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 c
all 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.
We investigate various module-theoretic properties of Koszul homology under mild conditions. These include their depth, $S_2$-property and their Bass numbers
Let $R$ be a Cohen-Macaulay local ring with a canonical module $omega_R$. Let $I$ be an $m$-primary ideal of $R$ and $M$, a maximal Cohen-Macaulay $R$-module. We call the function $nlongmapsto ell (Hom_R(M,{omega_R}/{I^{n+1} omega_R}))$ the dual Hilb
ert-Samuel function of $M$ with respect to $I$. By a result of Theodorescu this function is a polynomial function. We study its first two normalized coefficients.
Let $Q$ be a Noetherian ring with finite Krull dimension and let $mathbf{f}= f_1,... f_c$ be a regular sequence in $Q$. Set $A = Q/(mathbf{f})$. Let $I$ be an ideal in $A$, and let $M$ be a finitely generated $A$-module with $projdim_Q M$ finite. Set
$R = bigoplus_{ngeq 0}I^n$, the Rees-Algebra of $I$. Let $N = bigoplus_{j geq 0}N_j$ be a finitely generated graded $R$-module. We show that [bigoplus_{jgeq 0}bigoplus_{igeq 0} Ext^{i}_{A}(M,N_j) ] is a finitely generated bi-graded module over $Sc = R[t_1,...,t_c]$. We give two applications of this result to local complete intersection rings.
Let $(A,m)$ be a Noetherian local ring, let $M$ be a finitely generated CM $A$-module of dimension $r geq 2$ and let $I$ be an ideal of definition for $M$. Set $L^I(M) = bigoplus_{ngeq 0}M/I^{n+1}M$. In part one of this paper we showed that $L^I(M)
$ is a module over $R$, the Rees algebra of $I$ and we gave many applications of $L^I(M)$ to study the associated graded module, $G_I(M)$. In this paper we give many further applications of our technique; most notable is a reformulation of a classical result due to Narita in terms of the Ratliff-Rush filtration. This reformulation can be extended to all dimensions $geq 2$.
We give an elementary proof of Grothendiecks non-vanishing Theorem: For a finitely generated non-zero module $M$ over a Noetherian local ring $A$ with maximal ideal $m$, the local cohomology module $H^{dim M}_{m}(M)$ is non-zero.
