathrm{id}_L$. Denote by $\mathbf{U}^0(A,\sigma)$ the neutral connected component of $\mathbf{U}(A,\sigma)$.

Let $\overline{K}$ be an algebraic closure of $K$. Using the fact that $A_{\overline{K}}$ is a product of matrix algebras over $\overline{K}$, it is not difficult to see that:

  • $\mathbf{U}(A,\sigma)$ is a form a prodcut of copies of $\mathbf{GL}_m$, $\mathbf{Sp}_{2n}$, $\mathbf{O}_k$ ($m,n,k$ can vary).

  • $\mathbf{U}^0(A,\sigma)$ is a form a prodcut of copies of $\mathbf{GL}_m$, $\mathbf{Sp}_{2n}$, $\mathbf{SO}_k$ ($m,n,k$ can vary).

A "folklore" fact asserts that the converse also holds. That is, for all $m_1,\dots,m_r,n_1,\dots,n_s,k_1,\dots,k_t\in \mathbb{N}$, any $K$-form of $\mathbf{GL}_{m_1}\times\dots\times \mathbf{GL}_{m_r}\times \mathbf{Sp}_{2n_1}\times\dots\times\mathbf{Sp}_{2n_s}\times\mathbf{O}_{k_1}\times\dots\times \mathbf{O}_{k_t}$ is of the form $\mathbf{U}(A,\sigma)$, with $(A,\sigma)$ uniquely determined up to isomorphism, and similarly with $\mathbf{U}^0(A,\sigma)$ when one replaces $\mathbf{O}$ with $\mathbf{SO}$.

My first question is whether there is an explicit reference for this statement in the literature? Notice that it should be possible to deduce this statement from section 26 in the Book of Involutions, say, by passing to the simply-connected covering. This will presumably require some work to bridge the difference between $\mathbf{SL}_m$ and $\mathbf{GL}_m$, and also to eliminate the tritalitarian forms of $\mathbf{Spin}_8$. I am asking for a reference which will require less adjustments.

My second question is whether the scheme version of the "fact" above is known in the literature? In more detail, we can replace $K$ with a scheme $S$ (with $2$ invertible on $S$) and assume that $A$ is a locally-free separable $\mathcal{O}_S$-algebra. (This is same as saying that there are $t\geq 0$ and $n_1,\dots,n_t\in\Gamma(S,\mathbb{N})$ such that $A$ and $\prod_{i=1}^t\mathrm{Mat}_{n_i\times n_i}(\mathcal{O}_S)$ are locally isomorphic relative to the etale topology.) Is it true that any (etale) form of a product of copies of the group $S$-schemes $\mathbf{GL}_m$, $\mathbf{Sp}_{2n}$, $\mathbf{O}_k$ (where $m,n,k\in\Gamma(S,\mathbb{N})$ can vary) is of the form $\mathbf{U}(A,\sigma)\to \mathrm{Spec}S$? I would also be happy for a proof in case a reference cannot be found.

  • 2
    $\begingroup$ You are trying to compare twisted forms of matrix algebras with involution and twisted forms of reductive group schemes. Both are parametrized by $H^1_{fppf}$ with coefficients in the automorphism group, so the question reduces to computation of the latter. For reductive groups schemes is is well known (the common reference for all this stuff is SGA 3) that the automorphism group is just the adjoint group up to some (naturally seen from the Dynkin diagram) finite constant group scheme. For matrix algebras with involution it is not hard to see that the answer is (up to triality issues) the same $\endgroup$ – Victor Petrov Sep 7 '18 at 16:54
  • $\begingroup$ @Victor Thanks! This indeed solves it. I should have seen it... $\endgroup$ – Uriya First Sep 9 '18 at 6:57

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