  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.
