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On deformations of Azumaya algebras with quadratic pair

Eoin Mackall, Cameron Ruether

TL;DR

This work develops a tangent-obstruction theory for Azumaya algebras equipped with a quadratic pair and shows that, under either 2 being a global unit or deg(A)=2, deformation theory for the quadratic pair reduces to that of the underlying Azumaya algebra. It introduces an intermediate obstruction Ω′(A,σ) in characteristic 2, which sits between the strong and weak obstructions and governs when a canonical Lie-algebra extension splits. An explicit Igusa-surface example demonstrates that the quadratic pair can be obstructed even when the underlying Azumaya algebra deforms unobstructed, highlighting subtle characteristic-2 phenomena. Additionally, a relative unobstructedness result is established in two key cases, and the norm-functor framework (A1×A1 ≅ D2) relates quaternion and degree-4 quadratic triple moduli in this deformation setting. The results illuminate when quadratic-pair structures influence deformations and provide a concrete obstructed example in positive characteristic with potential implications for moduli theory and algebraic group actions.

Abstract

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.

On deformations of Azumaya algebras with quadratic pair

TL;DR

This work develops a tangent-obstruction theory for Azumaya algebras equipped with a quadratic pair and shows that, under either 2 being a global unit or deg(A)=2, deformation theory for the quadratic pair reduces to that of the underlying Azumaya algebra. It introduces an intermediate obstruction Ω′(A,σ) in characteristic 2, which sits between the strong and weak obstructions and governs when a canonical Lie-algebra extension splits. An explicit Igusa-surface example demonstrates that the quadratic pair can be obstructed even when the underlying Azumaya algebra deforms unobstructed, highlighting subtle characteristic-2 phenomena. Additionally, a relative unobstructedness result is established in two key cases, and the norm-functor framework (A1×A1 ≅ D2) relates quaternion and degree-4 quadratic triple moduli in this deformation setting. The results illuminate when quadratic-pair structures influence deformations and provide a concrete obstructed example in positive characteristic with potential implications for moduli theory and algebraic group actions.

Abstract

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.

Paper Structure

This paper contains 25 sections, 35 theorems, 180 equations.

Table of Contents

  1. Preliminaries
  2. Azumaya algebras with additional structure
  3. Azumaya algebras
  4. Involutions on Azumaya algebras
  5. Quadratic triples
  6. Group schemes and Lie algebra sheaves
  7. Algebraic groups
  8. Lie algebra sheaves
  9. $\mathfrak{A}_1\times \mathfrak{A}_1$, $\mathfrak{D}_2$, and the norm functor
  10. Deformations and obstructions
  11. Relative deformations
  12. Infinitesimal automorphisms
  13. Obstructions and classifying deformations
  14. Tangent-obstruction theories
  15. Igusa surfaces
  16. ...and 10 more sections

Key Result

Theorem A

Let $X_C \rightarrow X_{C'}$ be a square-zero thickening of schemes associated to a small extension of local Artinian $k$--algebras as above. Let $(\mathcal{A},\sigma,f)$ be an Azumaya algebra with quadratic pair on $X_C$. Then, there exists a deformation $(\mathcal{A}',\sigma',f')$ on $X_{C'}$ of t

Theorems & Definitions (77)

  • Theorem A
  • Theorem B
  • Theorem C
  • Remark A
  • Theorem B: GNR2024TheNormFunctor
  • Definition C
  • Definition D
  • Definition E
  • Definition F
  • Definition G
  • ...and 67 more