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

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• 作者：Alexander I. Efimov^a ; ^b ; ¹ ; ^{efimov@mccme.ru" class="auth_mail" title="E-mail the corresponding author}
• 关键词：16E40 ; 18G40 ; 58K05 ; 11R04
• 刊名：Journal of Algebra
• 出版年：2016
• 出版时间：1 December 2016
• 年：2016
• 卷：467
• 期：Complete
• 页码：80-154
• 全文大小：947 K

文摘

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 $HM{L}^{\mathit{II},0}(\mathbb{Z}[\frac{1}{n!}],w)\ne \mathbb{Q}$.

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 [26]) on its Mac Lane homology.