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Register a new userChromatic Complexity of the Algebraic K-theory of $y(n)$
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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.
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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.
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.
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 $K(KU_p)$.
In this paper, we study the global structure of an algebraic avatar of the derived category of ind-coherent sheaves on the moduli stack of formal groups. In analogy with the stable homotopy category, we prove a version of the nilpotence theorem as well as the chromatic convergence theorem, and construct a generalized chromatic spectral sequence. Furthermore, we discuss analogs of the telescope conjecture and chromatic splitting conjecture in this setting, using the local duality techniques established earlier in joint work with Valenzuela.
These are some notes on the basic properties of algebraic K-theory and G-theory of derived algebraic spaces and stacks, and the theory of fundamental classes in this setting.
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