A classification of nullity classes in the derived category of a ring

TL;DR

This paper classifies nullity classes and t-structures in a specific derived category of a Noetherian ring using invariants related to prime spectrum and perversity functions.

Contribution

It provides a complete invariant for nullity classes and t-structures in the derived category of finitely generated cofibrant objects over a Noetherian ring.

Findings

Complete classification of nullity classes in the category.

Establishment of invariants based on prime spectrum and perversity functions.

Connection between nullity classes and t-structures in the derived category.

Abstract

For a commutative Noetherian ring $R$ with finite Krull dimension, we study the nullity classes in $D^c_{fg}(R)$, the full triangulated subcategory $D^c_{fg}(R)$ of the derived category $D(R)$ consisting of objects which can be represented by cofibrant objects with each degree finitely generated. In the light of perversity functions over the prime spectrum $\mathrm{Spec} R$, we prove that there is a complete invariant of nullity classes thus that of aisles (or equivalently, $t$-structures) in $D^c_{fg}(R)$.