On Flatness of Generic Projections

Let R be a commutative noetherian ring and let I be an ideal of R[x1,…,xn = R [x]. The morphism ψ R → R [x]/I defines a family of algebraic varieties as follows: Let p be a prime ideal of R (or an element of SpecR) and let K(p) be the quotient field of the localization Rp of R atp, then we have an algebraic variety in AnK(p) defined by K (p)[x]/I(p) where I(p) = I.K (p)[x]. When p varies, these varieties are called fibers of ψ. On the other hand, when ψ is flat, many properties are preserved in the fibers. The main objective of this paper is to characterize flatness of ψ by studying the relationship with the notions of Gröbner and standard bases. When R is principal, we obtain an algorithm to compute the maximal generic open set of flatness of SpecR and then we give some applications related to this situation.