On the K-theory of truncated polynomial algebras over the integers

We show that the K_{2i}(Z[x]/(x^m),(x)) is finite of order (mi)!(i!)^{m-2} and that K_{2i+1}(Z[x]/(x^m),(x)) is free abelian of rank m-1. This is accomplished by showing that the equivariant homotopy groups of the topological Hochschild spectrum THH(Z) are finite, in odd degrees, and free abelian, in even degrees, and by evaluating their orders and ranks, respectively.Comment: Journal of Topology (to appear

On the K-theory of planar cuspical curves and a new family of polytopes

Let k be a regular F_p-algebra, let A = k[x,y]/(x^b - y^a) be the coordinate ring of a planar cuspical curve, and let I = (x,y) be the ideal that defines the cusp point. We give a formula for the relative K-groups K_q(A,I) in terms of the groups of de Rham-Witt forms of the ring k. At present, the validity of the formula depends on a conjecture that concerns the combinatorial structure of a new family of polytopes that we call stunted regular cyclic polytopes. The polytopes in question appear as the intersections of regular cyclic polytopes with (certain) linear subspaces. We verify low-dimensional cases of the conjecture. This leads to unconditional new results on K_2 and K_3 which extend earlier results by Krusemeyer for K_0 and K_1.Comment: 38 page

Topological Hochschild homology and the Hasse-Weil zeta function

We consider the Tate cohomology of the circle group acting on the topological Hochschild homology of schemes. We show that in the case of a scheme smooth and proper over a finite field, this cohomology theory naturally gives rise to the cohomological interpretation of the Hasse-Weil zeta function by regularized determinants envisioned by Deninger. In this case, the periodicity of the zeta function is reflected by the periodicity of said cohomology theory, whereas neither is periodic in general

On the K-theory of local fields

The authors establish a connection between the Quillen K-theory of certain local fields and the de Rham-Witt complex of their rings of integers with logarithmic poles at the maximal ideal. They consider fields K that are complete discrete valuation fields of characteristic zero with perfect residue fields k of characteristic p > 2. They evaluate the K-theory with Z/p^v-coefficients of K, and verify the Lichtenbaum-Quillen conjecture for K.Comment: Abstract added in migration; 113 pages, published versio

Bi-relative algebraic K-theory and topological cyclic homology

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