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

Abstract

Abstract Let $(A,\mathfrak{m})$ be a Cohen–Macaulay local ring, and then the notion of a $T$ -split sequence was introduced in the part-1 of this paper for the $\mathfrak{m}$ -adic filtration with the help of the numerical function $e^T_A$ . In this article, we explore the relation between Auslander–Reiten (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 $\underline{\mathrm{CM}}(A)$ . This category preserves isomorphism, that is, $M\cong N$ in $\mathcal{D}_A$ if and only if $M\cong N$ in $\underline{\mathrm{CM}}(A)$ .This article has two objectives: first objective is to extend the notion of $T$ -split sequences, and second objective is to explore the function $e^T_A$ and $T$ -split sequences. When $(A,\mathfrak{m})$ is an analytically unramified Cohen–Macaulay local ring and $I$ is 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.