cohomology

# Contents

## Idea

Algebraic K-theory was initially a body of theory that attempted to generalise parts of linear algebra, notably the theory of dimension of vector spaces, and determinants,to modules over arbitrary rings. It has grown into a well developed tool for studying a wide range of algebraic, geometric and even analytic situations from a variety of points of view. It is thus difficult to give a single idea of what the subject is about. It has a side that looks at the manipulation and rewriting of the elementary operations of linear algebra, but also a definite infinity category aspect. Its development uses a lot of algebraic topology, particularly homotopy theory, both stable homotopy theory and the more simplicial parts of that area, and more recently has interacted with infinity category theory in various forms.

Some idea of its history is given in Algebraic K-theory, a historical perspective. A brief summary of the topics that are subsumed under the title of Algebraic K-theory can be gleaned from the list of subtopic pages and sections listed below:

## More recent history and the nPOV.

From the most recent nPOV, algebraic K-theory is an (∞,1)-functor that sends stable (∞,1)-categories to certain spectra – their K-theory spectra – characterized roughly speaking as the universal finitary Morita invariant of stable (∞,1)-categories under which all exact sequences of stable (∞,1)-categories split.

Traditionally the stable $(\infty,1)$-categories that have been considered are those coming from categories of chain complexes, and in this sense algebraic K-theory is the study of K-theory of categories more general than that of (bounded chain complexes of) vector bundles on a topological space, which is the topic of topological K-theory.

In its simplest initial form algebraic K-theory provides tools for computing the Grothendieck group of suitable categories. In its more refined form it studies the K-theory spectrum assigned to these categories. Crucial tools for this include the Q-construction? and Waldhausen S-construction.

Types of categories for which a theory of algebraic K-theory exist include notably the notions

Concrete examples of interest include for instance

and the like.

## More Details

There is a universal characterization of the construction of the K-theory spectrum $K(A)$ of a stable $(\infty,1)$-category $A$:

there is an $(\infty,1)$-functor

$U : (\infty,1)StabCat \to N$

to a stable $(\infty,1)$-category which is universal with the property that it respects colimits and exact sequences in a suitable way. Given any stable $(\infty,1)$-category $A$, its (connective or non-connective, depending on details) algebraic K-theory spectrum is the hom-object

$K(A) \simeq Hom(U(Sp), U(A)) \,,$

where $Sp$ denotes the stable $(\infty,1)$-category of compact spectra. (See the references below.)

When one talks about the algebraic $K$-theory of rings, one means the algebraic $K$-theory of the corresponding category of (one-sided) finitely-generated projective modules.

A K-theory should be given by a sequence of functors $K_i$ from some class of categories as above to abelian groups having some similarity to derived functors and cohomology theory for spaces. $K_0$ and $K_1$ are rather classical objects from the 1950s; higher $K$-groups are defined by Quillen in two steps: to a Quillen exact category $C$ one first associates a $K$-theory space $\mathcal{K}(C)$ (or in better versions, a $K$-theory spectrum) and then defines $K$-groups as homotopy groups of that space:

$K_i(C)=\pi_i(\mathcal{K}(C)).$

The $K$-theory space of $C$ in Quillen’s version was obtained as a classifying space of the Quillen $Q$-construction applied to $C$. The $Q$-construction has been refined to more sophisticated delooping methods by Waldhausen, Karoubi and others.

## Properties

### Red-shift conjecture

chromatic levelcomplex oriented cohomology theoryE-∞ ring/A-∞ ring
0ordinary cohomologyEilenberg-MacLane spectrum $H \mathbb{Z}$
0th Morava K-theory$K(0)$
1complex K-theorycomplex K-theory spectrum $KU$
first Morava K-theory$K(1)$
first Morava E-theory$E(1)$
2elliptic cohomology$Ell_E$
tmftmf spectrum
second Morava K-theory$K(2)$
second Morava E-theory$E(2)$
algebraic K-theory of KU$K(KU)$
$n$$n$th Morava K-theory$K(n)$
$n$th Morava E-theory$E(n)$
$n+1$algebraic K-theory applied to chrom. level $n$$K(E_n)$ (red-shift conjecture)
$\infty$
geometric contextuniversal additive bivariant (preserves split exact sequences)universal localizing bivariant (preserves all exact sequences in the middle)universal additive invariantuniversal localizing invariant
noncommutative algebraic geometrynoncommutative motives $Mot_{add}$noncommutative motives $Mot_{loc}$algebraic K-theorynon-connective algebraic K-theory
noncommutative topologyKK-theoryE-theoryoperator K-theory

## References

A reference for classical constructions is

The (infinity,1)-category theory picture is discussed in

(in terms of noncommutative motives) and in

For discussion of stable phenomena in algebraic K-theory, see section 4 of

• Ralph Cohen, Stability phenomena in the topology of moduli spaces (pdf)
Revised on December 12, 2013 00:11:58 by David Corfield (87.115.153.70)