Subject area in mathematics
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 "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 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 "K"0("F") is isomorphic to the integers Z and is closely related to the notion of vector space dimension. For a commutative ring "R", the group "K"0("R") 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 "R"×, 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 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. 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:
formula_1
Grothendieck imposed the relation ["V"] = ["V′"] + ["V″"]. 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 "f" : "X" → "Y" to a smooth variety "Y" determines a homomorphism "f"* : "K"("X") → "K"("Y") 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 representations. 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 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 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. 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 "K"1(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 "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 "GL"("R") / "E"("R"), 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 defined "K"2("R") to be the kernel of the homomorphism St("R") → "E"("R"). 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's 1968 thesis showed that for a field "F", "K"2("F") was isomorphic to:
formula_2
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.
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