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 $A$, $G_{\mathfrak{a}}(A)={⨁}_{n\ge 0}{\mathfrak{a}}^n\mathrm{/}{\mathfrak{a}}^{n+1}$ 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}$}$ofthedual$Hom_A(M,A)$. 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.