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Algebraic ''K''-theory is a subject area in mathematics with connections to
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
,
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...
, ring theory, and
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...
. Geometric, algebraic, and arithmetic objects are assigned objects called ''K''-groups. These are groups in the sense of
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The te ...
. They contain detailed information about the original object but are notoriously difficult to compute; for example, an important outstanding problem is to compute the ''K''-groups of the
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s. ''K''-theory was discovered in the late 1950s by Alexander Grothendieck in his study of
intersection theory In mathematics, intersection theory is one of the main branches of algebraic geometry, where it gives information about the intersection of two subvarieties of a given variety. The theory for varieties is older, with roots in Bézout's theorem o ...
on
algebraic varieties Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex number ...
. In the modern language, Grothendieck defined only ''K''0, the zeroth ''K''-group, but even this single group has plenty of applications, such as the Grothendieck–Riemann–Roch theorem. Intersection theory is still a motivating force in the development of (higher) algebraic ''K''-theory through its links with motivic cohomology and specifically Chow groups. The subject also includes classical number-theoretic topics like
quadratic reciprocity In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. Due to its subtlety, it has many formulations, but the most standard s ...
and embeddings of number fields into the
real number In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
s and
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s, as well as more modern concerns like the construction of higher regulators and special values of ''L''-functions. The lower ''K''-groups were discovered first, in the sense that adequate descriptions of these groups in terms of other algebraic structures were found. For example, if ''F'' is a field, then is isomorphic to the integers Z and is closely related to the notion of vector space dimension. For a commutative ring ''R'', the group is related to the Picard group of ''R'', and when ''R'' is the ring of integers in a number field, this generalizes the classical construction of the class group. The group ''K''1(''R'') is closely related to the group of units , and if ''R'' is a field, it is exactly the group of units. For a number field ''F'', the group ''K''2(''F'') is related to class field theory, the Hilbert symbol, and the solvability of quadratic equations over completions. In contrast, finding the correct definition of the higher ''K''-groups of rings was a difficult achievement of Daniel Quillen, and many of the basic facts about the higher ''K''-groups of algebraic varieties were not known until the work of Robert Thomason.


History

The history of ''K''-theory was detailed by Charles Weibel.


The Grothendieck group ''K''0

In the 19th century, Bernhard Riemann and his student
Gustav Roch Gustav Adolph Roch (; 9 December 1839 – 21 November 1866) was a German mathematician who made significant contributions to the theory of Riemann surfaces. His promising career was cut short by untimely death at the age of 26. Biography Bor ...
proved what is now known as the
Riemann–Roch theorem The Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeros and allowed poles. It ...
. If ''X'' is a Riemann surface, then the sets of
meromorphic function In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are poles of the function. ...
s and meromorphic
differential form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many application ...
s on ''X'' form vector spaces. A line bundle on ''X'' determines subspaces of these vector spaces, and if ''X'' is projective, then these subspaces are finite dimensional. The Riemann–Roch theorem states that the difference in dimensions between these subspaces is equal to the degree of the line bundle (a measure of twistedness) plus one minus the genus of ''X''. In the mid-20th century, the Riemann–Roch theorem was generalized by Friedrich Hirzebruch to all algebraic varieties. In Hirzebruch's formulation, the Hirzebruch–Riemann–Roch theorem, the theorem became a statement about
Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological spac ...
s: The Euler characteristic of a
vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to ev ...
on an algebraic variety (which is the alternating sum of the dimensions of its cohomology groups) equals the Euler characteristic of the trivial bundle plus a correction factor coming from characteristic classes of the vector bundle. This is a generalization because on a projective Riemann surface, the Euler characteristic of a line bundle equals the difference in dimensions mentioned previously, the Euler characteristic of the trivial bundle is one minus the genus, and the only nontrivial characteristic class is the degree. The subject of ''K''-theory takes its name from a 1957 construction of Alexander Grothendieck which appeared in the Grothendieck–Riemann–Roch theorem, his generalization of Hirzebruch's theorem. Let ''X'' be a smooth algebraic variety. To each vector bundle on ''X'', Grothendieck associates an invariant, its ''class''. The set of all classes on ''X'' was called ''K''(''X'') from the German ''Klasse''. By definition, ''K''(''X'') is a quotient of the free abelian group on isomorphism classes of vector bundles on ''X'', and so it is an abelian group. If the basis element corresponding to a vector bundle ''V'' is denoted 'V'' then for each short exact sequence of vector bundles: :0 \to V' \to V \to V'' \to 0, Grothendieck imposed the relation . These generators and relations define ''K''(''X''), and they imply that it is the universal way to assign invariants to vector bundles in a way compatible with exact sequences. Grothendieck took the perspective that the Riemann–Roch theorem is a statement about morphisms of varieties, not the varieties themselves. He proved that there is a homomorphism from ''K''(''X'') to the Chow groups of ''X'' coming from the Chern character and
Todd class In mathematics, the Todd class is a certain construction now considered a part of the theory in algebraic topology of characteristic classes. The Todd class of a vector bundle can be defined by means of the theory of Chern classes, and is enco ...
of ''X''. Additionally, he proved that a proper morphism to a smooth variety ''Y'' determines a homomorphism called the ''pushforward''. This gives two ways of determining an element in the Chow group of ''Y'' from a vector bundle on ''X'': Starting from ''X'', one can first compute the pushforward in ''K''-theory and then apply the Chern character and Todd class of ''Y'', or one can first apply the Chern character and Todd class of ''X'' and then compute the pushforward for Chow groups. The Grothendieck–Riemann–Roch theorem says that these are equal. When ''Y'' is a point, a vector bundle is a vector space, the class of a vector space is its dimension, and the Grothendieck–Riemann–Roch theorem specializes to Hirzebruch's theorem. The group ''K''(''X'') is now known as ''K''0(''X''). Upon replacing vector bundles by projective modules, ''K''0 also became defined for non-commutative rings, where it had applications to
group representation In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used t ...
s. Atiyah and Hirzebruch quickly transported Grothendieck's construction to topology and used it to define topological K-theory. Topological ''K''-theory was one of the first examples of an extraordinary cohomology theory: It associates to each topological space ''X'' (satisfying some mild technical constraints) a sequence of groups ''K''''n''(''X'') which satisfy all the
Eilenberg–Steenrod axioms In mathematics, specifically in algebraic topology, the Eilenberg–Steenrod axioms are properties that homology theories of topological spaces have in common. The quintessential example of a homology theory satisfying the axioms is singular hom ...
except the normalization axiom. The setting of algebraic varieties, however, is much more rigid, and the flexible constructions used in topology were not available. While the group ''K''0 seemed to satisfy the necessary properties to be the beginning of a cohomology theory of algebraic varieties and of non-commutative rings, there was no clear definition of the higher ''K''''n''(''X''). Even as such definitions were developed, technical issues surrounding restriction and gluing usually forced ''K''''n'' to be defined only for rings, not for varieties.


''K''0, ''K''1, and ''K''2

A group closely related to ''K''1 for group rings was earlier introduced by J.H.C. Whitehead.
Henri Poincaré Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The ...
had attempted to define the Betti numbers of a manifold in terms of a triangulation. His methods, however, had a serious gap: Poincaré could not prove that two triangulations of a manifold always yielded the same Betti numbers. It was clearly true that Betti numbers were unchanged by subdividing the triangulation, and therefore it was clear that any two triangulations that shared a common subdivision had the same Betti numbers. What was not known was that any two triangulations admitted a common subdivision. This hypothesis became a conjecture known as the '' Hauptvermutung'' (roughly "main conjecture"). The fact that triangulations were stable under subdivision led J.H.C. Whitehead to introduce the notion of
simple homotopy type In mathematics, particularly the area of topology, a simple-homotopy equivalence is a refinement of the concept of homotopy equivalence. Two CW-complexes are simple-homotopy equivalent if they are related by a sequence of collapses and expansion ...
. A simple homotopy equivalence is defined in terms of adding simplices or cells to a
simplicial complex In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial ...
or cell complex in such a way that each additional simplex or cell deformation retracts into a subdivision of the old space. Part of the motivation for this definition is that a subdivision of a triangulation is simple homotopy equivalent to the original triangulation, and therefore two triangulations that share a common subdivision must be simple homotopy equivalent. Whitehead proved that simple homotopy equivalence is a finer invariant than homotopy equivalence by introducing an invariant called the ''torsion''. The torsion of a homotopy equivalence takes values in a group now called the ''Whitehead group'' and denoted ''Wh''(''π''), where ''π'' is the fundamental group of the two complexes. Whitehead found examples of non-trivial torsion and thereby proved that some homotopy equivalences were not simple. The Whitehead group was later discovered to be a quotient of ''K''1(Z''π''), where Z''π'' is the integral
group ring In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the gi ...
of ''π''. Later
John Milnor John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology, algebraic K-theory and low-dimensional holomorphic dynamical systems. Milnor is a distinguished professor at Stony Brook Un ...
used
Reidemeister torsion In mathematics, Reidemeister torsion (or R-torsion, or Reidemeister–Franz torsion) is a topological invariant of manifolds introduced by Kurt Reidemeister for 3-manifolds and generalized to higher dimensions by and . Analytic torsion (or Ray– ...
, an invariant related to Whitehead torsion, to disprove the Hauptvermutung. The first adequate definition of ''K''1 of a ring was made by Hyman Bass and Stephen Schanuel. In topological ''K''-theory, ''K''1 is defined using vector bundles on a suspension of the space. All such vector bundles come from the clutching construction, where two trivial vector bundles on two halves of a space are glued along a common strip of the space. This gluing data is expressed using the general linear group, but elements of that group coming from elementary matrices (matrices corresponding to elementary row or column operations) define equivalent gluings. Motivated by this, the Bass–Schanuel definition of ''K''1 of a ring ''R'' is , where ''GL''(''R'') is the infinite general linear group (the union of all ''GL''''n''(''R'')) and ''E''(''R'') is the subgroup of elementary matrices. They also provided a definition of ''K''0 of a homomorphism of rings and proved that ''K''0 and ''K''1 could be fit together into an exact sequence similar to the relative homology exact sequence. Work in ''K''-theory from this period culminated in Bass' book ''Algebraic ''K''-theory''. In addition to providing a coherent exposition of the results then known, Bass improved many of the statements of the theorems. Of particular note is that Bass, building on his earlier work with Murthy, provided the first proof of what is now known as the fundamental theorem of algebraic ''K''-theory. This is a four-term exact sequence relating ''K''0 of a ring ''R'' to ''K''1 of ''R'', the polynomial ring ''R'' 't'' and the localization ''R'' 't'', ''t''−1 Bass recognized that this theorem provided a description of ''K''0 entirely in terms of ''K''1. By applying this description recursively, he produced negative ''K''-groups ''K''−n(''R''). In independent work, Max Karoubi gave another definition of negative ''K''-groups for certain categories and proved that his definitions yielded that same groups as those of Bass. The next major development in the subject came with the definition of ''K''2. Steinberg studied the universal central extensions of a Chevalley group over a field and gave an explicit presentation of this group in terms of generators and relations. In the case of the group E''n''(''k'') of elementary matrices, the universal central extension is now written St''n''(''k'') and called the ''Steinberg group''. In the spring of 1967,
John Milnor John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology, algebraic K-theory and low-dimensional holomorphic dynamical systems. Milnor is a distinguished professor at Stony Brook Un ...
defined ''K''2(''R'') to be the kernel of the homomorphism . The group ''K''2 further extended some of the exact sequences known for ''K''1 and ''K''0, and it had striking applications to number theory.
Hideya Matsumoto Hideya Matsumoto(英也, 松本)Vol.39 No.1 P.46 Journal of the Mathematical Society of Japan https://www.jstage.jst.go.jp/article/sugaku1947/39/1/39_1_43/_pdf/-char/ja is a Japanese mathematician who works on algebraic groups, who proved Mats ...
's 1968 thesis showed that for a field ''F'', ''K''2(''F'') was isomorphic to: :F^\times \otimes_ F^\times / \langle x \otimes (1 - x) \colon x \in F \setminus \ \rangle. This relation is also satisfied by the Hilbert symbol, which expresses the solvability of quadratic equations over local fields. In particular, John Tate was able to prove that ''K''2(Q) is essentially structured around the law of
quadratic reciprocity In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. Due to its subtlety, it has many formulations, but the most standard s ...
.


Higher ''K''-groups

In the late 1960s and early 1970s, several definitions of higher ''K''-theory were proposed. Swan and Gersten both produced definitions of ''K''''n'' for all ''n'', and Gersten proved that his and Swan's theories were equivalent, but the two theories were not known to satisfy all the expected properties. Nobile and Villamayor also proposed a definition of higher ''K''-groups. Karoubi and Villamayor defined well-behaved ''K''-groups for all ''n'', but their equivalent of ''K''1 was sometimes a proper quotient of the Bass–Schanuel ''K''1. Their ''K''-groups are now called ''KV''''n'' and are related to homotopy-invariant modifications of ''K''-theory. Inspired in part by Matsumoto's theorem, Milnor made a definition of the higher ''K''-groups of a field. He referred to his definition as "purely ''ad hoc''", and it neither appeared to generalize to all rings nor did it appear to be the correct definition of the higher ''K''-theory of fields. Much later, it was discovered by Nesterenko and Suslin and by Totaro that Milnor ''K''-theory is actually a direct summand of the true ''K''-theory of the field. Specifically, ''K''-groups have a filtration called the ''weight filtration'', and the Milnor ''K''-theory of a field is the highest weight-graded piece of the ''K''-theory. Additionally, Thomason discovered that there is no analog of Milnor ''K''-theory for a general variety. The first definition of higher ''K''-theory to be widely accepted was Daniel Quillen's. As part of Quillen's work on the Adams conjecture in topology, he had constructed maps from the
classifying space In mathematics, specifically in homotopy theory, a classifying space ''BG'' of a topological group ''G'' is the quotient of a weakly contractible space ''EG'' (i.e. a topological space all of whose homotopy groups are trivial) by a proper free ac ...
s ''BGL''(F''q'') to the homotopy fiber of , where ''ψ''''q'' is the ''q''th Adams operation acting on the classifying space ''BU''. This map is acyclic, and after modifying ''BGL''(F''q'') slightly to produce a new space ''BGL''(F''q'')+, the map became a homotopy equivalence. This modification was called the plus construction. The Adams operations had been known to be related to Chern classes and to ''K''-theory since the work of Grothendieck, and so Quillen was led to define the ''K''-theory of ''R'' as the homotopy groups of ''BGL''(''R'')+. Not only did this recover ''K''1 and ''K''2, the relation of ''K''-theory to the Adams operations allowed Quillen to compute the ''K''-groups of finite fields. The classifying space ''BGL'' is connected, so Quillen's definition failed to give the correct value for ''K''0. Additionally, it did not give any negative ''K''-groups. Since ''K''0 had a known and accepted definition it was possible to sidestep this difficulty, but it remained technically awkward. Conceptually, the problem was that the definition sprung from ''GL'', which was classically the source of ''K''1. Because ''GL'' knows only about gluing vector bundles, not about the vector bundles themselves, it was impossible for it to describe ''K''0. Inspired by conversations with Quillen, Segal soon introduced another approach to constructing algebraic ''K''-theory under the name of Γ-objects. Segal's approach is a homotopy analog of Grothendieck's construction of ''K''0. Where Grothendieck worked with isomorphism classes of bundles, Segal worked with the bundles themselves and used isomorphisms of the bundles as part of his data. This results in a
spectrum A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of color ...
whose homotopy groups are the higher ''K''-groups (including ''K''0). However, Segal's approach was only able to impose relations for split exact sequences, not general exact sequences. In the category of projective modules over a ring, every short exact sequence splits, and so Γ-objects could be used to define the ''K''-theory of a ring. However, there are non-split short exact sequences in the category of vector bundles on a variety and in the category of all modules over a ring, so Segal's approach did not apply to all cases of interest. In the spring of 1972, Quillen found another approach to the construction of higher ''K''-theory which was to prove enormously successful. This new definition began with an exact category, a category satisfying certain formal properties similar to, but slightly weaker than, the properties satisfied by a category of modules or vector bundles. From this he constructed an auxiliary category using a new device called his " ''Q''-construction." Like Segal's Γ-objects, the ''Q''-construction has its roots in Grothendieck's definition of ''K''0. Unlike Grothendieck's definition, however, the ''Q''-construction builds a category, not an abelian group, and unlike Segal's Γ-objects, the ''Q''-construction works directly with short exact sequences. If ''C'' is an abelian category, then ''QC'' is a category with the same objects as ''C'' but whose morphisms are defined in terms of short exact sequences in ''C''. The ''K''-groups of the exact category are the homotopy groups of Ω''BQC'', the
loop space In topology, a branch of mathematics, the loop space Ω''X'' of a pointed topological space ''X'' is the space of (based) loops in ''X'', i.e. continuous pointed maps from the pointed circle ''S''1 to ''X'', equipped with the compact-open topo ...
of the geometric realization (taking the loop space corrects the indexing). Quillen additionally proved his " theorem" that his two definitions of ''K''-theory agreed with each other. This yielded the correct ''K''0 and led to simpler proofs, but still did not yield any negative ''K''-groups. All abelian categories are exact categories, but not all exact categories are abelian. Because Quillen was able to work in this more general situation, he was able to use exact categories as tools in his proofs. This technique allowed him to prove many of the basic theorems of algebraic ''K''-theory. Additionally, it was possible to prove that the earlier definitions of Swan and Gersten were equivalent to Quillen's under certain conditions. ''K''-theory now appeared to be a homology theory for rings and a cohomology theory for varieties. However, many of its basic theorems carried the hypothesis that the ring or variety in question was regular. One of the basic expected relations was a long exact sequence (called the "localization sequence") relating the ''K''-theory of a variety ''X'' and an open subset ''U''. Quillen was unable to prove the existence of the localization sequence in full generality. He was, however, able to prove its existence for a related theory called ''G''-theory (or sometimes ''K''′-theory). ''G''-theory had been defined early in the development of the subject by Grothendieck. Grothendieck defined ''G''0(''X'') for a variety ''X'' to be the free abelian group on isomorphism classes of coherent sheaves on ''X'', modulo relations coming from exact sequences of coherent sheaves. In the categorical framework adopted by later authors, the ''K''-theory of a variety is the ''K''-theory of its category of vector bundles, while its ''G''-theory is the ''K''-theory of its category of coherent sheaves. Not only could Quillen prove the existence of a localization exact sequence for ''G''-theory, he could prove that for a regular ring or variety, ''K''-theory equaled ''G''-theory, and therefore ''K''-theory of regular varieties had a localization exact sequence. Since this sequence was fundamental to many of the facts in the subject, regularity hypotheses pervaded early work on higher ''K''-theory.


Applications of algebraic ''K''-theory in topology

The earliest application of algebraic ''K''-theory to topology was Whitehead's construction of Whitehead torsion. A closely related construction was found by C. T. C. Wall in 1963. Wall found that a space ''π'' dominated by a finite complex has a generalized Euler characteristic taking values in a quotient of ''K''0(Z''π''), where ''π'' is the fundamental group of the space. This invariant is called ''Wall's finiteness obstruction'' because ''X'' is homotopy equivalent to a finite complex if and only if the invariant vanishes.
Laurent Siebenmann Laurent Carl Siebenmann (the first name is sometimes spelled Laurence or Larry) (born 1939) is a Canadian mathematician based at the Université de Paris-Sud at Orsay, France. After working for several years as a Professor at Orsay he became a ...
in his thesis found an invariant similar to Wall's that gives an obstruction to an open manifold being the interior of a compact manifold with boundary. If two manifolds with boundary ''M'' and ''N'' have isomorphic interiors (in TOP, PL, or DIFF as appropriate), then the isomorphism between them defines an ''h''-cobordism between ''M'' and ''N''. Whitehead torsion was eventually reinterpreted in a more directly ''K''-theoretic way. This reinterpretation happened through the study of ''h''-cobordisms. Two ''n''-dimensional manifolds ''M'' and ''N'' are ''h''-cobordant if there exists an -dimensional manifold with boundary ''W'' whose boundary is the disjoint union of ''M'' and ''N'' and for which the inclusions of ''M'' and ''N'' into ''W'' are homotopy equivalences (in the categories TOP, PL, or DIFF).
Stephen Smale Stephen Smale (born July 15, 1930) is an American mathematician, known for his research in topology, dynamical systems and mathematical economics. He was awarded the Fields Medal in 1966 and spent more than three decades on the mathematics facult ...
's ''h''-cobordism theorem asserted that if , ''W'' is compact, and ''M'', ''N'', and ''W'' are simply connected, then ''W'' is isomorphic to the cylinder (in TOP, PL, or DIFF as appropriate). This theorem proved the
Poincaré conjecture In the mathematical field of geometric topology, the Poincaré conjecture (, , ) is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space. Originally conjectured b ...
for . If ''M'' and ''N'' are not assumed to be simply connected, then an ''h''-cobordism need not be a cylinder. The ''s''-cobordism theorem, due independently to Mazur, Stallings, and Barden, explains the general situation: An ''h''-cobordism is a cylinder if and only if the Whitehead torsion of the inclusion vanishes. This generalizes the ''h''-cobordism theorem because the simple connectedness hypotheses imply that the relevant Whitehead group is trivial. In fact the ''s''-cobordism theorem implies that there is a bijective correspondence between isomorphism classes of ''h''-cobordisms and elements of the Whitehead group. An obvious question associated with the existence of ''h''-cobordisms is their uniqueness. The natural notion of equivalence is isotopy.
Jean Cerf Jean Cerf (born in 1928) is a French mathematician, specializing in topology. Education and career Jean Cerf was born in Strasbourg, France, in 1928. He studied at the École Normale Supérieure, graduating in sciences in 1947. After passing his ...
proved that for simply connected smooth manifolds ''M'' of dimension at least 5, isotopy of ''h''-cobordisms is the same as a weaker notion called pseudo-isotopy. Hatcher and Wagoner studied the components of the space of pseudo-isotopies and related it to a quotient of ''K''2(Z''π''). The proper context for the ''s''-cobordism theorem is the classifying space of ''h''-cobordisms. If ''M'' is a CAT manifold, then ''H''CAT(''M'') is a space that classifies bundles of ''h''-cobordisms on ''M''. The ''s''-cobordism theorem can be reinterpreted as the statement that the set of connected components of this space is the Whitehead group of ''π''1(''M''). This space contains strictly more information than the Whitehead group; for example, the connected component of the trivial cobordism describes the possible cylinders on ''M'' and in particular is the obstruction to the uniqueness of a homotopy between a manifold and . Consideration of these questions led Waldhausen to introduce his algebraic ''K''-theory of spaces. The algebraic ''K''-theory of ''M'' is a space ''A''(''M'') which is defined so that it plays essentially the same role for higher ''K''-groups as ''K''1(Zπ1(''M'')) does for ''M''. In particular, Waldhausen showed that there is a map from ''A''(''M'') to a space Wh(''M'') which generalizes the map and whose homotopy fiber is a homology theory. In order to fully develop ''A''-theory, Waldhausen made significant technical advances in the foundations of ''K''-theory. Waldhausen introduced Waldhausen categories, and for a Waldhausen category ''C'' he introduced a simplicial category ''S''''C'' (the ''S'' is for Segal) defined in terms of chains of cofibrations in ''C''. This freed the foundations of ''K''-theory from the need to invoke analogs of exact sequences.


Algebraic topology and algebraic geometry in algebraic ''K''-theory

Quillen suggested to his student Kenneth Brown that it might be possible to create a theory of sheaves of spectra of which ''K''-theory would provide an example. The sheaf of ''K''-theory spectra would, to each open subset of a variety, associate the ''K''-theory of that open subset. Brown developed such a theory for his thesis. Simultaneously, Gersten had the same idea. At a Seattle conference in autumn of 1972, they together discovered a spectral sequence converging from the sheaf cohomology of \mathcal K_n, the sheaf of ''K''''n''-groups on ''X'', to the ''K''-group of the total space. This is now called the Brown–Gersten spectral sequence. Spencer Bloch, influenced by Gersten's work on sheaves of ''K''-groups, proved that on a regular surface, the cohomology group H^2(X, \mathcal K_2) is isomorphic to the Chow group ''CH''2(''X'') of codimension 2 cycles on ''X''. Inspired by this, Gersten conjectured that for a regular local ring ''R'' with fraction field ''F'', ''K''''n''(''R'') injects into ''K''''n''(''F'') for all ''n''. Soon Quillen proved that this is true when ''R'' contains a field, and using this he proved that :H^p(X, \mathcal K_p) \cong \operatorname^p(X) for all ''p''. This is known as ''Bloch's formula''. While progress has been made on Gersten's conjecture since then, the general case remains open. Lichtenbaum conjectured that special values of the zeta function of a number field could be expressed in terms of the ''K''-groups of the ring of integers of the field. These special values were known to be related to the
étale cohomology In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conject ...
of the ring of integers. Quillen therefore generalized Lichtenbaum's conjecture, predicting the existence of a spectral sequence like the
Atiyah–Hirzebruch spectral sequence In mathematics, the Atiyah–Hirzebruch spectral sequence is a spectral sequence for calculating generalized cohomology, introduced by in the special case of topological K-theory. For a CW complex X and a generalized cohomology theory E^\bullet ...
in topological ''K''-theory. Quillen's proposed spectral sequence would start from the étale cohomology of a ring ''R'' and, in high enough degrees and after completing at a prime invertible in ''R'', abut to the -adic completion of the ''K''-theory of ''R''. In the case studied by Lichtenbaum, the spectral sequence would degenerate, yielding Lichtenbaum's conjecture. The necessity of localizing at a prime suggested to Browder that there should be a variant of ''K''-theory with finite coefficients. He introduced ''K''-theory groups ''K''''n''(''R''; Z/Z) which were Z/Z-vector spaces, and he found an analog of the Bott element in topological ''K''-theory. Soule used this theory to construct "étale Chern classes", an analog of topological Chern classes which took elements of algebraic ''K''-theory to classes in
étale cohomology In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conject ...
. Unlike algebraic ''K''-theory, étale cohomology is highly computable, so étale Chern classes provided an effective tool for detecting the existence of elements in ''K''-theory. William G. Dwyer and Eric Friedlander then invented an analog of ''K''-theory for the étale topology called étale ''K''-theory. For varieties defined over the complex numbers, étale ''K''-theory is isomorphic to topological ''K''-theory. Moreover, étale ''K''-theory admitted a spectral sequence similar to the one conjectured by Quillen. Thomason proved around 1980 that after inverting the Bott element, algebraic ''K''-theory with finite coefficients became isomorphic to étale ''K''-theory. Throughout the 1970s and early 1980s, ''K''-theory on singular varieties still lacked adequate foundations. While it was believed that Quillen's ''K''-theory gave the correct groups, it was not known that these groups had all of the envisaged properties. For this, algebraic ''K''-theory had to be reformulated. This was done by Thomason in a lengthy monograph which he co-credited to his dead friend Thomas Trobaugh, who he said gave him a key idea in a dream. Thomason combined Waldhausen's construction of ''K''-theory with the foundations of intersection theory described in volume six of Grothendieck's Séminaire de Géométrie Algébrique du Bois Marie. There, ''K''0 was described in terms of complexes of sheaves on algebraic varieties. Thomason discovered that if one worked with in
derived category In mathematics, the derived category ''D''(''A'') of an abelian category ''A'' is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on ''A''. The construction pro ...
of sheaves, there was a simple description of when a complex of sheaves could be extended from an open subset of a variety to the whole variety. By applying Waldhausen's construction of ''K''-theory to derived categories, Thomason was able to prove that algebraic ''K''-theory had all the expected properties of a cohomology theory. In 1976, Keith Dennis discovered an entirely novel technique for computing ''K''-theory based on Hochschild homology. This was based around the existence of the Dennis trace map, a homomorphism from ''K''-theory to Hochschild homology. While the Dennis trace map seemed to be successful for calculations of ''K''-theory with finite coefficients, it was less successful for rational calculations. Goodwillie, motivated by his "calculus of functors", conjectured the existence of a theory intermediate to ''K''-theory and Hochschild homology. He called this theory topological Hochschild homology because its ground ring should be the sphere spectrum (considered as a ring whose operations are defined only up to homotopy). In the mid-1980s, Bokstedt gave a definition of topological Hochschild homology that satisfied nearly all of Goodwillie's conjectural properties, and this made possible further computations of ''K''-groups. Bokstedt's version of the Dennis trace map was a transformation of spectra . This transformation factored through the fixed points of a circle action on ''THH'', which suggested a relationship with cyclic homology. In the course of proving an algebraic ''K''-theory analog of the Novikov conjecture, Bokstedt, Hsiang, and Madsen introduced topological cyclic homology, which bore the same relationship to topological Hochschild homology as cyclic homology did to Hochschild homology. The Dennis trace map to topological Hochschild homology factors through topological cyclic homology, providing an even more detailed tool for calculations. In 1996, Dundas, Goodwillie, and McCarthy proved that topological cyclic homology has in a precise sense the same local structure as algebraic ''K''-theory, so that if a calculation in ''K''-theory or topological cyclic homology is possible, then many other "nearby" calculations follow.


Lower ''K''-groups

The lower ''K''-groups were discovered first, and given various ad hoc descriptions, which remain useful. Throughout, let ''A'' be a ring.


''K''0

The functor ''K''0 takes a ring ''A'' to the
Grothendieck group In mathematics, the Grothendieck group, or group of differences, of a commutative monoid is a certain abelian group. This abelian group is constructed from in the most universal way, in the sense that any abelian group containing a homomorphic ...
of the set of isomorphism classes of its finitely generated projective modules, regarded as a monoid under direct sum. Any ring homomorphism ''A'' → ''B'' gives a map ''K''0(''A'') → ''K''0(''B'') by mapping (the class of) a projective ''A''-module ''M'' to ''M'' ⊗''A'' ''B'', making ''K''0 a covariant functor. If the ring ''A'' is commutative, we can define a subgroup of ''K''0(''A'') as the set : \tilde_0\left(A\right) = \bigcap\limits_\mathrm\dim_, where : :\dim_:K_0\left(A\right)\to \mathbf is the map sending every (class of a) finitely generated projective ''A''-module ''M'' to the rank of the
free Free may refer to: Concept * Freedom, having the ability to do something, without having to obey anyone/anything * Freethought, a position that beliefs should be formed only on the basis of logic, reason, and empiricism * Emancipate, to procur ...
A_-module M_ (this module is indeed free, as any finitely generated projective module over a local ring is free). This subgroup \tilde_0\left(A\right) is known as the ''reduced zeroth K-theory'' of ''A''. If ''B'' is a ring without an identity element, we can extend the definition of K0 as follows. Let ''A'' = ''B''⊕Z be the extension of ''B'' to a ring with unity obtaining by adjoining an identity element (0,1). There is a short exact sequence ''B'' → ''A'' → Z and we define K0(''B'') to be the kernel of the corresponding map ''K''0(''A'') → K0(Z) = Z.Rosenberg (1994) p.30


Examples

* (Projective) modules over a field ''k'' are
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
s and ''K''0(''k'') is isomorphic to Z, by
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
. * Finitely generated projective modules over a local ring ''A'' are free and so in this case once again ''K''0(''A'') is isomorphic to Z, by rank.Milnor (1971) p.5 * For ''A'' a
Dedekind domain In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessari ...
, ''K''0(''A'') = Pic(''A'') ⊕ Z, where Pic(''A'') is the Picard group of ''A'',Milnor (1971) p.14 An algebro-geometric variant of this construction is applied to the category of
algebraic varieties Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex number ...
; it associates with a given algebraic variety ''X'' the Grothendieck's ''K''-group of the category of locally free sheaves (or coherent sheaves) on ''X''. Given a
compact topological space In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i.e ...
''X'', the topological ''K''-theory ''K''top(''X'') of (real)
vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to ev ...
s over ''X'' coincides with ''K0'' of the ring of continuous real-valued functions on ''X''.


Relative ''K''0

Let ''I'' be an ideal of ''A'' and define the "double" to be a subring of the
Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\ ...
''A''×''A'':Rosenberg (1994) 1.5.1, p.27 :D(A,I) = \ \ . The ''relative K-group'' is defined in terms of the "double"Rosenberg (1994) 1.5.3, p.27 :K_0(A,I) = \ker \left(\right) \ . where the map is induced by projection along the first factor. The relative ''K''0(''A'',''I'') is isomorphic to ''K''0(''I''), regarding ''I'' as a ring without identity. The independence from ''A'' is an analogue of the Excision theorem in homology.


''K''0 as a ring

If ''A'' is a commutative ring, then the
tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same Field (mathematics), field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an e ...
of projective modules is again projective, and so tensor product induces a multiplication turning K0 into a commutative ring with the class 'A''as identity. The exterior product similarly induces a
λ-ring In algebra, a λ-ring or lambda ring is a commutative ring together with some operations λ''n'' on it that behave like the exterior powers of vector spaces. Many rings considered in K-theory carry a natural λ-ring structure. λ-rings also provide ...
structure. The Picard group embeds as a subgroup of the group of units ''K''0(''A'').Milnor (1971) p.15


''K''1

Hyman Bass provided this definition, which generalizes the group of units of a ring: ''K''1(''A'') is the abelianization of the
infinite general linear group In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
: :K_1(A) = \operatorname(A)^ = \operatorname(A) / operatorname(A),\operatorname(A)/math> Here :\operatorname(A) = \operatorname \operatorname(n, A) is the direct limit of the GL(''n''), which embeds in GL(''n'' + 1) as the upper left
block matrix In mathematics, a block matrix or a partitioned matrix is a matrix that is '' interpreted'' as having been broken into sections called blocks or submatrices. Intuitively, a matrix interpreted as a block matrix can be visualized as the original ma ...
, and operatorname(A), \operatorname(A)/math> is its commutator subgroup. Define an ''elementary matrix'' to be one which is the sum of an identity matrix and a single off-diagonal element (this is a subset of the elementary matrices used in linear algebra). Then
Whitehead's lemma : Whitehead's lemma is a technical result in abstract algebra used in algebraic K-theory. It states that a matrix of the form : \begin u & 0 \\ 0 & u^ \end is equivalent to the identity matrix by elementary transformations (that is, transvec ...
states that the group ''E''(''A'') generated by elementary matrices equals the commutator subgroup
L(''A''), GL(''A'') L, or l, is the twelfth letter in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''el'' (pronounced ), plural ''els''. History Lamedh ...
Indeed, the group GL(''A'')/E(''A'') was first defined and studied by Whitehead, and is called the Whitehead group of the ring ''A''.


Relative ''K''1

The ''relative K-group'' is defined in terms of the "double"Rosenberg (1994) 2.5.1, p.92 :K_1(A,I) = \ker \left(\right) \ . There is a natural
exact sequence An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the conte ...
Rosenberg (1994) 2.5.4, p.95 : K_1(A,I) \rightarrow K_1(A) \rightarrow K_1(A/I) \rightarrow K_0(A,I) \rightarrow K_0(A) \rightarrow K_0(A/I) \ .


Commutative rings and fields

For ''A'' a commutative ring, one can define a determinant det: GL(''A'') → ''A*'' to the group of units of ''A'', which vanishes on E(''A'') and thus descends to a map det : ''K''1(''A'') → ''A*''. As E(''A'') ◅ SL(''A''), one can also define the special Whitehead group S''K''1(''A'') := SL(''A'')/E(''A''). This map splits via the map ''A*'' → GL(1, ''A'') → ''K''1(''A'') (unit in the upper left corner), and hence is onto, and has the special Whitehead group as kernel, yielding the
split short exact sequence An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the context ...
: :1 \to SK_1(A) \to K_1(A) \to A^* \to 1, which is a quotient of the usual split short exact sequence defining the special linear group, namely :1 \to \operatorname(A) \to \operatorname(A) \to A^* \to 1. The determinant is split by including the group of units ''A*'' = GL1(''A'') into the general linear group GL''(A)'', so ''K''1(''A'') splits as the direct sum of the group of units and the special Whitehead group: ''K''1(''A'') ≅ ''A*'' ⊕ SK1 (''A''). When ''A'' is a Euclidean domain (e.g. a field, or the integers) S''K''1(''A'') vanishes, and the determinant map is an isomorphism from ''K''1(''A'') to ''A''.Rosenberg (1994) Theorem 2.3.2, p.74 This is ''false'' in general for PIDs, thus providing one of the rare mathematical features of Euclidean domains that do not generalize to all PIDs. An explicit PID such that SK1 is nonzero was given by Ischebeck in 1980 and by Grayson in 1981.Rosenberg (1994) p.75 If ''A'' is a Dedekind domain whose quotient field is an
algebraic number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a ...
(a finite extension of the rationals) then shows that S''K''1(''A'') vanishes.Rosenberg (1994) p.81 The vanishing of SK1 can be interpreted as saying that K1 is generated by the image of GL1 in GL. When this fails, one can ask whether K1 is generated by the image of GL2. For a Dedekind domain, this is the case: indeed, K1 is generated by the images of GL1 and SL2 in GL. The subgroup of SK1 generated by SL2 may be studied by Mennicke symbols. For Dedekind domains with all quotients by maximal ideals finite, SK1 is a torsion group.Rosenberg (1994) p.78 For a non-commutative ring, the determinant cannot in general be defined, but the map GL(''A'') → ''K''1(''A'') is a generalisation of the determinant.


Central simple algebras

In the case of a central simple algebra ''A'' over a field ''F'', the reduced norm provides a generalisation of the determinant giving a map ''K''1(''A'') → ''F'' and S''K''1(''A'') may be defined as the kernel. Wang's theorem states that if ''A'' has prime degree then S''K''1(''A'') is trivial,Gille & Szamuely (2006) p.47 and this may be extended to square-free degree.Gille & Szamuely (2006) p.48
Wang Wang may refer to: Names * Wang (surname) (王), a common Chinese surname * Wāng (汪), a less common Chinese surname * Titles in Chinese nobility * A title in Korean nobility * A title in Mongolian nobility Places * Wang River in Thai ...
also showed that S''K''1(''A'') is trivial for any central simple algebra over a number field, but Platonov has given examples of algebras of degree prime squared for which S''K''1(''A'') is non-trivial.


''K''2

John Milnor John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology, algebraic K-theory and low-dimensional holomorphic dynamical systems. Milnor is a distinguished professor at Stony Brook Un ...
found the right definition of ''K''2: it is the center of the Steinberg group St(''A'') of ''A''. It can also be defined as the
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine lea ...
of the map :\varphi\colon\operatorname(A)\to\mathrm(A), or as the Schur multiplier of the group of
elementary matrices In mathematics, an elementary matrix is a matrix which differs from the identity matrix by one single elementary row operation. The elementary matrices generate the general linear group GL''n''(F) when F is a field. Left multiplication (pre-multip ...
. For a field, K2 is determined by Steinberg symbols: this leads to Matsumoto's theorem. One can compute that K2 is zero for any finite field.Lam (2005) p.139Lemmermeyer (2000) p.66 The computation of K2(Q) is complicated: Tate provedMilnor (1971) p.101 :K_2(\mathbf) = (\mathbf/4)^* \times \prod_ (\mathbf/p)^* \ and remarked that the proof followed
Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
's first proof of the Law of Quadratic Reciprocity.Milnor (1971) p.102Gras (2003) p.205 For non-Archimedean local fields, the group K2(''F'') is the direct sum of a finite
cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bi ...
of order ''m'', say, and a divisible group K2(''F'')''m''.Milnor (1971) p.175 We have K2(Z) = Z/2,Milnor (1971) p.81 and in general K2 is finite for the ring of integers of a number field.Lemmermeyer (2000) p.385 We further have K2(Z/''n'') = Z/2 if ''n'' is divisible by 4, and otherwise zero.Silvester (1981) p.228


Matsumoto's theorem

Matsumoto's theorem states that for a field ''k'', the second ''K''-group is given byRosenberg (1994) Theorem 4.3.15, p.214 :K_2(k) = k^\times\otimes_ k^\times/\langle a\otimes(1-a)\mid a\not=0,1\rangle. Matsumoto's original theorem is even more general: For any
root system In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representat ...
, it gives a presentation for the unstable K-theory. This presentation is different from the one given here only for symplectic root systems. For non-symplectic root systems, the unstable second K-group with respect to the root system is exactly the stable K-group for GL(''A''). Unstable second K-groups (in this context) are defined by taking the kernel of the universal central extension of the Chevalley group of universal type for a given root system. This construction yields the kernel of the Steinberg extension for the root systems ''A''''n'' (''n'' > 1) and, in the limit, stable second ''K''-groups.


Long exact sequences

If ''A'' is a
Dedekind domain In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessari ...
with
field of fractions In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field ...
''F'' then there is a long exact sequence : K_2F \rightarrow \oplus_ K_1 A/ \rightarrow K_1 A \rightarrow K_1 F \rightarrow \oplus_ K_0 A/ \rightarrow K_0 A \rightarrow K_0 F \rightarrow 0 \ where ''p'' runs over all prime ideals of ''A''.Milnor (1971) p.123 There is also an extension of the exact sequence for relative K1 and K0:Rosenberg (1994) p.200 :K_2(A) \rightarrow K_2(A/I) \rightarrow K_1(A,I) \rightarrow K_1(A) \cdots \ .


Pairing

There is a pairing on K1 with values in K2. Given commuting matrices ''X'' and ''Y'' over ''A'', take elements ''x'' and ''y'' in the Steinberg group with ''X'',''Y'' as images. The commutator x y x^ y^ is an element of K2.Milnor (1971) p.63 The map is not always surjective.Milnor (1971) p.69


Milnor ''K''-theory

The above expression for ''K''2 of a field ''k'' led Milnor to the following definition of "higher" ''K''-groups by : K^M_*(k) := T^*(k^\times)/(a\otimes (1-a)), thus as graded parts of a quotient of the
tensor algebra In mathematics, the tensor algebra of a vector space ''V'', denoted ''T''(''V'') or ''T''(''V''), is the algebra of tensors on ''V'' (of any rank) with multiplication being the tensor product. It is the free algebra on ''V'', in the sense of bein ...
of the multiplicative group ''k''× by the two-sided ideal, generated by the :\left \. For ''n'' = 0,1,2 these coincide with those below, but for ''n'' ≧ 3 they differ in general. For example, we have ''K'(Fq) = 0'' for ''n'' ≧ 2 but ''KnFq'' is nonzero for odd ''n'' (see below). The tensor product on the tensor algebra induces a product K_m \times K_n \rightarrow K_ making K^M_*(F) a graded ring which is graded-commutative.Gille & Szamuely (2006) p.184 The images of elements a_1 \otimes \cdots \otimes a_n in K^M_n(k) are termed ''symbols'', denoted \. For integer ''m'' invertible in ''k'' there is a map :\partial : k^* \rightarrow H^1(k,\mu_m) where \mu_m denotes the group of ''m''-th roots of unity in some separable extension of ''k''. This extends to :\partial^n : k^* \times \cdots \times k^* \rightarrow H^n\left(\right) \ satisfying the defining relations of the Milnor K-group. Hence \partial^n may be regarded as a map on K^M_n(k), called the ''Galois symbol'' map.Gille & Szamuely (2006) p.108 The relation between étale (or Galois) cohomology of the field and Milnor K-theory modulo 2 is the
Milnor conjecture In mathematics, the Milnor conjecture was a proposal by of a description of the Milnor K-theory (mod 2) of a general field ''F'' with characteristic different from 2, by means of the Galois (or equivalently étale) cohomology of ''F'' w ...
, proven by Vladimir Voevodsky. The analogous statement for odd primes is the
Bloch-Kato conjecture In mathematics, the norm residue isomorphism theorem is a long-sought result relating Milnor ''K''-theory and Galois cohomology. The result has a relatively elementary formulation and at the same time represents the key juncture in the proofs o ...
, proved by Voevodsky, Rost, and others.


Higher ''K''-theory

The accepted definitions of higher ''K''-groups were given by , after a few years during which several incompatible definitions were suggested. The object of the program was to find definitions of K(''R'') and K(''R'',''I'') in terms of
classifying space In mathematics, specifically in homotopy theory, a classifying space ''BG'' of a topological group ''G'' is the quotient of a weakly contractible space ''EG'' (i.e. a topological space all of whose homotopy groups are trivial) by a proper free ac ...
s so that ''R'' ⇒ K(''R'') and (''R'',''I'') ⇒ K(''R'',''I'') are functors into a homotopy category of spaces and the long exact sequence for relative K-groups arises as the
long exact homotopy sequence In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homotop ...
of a fibration K(''R'',''I'') → K(''R'') → K(''R''/''I'').Rosenberg (1994) pp. 245–246 Quillen gave two constructions, the "plus-construction" and the "''Q''-construction", the latter subsequently modified in different ways.Rosenberg (1994) p.246 The two constructions yield the same K-groups.Rosenberg (1994) p.289


The +-construction

One possible definition of higher algebraic ''K''-theory of rings was given by Quillen : K_n(R) = \pi_n(B\operatorname(R)^+), Here π''n'' is a
homotopy group In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homot ...
, GL(''R'') is the direct limit of the general linear groups over ''R'' for the size of the matrix tending to infinity, ''B'' is the classifying space construction of
homotopy theory In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topol ...
, and the + is Quillen's plus construction. He originally found this idea while studying the group cohomology of GL_n(\mathbb_q) and noted some of his calculations were related to K_1(\mathbb_q). This definition only holds for ''n'' > 0 so one often defines the higher algebraic ''K''-theory via : K_n(R) = \pi_n(B\operatorname(R)^+\times K_0(R)) Since ''BGL''(''R'')+ is path connected and ''K''0(''R'') discrete, this definition doesn't differ in higher degrees and also holds for ''n'' = 0.


The ''Q''-construction

The ''Q''-construction gives the same results as the +-construction, but it applies in more general situations. Moreover, the definition is more direct in the sense that the ''K''-groups, defined via the ''Q''-construction are functorial by definition. This fact is not automatic in the plus-construction. Suppose P is an exact category; associated to P a new category QP is defined, objects of which are those of P and morphisms from ''M''′ to ''M''″ are isomorphism classes of diagrams : M'\longleftarrow N\longrightarrow M'', where the first arrow is an admissible epimorphism and the second arrow is an admissible monomorphism. Note the morphisms in QP are analogous to the definitions of morphisms in the category of motives, where morphisms are given as correspondences Z \subset X \times Y such that
X \leftarrow Z \rightarrow Y
is a diagram where the arrow on the left is a covering map (hence surjective) and the arrow on the right is injective. This category can then be turned into a topological space using the classifying space construction BQP , which is defined to be the
geometric realisation In mathematics, a simplicial set is an object composed of ''simplices'' in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and categories. Formally, a simplicial set may be defined a ...
of the ''
nerve A nerve is an enclosed, cable-like bundle of nerve fibers (called axons) in the peripheral nervous system. A nerve transmits electrical impulses. It is the basic unit of the peripheral nervous system. A nerve provides a common pathway for the ...
'' of QP. Then, the i-th ''K''-group of the exact category P is then defined as : K_i(P)=\pi_(\mathrmP,0) with a fixed zero-object 0. Note the classifying space of a groupoid B\mathcal moves the homotopy groups up one degree, hence the shift in degrees for K_i being \pi_ of a space. This definition coincides with the above definition of ''K''0(''P''). If ''P'' is the category of finitely generated projective ''R''-modules, this definition agrees with the above ''BGL+'' definition of ''K''''n''(''R'') for all ''n''. More generally, for a scheme ''X'', the higher ''K''-groups of ''X'' are defined to be the ''K''-groups of (the exact category of) locally free coherent sheaves on ''X''. The following variant of this is also used: instead of finitely generated projective (= locally free) modules, take finitely generated modules. The resulting ''K''-groups are usually written ''G''''n''(''R''). When ''R'' is a noetherian
regular ring In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension. In symbols, let ''A'' be a Noetherian local ring with maximal i ...
, then ''G''- and ''K''-theory coincide. Indeed, the global dimension of regular rings is finite, i.e. any finitely generated module has a finite projective resolution ''P''* → ''M'', and a simple argument shows that the canonical map ''K''0(R) → ''G''0(R) is an
isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
, with 'M''Σ ± 'P''''n'' This isomorphism extends to the higher ''K''-groups, too.


The ''S''-construction

A third construction of ''K''-theory groups is the ''S''-construction, due to
Waldhausen Waldhausen is a municipality in the district of Zwettl in the Austrian state of Lower Austria Lower Austria (german: Niederösterreich; Austro-Bavarian: ''Niedaöstareich'', ''Niedaestareich'') is one of the nine states of Austria, located in ...
. It applies to categories with cofibrations (also called Waldhausen categories). This is a more general concept than exact categories.


Examples

While the Quillen algebraic ''K''-theory has provided deep insight into various aspects of algebraic geometry and topology, the ''K''-groups have proved particularly difficult to compute except in a few isolated but interesting cases. (See also: K-groups of a field.)


Algebraic ''K''-groups of finite fields

The first and one of the most important calculations of the higher algebraic ''K''-groups of a ring were made by Quillen himself for the case of
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subt ...
s: If F''q'' is the finite field with ''q'' elements, then: * ''K''0(F''q'') = Z, * ''K''2''i''(F''q'') = 0 for ''i'' ≥1, * ''K''2''i''–1(F''q'') = Z/(''q'' ''i'' − 1)Z for ''i'' ≥ 1. reproved Quillen's computation using different methods.


Algebraic ''K''-groups of rings of integers

Quillen proved that if ''A'' is the ring of algebraic integers in an algebraic number field ''F'' (a finite extension of the rationals), then the algebraic K-groups of ''A'' are finitely generated.
Armand Borel Armand Borel (21 May 1923 – 11 August 2003) was a Swiss mathematician, born in La Chaux-de-Fonds, and was a permanent professor at the Institute for Advanced Study in Princeton, New Jersey, United States from 1957 to 1993. He worked in ...
used this to calculate ''K''''i''(''A'') and K''i''(''F'') modulo torsion. For example, for the integers Z, Borel proved that (modulo torsion) * ''K''i (Z)/tors.=0 for positive ''i'' unless ''i=4k+1'' with ''k'' positive * ''K''4''k''+1 (Z)/tors.= Z for positive ''k''. The torsion subgroups of K2''i''+1(Z), and the orders of the finite groups K4''k''+2(Z) have recently been determined, but whether the latter groups are cyclic, and whether the groups ''K''4''k''(Z) vanish depends upon Vandiver's conjecture about the class groups of cyclotomic integers. See
Quillen–Lichtenbaum conjecture In mathematics, the Quillen–Lichtenbaum conjecture is a conjecture relating étale cohomology to algebraic K-theory introduced by , who was inspired by earlier conjectures of . and proved the Quillen–Lichtenbaum conjecture at the prime 2 for s ...
for more details.


Applications and open questions

Algebraic ''K''-groups are used in conjectures on special values of L-functions and the formulation of a
non-commutative main conjecture of Iwasawa theory In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
and in construction of
higher regulator In mathematics, Dirichlet's unit theorem is a basic result in algebraic number theory due to Peter Gustav Lejeune Dirichlet. It determines the rank of the group of units in the ring of algebraic integers of a number field . The regulator is a ...
s.Lemmermeyer (2000) p.385
Parshin's conjecture In mathematics, more specifically in algebraic geometry, Parshin's conjecture (also referred to as the Beilinson–Parshin conjecture) states that for any smooth morphism, smooth projective variety ''X'' defined over a finite field, the higher algeb ...
concerns the higher algebraic ''K''-groups for smooth varieties over finite fields, and states that in this case the groups vanish up to torsion. Another fundamental conjecture due to Hyman Bass (
Bass' conjecture In mathematics, especially algebraic geometry, the Bass conjecture says that certain algebraic ''K''-groups are supposed to be finitely generated. The conjecture was proposed by Hyman Bass. Statement of the conjecture Any of the following equivale ...
) says that all of the groups ''Gn''(''A'') are finitely generated when ''A'' is a finitely generated Z-algebra. (The groups ''Gn''(''A'') are the ''K''-groups of the category of finitely generated ''A''-modules) , Lecture VI


See also

*
Additive K-theory In mathematics, additive K-theory means some version of algebraic K-theory in which, according to Spencer Bloch, the general linear group ''GL'' has everywhere been replaced by its Lie algebra ''gl''. It is not, therefore, one theory but a way of cr ...
*
Bloch's formula In algebraic K-theory, a branch of mathematics, Bloch's formula, introduced by Spencer Bloch for K_2, states that the Chow group of a smooth variety ''X'' over a field is isomorphic to the cohomology of ''X'' with coefficients in the K-theory of t ...
* Fundamental theorem of algebraic ''K''-theory * Basic theorems in algebraic ''K''-theory * ''K''-theory * ''K''-theory of a category * ''K''-group of a field * ''K''-theory spectrum *
Redshift conjecture In mathematics, more specifically in chromatic homotopy theory, the redshift conjecture states, roughly, that algebraic K-theory K(R) has chromatic level one higher than that of a complex-oriented ring spectrum ''R''. It was formulated by John Ro ...
* Topological ''K''-theory * Rigidity (''K''-theory)


Notes


References

* * * * * * * * * * (lower K-groups) * * (Quillen's Q-construction) * (relation of Q-construction to plus-construction) *.
Errata
* * * (survey article) *


Further reading

* * * *


Pedagogical references


Higher Algebraic K-Theory: an overview
*.
Errata
*


Historical references

* * * * * * * Bokstedt, M., ''Topological Hochschild homology''. Preprint, Bielefeld, 1986. * Bokstedt, M., Hsiang, W. C., Madsen, I., ''The cyclotomic trace and algebraic ''K''-theory of spaces''. Invent. Math., 111(3) (1993), 465–539. * * * Brown, K., Gersten, S., ''Algebraic ''K''-theory as generalized sheaf cohomology'', Algebraic K-theory I, Lecture Notes in Math., vol. 341, Springer-Verlag, 1973, pp. 266–292. * * Dennis, R. K., ''Higher algebraic ''K''-theory and Hochschild homology'', unpublished preprint (1976). * * Grothendieck, Alexander, ''Classes de fasiceaux et theoreme de Riemann–Roch'', mimeographed notes, Princeton 1957. * * * * * * * Milnor, J., ''Introduction to Algebraic ''K''-theory'', Princeton Univ. Press, 1971. * Nobile, A., Villamayor, O., ''Sur la ''K''-theorie algebrique'',
Annales Scientifiques de l'École Normale Supérieure ''Annales Scientifiques de l'École Normale Supérieure'' is a French scientific journal of mathematics published by the Société Mathématique de France. It was established in 1864 by the French chemist Louis Pasteur and published articles in m ...
, 4e serie, 1, no. 3, 1968, 581–616. * Quillen, Daniel, ''Cohomology of groups'', Proc. ICM Nice 1970, vol. 2, Gauthier-Villars, Paris, 1971, 47–52. * Quillen, Daniel, ''Higher algebraic ''K''-theory I'', Algebraic ''K''-theory I, Lecture Notes in Math., vol. 341, Springer Verlag, 1973, 85–147. * Quillen, Daniel, ''Higher algebraic ''K''-theory'', Proc. Intern. Congress Math., Vancouver, 1974, vol. I, Canad. Math. Soc., 1975, pp. 171–176. * * Siebenmann, Larry,
The Obstruction to Finding a Boundary for an Open Manifold of Dimension Greater than Five
', Thesis, Princeton University (1965). * * Steinberg, R., ''Generateurs, ,relations et revetements de groupes algebriques'', ́Colloq. Theorie des Groupes Algebriques, Gauthier-Villars, Paris, 1962, pp. 113–127. (French) * Swan, Richard,
Nonabelian homological algebra and K-theory
', Proc. Sympos. Pure Math., vol. XVII, 1970, pp. 88–123. * Thomason, R. W., ''Algebraic ''K''-theory and étale cohomology'', Ann. Scient. Ec. Norm. Sup. 18, 4e serie (1985), 437–552; erratum 22 (1989), 675–677. * Thomason, R. W., ''Le principe de sciendage et l'inexistence d'une ''K''-theorie de Milnor globale'',
Topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...
31, no. 3, 1992, 571–588. * * Waldhausen, F.,
Algebraic ''K''-theory of topological spaces
I'', in Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 1, pp. 35–60, Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978. * Waldhausen, F., ''Algebraic ''K''-theory of spaces'', in ''Algebraic and geometric topology (New Brunswick, N.J., 1983)'', Lecture Notes in Mathematics, vol. 1126 (1985), 318–419. * * * *


External links


K theory preprint archive
{{DEFAULTSORT:Algebraic K-Theory Algebraic geometry