Tilting complexes and codimension functions over commutative noetherian rings
Michal Hrbek, Tsutomu Nakamura, Jan Šťovíček
TL;DR
This work constructs explicit silting objects in the derived category of a commutative noetherian ring from slice sp-filtrations, using local cohomology to decompose objects into computable pieces. It shows that the resulting silting object T_Φ is always silting for a slice filtration, and it detangles when T_Φ is tilting precisely by requiring Φ to be a codimension filtration, with the tilting property obtainable under dualizing-complex or Cohen–Macaulay image assumptions. The dual cosilting theory is developed in parallel, yielding cosilting objects C_Φ and a Cohen–Macaulay heart, and endomorphism-ring flatness is established for codimension-function-induced objects. These results bridge silting/cosilting theory with depth/width, sp-filtrations, and Kawasaki’s CM-imaging theory, offering a robust tilting framework for commutative noetherian rings and illuminating when such rings arise as homomorphic images of CM rings. The work further explicates endomorphism structures and provides dimension-specific refinements, including a complete two-dimensional local-ring case where tilting/cotilting equivalence aligns with CM-ness.
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.