# nLab iterated algebraic K-theory

Contents

cohomology

### Theorems

#### Higher algebra

higher algebra

universal algebra

# Contents

## Idea

### General

The construction of algebraic K-theory $K(R)$, originally defined for rings $R$, 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 $K(K(R))$, $K(K(K(R)))$, 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 $R =$ ku is the connective ring spectrum representing complex topological K-theory.

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

The tower $K^{2r}(ku)$ 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).

## 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 $K(R)$ of a ring spectrum $R$ as the Grothendieck group of (∞,1)-module bundles over $R$:

### Algebraic K-theory of topological K-theory

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

Interpretation of $K(ku)$ 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 $K(K(\mathbb{F}))$:

• Gabe Angelini-Knoll, Detecting the $\beta$-family in iterated algebraic K-theory of finite fields, Wayne State University 2017 (arXiv:1810.10088, oa_dissertations:1778)

### Higher iterated algebraic K-theory of topological K-theory

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

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