Cyclotomic complexes

TL;DR

This paper develops a new triangulated category of cyclotomic complexes, linking homological algebra with cyclotomic spectra, and relates it to topological cyclic homology and syntomic cohomology.

Contribution

It constructs a triangulated category of cyclotomic complexes and establishes its connection with cyclotomic spectra and topological cyclic homology.

Findings

Category of cyclotomic complexes is a twisted 2-periodic derived category of filtered Dieudonne modules.

The TC functor on cyclotomic complexes corresponds to syntomic cohomology under certain conditions.

Provides a homological framework for understanding cyclotomic spectra and their invariants.

Abstract

We construct a triangulated category of cyclotomic complexes, a homological counterpart of cyclotomic spectra of Bokstedt and Madsen. We also construct a version of the Topological Cyclic Homology functor TC for cyclotomic complexes, and an equivariant homology functor from cycloctomic spectra to cyclotomic complexes which commutes with TC. Then on the other hand, we prove that the category of cyclotomic complexes is essentially a twisted 2-periodic derived category of the category of filtered Dieudonne modules of Fontaine and Lafaille. We also show that under some mild conditions, the functor TC on cyclotomic complexes is the syntomic cohomology functor.