Generic flatness for strongly Noetherian algebras
The central result of this paper is the following theorem: Let \(R\) be a commutative domain which is affine over a field or an excellent Dedekind domain. Let \(A\) be a strongly right Noetherian \(R\)-algebra. Then every finitely generated right \(A\)-module is generically flat as an \(R\)-module. Here, ``strongly right Noetherian'' means that \(A\otimes_RS\) is right Noetherian for every commutative Noetherian \(R\)-algebra \(S\); and an \(R\)-module \(M\) is ``generically flat'' if there exists \(0\neq r\in R\) such that \(M[r^{-1}]\) is a flat \(R[r^{-1}]\)-module. The latter concept was introduced by Grothendieck in the 60s, in the setting of a commutative \(R\)-algebra \(A\), and has been much studied since then for various classes of noncommutative Noetherian algebras \(A\), in part because of its relevance for the Jacobson condition and the noncommutative Nullstellensatz -- see, for example, Chapter 9 of [\textit{J. C. McConnell} and \textit{J. C. Robson}, Noncommutative Noetherian rings, Wiley, Chichester (1987; Zbl 0644.16008)] and the references cited there. These topics are also revisited in the present paper, which also discusses a number of interesting examples and counterexamples, and which develops methods which may well prove useful in other contexts.
scientific article; zbMATH DE number 1400110
scientific article