Algebraic K-theory of topological K-theory

By Christian Ausoni, John Rognes. 2002

107 citations recorded by the source corpus.

We are interested in the arithmetic of ring spectra. To make sense of this we must work with structured ring spectra, such as S-algebras [EKMM], symmetric ring spectra [HSS] or Γ-rings [Ly]. We will refer to these as Salgebras. The commutative objects are then commutative S-algebras. The category of rings is embedded in the category of S-algebras by the Eilenberg– MacLane functor R →HR. We may therefore view an S-algebra as a generalization of a ring in the algebraic sense. The added flexibility of S-algebras provides room for new examples and constructions, which may eventually also shed light upon the category of rings itself. In algebraic number theory the arithmetic of the ring of integers in a number field is largely captured by its Picard group, its unit group and its Brauer group. These are

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

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Citations in the corpus

The localization sequence for the algebraic K-theory of topological K-theory (2006, methodology): “Hesselholt observed that the Ausoni-Rognes and Ausoni calculations of T HH(l) and T HH(ku) in [ 2 , 1] are explained by the existence of a T HH version of this localization sequence along with a conjecture about the behavior of T HH for “tamely ramified” extensions of ring spectra.”
Units of ring spectra and their traces in algebraic K-theory (2004, background): “This example is interesting in view of the attempts [1], [2], to relate algebraic Ktheory to elliptic cohomology and the chromatic filtration of homotopy theory.”
The Local Structure of Algebraic K-Theory (2012, methodology): “So, in Ar([11]) there is a unique morphism from the object (2 ≤ 4) to (3 ≤ 7) and no morphism the other way.”
Hopf algebra structure on topological Hochschild homology (2005, background): “these ﬁelds. Particular computations for other S-algebras, related to topological K-theory, have revealed a more general pattern of how algebraic K-theory creates a “red-shift” in chromatic ﬁltration [AR02], and satisﬁes a Galois descent property [Au]. These results indicate that the algebraic K-the”
On the algebraic K‐theory of higher categories (2012, background): “We now consider the following simplicial functor U ′ : Pair∞ Fun([1],Cat Δ ∞).”
Higher traces, noncommutative motives, and the categorified Chern character (2015, methodology): “From this perspective, the analogy with elliptic cohomology is made precise by the work of Ausoni and Rognes [3,2], who showed that K(K(C)) is a spectrum of telescopic complexity 2 at primes ≥ 5.”