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 [29], we define the Mac Lane (co)homology of the second kind of an associative ring with a central element. We compute these invariants for finite localizations of global number rings with an element w and obtain that the result is closely related to 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 HMLII,0(Z[1n!],w)≠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 [26]) on its Mac Lane homology.
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