On locally quasi \(\mathbb A^*\) algebras in codimension-one over a Noetherian normal domain
Let \(R\) be a Noetherian ring and let \(A\) be an \(R\)-algebra. Let \(P\) be a prime ideal of \(R\). One says that \(A\) is \(\mathbb{A}^1\) over \(R\) at \(P\), if \(A_P\) is isomorphic as an \(R_P\)-algebra to \(R_P[X]\) for an indeterminate \(X\). Similarly, it is an \(\mathbb{A}^*\) if it is isomorphic to \(R_P[X,x^{-1}]\). It is a quasi \(\mathbb{A}^*\) if it is isomorphic to \(R_p[X,(aX+b)^{-1}]\) (at least when \(R\) is a domain) with \(a\neq 0\) and \(a,b\) comaximal in \(R\). The \(\mathbb{A}^1\) case has been extensively studied and one of the seminal works in this direction is due to \textit{P.~Russell} and \textit{A.~Sathaye} [J. Alg. 57, 151--166 (1979; Zbl 0411.13011)]. The first author of the paper under review has made several contributions in this direction and initiated work on \(\mathbb{A}^*\) and quasi \(\mathbb{A}^*\) fibrations. The authors prove the following two theorems. Theorem A. Let \((R,\mathfrak{m})\) be a Noetherian normal local domain and \(A\) a faithfully flat algebra over \(R\) which is quasi \(\mathbb{A}^*\) at every height at most one primes of \(R\). Then \(A\) is a quasi \(\mathbb{A}^*\) algebra if and only if \(A\) is finitely generated over \(R\) if and only if \(R/\mathfrak{m}\subsetneq A/\mathfrak{m}A\). Moreover, if \(R\) is complete and \(A\) is Noetherian, then the above equivalent conditions are satisfied. Theorem B. Let \(R\) be a Noetherian normal domain and \(A\) a faithfully flat \(R\)-algebra which is \(\mathbb{A}^*\) at all primes of height at most one in \(R\). Then \(A=\oplus_{n\in\mathbb{Z}} I^nu^n\) for an invertible ideal \(I\) of \(R\).
scientific article; zbMATH DE number 5899131