A sub-functor for Ext and Cohen-Macaulay associated graded modules with bounded multiplicity-II

Let $(A,\mathfrak{m})$ be a \CM\ local ring, then notion of $T$-split sequence was introduced in part-1 of this paper for $\mathfrak{m}$-adic filtration with the help of numerical function $e^T_A$. We have explored the relation between AR-sequences and $T$-split sequences. For a Gorenstein ring $(A,\mathfrak{m})$ we define a Hom-finite Krull-Remak-Schmidt category $\mathcal{D}_A$ as a quotient of the stable category $\CMS(A)$. This category preserves isomorphism, i.e. $M\cong N$ in $\mathcal{D}_A$ if and only if $M\cong N$ in $\CMS(A)$.This article has two objectives; first objective is to extend the notion of $T$-split sequence, and second objective is to explore function $e^T_A$ and $T$-split sequence. When $(A,\mathfrak{m})$ be an \anram\ \CM\ local ring and $I$ be an $\mathfrak{m}$-primary ideal then we extend the techniques in part-1 of this paper to the integral closure filtration with respect to $I$ and prove a version of Brauer-Thrall-II for a class of such rings.