Some patching results on algebras over two-dimensional factorial domains (Q714118)

Let \(R\) be a Noetherian ring (always commutative) and let \(A\) be a flat algebra which locally over \(R\) is a polynomial algebra. It is of great interest then to study the structure of \(A\) globally. In the present paper, the authors look at the following special case. Let \(R\) be a unique factorization domain with a maximal ideal generated by a regular sequence \(x,y\) and let \(A\) be a domain containing \(R\) with \(Ax,A_y\) both polynomial algebras in one variable over \(R_x,R_y\) respectively. Further assume that \(A=A_x\cap A_y\). The authors completely classify these rings into three distinct classes.