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Mac Lane (co)homology of the second kind and Wieferich primes

Alexander I. Efimov

TL;DR

The paper establishes a deep link between Mac Lane (co)homology of the second kind and Wieferich primes in finite localizations of global number rings, providing explicit computations of HML^{II,0} and revealing a criterion tying infinitude of Wieferich primes to the nontriviality of HML^{II,0} over localized bases. It develops a comprehensive DG-categorical framework using the generalized cubical construction Q^n_{ullet} and DG quotients to compute Mac Lane (co)homology of the second kind, including localization and completion scenarios, discrete valuation rings, and finite fields. Moreover, it identifies the ring structure on HML^{ullet} for number rings and derives Adams operations on Mac Lane homology, connecting classical algebraic K-theory concepts (stable K-theory) to these second-kind invariants. The results illuminate how arithmetic properties of primes influence homological invariants of localized rings, with potential implications for understanding stable homotopy types and arithmetic geometry through Mac Lane theory.

Abstract

In this paper we investigate the connection between the Mac Lane (co)homology and Wieferich primes in finite localizations of global number rings. Following the ideas of Polishchuk-Positselski \cite{PP}, we define the Mac Lane (co)homology of the second kind of an associative ring with a central element. We compute this invariants for finite localizations of global number rings with an element $w$ and obtain that the result is closely related with the Wieferich primes to the base $w.$ In particular, for a given non-zero integer $w,$ the infiniteness of Wieferich primes to the base $w$ turns out to be equivalent to the following: for any positive integer $n,$ we have $HML^{II,0}(\mathbb{Z}[\frac1{n!}],w)\ne\mathbb{Q}.$ As an application of our technique, we identify the ring structure on the Mac Lane cohomology of a global number ring and compute the Adams operations (introduced in this case by McCarthy \cite{McC}) on its Mac Lane homology.

Mac Lane (co)homology of the second kind and Wieferich primes

TL;DR

The paper establishes a deep link between Mac Lane (co)homology of the second kind and Wieferich primes in finite localizations of global number rings, providing explicit computations of HML^{II,0} and revealing a criterion tying infinitude of Wieferich primes to the nontriviality of HML^{II,0} over localized bases. It develops a comprehensive DG-categorical framework using the generalized cubical construction Q^n_{ullet} and DG quotients to compute Mac Lane (co)homology of the second kind, including localization and completion scenarios, discrete valuation rings, and finite fields. Moreover, it identifies the ring structure on HML^{ullet} for number rings and derives Adams operations on Mac Lane homology, connecting classical algebraic K-theory concepts (stable K-theory) to these second-kind invariants. The results illuminate how arithmetic properties of primes influence homological invariants of localized rings, with potential implications for understanding stable homotopy types and arithmetic geometry through Mac Lane theory.

Abstract

In this paper we investigate the connection between the Mac Lane (co)homology and Wieferich primes in finite localizations of global number rings. Following the ideas of Polishchuk-Positselski \cite{PP}, we define the Mac Lane (co)homology of the second kind of an associative ring with a central element. We compute this invariants for finite localizations of global number rings with an element and obtain that the result is closely related with the Wieferich primes to the base In particular, for a given non-zero integer the infiniteness of Wieferich primes to the base turns out to be equivalent to the following: for any positive integer we have As an application of our technique, we identify the ring structure on the Mac Lane cohomology of a global number ring and compute the Adams operations (introduced in this case by McCarthy \cite{McC}) on its Mac Lane homology.

Paper Structure

This paper contains 21 sections, 83 theorems, 372 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on DG categories
  3. Cross-effects and polynomial functors
  4. Generalized cubical construction
  5. DG quotients and the functors $Q^n_{\bullet}(-).$
  6. Mixed complexes
  7. Hochschild and Mac Lane (co)homology of the second kind
  8. Hochschild (co)homology
  9. Hochschild (co)homology of the second kind
  10. Mac Lane (co)homology
  11. Change of rings spectral sequence
  12. Mac Lane (co)homology of finite fields
  13. Mac Lane (co)homology of the second kind of discrete valuation rings.
  14. Ramified case
  15. Non-ramified case
  16. ...and 6 more sections

Key Result

Theorem 1.2

Within the above notation, let $w\in S^{-1}A$ an element. We always have $HML^{II,1}(S^{-1}A,w)=0.$ Assume that and put Then we have

Theorems & Definitions (191)

  • Conjecture 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • ...and 181 more