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Iterative Derivations on Central Simple Algebras

Manujith K. Michel, Varadharaj R. Srinivasan

TL;DR

This work addresses extending iterative derivations from a δ-field F to central simple F-algebras A in positive characteristic, under the condition that char(F) does not divide the exponent exp_F(A) in Br(F). It develops an iterative differential module framework and uses a filtration via crossed products to construct δ-extensions to A, establishing the existence of a δ-extension for any A with p ∤ exp_F(A). The δ-Galois theoretic viewpoint then classifies δ-F-central simple algebras split by a Picard-Vessiot field K through projective representations G(K|F) → PGL_n(F^δ) and torsor pushforwards, connecting A to the action on its δ-constants via (A ⊗_F K)^δ ≅ M_n(F^δ). The results yield a unique Picard-Vessiot splitting field for such algebras, a Brauer-structure description via δ-A-right ideals, and a characteristic-free generalization of MMVRS-1, providing a cohesive bridge between iterative derivations, splitting fields, torsors, and central simple algebras.

Abstract

We prove that an iterative derivation $δ_F$ on a field $F$ can be extended to an iterative derivation $δ_A$ on a central simple $F-$algebra $A$ if the characteristic of $F$ does not divide the exponent of $A$ in the Brauer group of $F.$ For a central simple $F-$algebra with an iterative derivation, we show the existence of a unique (up to isomorphism) Picard-Vessiot splitting field and from the nature its Galois group, we also describe the structure of the central simple algebra in terms of its $δ_A-$right ideals.

Iterative Derivations on Central Simple Algebras

TL;DR

This work addresses extending iterative derivations from a δ-field F to central simple F-algebras A in positive characteristic, under the condition that char(F) does not divide the exponent exp_F(A) in Br(F). It develops an iterative differential module framework and uses a filtration via crossed products to construct δ-extensions to A, establishing the existence of a δ-extension for any A with p ∤ exp_F(A). The δ-Galois theoretic viewpoint then classifies δ-F-central simple algebras split by a Picard-Vessiot field K through projective representations G(K|F) → PGL_n(F^δ) and torsor pushforwards, connecting A to the action on its δ-constants via (A ⊗_F K)^δ ≅ M_n(F^δ). The results yield a unique Picard-Vessiot splitting field for such algebras, a Brauer-structure description via δ-A-right ideals, and a characteristic-free generalization of MMVRS-1, providing a cohesive bridge between iterative derivations, splitting fields, torsors, and central simple algebras.

Abstract

We prove that an iterative derivation on a field can be extended to an iterative derivation on a central simple algebra if the characteristic of does not divide the exponent of in the Brauer group of For a central simple algebra with an iterative derivation, we show the existence of a unique (up to isomorphism) Picard-Vessiot splitting field and from the nature its Galois group, we also describe the structure of the central simple algebra in terms of its right ideals.
Paper Structure (6 sections, 6 theorems, 18 equations)

This paper contains 6 sections, 6 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Extension of iterative derivations on fields to central simple algebras
  3. Splittings of $\delta-$central simple algebras
  4. Pushforward of Torsors.
  5. Splittings of $\delta-F-$algebras
  6. structure of $\delta-F-$central simple algebras

Key Result

Proposition 2.1

Let $F$ be a field of characteristic $p>0$ and $B=(L|F, G(L|F), f)$ be a crossed product algebra over $F$ such that $p$ does not divide $\mathrm{exp}(B)$. Then every iterative derivation $\delta_F$ on $F$ extends to an iterative derivation $\delta_B$ on $B$ such that $\delta_B$ restricts to an itera

Theorems & Definitions (14)

  • Proposition 2.1
  • proof
  • Theorem 2.2
  • proof
  • Remark 2.3
  • Theorem 3.1
  • proof
  • Remark 3.2
  • Lemma 4.1
  • proof
  • ...and 4 more