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 A n = A [ x ] / x n . This description involves the projection and transfer maps relating the relative K-theory spectra K ˜ ( A n ) when n varies. From this point of view the cyclotomic trace corresponds to multiplication by the units 1 + x + ⋯ + x n - 1 in K ˜ 1 ( Z [ x ] / x n ) .