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# Complete intersection Approximation, Dual Filtrations and Applications

Tony J. Puthenpurakal
Arxiv ID: 0807.0471Last 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.