Gorenstein 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 Gorenstein 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 Gorenstein 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.