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On the Tannakian group scheme of $\mathcal{D}$-modules

Xiaodong Yi

Abstract

We prove a Lefschetz theorem for the Tannakian group scheme of $\mathcal{D}$-modules, in arbitrary characteristic. In characteristic $0$, We prove a Künneth formula for the Tannakian group scheme of regular singular integrable connections, and disprove it for the Tannakian group scheme of all integrable connections without any regularity assumption.

On the Tannakian group scheme of $\mathcal{D}$-modules

Abstract

We prove a Lefschetz theorem for the Tannakian group scheme of -modules, in arbitrary characteristic. In characteristic , We prove a Künneth formula for the Tannakian group scheme of regular singular integrable connections, and disprove it for the Tannakian group scheme of all integrable connections without any regularity assumption.

Paper Structure

This paper contains 11 sections, 10 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Characterisitc $0$
  4. Characteristic $p$
  5. A Lefschetz theorem
  6. Characteristic $0$
  7. Characteristic $p$
  8. On the Künneth formula
  9. Some general dicussion
  10. Künneth formula for $\pi_{1}^{alg}$
  11. A counterexample for $\pi_{1}^{diff}$

Key Result

Theorem 1.1

Assume $k$ is a field of characteristic $0$.

Theorems & Definitions (28)

  • Theorem 1.1: Proposition \ref{['lef_alg']}, Proposition \ref{['kun_alg']}, Example \ref{['kun_diff']}
  • Theorem 1.2: Proposition \ref{['lef_str']}
  • Definition 2.1: PMIHES_1960__4__5_0
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Theorem 2.8: deligne1970equations, 10.1007/978-1-4613-9649-9_3
  • Example 2.9
  • ...and 18 more