On Flatness of Generic Projections

✍ Scribed by Abdallah Assi

• Publisher: Elsevier Science
• Year: 1994
• Tongue: English
• Weight: 497 KB
• Volume: 18
• Category: Article
• ISSN: 0747-7171
• DOI: 10.1006/jsco.1994.1058

✦ Synopsis

Let (R) be a commutative noetherian ring and let (I) be an ideal of (R\left[x_{1}, \ldots, x_{n}\right]=R[x]). The morphism (\psi: R \longmapsto R[x] / I) defines a family of algebraic varieties as follows: Let (p) be a prime ideal of (R) (or an element of (\operatorname{Spec} R) ) and let (K(p)) be the quotient field of the localization (R_{p}) of (R) at (p), then we have an algebraic variety in (A_{K(p)}^{n}) defined by (K(p)[x] / I(p)) where (I(p)=I . K(p)[x]). When (p) varies, these varieties are called fibers of (\psi). On the other hand, when (\psi) is flat, many properties are preserved in the fibers. The main objective of this paper is to characterize flatness of (\psi) 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 Spec (R) and then we give some applications related to this situation.