On the dual of Burch's inequality (Q1339390)

The author investigates quantities dual to those occurring in an inequality of \textit{L. Burch} and proves for them the following corresponding inequality. Let \({\mathfrak a}\subset{\mathfrak b}\) be ideals of a noethering ring \(A\) and \(M \neq 0\) an artinian \(A\)-module. Assume that \(A/{\mathfrak b}\) is artinian and \({\mathfrak b}\) is contained in some associated prime of \(M\). Then, for all sufficiently large \(i \in \mathbb{N}\), \[ s_{\mathfrak b} ({\mathfrak a}, M) \leq K \dim_ A (M)- \text{width}_{\mathfrak b} (0:_ M {\mathfrak a}^ i). \] Here \(K\dim\) denotes Roberts' dual Krull dimension [\textit{R. N. Roberts}, Q. J. Math., Oxf. II. Ser. 26, 269-273 (1975; Zbl 0311.13006)]. width is defined as by \textit{A. Ooishi} [Hiroshima Math. J. 6, 573-587 (1976; Zbl 0437.13007)], and the ``dual analytic spread'' \(s_{\mathfrak b} ({\mathfrak a},M)\) is the dual Krull dimension of \(\bigoplus_{i \geq 0} (0:_ M {\mathfrak a}^ i{\mathfrak b})/(0:_ M {\mathfrak a}^ i)\) over the Rees ring \(\bigoplus_{i \geq 0} {\mathfrak a}^ i\).