The Finiteness of I When R[ X ]/I is Flat

Abstract

Let R be a commutative ring with identity, let X be an indeterminate, and let I be an ideal of the polynomial ring R[X]. Let min I denote the set of elements of I of minimal degree and assume henceforth that min I contains a regular element. Then R[XI/I is a flat R-module implies I is a finitely generated ideal. Under the additional hypothesis that R is quasi-local integrally closed, the stronger conclusion that I is principal holds. (An example shows that the first statement is no longer valid when min I does not contain a regular element.) Let c(I) denote the content ideal of I, i.e. c(I) is the ideal of R generated by the coefficients of the elements of I. A corollary to the above theorem asserts that R[X]/I is a flat R-module if and only if I is an invertible ideal of R[X] and c(I) = R. Moreover, if R is quasi-local integrally closed, then the following are equivalent: (i) R[X1/I is a flat R-module; (ii) R[X]/I is a torsion free R-module and c(I) = R; (iii) I is principal and c(I) = R. Let 6 denote the equivalence class of X in R[XI/I, and let (1, 6 . 4t denote the R-module generated by 1, 6, ***, ft. The following statements are also equivalent: (i) (1, 4, .., t) is flat for all t ? 0; (ii) (1, , ..., t) is flat for some t > 0 for which 1 -, .. t are linearly dependent over R; (iii) I = (f , , fn), fi e min I, and c(I) R; (iv) c(min I) = R. Moreover, if R is integrally closed, these are equivalent to R[X]/I being a flat R-module. A certain symmetry enters in when f is regular in R[L], and in this case (i)-(iv) are also equivalent to the assertion that R[L] and R[1/f are flat R-modules. The main results of this paper are found in ??2 and 3. Many of the difficulties of these sections already occur when the ring R is an integral domain, and the reader might benefit by first confining his attention to this case. Additional technical difficulties arise when one proceeds to the case that R is an arbitrary ring and I is subject to the restriction that min I contain a regular element. ?4 is devoted to a discussion of what happens when one removes this condition on min I. In particular, we make there a conjecture as to the class of rings R with the property that for every ideal I of R[X], if R[X]/I is R-flat, then I is finitely generated. The corresponding question for finitely generated modules is easily answered as follows: The class of rings R with the property that R[X]/I is a finite flat R-module implies I is finitely generated is exactly Presented to the Society, March 27, 1971; received by the editors May 10,1971. AMS 1970 subject classifications. Primary 13A15, 13B25, 13C10; Secondary 13B20.