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Smooth profinite groups, III: the Smoothness Theorem

Charles De Clercq, Mathieu Florence

TL;DR

This work proves the Smoothness Theorem, showing that a $(1,1)$-cyclotomic pair lifts mod $p$ cohomology to mod $p^2$ and, consequently, yields the Norm Residue Isomorphism Theorem without motivic cohomology. The authors build a comprehensive framework around versal cohomology using ring schemes, Hochschild cohomology, and uplifting/descent techniques, and they extend the results from absolute Galois groups to algebraic fundamental groups of curves. The cyclotomic-case development centers on constructing a universal class $C_{\mathrm{vers}}$ and showing how any given cohomology class can be realized via specializations, with a crucial uplifting pattern to manage lifts to $\mathbf{R}$-coefficients. The general case then follows by incorporating scheme-theoretic cyclotomic twists, preserving the lifting/descent mechanism. Together, these results broaden the applicability of norm-residue phenomena to broader geometric settings and lay groundwork for the Symbols conjecture and related questions about curve fundamental groups.

Abstract

Let $p$ be a prime. In this article, we prove the Smoothness Theorem, which asserts that a $(1,1)$-cyclotomic pair is $(n,1)$-cyclotomic, for all $n \geq 1$. In the particular case of Galois cohomology, the Smoothness Theorem provides a new proof of the Norm Residue Isomorphism Theorem, entirely disjoint from motivic cohomology. A byproduct of this approach, is that the latter Theorem follows from mod $p^2$ Kummer theory for fields alone. We moreover extend it, from absolute Galois groups of fields, to algebraic fundamental groups of (not necessarily smooth, nor proper) curves over algebraically closed fields.

Smooth profinite groups, III: the Smoothness Theorem

TL;DR

This work proves the Smoothness Theorem, showing that a -cyclotomic pair lifts mod cohomology to mod and, consequently, yields the Norm Residue Isomorphism Theorem without motivic cohomology. The authors build a comprehensive framework around versal cohomology using ring schemes, Hochschild cohomology, and uplifting/descent techniques, and they extend the results from absolute Galois groups to algebraic fundamental groups of curves. The cyclotomic-case development centers on constructing a universal class and showing how any given cohomology class can be realized via specializations, with a crucial uplifting pattern to manage lifts to -coefficients. The general case then follows by incorporating scheme-theoretic cyclotomic twists, preserving the lifting/descent mechanism. Together, these results broaden the applicability of norm-residue phenomena to broader geometric settings and lay groundwork for the Symbols conjecture and related questions about curve fundamental groups.

Abstract

Let be a prime. In this article, we prove the Smoothness Theorem, which asserts that a -cyclotomic pair is -cyclotomic, for all . In the particular case of Galois cohomology, the Smoothness Theorem provides a new proof of the Norm Residue Isomorphism Theorem, entirely disjoint from motivic cohomology. A byproduct of this approach, is that the latter Theorem follows from mod Kummer theory for fields alone. We moreover extend it, from absolute Galois groups of fields, to algebraic fundamental groups of (not necessarily smooth, nor proper) curves over algebraically closed fields.

Paper Structure

This paper contains 15 sections, 13 theorems, 124 equations.

Table of Contents

  1. Introduction.
  2. The Smoothness Theorem.
  3. Proof in the cyclotomic case, and assuming $\mathbb{Z}/p^2(1)=\mathbb{Z}/p^2$.
  4. Resolutions in homological algebra.
  5. The ring scheme $\mathbf{R}$.
  6. Hochschild cohomology of $\mathbb{Z}$-group schemes.
  7. Group-change $\Gamma_{\mathrm{vers} } \xrightarrow{\gamma} \Gamma$, and versal class $C_{\mathrm{vers} }$.
  8. Specialising $C_{\mathrm{vers} }$.
  9. Uplifting...
  10. ... then descent.
  11. Proof of the Smoothness Theorem, general case.
  12. What next?
  13. The Symbols conjecture
  14. Demushkin groups.
  15. Curves over algebraically closed fields.

Key Result

Theorem 2.1

(The Smoothness Theorem.) Let $G$ be a $(1,1)$-smooth profinite group. Then, for every $n \geq 1$, $G$ is $(n,1)$-smooth. Moreover, assume that $(G,\mathbb{Z}/p^2(1))$ is a $(1,1)$-cyclotomic pair. Then, for every $n \geq 1$, $(G,\mathbb{Z}/p^2(1))$ is $(n,1)$-cyclotomic.

Theorems & Definitions (40)

  • Theorem 2.1
  • Remark 2.2
  • Corollary 2.3: The Norm Residue Isomorphism Theorem
  • Definition 3.1
  • Lemma 3.2
  • Definition 3.3
  • Definition 3.4
  • Definition 3.5
  • Definition 3.6
  • Definition 3.7
  • ...and 30 more