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The integral (log) cotangent complex of extensions of valued fields

Michaël Maex

Abstract

Let $(L, v_L) / (K, v_K)$ be a finite or purely transcendental extension of real valued fields. We construct the associated integral cotangent and log cotangent complexes in terms of a MacLane-Vaquié chain approximating $v_L$. This leads to explicit formulas for associated invariants such as the (absolute) (log) different, weight norm and Kähler norm. As a corollary of our methods we obtain strong control of the higher homology of the integral (log) cotangent complex, generalizing an important result of Gabber and Ramero to the logarithmic setting.

The integral (log) cotangent complex of extensions of valued fields

Abstract

Let be a finite or purely transcendental extension of real valued fields. We construct the associated integral cotangent and log cotangent complexes in terms of a MacLane-Vaquié chain approximating . This leads to explicit formulas for associated invariants such as the (absolute) (log) different, weight norm and Kähler norm. As a corollary of our methods we obtain strong control of the higher homology of the integral (log) cotangent complex, generalizing an important result of Gabber and Ramero to the logarithmic setting.

Paper Structure

This paper contains 29 sections, 37 theorems, 132 equations, 1 table.

Table of Contents

  1. Introduction
  2. The integral (log) differentials and cotangent complex
  3. Motivation
  4. Main results
  5. The approach
  6. Overview of the paper
  7. Further questions
  8. Relation to other works
  9. Acknowledgements
  10. Valuations and augmentations
  11. Conventions on valuations
  12. A brief overview of MacLane-Vaquié theory
  13. Steps and w-optimal augmentations
  14. The (log) cotangent complex associated to augmentations
  15. Notation and helpful definitions
  16. ...and 14 more sections

Key Result

Theorem 2.2.19

Any semi-valuation $w$ on $K[T]$ falls into exactly one of the following cases, denoted (a), (b), and (c) following loc. cit. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (113)

  • Definition 2.1.1
  • Definition 2.1.3
  • Definition 2.2.3
  • Definition 2.2.4
  • Definition 2.2.6
  • Definition 2.2.7
  • Definition 2.2.9
  • Definition 2.2.10
  • Definition 2.2.11
  • Remark 2.2.12
  • ...and 103 more