Dimension filtration of the bounded Derived category of a Noetherian ring

TL;DR

This paper introduces a dimension filtration of the bounded derived category of a Noetherian ring, analyzing the structure of the resulting quotient categories and their properties, including hearts and AR-triangles in the regular case.

Contribution

It provides a detailed structural analysis of the dimension-filtered subcategories and their quotients in the derived category of a Noetherian ring, including identification of hearts and conditions for AR-triangles.

Findings

Each quotient category $ ext{C}_i(A)$ is a Krull-Remak-Schmidt triangulated category with a bounded t-structure.

The heart of $ ext{C}_i(A)$ is explicitly identified.

If $A$ is regular, then $ ext{C}_i(A)$ has Auslander-Reiten triangles.

Abstract

Let $A$ be a Noetherian ring of dimension $d$ and let $\mathcal{D}^b(A)$ be the bounded derived category of $A$. Let $\mathcal{D}_i^b(A)$ denote the thick subcategory of $\mathcal{D}^b(A)$ consisting of complexes $\mathbf{X}_\bullet$ with $\dim H^n(\mathbf{X}_\bullet) \leq i$ for all $n$. Set $\mathcal{D}_{-1}^b(A) = 0$. Consider the Verdier quotients $\mathcal{C}_i(A) = \mathcal{D}_i^b(A)/\mathcal{D}_{i-1}^b(A)$. We show for $i = 0, \ldots, d$, $\mathcal{C}_i(A)$ is a Krull-Remak-Schmidt triangulated category with a bounded $t$-structure. We identify its heart. We also prove that if $A$ is regular then $\mathcal{C}_i(A)$ has AR-triangles. We also prove that $$ \mathcal{C}_i(A) \cong \bigoplus_{\stackrel{P}{\dim A/P = i}} \mathcal{D}_0^b(A_P). $$