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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 $K(KU_p)$.
Let $p$ be a prime, $n geq 1$, $K(n)$ the $n$th Morava $K$-theory spectrum, $mathbb{G}_n$ the extended Morava stabilizer group, and $K(A)$ the algebraic $K$-theory spectrum of a commutative $S$-algebra $A$. For a type $n+1$ complex $V_n$, Ausoni and
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 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 positiv
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.
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