Krull and valuative dimension of the Serre conjecture ring \(R\langle n \rangle\) (Q2785927)

Let \(R\) be a commutative ring with unit, \(n\in \mathbb{N}\), \(U\) the multiplicative set of monic polynomials in \(R[X]\), \(R\langle X\rangle =U^{-1} R[X]\) and \(R\langle X_1, \dots, X_n \rangle =R \langle X_1, \dots, X_{n-1} \rangle\) \(\langle X_n \rangle\) denoted also by \(R \langle n\rangle\). Let \(S\) be the multiplicative set of the polynomials in \(R[X_1, \dots, X_n]\) whose coefficients generate \(R\) and \(R(n) =S^{-1} R[X_1, \dots, X_n]\). Let also denote, for a ring \(A\) by \(\dim A\) the Krull dimension of \(A\) and by \(\dim_v R= \lim_{n \to\infty} (\dim R[X_1, \dots, X_n] -n)\) the valuative dimension of \(R\).NEWLINENEWLINENEWLINEThe first task of the paper is to establish the Krull and valuative dimension of \(R\langle n\rangle\) and \(R(n)\). The second is the study of the transfer of the Jaffard property from \(R(n)\) to \(R\langle n\rangle\) and conversely. If \(R(\infty) =\bigcup_{m \geq 0} R(n)\) and \(R \langle \infty \rangle =\bigcup_{m\geq 0} R\langle n \rangle\) it is shown that the Krull dimension of these rings is the valuative dimension of \(R\). Finally it is shown that \(R(\infty)\) and \(R\langle \infty \rangle\) are stably strong \(S\)-rings. Several open questions are presented.NEWLINENEWLINEFor the entire collection see [Zbl 0855.00015].