The cyclotomic trace and curves on K-theory

We give a functorial description of the topological cyclic homology of a ring A in terms of the relative algebraic K-theory of the truncated polynomial rings An=A[x]/xn. This description involves the projection and transfer maps relating the relative K-theory spectra K˜(An) when n varies. From this point of view the cyclotomic trace corresponds to multiplication by the units 1+x+⋯+xn-1 in K˜1(Z[x]/xn).