| If B is a polynomial ring in one variable (or more generally locally polynomial) over a Noetherian domain R, then B has various properties as an R-algebra. To mention some of them:
(i) B is Noetherian [Hilbert's Basis Theorem];
(ii) B is finitely generated;
(iii) B has a retract to R;
(iv) B is faithfully flat over R; and, of course,
(v) B_P is a polynomial ring over R_P for every prime ideal P of R.
In this talk, we shall discuss the converse problem of determining minimal sufficient conditions (of the type listed above) for an R-algebra to be locally polynomial. Recall that, by a result of Bass-Connell-Wright (1977), a finitely generated locally polynomial algebra in n variables over a Noetherian domain R is the symmetric algebra of a projective R-module of rank n. The initial results in this direction by Kambayashi-Miyanishi (1978) over a Noetherian UFD and Kambayashi-Wright (1985) over a Noetherian normal (i.e., integrally closed) domain assumed condition (v) for ALL prime ideals. Subsequent results of Bhatwadekar-Dutta (1995) and Dutta (1995) showed that, for the converse problem, it often suffices to consider algebras which are ``locally polynomial in codimension-one'' i.e., algbras for which condition (v) is assumed only for prime ideals of codimension-one.
Recently, Dutta-Onoda (2007) explored the general structure of faithfully flat algebras (not necessarily finitely generated) over a Noetherian UFD R which are locally polynomial in codimension-one. In an ongoing work by Bhatwadekar-Dutta-Onoda, the more intricate case of algebras over Noetherian normal domains is being investigated. The results obtained shed new light on the converse problem mentioned earlier. |