nLab algebraic K-theory

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, whence 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 Whitehead-generalized 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 Whitehead-generalized 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

• the maximal subgroupoid $i C \subset C$,
• the classifying space $B(i C)$ (a topological monoid),
• the group completion $\Omega B B (i C)$.

See at

• K-theory of a symmetric monoidal (∞,1)-category

• K-theory of a permutative category

• K-theory of a bipermutative category

Quillen Q-construction (for exact categories)

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

• the Quillen Q-construction $Q(E)$,
• the group completion $\Omega B Q(E)$.

See at

• Quillen Q-construction

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

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

• the Waldhausen S-construction $w S_\bullet C$ (a simplicial category),
• the nerve (a bisimplicial set),
• the colimit (a simplicial set),
• the geometric realization (a space).

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

See at

• Waldhausen S-construction
• K-theory of a stable (infinity,1)-category

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

• the general linear groups $GL_n(R)$ for $n \ge 0$,
• their classifying spaces,
• the colimit $BGL(R) = colim_n BGL_n(R)$,
• the Quillen plus construction.

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

• algebraic K-theory of smooth manifolds

• differential cohomology hexagon, section Algebraic K-theory of smooth manifolds and the e-invariant

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

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.

Comparison map and Relation to topological K-theory

• comparison map between algebraic and topological K-theory

Descent

See also

• Moritz Kerz, Florian Strunk, Georg Tamme, Algebraic K-theory and descent for blow-ups (arXiv:1611.08466)

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 context	universal additive bivariant (preserves split exact sequences)	universal localizing bivariant (preserves all exact sequences in the middle)	universal additive invariant	universal localizing invariant
noncommutative algebraic geometry	noncommutative motives $Mot_{add}$	noncommutative motives $Mot_{loc}$	algebraic K-theory	non-connective algebraic K-theory
noncommutative topology	KK-theory	E-theory	operator K-theory	…

Red-shift conjecture

• red-shift conjecture

chromatic homotopy theory

chromatic level	complex oriented cohomology theory	E-∞ ring/A-∞ ring	real oriented cohomology theory
0	ordinary cohomology	Eilenberg-MacLane spectrum $H \mathbb{Z}$	HZR-theory
	0th Morava K-theory	$K(0)$
1	complex K-theory	complex K-theory spectrum $KU$	KR-theory
	first Morava K-theory	$K(1)$
	first Morava E-theory	$E(1)$
2	elliptic cohomology	elliptic spectrum $Ell_E$
	second Morava K-theory	$K(2)$
	second Morava E-theory	$E(2)$
	algebraic K-theory of KU	$K(KU)$
3 …10	K3 cohomology	K3 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 cohomology	MU	MR-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.

Related concepts

• Eilenberg swindle

• equivariant algebraic K-theory

• iterated algebraic K-theory

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

• Quillen exact category,

• abelian category,

• Waldhausen category,

• triangulated category.

Concrete examples of interest include for instance

• the category of finitely generated projective objects over a unital $k$-algebra,

• the category of coherent sheaves over a noetherian scheme,

• the category of locally free sheaves over a scheme,

• Gersten resolution

• Milnor's K2? (Steinberg group, universal central extension)

• higher algebraic K-theory?Quillen exact category, Quillen's Q-construction, Waldhausen S-construction, Volodin spaces;

• topological cyclic homology, algebraic K-theory of operator algebras

• Beilinson regulator

• differential algebraic K-theory

• non-connective algebraic K-theory

• real algebraic K-theory

• bivariant algebraic K-theory

• K-motive

• red-shift conjecture

• filtrations on algebraic K-theory

geometric context	universal additive bivariant (preserves split exact sequences)	universal localizing bivariant (preserves all exact sequences in the middle)	universal additive invariant	universal localizing invariant
noncommutative algebraic geometry	noncommutative motives $Mot_{add}$	noncommutative motives $Mot_{loc}$	algebraic K-theory	non-connective algebraic K-theory
noncommutative topology	KK-theory	E-theory	operator K-theory	…

References

Introductions

Surveys with accounts of the historical development include

• Dominique Arlettaz, Algebraic K-theory of rings from a topological viewpoint (pdf)

• Daniel Grayson, Quillen’s work in algebraic K-theory, J. K-Theory 11 (2013), 527–547 pdf

An introductory textbook account is in

• Charles Weibel, The K-Book: An introduction to algebraic K-theory (web)

Further review includes

• Olivier Isely, Algebraic $K$-theory, 2005-06 (pdf)

• Teena Gerhardt, Computations in algebraic K-theory, talk at CUNY Workshop on differential cohomologies 2014 (video recording)

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

• Bjørn Dundas, Thomas Goodwillie, Randy McCarthy, The local structure of algebraic K-theory, Springer (2013) [doi:10.1007/978-1-4471-4393-2]

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.

  (pdf)

  also: Daniel Grayson, Higher algebraic K-theory II, [after Daniel Quillen] (pdf)

• 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.

For complex varieties:

• 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 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:

• Ulrich Bunke, Thomas Nikolaus, Michael Völkl, Differential cohomology theories as sheaves of spectra, Journal of Homotopy and Related Structures October 2014 (arXiv:1311.3188)

• Ulrich Bunke, A regulator for smooth manifolds and an index theorem (arXiv:1407.1379)

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)

See also at universal Chern-Simons 3-bundle – For reductive groups.

Algebraic K-theory of ring spectra

The algebraic K-theory of ring spectra:

• Anthony Elmendorf, Igor Kriz, Michael Mandell, Peter May, chapter VI of Rings, modules and algebras in stable homotopy theory, AMS Mathematical Surveys and Monographs Volume 47 (1997) (pdf)

• Andrew Blumberg, David Gepner, Gonçalo Tabuada, Section 9.5 of: A universal characterization of higher algebraic K-theory, Geom. Topol. 17 (2013) 733-838 (arXiv:1001.2282, doi:10.2140/gt.2013.17.733)

• Jacob Lurie, Algebraic K-Theory of Ring Spectra, Lecture 19 of Algebraic K-Theory and Manifold Topology, 2014 (pdf)

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)

See also:

(Thomason-Trobaugh 90)

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

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

• Andrew Blumberg, David Gepner, Gonçalo Tabuada, A universal characterization of higher algebraic K-theory, Geom. Topol. 17 (2013) 733-838 (arXiv:1001.2282, doi:10.2140/gt.2013.17.733)

(in terms of noncommutative motives) and in

• Clark Barwick, On the algebraic K-theory of higher categories, I. The universal property of Waldhausen K-theory (arXiv:1204.3607)

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

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

• Ulrich Bunke, Georg Tamme, section 2.1 of Regulators and cycle maps in higher-dimensional differential algebraic K-theory (arXiv:1209.6451)

• Thomas NikolausAlgebraic K-Theory of $\infty$-Operads (arXiv:1303.2198)

• Ulrich Bunke, Georg Tamme, Multiplicative differential algebraic K-theory and applications (arXiv:1311.1421)

• Ulrich Bunke, Thomas Nikolaus, Michael Völkl, def. 6.1 in Differential cohomology theories as sheaves of spectra (arXiv:1311.3188)

• David Gepner, Moritz Groth, Thomas Nikolaus, Universality of multiplicative infinite loop space machines, arXiv:1305.4550.

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

• Sho Saito, Higher Tate central extensions via K-theory and infinity-topos theory (arXiv:1405.0923)

implementing a suggestion stated in

• Vladimir Drinfeld, Infinite-dimensional vector bundles in algebraic geometry (an introduction) (arXiv:math/0309155)

Equivariant versions

Refinement to global equivariant stable homotopy theory:

• Stefan Schwede, Global algebraic K-theory (arXiv:1912.08872)

Examples

• Benjamin Antieau, Achim Krause, Thomas Nikolaus. On the K-theory of $\mathbb{Z}/p^n$ (2024). (arXiv:2405.04329).

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

On the algebraic K-theory of rings being encoded in the respective derived categories:

• Daniel Dugger, Brooke Shipley, K-theory and derived equivalences, Duke Math. J. 124 3 (2004) 587-617 [arXiv:math/0209084, doi:10.1215/S0012-7094-04-12435-2]