On \(\mathbb{Z}_p\)-extensions of commutative rings
type: Item
instance of: scholarly article
title: On \(\mathbb{Z}_p\)-extensions of commutative rings
zbMATH Open document ID: 0748.13001
DOI: 10.1016/0022-4049(91)90125-L
author: Miguel Ferrero
author: Antonio Paques
author: Andrzej Solecki
published in: Journal of Pure and Applied Algebra
publication date: 1992-06-25T00:00:00Z
review text: ``In this paper we mainly study \(\mathbb{Z}_ p\)-extensions of commutative rings of characteristic \(p\) where \(\mathbb{Z}_ p\) denotes the additive group of \(p\)-adic integers. In section 1 we prove some general properties of \(\mathbb{Z}_ p\)-extensions of any commutative ring \(R\). In particular, we show that the set \(T(\mathbb{Z}_ p,R)\) of all isomorphism classes of \(\mathbb{Z}_ p\)-extensions of \(R\) can always be endowed with a \(\mathbb{Z}_ p\)- module structure. This result was firstly proved by Kersten and Michalicek in the case where \(R\) is a connected ring. In section 2 we describe the structure of a \(\mathbb{Z}_ p\)-extension of a commutative ring \(R\) of characteristic \(p\), via Witt vectors. It follows that a \(\mathbb{Z}_ p\)-extension of \(R\) is always a quotient of a polynomial ring \(R[X]\) in an infinite set of indeterminates. Also, we show that \(T(\mathbb{Z}_ p,R)\) is isomorphic as \(\mathbb{Z}_ p\)-module to \(w(R)=W(R)/PW(R)\), where \(W(R)\) denotes the ring of Witt vectors with infinitely many components and \(PW(R) = \{a^ \pi-a\mid a\in W(R)\}\), \(\pi\) denoting the \(p\)-th power operator on \(W(R)\). These results correspond to the classical results for cyclic \(p^ n\)-extensions of fields proved for rings. Nevertheless our proofs differ from those ones. In section 3 we consider cyclic \(p^ n\)-extensions of rings of characteristic \(p\). The main result here states that the Harrison group \(T(\mathbb{Z}/p^ n\mathbb{Z},R)\) of all the isomorphism classes of \(\mathbb{Z}/p^ n\mathbb{Z}\)-extensions of \(R\) is a free \(\mathbb{Z}/p^ n\mathbb{Z}\)-module with essentially the same basis of \(T(\mathbb{Z}/p\mathbb{Z},R)\simeq R/{\mathcal P}R\) as \(\mathbb{Z}/p\mathbb{Z}\)-vector space \(({\mathcal P}R=\{a^ p-a\mid a\in \mathbb{R}\})\). In particular \(T(\mathbb{Z} /p^ n\mathbb{Z} ,\mathbb{F} )\simeq \mathbb{Z} /p^ n\mathbb{Z}\) for any finite field \(\mathbb{F}\) of characteristic \(p\). Finally, in section 4 we come back to …
reviewed by: John N. Mordeson
Mathematics Subject Classification ID: 13B02
Mathematics Subject Classification ID: 12F05
zbMATH DE Number: 11171
zbMATH Keywords: \(\mathbb{Z}_p\)-extensions
zbMATH Keywords: Witt vectors
zbMATH Keywords: Harrison group
zbMATH Keywords: \(p\)-adic completion
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 11171