# Complete intersection Approximation, Dual Filtrations and Applications

Tony J. Puthenpurakal

Arxiv ID: 0807.0471•Last updated: 6/25/2021

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.

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