Some patching results on algebras over two-dimensional factorial domains

Let R=k[x,y] be the polynomial ring in two variables over a field k. We investigate the structure and properties of R-algebras A which are obtained as A=Ax∩Ay where Ax and Ay are polynomial algebras in one variable over Rx and Ry respectively. Most of our results hold when R is a two-dimensional UFD and x,y is an R-regular sequence generating a maximal ideal of R.