nLab
iterated algebraic K-theory

Contents

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Higher algebra

Contents

Idea

General

The construction of algebraic K-theory K(R)K(R), originally defined for rings RR, generalizes to ring spectra. But algebraic K-theory is itself represented by a ring spectrum, so that this construction may then be iterated to yield iterated algebraic K-theories K(K(R))K(K(R)), K(K(K(R)))K(K(K(R))), etc.

The red-shift conjecture says that this iteration plays a special role in chromatic homotopy theory.

On topological K-theory

The construction of iterated algebraic K-theory has received particular attention for the case that R=R = ku is the connective ring spectrum representing complex topological K-theory.

Here the first iterated stage K(ku)K(ku) is related to BDR 2-vector bundles essentially like ku is related to ordinary complex vector bundles.

The tower K 2r(ku)K^{2r}(ku) of higher iterated algebraic K-theories of topological K-theory has been shown to accommodate a generalization of the Fourier-Mukai-type transform on twisted K-theory that is given by topological T-duality, generalizing it to spherical T-duality (Lind-Sati-Westerland 16).

Properties

Red-shift conjecture

References

Algebraic K-theory of ring spectra

On the algebraic K-theory of ring spectra:

The algebraic K-theory of specifically of suspension spectra of loop spaces (Waldhausen’s A-theory) is originally due to

  • Friedhelm Waldhausen, Algebraic K-theory of spaces, In: A. Ranicki N., Levitt, F. Quinn (eds.), Algebraic and Geometric Topology, Lecture Notes in Mathematics, vol 1126. Springer, Berlin, Heidelberg (1985) (doi:10.1007/BFb0074449)

On the algebraic K-theory K(R)K(R) of a ring spectrum RR as the Grothendieck group of (∞,1)-module bundles over RR:

Algebraic K-theory of topological K-theory

On the first algebraic K-theory K(ku)K(ku) of connective topological K-theory:

Interpretation of K(ku)K(ku) as the K-theory of BDR 2-vector bundles:

Algebraic K-theory of algebraic K-theory

On the algebraic K-theory of algebraic K-theory of finite fields K(K(𝔽))K(K(\mathbb{F})):

Higher iterated algebraic K-theory of topological K-theory

Discussion of higher and of twisted iterated K-theory on kuku, and realization of the spherical T-duality on twisted K 2r(ku)K^{2r}(ku):

Last revised on September 4, 2020 at 09:33:38. See the history of this page for a list of all contributions to it.