We construct localization cofibration sequences for the topological Hochschild homology (T H H) and topological cyclic homology (T C) of small spectral categories. Using a global construction of the T H H and T C of a scheme in terms of the perfect complexes in a spectrally enriched version of the category of unbounded complexes, the sequences specialize to localization cofibration sequences associated to the inclusion of an open subscheme. These are the targets of the cyclotomic trace from the localization sequence of Thomason–Trobaugh in K–theory. We also deduce versions of Thomason’s blow-up formula and the projective bundle formula for T H H and T C.