Complete intersection Approximation, Dual Filtrations and Applications

We give a two step method to study certain questions regarding associated graded module of a Cohen-Macaulay (CM) module $M$ w.r.t an $mathfrak{m}$-primary ideal $mathfrak{a}$ in a complete Noetherian local ring $(A,mathfrak{m})$. The first step, we call it complete intersection approximation, enables us to reduce to the case when both $A$, $ G_mathfrak{a}(A) = bigoplus_{n geq 0} mathfrak{a}^n/mathfrak{a}^{n+1} $ are complete intersections and $M$ is a maximal CM $A$-module. The second step consists of analyzing the classical filtration ${Hom_A(M,mathfrak{a}^n) }_{mathbb{Z}}$ of the dual $Hom_A(M,A)$. We give many applications of this point of view. For instance let $(A,mathfrak{m})$ be equicharacteristic and CM. Let $a(G_mathfrak{a}(A))$ be the $a$-invariant of $G_mathfrak{a}(A)$. We prove: 1. $a(G_mathfrak{a}(A)) = -dim A$ iff $mathfrak{a}$ is generated by a regular sequence. 2. If $mathfrak{a}$ is integrally closed and $a(G_mathfrak{a}(A)) = -dim A + 1$ then $mathfrak{a}$ has minimal multiplicity. We extend to modules a result of Ooishi relating symmetry of $h$-vectors. As another application we prove a conjecture of Itoh, if $A$ is a CM local ring and $mathfrak{a}$ is a normal ideal with $e_3^mathfrak{a}(A) = 0$ then $G_mathfrak{a}(A)$ is CM.