Tilting complexes and codimension functions over commutative noetherian rings

TL;DR

This paper constructs silting objects in derived categories of commutative noetherian rings based on sp-filtrations, exploring conditions under which these objects are tilting, especially involving dualizing complexes and codimension functions.

Contribution

It introduces a new explicit construction of silting objects linked to sp-filtrations and characterizes when they are tilting using dualizing complexes and Cohen-Macaulay conditions.

Findings

Silting objects are explicitly constructed from sp-filtrations.

Tilting occurs when the sp-filtration comes from a codimension function with a dualizing complex.

In absence of a dualizing complex, tilting relates to the ring being a Cohen-Macaulay homomorphic image.

Abstract

In the derived category of a commutative noetherian ring, we explicitly construct a silting object associated with each sp-filtration of the Zariski spectrum satisfying the "slice" condition. Our new construction is based on local cohomology and it allows us to study when the silting object is tilting. For a ring admitting a dualizing complex, this occurs precisely when the sp-filtration arises from a codimension function on the spectrum. In the absence of a dualizing complex, the situation is more delicate and the tilting property is closely related to the condition that the ring is a homomorphic image of a Cohen-Macaulay ring. We also provide dual versions of our results in the cosilting case.