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Let $K$ be a complete discrete valuation field with finite residue field of characteristic $p$, and let $D$ be a central division algebra over $K$ of finite index $d$. Thirty years ago, Suslin and Yufryakov showed that for all prime numbers $\ell$ different from $p$ and integers $j \geq 1$ , there exists a "reduced norm" isomorphism of $\ell$-adic $K$-groups $\operatorname{Nrd}_{D/K} \colon K_j(D,\mathbb{Z}_{\ell}) \to K_j(K,\mathbb{Z}_{\ell})$ such that $d \cdot \operatorname{Nrd}_{D/K}$ is equal to the norm homomorphism $N_{D/K}$. The purpose of this paper is to prove the analogous result for the $p$-adic $K$-groups. To do so, we employ the cyclotomic trace map to topological cyclic homology and show that there exists a "reduced trace" equivalence $\operatorname{Trd}_{A/S} \colon \operatorname{THH}(A\,|\,D,\mathbb{Z}_p) \to \operatorname{THH}(S\,|\,K,\mathbb{Z}_p)$ between two $p$-complete cyclotomic spectra associated with $D$ and $K$, respectively. Interestingly, we show that if $p$ divides $d$, then it is not possible to choose said equivalence such that, as maps of cyclotomic spectra, $d \cdot \operatorname{Trd}_{A/S}$ agrees with the trace $\operatorname{Tr}_{A/S}$, although this is possible as maps of spectra with $\mathbb{T}$-action.

Let $K$ be a complete discrete valuation field of characteristic zero with residue field $k_K$ of characteristic $p\gt 0$ . Let $L/K$ be a finite Galois extension with Galois group $G\,{=}\,G_{L/K}$ and suppose that the induced extension of residue fields $k_L/k_K$ is separable. By the normal basis theorem, $L$ is always a projective $K[G]$ -module. But the ring of integers $\mathcal{O}_L$ is a projective $\mathcal{O}_K[G]$ -module if and only $L/K$ is tamely ramified. For wildly ramified extensions, the structure of $\mathcal{O}_L$ as an $\mathcal{O}_K[G]$ -module is very complicated and quite far from understood. We propose that the ring of Witt vectors $W\s(\mathcal{O}_L)$ is a more well-behaved object. Indeed, we show that for a large class of extensions $L/K$ , the pro-abelian group $H^1(G,W\s(\mathcal{O}_L))$ is zero. We conjecture that this is true in general.

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.

This paper calculates the relative algebraic K -theory K∗(k [x ]/(x n ), (x )) of a truncated polynomial algebra over a perfect field k of positive characteristic p. Since the ideal generated by x is nilpotent, we can apply McCarthy’s theorem: the relative algebraic K -theory is isomorphic to the relative topological cyclic homology, [Mc], and it is the latter groups we actually evaluate. The result is best expressed in terms of big Witt vectors. Let Wm (k ) denote the big Witt vectors in k of length m , i.e. the multiplicative group Wm (k ) = (1 + xk [[x ]])×/(1 + x k [[x ]])×

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)

Let A a regular F_p-algebra. The relative K-groups K_q(A[x]/(x^m),(x)) and the Nil-groups Nil_q(A[x]/(x^m)) were evaluated earlier by the author and Madsen in terms of the big de Rham-Witt groups of the ring A. In this paper, we evaluate the maps of relative K-groups and Nil-groups induced by the canonical projection from A[x]/(x^m) to A[x]/(x^n). The result depends strongly on the prime p. It generalizes work of Stienstra on the groups in degrees 2 and 3.

Let k a regular noetherian p-algebra, let A = k[x, y]/(xy) be the coordinate ring of the coordinate axes in the affine k-plane, and let I = (x,y) be the ideal that defines the intersection point. We evaluate the relative K-groups Kq(A, I) completely in terms of the big de Rham-Witt groups of k. This generalizes a formula for K1(A, I) and K2(A, I) by Dennis and Krusemeyer.