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

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 paper constructs explicit two-term cotangent complexes with vanishing higher homology in separable cases.
• It employs MacLane–Vaquié chains and scalable rings to compute key invariants like the absolute (log) different and Kähler norms.
• The work extends classical results to the logarithmic context, enabling new computational tools for non-archimedean and birational geometry.

The Integral (Log) Cotangent Complex of Extensions of Valued Fields

Introduction and Motivation

The paper under analysis provides a comprehensive and technically advanced study of the integral and logarithmic cotangent complexes associated with finite and purely transcendental extensions of real valued fields. The work is situated at the interface of valuation theory, ramification theory, birational and non-archimedean analytic geometry, and (log-)motivic integration. The main goal is to construct and analyze the integral cotangent and log cotangent complexes of such field extensions, delivering explicit formulas for associated birational invariants: the absolute (log) different, the weight and Kähler norms, as well as highly explicit control on the higher homology of the (log) cotangent complex. The results generalize classical theorems due to Gabber and Ramero to the logarithmic context.

The motivation arises from the need to effectively calibrate subtle arithmetic and birational invariants on Berkovich analytic spaces, such as those needed to define and analyze weight and discrepancy functions on skeleta—a key tool for both non-archimedean and birational methods. By moving from the classical discretely valued context to real valued fields, the work enables applications to non-Noetherian situations which are common in non-archimedean geometry and birational geometry over real-valued fields.

Methodological Framework: MacLane–Vaquié Chains and Scalable Rings

A cornerstone of the paper is the use of MacLane–Vaquié chains, which provide an inductive framework for approximating any real extension semi-valuation on $K[T]$ by a chain of ordinary and limit augmentations, with each step governed by the selection of key polynomials. The author leverages the enhancements of valuation theory offered by Vaquié, removing both completeness and residue class field restrictions, and working in full rank-one generality.

At each augmentation step, the unit ball of the valuation ring is described as an explicit increasing union (“enlargement”) of scalable rings—rings admitting division by constants of arbitrary valuation. This allows the reduction of the computation of the cotangent complex to a sequence of well-controlled complete intersection presentations, with explicit control of torsion and content invariants via the computation of regular sequences and their determinants.

A technical innovation is the systematic use of content (in the sense of Temkin) as a measure of torsion in non-Noetherian modules, enabling uniform computation of lengths in both discretely and real-valued, possibly non-Noetherian, settings.

Main Results: Computations of (Log) Cotangent Complexes and Invariants

The central technical achievement is the explicit construction, up to filtered colimit, of the (log) cotangent complex of $L^\circ/K^\circ$ and $L^\circ/K^\circ$ (with suitable log structures) for any finite or purely transcendental valued field extension. The explicit nature of the construction is manifested in the following key outcomes:

• The cotangent and log cotangent complex can always be presented as a two-term complex, with all higher homology vanishing in the separable case.
• The length/content of the torsion part of the module of (log) differentials is expressed via the “step” function, which is an explicit sum of the increments of log radii and evaluations of key polynomials along the MacLane–Vaquié chain.
• For finite separable extensions, the different and log different admit explicit descriptions involving the valuations of monic minimal polynomials and their derivatives.
• For purely transcendental extensions, the (log) discrepancy and associated invariants arise as contents of specific cyclic modules generated by key polynomials and their differentials/logarithmic differentials.

Precise Description of Homological Behavior

A significant technical output is the demonstration that all higher homology of the (log) cotangent complex vanishes above degree one, and the first homology is torsion-free, vanishing precisely in the separable case. This extends classical results on the structure of the cotangent complex in the discretely valued setting (Gabber–Ramero), but the methods are effective in the much more general context enabled by real valuations and the logarithmic setting.

Kähler Norms and Log Structures

A notable application is the precise relationship established between key polynomials and the Kähler semi-valuation. The minimal non-expansive seminorm on Kähler differentials is controlled precisely by valuations on key polynomials in the MacLane–Vaquié chain. Moreover, a direct generator description of the module of integral log differentials in terms of $d \log \phi$ for key polynomials $\phi$ is realized, and explicit formulas for Kähler and weight norms are given. These generalize and sharpen constructions previously only available in the discretely valued case and allow for new interpretations of canonical metrics and weight functions in non-archimedean settings.

Logarithmic Invariants and Absolute Log Different

The framework is extended to the computation of the absolute log different, which, beyond purely finite extensions, quantifies the torsion in the module of integral log differentials for arbitrary transcendence degrees. The paper shows that this invariant is controlled by the content of torsion differentials/log differentials, and relates it to the Kähler valuation and the explicit steps along the MacLane–Vaquié chain. The results facilitate new, computable forms of invariants such as the log canonical threshold and discrepancy functions, with applications to birational and non-archimedean geometry.

Theoretical and Practical Implications

The theoretical implications are substantial: the explicit formulas and structure theorems provided clarify the precise homological and valuation-theoretic nature of (log) differentials and related modules in situations relevant to the birational and non-archimedean theories. The methodology establishes a new paradigm for reducing global problems about complex invariants to computations along explicit augmentation chains, suitable both for concrete computation and further structural analysis. The extension to the logarithmic context opens the way for arithmetic and motivic applications, including finer ramification analysis and the construction of canonical metrics and weights in non-archimedean geometry.

Practically, the work creates a toolkit enabling explicit calculation of birational and arithmetic invariants for valued field extensions outside the strictly discretely valued setting. This is expected to impact the ongoing study of weight functions, Berkovich skeletons, and motivic invariants, and enables the analytical treatment of ramification and discriminants in highly general contexts.

The arguments, couched in the language of semi-valuation chains and scalable rings, offer a pathway for extending these techniques to value groups of arbitrary rank, for which the paper posits conjectural generalizations.

Outlook and Future Directions

The results in this work pave the way for several immediate developments. First, the explicit nature of the constructions suggests effective algorithms for the computation of birational invariants in both arithmetic and analytic geometry. Second, the author alludes to extension of these results to more general valuations, especially in the context of Berkovich spaces where non-real value groups may appear. The connection to the structure of valuative trees and their skeleta suggests further applications to the geometry of non-archimedean spaces and their motivic invariants. Moreover, there is potential for enriching the theory of weight and discrepancy functions, refining the metric and piecewise-linear geometry of Berkovich spaces.

Conclusion

This work provides a technically thorough and explicit construction of the integral and logarithmic cotangent complexes for extensions of real-valued fields, unveils new formulas for associated invariants including the (absolute) (log) different, Kähler and weight norms, and precisely delimits the homological structure of these complexes. The adoption of the MacLane–Vaquié chain framework, in combination with the theory of scalable rings and content, offers both theoretical insight and practical computational tools, extending the classical theory of ramification and differential invariants to the full generality required in non-archimedean and birational geometry. The results establish a new foundation for further exploration of differential invariants and their applications in arithmetic and geometry.