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Register a new userOn the algebraic K-theory of the coordinate axes over the integers
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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.
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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.
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.
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 $n$. We show that topological periodic cyclic homology of $y(n)$ has chromatic complexity $n+1$. This gives evidence that topological periodic cyclic homology shifts chromatic height at all chromatic heights, supporting a variant of the Ausoni--Rognes red-shift conjecture. We also show that relative algebraic K-theory, topological cyclic homology, and topological negative cyclic homology of $y(n)$ at least preserve chromatic complexity.
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 $L_{K(1)}S^0 mspace{-1.5mu}xrightarrow{mspace{-2mu}text{unit} , i}~mspace{-7mu}(KU)^{wedge}_p$ induces a map $K(L_{K(1)}S^0) wedge v_2^{-1}V(1) to K(KU_p)^{hmathbb{Z}^times_p} wedge v_2^{-1}V(1)$ that is an equivalence. Since the definition of this map is not well understood, we consider $K(L_{K(1)}S^0) wedge v_2^{-1}V(1) to (K(KU_p) wedge v_2^{-1}V(1))^{hmathbb{Z}^times_p}$, which is induced by $i$ and also should be an equivalence. We show that for any closed $G < mathbb{Z}^times_p$, $pi_ast((K(KU_p) wedge v_2^{-1}V(1))^{hG})$ is a direct sum of two pieces given by (co)invariants and a coinduced module, for $K(KU_p)_ast(V(1))[v_2^{-1}]$. When $G = mathbb{Z}^times_p$, the direct sum is, conjecturally, $K(L_{K(1)}S^0)_ast(V(1))[v_2^{-1}]$ and, by using $K(L_p)_ast(V(1))[v_2^{-1}]$, where $L_p = ((KU)^{wedge}_p)^{hmathbb{Z}/((p-1)mathbb{Z})}$, the summands simplify. The Ausoni-Rognes conjecture suggests that in [(-)^{hmathbb{Z}^times_p} wedge v_2^{-1}V(1) simeq (K(KU_p) wedge v_2^{-1}V(1))^{hmathbb{Z}^times_p},] $K(KU_p)$ fills in the blank; we show that for any $G$, the blank can be filled by $(K(KU_p))^mathrm{dis}_mathcal{O}$, a discrete $mathbb{Z}^times_p$-spectrum built out of $K(KU_p)$.
We prove that the theory of all modules over the ring of algebraic integers is decidable.
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