We construct a triangulated category of cyclotomic complexes (homological analogues of the cyclotomic spectra of B?kstedt and Madsen) along with a version of the topological cyclic homology functor TC for cyclotomic complexes and an equivariant homology functor (commuting with TC) from cyclotomic spectra to cyclotomic complexes. We also prove that the category of cyclotomic complexes essentially coincides with the twisted 2-periodic derived category of the category of filtered Dieudonné modules, which were introduced by Fontaine and Lafaille. Under certain conditions we show that the functor TC on cyclotomic complexes is the syntomic cohomology functor
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