Dimension filtration of the bounded Derived category of a Noetherian ring

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 $\dim H^n({\mathbf{X}}_{\bullet })\le i$ for all $n$. Set ${\mathcal{D}}_{-1}^b(A)=0$. Consider the Verdier quotients ${\mathcal{C}}_i(A)={\mathcal{D}}_i^b(A)\mathrm{/}{\mathcal{D}}_{i-1}^b(A)$. We show for $i=0,\dots ,d$, ${\mathcal{C}}_i(A)$ 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 $${\mathcal{C}}_i(A)\cong \underset{}{⨁}{\mathcal{D}}_0^b(A_P)\mathrm{.}$$