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Uriya First
Uriya First
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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 $\mathbf{SL}_m$ to 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.

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 $\mathbf{SL}_m$ to $\mathbf{GL}_m$, 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.

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.

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Uriya First
Uriya First
  • 2.9k
  • 14
  • 33

Classical reductive group schemes vs. unitary groups of separable algebras with involution --- reference request

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 $\mathbf{SL}_m$ to $\mathbf{GL}_m$, 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.