Classifying Compactly generated t-structures on the derived category of a Noetherian ring

TL;DR

This paper classifies compactly generated t-structures on the derived category of modules over a Noetherian ring using filtrations by supports, revealing their structure and conditions under which they correspond to all t-structures in certain cases.

Contribution

It provides a classification of compactly generated t-structures on derived categories of Noetherian rings via filtrations satisfying the weak Cousin condition, extending to rings with dualizing complexes.

Findings

Classification of t-structures via filtrations by supports.

Equivalence of all t-structures with those induced by filtrations satisfying the weak Cousin condition when a dualizing complex exists.

Extension of classification to rings with pointwise dualizing complexes.

Abstract

We classify complactly generated t-structures on the derived category of modules over a commutative Noetherian ring R in terms of decreasing filtrations by supports on Spec(R). A decreasing filtration by supports \phi : Z -> Spec(R) satisfies the weak Cousin condition if for any integer i \in Z, the set \phi(i) contains all the inmediate generalizations of each point in \phi(i+1). Every t-structure on D^b_fg(R) (equivalently, on D^-_fg(R)) is induced by complactly generated t-structures on D(R) whose associated filtrations by supports satisfy the weak Cousin condition. If the ring R has dualizing complex we prove that these are exactly the t-structures on D^b_fg(R). More generally, if R has a pointwise dualizing complex we classify all compactly generated t-structures on D_fg(R).