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Torsors in super-symmetry

Akira Masuoka, Takuya Oe, Yuta Takahashi

Abstract

Torsors under affine groups are generalized in the super context by super-torsors under affine super-groups. We investigate those super-torsors by using Hopf-algebra language and techniques. It is explicitly shown, under suitable assumptions, that every super-torsor arises from an ordinary torsor. Especially, the objects with affinity restriction, or namely, the affine super-torsors and the affine ordinary torsors are shown to be precisely in one-to-one correspondence. The results play substantial roles in ongoing construction of super-symmetric Picard-Vessiot theory.

Torsors in super-symmetry

Abstract

Torsors under affine groups are generalized in the super context by super-torsors under affine super-groups. We investigate those super-torsors by using Hopf-algebra language and techniques. It is explicitly shown, under suitable assumptions, that every super-torsor arises from an ordinary torsor. Especially, the objects with affinity restriction, or namely, the affine super-torsors and the affine ordinary torsors are shown to be precisely in one-to-one correspondence. The results play substantial roles in ongoing construction of super-symmetric Picard-Vessiot theory.

Paper Structure

This paper contains 27 sections, 33 theorems, 150 equations.

Table of Contents

  1. Introduction---basic notations and main theorems
  2. Very basic algebraic super-geometry
  3. The first main theorem
  4. The second main theorem
  5. Organization of the paper and additional remarks
  6. Preliminaries
  7. Super-linear algebra
  8. Super-(co)algebras
  9. Hopf super-algebras
  10. Smooth super-algebras
  11. Faithful flatness and finite presentation
  12. Shifting to the Hopf-algebra world
  13. Hopf-Galois extensions
  14. Inflation of Hopf-Galois extensions
  15. Main theorems re-formulated in the Hopf-algebra language
  16. ...and 12 more sections

Key Result

Theorem 1.4

Let $\mathsf{G}$ be a smooth affine algebraic super-group, and let $\mathsf{Y}$ be a Noetherian smooth affine super-scheme. If $\mathsf{X} \to \mathsf{Y}$ is a super $\mathsf{G}$-torsor, then the induced morphism $\mathsf{X}_{\mathsf{ev}}\to \mathsf{Y}_{\mathsf{ev}}$ of schemes is a $\mathsf{G}_{\ma onto

Theorems & Definitions (70)

  • Definition 1.1
  • Remark 1.2
  • Example 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Proposition 1.7
  • Corollary 1.8
  • Theorem 1.9
  • Remark 1.10
  • ...and 60 more