cohomology

# Contents

## Idea

Algebraic K-theory is about natural constructions of cohomology theories/spectra from algebraic data such as commutative rings, symmetric monoidal categories and various homotopy theoretic refinements of these.

From a modern perspective, the algebraic K-theory spectrum $\mathbf{K}(R)$ of a commutative ring is simply the ∞-group completion? of algebraic vector bundles on $Spec(R)$; this will be discussed in more detail below. In particular there is a natural concept of algebraic K-theory of “brave new rings”, i.e. of ring spectra/E-∞ rings.

Historically, the algebraic K-theory of a commutative ring $R$ (what today is the “0th” algebraic K-theory group) was originally defined to be the Grothendieck group of its symmetric monoidal category of projective modules (under tensor product of modules). Under the relation between modules and vector bundles, this is directly analogous to the basic definition of topological K-theory and hence the common term. (In fact when applied to the stack of vector bundles then algebraic K-theory subsumes topological K-theory and also differential K-theory, see below).

There are canonical maps $K_0(R)\to Pic(R)$ from the 0th algebraic K-theory of a ring to its Picard group and $K_1(R)\to GL_1(R)$ from the first algebraic K-theory group of $R$ to its group of units which are given in components by the determinant functor. This fact is sometimes used to motivate algebraic K-theory as a “generalization of linear algebra” (see e.g. this MO discussion). This is also how the traditional regulator of a number field relates to Beilinson regulators of algebraic K-theory.

More generally, following the axiomatics of generalized (Eilenberg-Steenrod) cohomology any algebraic K-theory should be given by a sequence of functors $K_i$ from some suitable class of categories of “algebraic nature” to abelian groups, satisfying some natural conditions. Moreover, following the Brown representability theorem these groups should arise as the homotopy groups of a spectrum, the algebraic K-theory spectrum. Classical constructions producing this by combinatorial means are known as the Quillen Q-construction defined on Quillen exact categories and more generally the Waldhausen S-construction defined on Waldhausen categories.

For more on the history of the subject see (Arlettaz 04, Grayson 13) and see at at Algebraic K-theory, a historical perspective.

There are two ways to think of the traditional algebraic K-theory of a commutative ring more conceptually: on the one hand this construction is the group completion of the direct sum symmetric monoidal-structure on the category of modules, on the other hand it is the group completion of the addition operation expressed by short exact sequences in that category. This leads to the two modern ways of expressing and viewing algebraic K-theory:

1. monoidal. The core of a symmetric monoidal category or more generally of a symmetric monoidal (∞,1)-category has a universal completion to an abelian ∞-group/connective spectrum optained by universally adjoining inverses to the symmetric monoidal operation – the ∞-group completion. This yields the concept of algebraic K-theory of a symmetric monoidal category and more generally that of algebraic K-theory of a symmetric monoidal (∞,1)-category;

2. exact/stable. Analogously, inverting the addition operation expressed by the exact sequences in an abelian category or more generally in a stable (∞,1)-category yields the algebraic K-theory of a stable (∞,1)-category. Explicit ways to express this are known as the Quillen Q-construction and the Waldhausen S-construction. This turns out to be a universal construction in the context of non-commutative motives.

Here the second construction may be understood as first splitting the exact sequences and then applying the first construction to the resulting direct sum monoidal structure. Typically the first construction here contains more information but is harder to compute, and vice versa (see also MO-discussion here and here).

Both of these constructions produce a spectrum (hence representing a generalized (Eilenberg-Steenrod) cohomology theory) – called the K-theory spectrum – and the algebraic K-theory groups are the homotopy groups of that spectrum.

The classical case of the algebraic K-theory of a commutative ring $R$ is a special case of this general concept of algebraic K-theory by either forming the symmetric monoidal category $(Mod(R), \oplus)$ and applying the abelian ∞-group-completion to that, or else forming the stable (∞,1)-category of chain complexes of $R$-modules and applyong the Waldhausen S-construction to that. In both cases the result is a spectrum whose degree-0 homotopy group is the ordinary algebraic K-theory of $R$ as given by the Grothendieck group and whose higher homotopy groups are its higher algebraic K-theory groups.

## Constructions

### Symmetric monoidal K-theory

For a symmetric monoidal category $C$, K-theory may be defined by taking

See at

### Quillen Q-construction (for exact categories)

Given an exact category $E$, one defines $K(E)$ by applying

See at

### Waldhausen $S_\bullet$-construction (for Waldhausen categories)

Given a Waldhausen category $(C, w C)$, one defines its $K$-theory by applying

There is also a Waldhausen S-construction for stable (infinity,1)-categories and, most generally, for Waldhausen (infinity,1)-categories?.

See at

## Examples

### For rings

We recall several constructions of the algebraic K-theory of a ring. See (Weibel, IV.4.8, IV.4.11.1) for details.

#### Plus construction

Given an associative unital ring $R$, one may define the algebraic K-theory space $K(R) = BGL(R)^+$ by taking

#### Direct sum K-theory

Consider the category $P(X)$ of finitely generated projective? (right) $R$-modules. It has a symmetric monoidal structure given by direct sum. The algebraic K-theory $K(R)$ may be described as the K-theory of a symmetric monoidal (infinity,1)-category of $P(R)$. That is, it is the group completion $K(R) = \Omega B B (i P(X))$ where $i P(X)$ denotes the maximal subgroupoid. See (Weibel, IV.4.8, IV.4.11.1).

#### Exact K-theory

Consider the category $P(X)$ of finitely generated projective (right) $R$-modules. This is an exact category and the K-theory $K(R)$ may be described via the Quillen Q-construction:

$K(R) = \Omega B (Q(P(R)).$

### For schemes

For schemes, there are two constructions which do not agree in full generality. See Thomason-Trobaugh 90.

#### Quillen K-theory

The Quillen K-theory of a scheme $X$ is defined as the algebraic K-theory of the exact category $Vect(X)$ of vector bundles on $X$ (using the Quillen Q-construction).

#### Thomason-Trobaugh K-theory

Let $Perf(X)$ be the category of perfect complexes on $X$. This admits the structure of a Waldhausen category, and the Thomason-Trobaugh K-theory of $X$ is defined via the Waldhausen S-construction.

It may also be defined as the K-theory of a stable (infinity,1)-category of $Perf(X)$ viewed as a stable (infinity,1)-category.

Thomason-Trobaugh K-theory coincides with Quillen K-theory for schemes that admit an ample family of line bundles, but has the advantage of better global descent properties.

### For smooth manifolds

Discussion of algebraic K-theory as a smooth spectrum $SmoothMfd^{op} \longrightarrow Spectra$ via $X \mapsto K(C^\infty(X))$ is in (Bunke-Nikolaus-Voelkl 13, Bunke 14).

For more on this see at

## Properties

### Chern characters

#### Regulators and relation to ordinary cohomology

See at Beilinson regulator.

#### Cyclotomic trace and relation to topological Hochschild homology

Given a ring $R$, then there is a natural morphism of spectra

$\array{ && \mathbf{TC}(R) \\ & \nearrow & \downarrow \\ \mathbf{K}(R) &\longrightarrow& \mathbf{THH}(R) }$

from the algebraic K-theory spectrum to the topological Hochschild homology spectrum and factoring through the topological cyclic homology spectrum called the cyclotomic trace which much like a Chern character map for algebraic K-theory.

### Descent

#### Zariski and Nisnevich descent

The algebraic K-theory spectrum $\mathbf{K}$ satisfies descent to give a sheaf of connective spectra on the Zariski site. For regular noetherian schemes this statement is due to (Brown Gersten 73). The generalization to finite dimensional noetherian schemes is due to (Thomason-Trobaugh 90).

Moreover, $\mathbf{K}$ satisfies descent with respect to the Nisnevich topology (which lies between Zariski and étale). This is due to (Nisnevich 89) and was generalized in turn to finite dimensional noetherian schemes in the same paper of Thomason.

Further generalization of the descent result to finite dimensional quasi-compact quasi-separated schemes is due to (Rosenschon 06).

#### Etale descent

The question of descent of $\mathbf{K}$ over the étale site is closely related to the Lichtenbaum-Quillen conjecture, see also (Thomason 85). This is now a theorem of Rost and Voevodsky and it implies that K-theory does satisfy etale descent in sufficient large degrees.

MO comment

#### Description of the K-theory sheaf via algebraic vector bundles

Let $Sch$ denote the gros Zariski site of regular, separated, noetherian schemes. It is explained in (Bunke-Tamme 12, section 3.3 that the presheaf of spectra on $Sch$ defined by algebraic K-theory admits the following description.

Regard the stack $\mathbf{Vect}^\oplus$ of algebraic vector bundles on $Sch$ as taking values in symmetric monoidal (∞,1)-categories, via the direct sum of vector bundles. Then apply the K-theory of a symmetric monoidal (∞,1)-category-construction $\mathcal{K}$ to this, yielding a sheaf of spectra. This identifies with the usual Thomason-Trobaugh K-theory sheaf,a fact that follows from

1. Zariski descent for Thomason-Trobaugh K-theory,

2. the Zariski-local equivalence between Thomason-Trobaugh K-theory, Quillen K-theory, and direct sum K-theory.

### Relation to non-commutative topology and non-commutative motives

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

### Red-shift conjecture

chromatic homotopy theory

chromatic levelcomplex oriented cohomology theoryE-∞ ring/A-∞ ringreal oriented cohomology theory
0ordinary cohomologyEilenberg-MacLane spectrum $H \mathbb{Z}$HZR-theory
0th Morava K-theory$K(0)$
1complex K-theorycomplex K-theory spectrum $KU$KR-theory
first Morava K-theory$K(1)$
first Morava E-theory$E(1)$
2elliptic cohomologyelliptic spectrum $Ell_E$
second Morava K-theory$K(2)$
second Morava E-theory$E(2)$
algebraic K-theory of KU$K(KU)$
3 …10K3 cohomologyK3 spectrum
$n$$n$th Morava K-theory$K(n)$
$n$th Morava E-theory$E(n)$BPR-theory
$n+1$algebraic K-theory applied to chrom. level $n$$K(E_n)$ (red-shift conjecture)
$\infty$complex cobordism cohomologyMUMR-theory

## Examples

### On monoidal stacks

Algebraic K-theory is traditionally applied to single symmetric monoidal/stable (∞,1)-categories, but to the extent that it is functorial it may just as well be applied to (∞,1)-sheaves with values in these.

Notably, applied to the monoidal stack of vector bundles (with connection) on the site of smooth manifolds, the K-theory of a monoidal category-functor produces a sheaf of spectra which is a form of differential K-theory and whose geometric realization is the topological K-theory spectrum. For more on this see at differential cohomology hexagon – Differential K-theory.

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

Concrete examples of interest include for instance

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

### Introductions

Surveys with accounts of the historical development include

An introductory textbook account is in

Further review includes

Review of the relation to Dennis trace, topological cyclic homology and topological Hochschild homology is in

### Classical

Original articles include

• Daniel Quillen, Higher algebraic K-theory, in Higher K-theories, pp. 85–147, Proc. Seattle 1972, Lec. Notes Math. 341, Springer 1973. A reference for classical constructions is

• Kenneth Brown, Stephen M. Gersten, Algebraic K-theory as generalized sheaf cohomology, Higher K-Theories, Lecture Notes in Mathematics Volume 341, 1973, pp 266-292.

• F. Waldhausen, Algebraic K-theory of spaces, Alg. and Geo. Top., Springer Lect. Notes Math. 1126 (1985), 318-419, pdf.

• R. W. Thomason, Algebraic K-theory and étale cohomology, Ann. Sci. Ecole Norm. Sup. 18 (4), 1985, pp. 437–552.

• Yevsey Nisnevich, The completely decomposed topology on schemes and associated descent spectral sequences in algebraic K-theory, Algebraic K-theory: connections with geometry and topology, 1989, pp 241-341.

• R. W. Thomason, Thomas Trobaugh, Higher algebraic K-theory of schemes and of derived categories, The Grothendieck Festschrift, 1990, 247-435.

• Andreas Rosenschon, P.A. Ostvær, Descent for K-theories, Journal of Pure and Applied Algebra 206, 2006, pp 141–152.

• Claudio Pedrini, Charles Weibel, The higher K-theory of complex varieties, K-theory 21 (2001), 367-385 (web)

• Michael Paluch, Algebraic K-theory and topological spaces (pdf)

For ring spectra

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

• Ralph Cohen, Stability phenomena in the topology of moduli spaces (pdf)

Discussion of the comparison map between algebraic and topological K-theory includes

• Jonathan Rosenberg, Comparison between algebraic and topological K-theory for Banach algebras and $C^\ast$-algebras (pdf)

For smooth manifolds:

### Algebraic K-theory of quotient stacks

Discussion of algebraic K-theory for algebraic stacks (generalizing algebraic equivariant K-theory) is in

• Robert Thomason, Algebraic K-theory of group scheme actions, Algebraic Topology and Algebraic K-theory, Ann. Math. Stud., Princeton, 113, (1987), 539-563.

• Amalendu Krishna, Charanya Ravi, On the K-theory of schemes with group scheme actions (arXiv:1509.05147)

### Algebraic K-theory of ring spectra

Waldhausen’s A-theory is the algebraic K-theory of suspension spectra of loop spaces.

### Via stable $(\infty,1)$-categories

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

(in terms of noncommutative motives) and in

### Via symmetric monoidal $(\infty,1)$-categories

The perspective of algebraic K-theory of a symmetric monoidal (∞,1)-category is developed in

### K-theory stacks

The system of infinite loop spaces of the algebraic K-theory spectrum regarded as an ∞-stack on the Nisnevich site and the principal ∞-bundles over it is considered in

implementing a suggestion stated in

### Examples

• Alexey Ananyevskiy, On the algebraic $K$-theory of some homogeneous varieties (pdf)

Revised on June 9, 2016 11:00:41 by Urs Schreiber (131.220.184.222)