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

TL;DR

This paper constructs integral and log cotangent complexes for valued field extensions, providing explicit formulas for invariants and controlling their higher homology, extending Gabber and Ramero's results.

Contribution

It introduces a new construction of cotangent complexes for valued field extensions using MacLane-Vaquié chains, with explicit formulas and homology control.

Findings

Explicit formulas for (log) different, weight norm, and Kähler norm.

Control of higher homology of the integral (log) cotangent complex.

Generalization of Gabber and Ramero's result to the logarithmic setting.

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-Vaqui\'e chain approximating $v_L$. This leads to explicit formulas for associated invariants such as the (absolute) (log) different, weight norm and K\"ahler 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.