Mac Lane (co)homology of the second kind and Wieferich primes
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 PolishchukPositselski \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 nonzero 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.
 Publication:

arXiv eprints
 Pub Date:
 May 2015
 arXiv:
 arXiv:1506.00257
 Bibcode:
 2015arXiv150600257E
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  KTheory and Homology;
 Mathematics  Number Theory;
 16E40;
 18G40;
 58K05;
 11R04
 EPrint:
 67 pages, no figures