Algebraic K-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called K-groups. These are groups in the sense of abstract algebra. 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 integers.
K-theory was discovered in the late 1950s by Alexander Grothendieck in his study of intersection theory on algebraic varieties. In the modern language, Grothendieck defined only K0, 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 and embeddings of number fields into the real numbers and complex numbers, 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 K1(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 K2(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.Weibel 1999
The Grothendieck group K0
In the 19th century, Bernhard Riemann and his student Gustav Roch proved what is now known as the Riemann–Roch theorem. If X is a Riemann surface, then the sets of meromorphic functions and meromorphic differential forms 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 characteristics: The Euler characteristic of a vector bundle 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.Grothendieck 1957, Borel–Serre 1958 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:
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 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 K0(X). Upon replacing vector bundles by projective modules, K0 also became defined for non-commutative rings, where it had applications to group representations. Atiyah and Hirzebruch quickly transported Grothendieck's construction to topology and used it to define topological K-theory.Atiyah–Hirzebruch 1961 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 Kn(X) which satisfy all the Eilenberg–Steenrod axioms 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 K0 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 Kn(X). Even as such definitions were developed, technical issues surrounding restriction and gluing usually forced Kn to be defined only for rings, not for varieties.
K0, K1, and K2
A group closely related to K1 for group rings was earlier introduced by J.H.C. Whitehead. Henri Poincaré 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.Whitehead 1939, Whitehead 1941, Whitehead 1950 A simple homotopy equivalence is defined in terms of adding simplices or cells to a simplicial complex 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 target complex. 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 K1(Zπ), where Zπ is the integral group ring of π. Later John Milnor used Reidemeister torsion, an invariant related to Whitehead torsion, to disprove the Hauptvermutung.
The first adequate definition of K1 of a ring was made by Hyman Bass and Stephen Schanuel.Bass–Schanuel 1962 In topological K-theory, K1 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 K1 of a ring R is , where GL(R) is the infinite general linear group (the union of all GLn(R)) and E(R) is the subgroup of elementary matrices. They also provided a definition of K0 of a homomorphism of rings and proved that K0 and K1 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.Bass 1968 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,Bass–Murthy 1967 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 K0 of a ring R to K1 of R, the polynomial ring R[t], and the localization R[t, t−1]. Bass recognized that this theorem provided a description of K0 entirely in terms of K1. 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.Karoubi 1968
The next major development in the subject came with the definition of K2. 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.Steinberg 1962 In the case of the group En(k) of elementary matrices, the universal central extension is now written Stn(k) and called the Steinberg group. In the spring of 1967, John Milnor defined K2(R) to be the kernel of the homomorphism .Milnor 1971 The group K2 further extended some of the exact sequences known for K1 and K0, and it had striking applications to number theory. Hideya Matsumoto's 1968 thesisMatsumoto 1969 showed that for a field F, K2(F) was isomorphic to:
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 K2(Q) is essentially structured around the law of quadratic reciprocity.
Higher K-groups