  We construct localization cofiber sequences for the topological Hochschild
homology (THH) and topological cyclic homology (TC) of spectral categories.
Using a global construction of the THH and TC 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 cofiber 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 THH and TC.
