Ratliff-Rush filtration, regularity and depth of higher associated graded modules. I (Q856342)

For a finitely generated module \(M\) over a noetherian local ring \((A,\mathfrak m)\) and \(I\) an ideal of definition for \(M\) one denotes by \(G_I(M)\) the associated graded module and by \(R(I)\) the Rees algebra. One of the main results of the paper gives a necessary and sufficient condition for depth\(\, G_{\mathfrak m ^c}(A) \geq 2\) for all \(c\gg 0\). It is also proved that if \(G_{I^c}(M)\) is Cohen-Macaulay for some positive \(c\), then \(G_I(M)\) is generalized Cohen-Macaulay. It is known that if grade\((I,M)\) is positive then the Ratliff-Rush filtration of \(M\) with respect to \(I\) coincides from some point on with the \(I\)-adic filtration. The paper contains an upper bound for the rank of the first coincidence. The proofs are based on a study of \(L^I(M)=\bigoplus _{n\geq 0} M/I^{n+1}M\), which is not finitely generated as a module over \(R(I)\). This novel technique is powerful in that it enables the author to generalize to modules some results of \textit{S. Huckaba} and \textit{T. Marley} [Nagoya Math. J. 133, 57--69 (1994; Zbl 0796.13004)] relating depths of \(R(I)\) and \(G_I(A)\), as well as a result of \textit{T. Marley} [Proc. Am. Math. Soc. 117, No. 2, 335--341 (1993; Zbl 0772.13006)] regarding local cohomology modules of \(G_I(A)\). The approach has the additional benefit of allowing for simpler proofs.