Structure of \(\mathbb A^2\)-fibrations over one-dimensional Noetherian domains
Let \(R\) be a commutative ring with unity. Let \(R^{[n]}\) denote a polynomial ring in \(n\) variables over \(R\). For a prime ideal \(P\) of \(R\), \(K(P)\) denotes the field \(R_P/ PR_P\). An \(R\)-algebra \(A\) is said to be an \(\mathbb{A}^r\)-fibration over \(R\) if (i) \(A\) is finitely generated over \(R\), (ii) \(A\) is flat over \(R\), (iii) \(A\otimes_RK(P)= K(P)^{[r]}\) for every prime ideal \(P\) of \(R\). If \(R\) is a discrete valuation ring containing \(\mathbb{Q}\) (the field of rationals) and \(A\) is an \(\mathbb{A}^2\)-fibration over \(R\), then \textit{A. Sathaye} [Invent. Math. 74, 159-168 (1983; Zbl 0538.13006)] has proved that \(A=R^{[2]}\). On the other hand, if a discrete valuation ring \(R\) does not contain \(\mathbb{Q}\) then \textit{T. Asanuma} [Invent. Math. 87, 101-127 (1987; Zbl 0607.13015)] has shown that there exists an \(\mathbb{A}^2\)-fibration \(A\) over \(R\) such that \(A\not\approx R^{[2]}\). -- In view of the above mentioned results, it is natural to ask: How does an \(\mathbb{A}^2\)-fibration arise over an arbitrary one-dimensional noetherian domain \(R\) containing \(\mathbb{Q}\)? Theorem. Let \(R\) be a one-dimensional noetherian domain containing the field \(\mathbb{Q}\) of rational numbers. Let \(A\) be an \(\mathbb{A}^2\)-fibration over \(R\). Then there exists \(H\in A\) such that \(A\) is an \(\mathbb{A}^1\)-fibration over \(R[H]\).
scientific article; zbMATH DE number 986722