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Auslander regularity of $p$-adic Banach algebras via almost mathematics

Andreas Bode

Abstract

We discuss an "almost" version of Auslander regularity and use it to prove the Auslander regularity of various Banach algebras over non-discretely valued fields appearing naturally in $p$-adic locally analytic representation theory: completed Weyl algebras, the completed enveloping algebra of a Lie algebra, and the Banach completion of the distribution algebra $D(G, K)$ for a compact $p$-adic Lie group $G$.

Auslander regularity of $p$-adic Banach algebras via almost mathematics

Abstract

We discuss an "almost" version of Auslander regularity and use it to prove the Auslander regularity of various Banach algebras over non-discretely valued fields appearing naturally in -adic locally analytic representation theory: completed Weyl algebras, the completed enveloping algebra of a Lie algebra, and the Banach completion of the distribution algebra for a compact -adic Lie group .

Paper Structure

This paper contains 16 sections, 57 theorems, 185 equations.

Table of Contents

  1. Introduction
  2. Background on almost mathematics
  3. Basic definitions
  4. Good filtrations and almost mathematics
  5. Almost Auslander regular rings
  6. Definition and relation to Auslander regularity on the generic fibre
  7. Preliminary results on the almost Auslander condition
  8. Almost Auslander regularity of polynomial rings
  9. Reduction to graded modules
  10. The almost Auslander condition for graded modules
  11. Dimension results
  12. Examples: Banach algebras with polynomial reduction
  13. Completed enveloping algebras
  14. Completed Weyl algebras
  15. Norm-completed distribution algebras
  16. ...and 1 more sections

Key Result

Theorem 1.1

Theorems & Definitions (124)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6
  • proof
  • ...and 114 more