  It is well-known that algebraic K-theory preserves products of rings.
However, in general, algebraic K-theory does not preserve fiber-products of
rings, and bi-relative algebraic K-theory measures the deviation. It was proved
by Cortinas that,rationally, bi-relative algebraic K-theory and bi-relative
cyclic homology agree. In this paper, we show that, with finite coefficients,
bi-relative algebraic K-theory and bi-relative topological cyclic homology
agree. As an application, we show that for a, possibly singular, curve over a
perfect field of positive characteristic p, the cyclotomic trace map induces an
isomorphism of the p-adic algebraic K-groups and the p-adic topological cyclic
homology groups in non-negative degrees. As a further application, we show that
the difference between the p-adic K-groups of the integral group ring of a
finite group and the p-adic K-groups of a maximal Z-order in the rational group
algebra can be expressed entirely in terms of topological cyclic homology.
