| An algebra A over a Noetherian normal domain R is called locally polynomial in codimension-one if A_P is a polynomial ring in one variable over R_P for every prime ideal P in R of height one. A result of A.K. Dutta says that if such an algebra A is finitely generated and faithfully flat, then A must be isomorphic to the symmetric algebra of some projective R-module L of rank one.
In this talk, I will indicate some recent results (obtained in my joint work with S.M. Bhatwadekar and A.K. Dutta) on the general structure of algebras (not necessarily finitely generated) which are locally polynomial in codimension-one, and its consequences. For example, one outcome of our work is a realization that if A is Noetherian and faithfully flat over R, then finite generation of A becomes equivalent to the existence of a retraction from A to R. Moreover we have constructed an intricate example to show that even when R is a nice local ring, such as a regular affine spot over an algebraically closed field of characteristic zero, there does exist a Noetherian faithfully flat R-algebra which is locally polynomial in codimension-one, but which does not have a retraction to R.
The talk will be focused on the basic ideas involved in the construction of such esoteric examples.
|