On \(\mathbb{Z}_p\)-extensions of commutative rings
``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 \(\mathbb{Z}_ p\)-extensions of rings of characteristic \(p\). We prove that \(T(\mathbb{Z}_ p,R)\) is isomorphic as \(\mathbb{Z}_ p\)-module to the \(p\)-adic completion of a free \(\mathbb{Z}_ p\)- module, in particular, \(T(\mathbb{Z}_ p,R)\) is free when \(\dim_{(\mathbb{Z}/p\mathbb{Z})}(R/{\mathcal P}R)<\infty\) and \(T(\mathbb{Z}_ p,\mathbb{F})\simeq\mathbb{Z}_ p\) for any finite field \(\mathbb{F}\) of characteristic \(p\)''.
scientific article; zbMATH DE number 11171
scientific article