arXiv subjects

Cameron Ruether

Publications and source records attributed to Cameron Ruether.

7 recordsLinked to original sources

arXiv2026

Outer Type Severi-Brauer Schemes

We introduce the notion of a lowered flag of $\mathcal{O}$--modules in order to define a sheaf of flags of ideals isomorphic to the sheaf of parabolic subgroups for the general linear group $\mathbf{GL}_{1,\mathcal{A}}$ of an Azumaya algebra over a general scheme $S$. This notion is extended to the outer type $A_n$ case and we define a suitable sheaf of flags of ideals isomorphic to the sheaf of parabolic subgroups for a unitary group over $S$. When the group is suitably split these are related to flags of submodules in a vector bundle or in a vector bundle with hermitian form, respectively. We also define a sheaf of tuples of idempotents in the associated algebra which is isomorphic to the sheaf of parabolic and Levi subgroup pairs. We show how the type morphism from parabolic subgroups to the Dynkin scheme can be defined in terms of these sheaves of flags. We review how the Severi-Brauer scheme associated to an Azumaya algebra $\mathcal{A}$ is isomorphic to a particular fiber of this type morphism and we generalize this idea to the outer case in order to define outer Severi-Brauer schemes. We provide a new approach to Quillen's construction which produces an Azumaya algebra from a Severi-Brauer scheme and we show that an outer version of Quillen's construction also exists for outer Severi-Brauer schemes which produces an algebra with unitary involution.

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arXiv2025

On deformations of Azumaya algebras with quadratic pair

We construct a tangent-obstruction theory for Azumaya algebras equipped with a quadratic pair. Under the assumption that either 2 is a global unit or the algebra is of degree 2, we show how the deformation theory of these objects reduces to the deformation theory of the underlying Azumaya algebra. Namely, if the underlying Azumaya algebra has unobstructed deformations then so does the quadratic pair. On the other hand, in the purely characteristic 2 setting, we construct an Azumaya algebra with unobstructed deformations which can be equipped with a quadratic pair such that the associated triple has obstructed deformations. Our example is a biquaternion Azumaya algebra on an Igusa surface. Independently from the above results, we also introduce a new obstruction for quadratic pairs, existing only in characteristic 2, which is intermediate to both the strong and weak obstructions that were recently introduced by Gille, Neher, and the second named author. This intermediate obstruction characterizes when a canonical extension of the Lie algebra sheaf of the automorphism group scheme of some quadratic triple is split.

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arXiv2024

Triality over Schemes

Working over an arbitrary base scheme, we provide an alternative development of triality which does not use Octonion algebras or symmetric composition algebras. Instead, we use the Clifford algebra of the split hyperbolic quadratic form of rank 8 and computations with Chevalley generators of groups of type $D_4$. Following the strategy of The Book of Involutions [KMRT], we then define the stack of trialitarian triples and show it is equivalent to the gerbe of $\mathbf{PGO}_8^+$--torsors. We show it has endomorphisms generating a group isomorphic to $\mathbb{S}_3$ and that several familiar cohomological properties of $\mathbf{PGO}_8^+$ follow in this setting as a result. Next, we define the stack of trialitarian algebras and show it is equivalent to the gerbe of $\mathbf{PGO}_8^+\rtimes \mathbb{S}_3$--torsors. Because of this, it is also equivalent to the gerbes of simply connected, respectively adjoint, groups of type $D_4$. We define $\mathbf{Spin}_\mathcal{T}$ and $\mathbf{PGO}^+_\mathcal{T}$ for a trialitarian algebra and define concrete functors $\mathcal{T} \mapsto \mathbf{Spin}_\mathcal{T}$ and $\mathcal{T} \mapsto \mathbf{PGO}^+_\mathcal{T}$ which realize these equivalences.

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arXiv2024

The Norm Functor over Schemes

We construct a globalization of Ferrand's norm functor over rings which generalizes it to the setting of a finite locally free morphism of schemes $T\to S$ of constant rank. It sends quasi-coherent modules over $T$ to quasi-coherent modules over $S$. These functors restrict to the category of quasi-coherent algebras. We also assemble these functors into a norm morphism from the stack of quasi-coherent modules over a finite locally free of constant rank extension of the base scheme into the stack of quasi-coherent modules. This morphism also restricts to the analogous stacks of algebras. Restricting our attention to finite \'etale covers, we give a cohomological description of the norm morphism in terms of the Segre embedding. Using this cohomological description, we show that the norm gives an equivalence of stacks of algebras $A_1^2 \equiv D_2$, akin to the result shown in The Book of Involutions.

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arXiv2023

The Canonical Quadratic Pair on Clifford Algebras over Schemes

Working over an arbitrary base scheme $S$, we define the canonical quadratic pair on the Clifford algebra associated to an Azumaya algebra with quadratic pair. Given an Azumaya algebra $\mathcal{A}$ with quadratic pair, i.e., with an orthogonal involution and a semi-trace, its associated Clifford algebra's canonical involution is only orthogonal in certain cases, namely when $\mathrm{deg}(\mathcal{A})$ is divisible by $8$ or when both $2=0$ over $S$ and $\mathrm{deg}(\mathcal{A})$ is divisible by $4$. When $\mathrm{deg}(\mathcal{A}) \geq 8$, our definition of the canonical quadratic pair on the Clifford algebra is extended from previous work of Dolphin and Qu\'eguiner-Mathieu, who worked over fields of characteristic $2$. When $\mathrm{deg}(\mathcal{A})=4$, we show that no canonical quadratic pair exists.

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arXiv2022

Azumaya Algebras and Obstructions to Quadratic Pairs over a Scheme

We investigate quadratic pairs for Azumaya algebras with involutions over a base scheme S as defined by Calm{\`e}s and Fasel, generalizing the case of quadratic pairs on central simple algebras over a field (Knus, Merkurjev, Rost, Tignol). We describe a cohomological obstruction for an Azumaya algebra over S with orthogonal involution to admit a quadratic pair. When S is affine this obstruction vanishes, however it is non-trivial in general. In particular, we construct explicit examples with non-trivial obstructions.

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arXiv2018

Rost Multipliers of Lifted Kronecker Tensor Products

We extend techniques employed by Garibaldi to construct various new injections involving the half-spin group, $\textbf{HSpin}$, induced by lifting the Kronecker tensor product to simply connected groups. We calculate the Rost multipliers of the maps we have constructed. Furthermore, we utilize our new map $\textbf{PSp}_{2n}\times \textbf{PSp}_{2m} \hookrightarrow \textbf{HSpin}_{4nm}$ to describe the structure of the normalized degree three cohomological invariants of $\textbf{HSpin}_{4n}$.

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