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

TL;DR

This paper explores the relationship between Mac Lane (co)homology of the second kind and Wieferich primes, revealing that the infiniteness of such primes relates to specific cohomology invariants of number rings.

Contribution

It introduces the concept of Mac Lane (co)homology of the second kind for associative rings with a central element and connects it to Wieferich primes, providing new computational techniques.

Findings

The infiniteness of Wieferich primes to a base w is equivalent to certain cohomology groups not being rational numbers.

Computed the ring structure on Mac Lane cohomology of global number rings.

Calculated Adams operations on Mac Lane homology of number rings.

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.