The finiteness of $I$ when $\textit {R}[\textit {X}]/\textit {I}$ 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 $R[X]/I$ 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 (An example shows that the first statement is no longer valid when $\min I$ does not contain a regular element.)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) $R[X]/I$ 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 $\xi$ denote the equivalence class of $X$ in $R[X]/I$, and let $\langle 1,\xi , \cdots ,{\xi ^t}\rangle$ denote the $R$-module generated by $1,\xi , \cdots ,{\xi ^t}$. The following statements are also equivalent: (i) $\langle 1,\xi , \cdots ,{\xi ^t}\rangle$ is flat for all $t \geqslant 0$; (ii) $\langle 1,\xi , \cdots ,{\xi ^t}\rangle$ is flat for some $t \geqslant 0$ for which $1,\xi , \cdots ,{\xi ^t}$ are linearly dependent over $R$; (iii) $I = ({f_1}, \cdots ,{f_n}),{f_i} \in \min I$, 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 $\xi$ is regular in $R[\xi ]$, and in this case (i)-(iv) are also equivalent to the assertion that $R[\xi ]$ and $R[1/\xi ]$ are flat $R$-modules.