ﻻ يوجد ملخص باللغة العربية
We consider the algebraic K-theory of a truncated polynomial algebra in several commuting variables, K(k[x_1, ..., x_n]/(x_1^a_1, ..., x_n^a_n)). This naturally leads to a new generalization of the big Witt vectors. If k is a perfect field of positive characteristic we describe the K-theory computation in terms of a cube of these Witt vectors on N^n. If the characteristic of k does not divide any of the a_i we compute the K-groups explicitly. We also compute the K-groups modulo torsion for k=Z. To understand this K-theory spectrum we use the cyclotomic trace map to topological cyclic homology, and write TC(k[x_1, ..., x_n]/(x_1^a_1, ..., x_n^a_n)) as the iterated homotopy cofiber of an n-cube of spectra, each of which is easier to understand. Updated: This is a substantial revision. We corrected several errors in the description of the Witt vectors on a truncation set on N^n and modified the key proofs accordingly. We also replaces several topological statement with purely algebraic ones. Most arguments have been reworked and streamlined.
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(
We show that K_{2i}(Z[x,y]/(xy),(x,y)) is free abelian of rank 1 and that K_{2i+1}(Z[x,y]/(xy),(x,y)) is finite of order (i!)^2. We also compute K_{2i+1}(Z[x,y]/(xy),(x,y)) in low degrees.
The family of Thom spectra $y(n)$ interpolate between the sphere spectrum and the mod two Eilenberg-MacLane spectrum. Computations of Mahowald, Ravenel, and Shick and the authors show that the $E_1$ ring spectrum $y(n)$ has chromatic complexity $n$.
For primes $pgeq 5 $, $K(KU_p)$ -- the algebraic $K$-theory spectrum of $(KU)^{wedge}_p$, Morava $K$-theory $K(1)$, and Smith-Toda complex $V(1)$, Ausoni and Rognes conjectured (alongside related conjectures) that $L_{K(1)}S^0 mspace{-1.5mu}xrightarr
In this note we prove the analogue of the Atiyah-Segal completion theorem for equivariant twisted K-theory in the setting of an arbitrary compact Lie group G and an arbitrary twisting of the usually considered type. The theorem generalizes a result b