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Galois theory of differential schemes

Ivan Tomašić, Behrang Noohi

Abstract

Since 1883, Picard-Vessiot theory had been developed as the Galois theory of differential field extensions associated to linear differential equations. Inspired by categorical Galois theory of Janelidze, and by using novel methods of precategorical descent applied to algebraic-geometric situations, we develop a Galois theory that applies to morphisms of differential schemes, and vastly generalises the linear Picard-Vessiot theory, as well as the strongly normal theory of Kolchin.

Galois theory of differential schemes

Abstract

Since 1883, Picard-Vessiot theory had been developed as the Galois theory of differential field extensions associated to linear differential equations. Inspired by categorical Galois theory of Janelidze, and by using novel methods of precategorical descent applied to algebraic-geometric situations, we develop a Galois theory that applies to morphisms of differential schemes, and vastly generalises the linear Picard-Vessiot theory, as well as the strongly normal theory of Kolchin.
Paper Structure (48 sections, 55 theorems, 294 equations)

This paper contains 48 sections, 55 theorems, 294 equations.

Table of Contents

  1. Introduction
  2. History
  3. Differential algebraic geometry
  4. Differential schemes as precategory actions and descent
  5. Differential schemes as precategory actions and descent
  6. Differential schemes as formal group actions and geometric quotients
  7. Galois theory of differential schemes
  8. Application: quasi-projective differential Galois theory
  9. Application: Parametric differential equations.
  10. Application: the Galois groupoid of a differential equation.
  11. Layout of the paper
  12. Descent
  13. Precategories and their actions
  14. Connected components of precategories
  15. Precategorical descent
  16. ...and 33 more sections

Key Result

Theorem 1

A pre-Picard-Vessiot morphism $f$ for $\mathscr{P}$ induces an equivalence between the category of objects of $\delta\text{\rm-}\mathscr{P}(Y)$ that are $C$-split by $f$ and the category of $\mathscr{P}$-actions of the precategory $\mathop{\rm Gal}[f]$. If $f$ is Picard-Vessiot for $U$, the latter becomes the category of $\mathscr{P}$-actions of the groupoid $\mathop{\rm G

Theorems & Definitions (153)

  • Definition
  • Definition
  • Theorem : Galois theorem for differential schemes, \ref{['scheme-dif-Galois']}
  • Corollary : Quasi-projective differential Galois correspondence \ref{['qproj-pv-corr']}
  • Theorem : Polarised quasi-projective differential Galois theory, \ref{['polarised-pv-thm']}
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • ...and 143 more