nLab iterated algebraic K-theory





Special and general types

Special notions


Extra structure



Higher algebra




The construction of algebraic K-theory K(R)K(R), originally defined for rings RR, generalizes to A A_\infty -ring spectra. When RR happens to be a connective E E_\infty -ring spectrum, then also the representing spectrum K(R)K(R) of its algebraic K-theory is a connective E E_\infty -ring spectrum (Schwänzl & Vogt 1994, Thm. 1, EKMM 1997, Thm. 6.1) so that this construction may then be iterated (Rognes 2014) 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).


Red-shift conjecture

See at red-shift conjecture.


Algebraic K-theory of ring spectra

On the algebraic K-theory of (connective) ring spectra:

The algebraic K-theory 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 iteration of the construction and the red-shift conjecture:

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:

which can also be understood as a special case of K-theory for 2-categories:

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 March 20, 2024 at 08:18:08. See the history of this page for a list of all contributions to it.