of functors from rings to abelian groups, for all nonnegative integers n. For historical reasons, the lower K-groups K0 and K1 are thought of in somewhat different terms from the higher algebraic K-groups Kn for n ≥ 2. Indeed, the lower groups are more accessible, and have more applications, than the higher groups. The theory of the higher K-groups is noticeably deeper, and certainly much harder to compute (even when R is the ring of integers).
The group K0(R) generalises the construction of the ideal class group of a ring, using projective modules. Its development in the 1960s and 1970s was linked to attempts to solve a conjecture of Serre on projective modules that now is the Quillen–Suslin theorem; numerous other connections with classical algebraic problems were found in this era. Similarly, K1(R) is a modification of the group of units in a ring, using elementary matrix theory. The group K1(R) is important in topology, especially when R is a group ring, because its quotient the Whitehead group contains the Whitehead torsion used to study problems in simple homotopy theory and surgery theory; the group K0(R) also contains other invariants such as the finiteness invariant. Since the 1980s, algebraic K-theory has increasingly had applications to algebraic geometry. For example, motivic cohomology is closely related to algebraic K-theory.
- 1 History
- 2 Lower K-groups
- 3 Milnor K-theory
- 4 Higher K-theory
- 5 Examples
- 6 Applications and open questions
- 7 See also
- 8 Notes
- 9 References
- 10 Further reading
- 11 External links
Alexander Grothendieck discovered K-theory in the mid-1950s as a framework to state his far-reaching generalization of the Riemann–Roch theorem. Within a few years, its topological counterpart was considered by Michael Atiyah and Friedrich Hirzebruch and is now known as topological K-theory.
A little later a branch of the theory for operator algebras was fruitfully developed, resulting in operator K-theory and KK-theory. It also became clear that K-theory could play a role in algebraic cycle theory in algebraic geometry (Gersten's conjecture): here the higher K-groups become connected with the higher codimension phenomena, which are exactly those that are harder to access. The problem was that the definitions were lacking (or, too many and not obviously consistent). Using Robert Steinberg's work on universal central extensions of classical algebraic groups, John Milnor defined the group K2(A) of a ring A as the center, isomorphic to H2(E(A),Z), of the universal central extension of the group E(A) of infinite elementary matrices over A. (Definitions below.) There is a natural bilinear pairing from K1(A) × K1(A) to K2(A). In the special case of a field k, with K1(k) isomorphic to the multiplicative group GL(1,k), computations of Hideya Matsumoto showed that K2(k) is isomorphic to the group generated by K1(A) × K1(A) modulo an easily described set of relations.
Eventually the foundational difficulties were resolved (leaving a deep and difficult theory) by Quillen (1973, 1974), who gave several definitions of Kn(A) for arbitrary non-negative n, via the plus-construction and the Q-construction.
The lower K-groups were discovered first, and given various ad hoc descriptions, which remain useful. Throughout, let A be a ring.
The functor K0 takes a ring A to the Grothendieck group 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 K0(A) → K0(B) by mapping (the class of) a projective A-module M to M ⊗A B, making K0 a covariant functor.
If the ring A is commutative, we can define a subgroup of K0(A) as the set
is the map sending every (class of a) finitely generated projective A-module M to the rank of the free -module (this module is indeed free, as any finitely generated projective module over a local ring is free). This subgroup 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 K0(A) → K0(Z) = Z.
- (Projective) modules over a field k are vector spaces and K0(k) is isomorphic to Z, by dimension.
- Finitely generated projective modules over a local ring A are free and so in this case again K0(A) is isomorphic to Z, by rank.
- For A a Dedekind domain,
- K0(A) = Pic(A) ⊕ Z,
An algebro-geometric variant of this construction is applied to the category of algebraic varieties; 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 X, the topological K-theory Ktop(X) of (real) vector bundles over X coincides with K0 of the ring of continuous real-valued functions on X.
The relative K-group is defined in terms of the "double"
where the map is induced by projection along the first factor.
K0 as a ring
If A is a commutative ring, then the tensor product 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 structure. The Picard group embeds as a subgroup of the group of units K0(A)∗.
is the direct limit of the GL(n), which embeds in GL(n + 1) as the upper left block matrix, and the commutator subgroup agrees with the group generated by elementary matrices E(A) = [GL(A), GL(A)], by Whitehead's lemma. Indeed, the group GL(A)/E(A) was first defined and studied by Whitehead, and is called the Whitehead group of the ring A.
The relative K-group is defined in terms of the "double"
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: K1(A) → A*. As E(A) ◅ SL(A), one can also define the special Whitehead group SK1(A) := SL(A)/E(A). This map splits via the map A* → GL(1, A) → K1(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:
which is a quotient of the usual split short exact sequence defining the special linear group, namely
The determinant is split by including the group of units A* = GL1(A) into the general linear group GL(A), so K1(A) splits as the direct sum of the group of units and the special Whitehead group: K1(A) ≅ A* ⊕ SK1 (A).
When A is a Euclidean domain (e.g. a field, or the integers) SK1(A) vanishes, and the determinant map is an isomorphism from K1(A) to A∗. 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. If A is a Dedekind domain whose quotient field is an algebraic number field (a finite extension of the rationals) then Milnor (1971, corollary 16.3) shows that SK1(A) vanishes.
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.
For a non-commutative ring, the determinant cannot in general be defined, but the map GL(A) → K1(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 K1(A) → F∗ and SK1(A) may be defined as the kernel. Wang's theorem states that if A has prime degree then SK1(A) is trivial, and this may be extended to square-free degree. Wang also showed that SK1(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 SK1(A) is non-trivial.
It can also be defined as the kernel of the map
For a field, K2 is determined by Steinberg symbols: this leads to Matsumoto's theorem.
We further have K2(Z/n) = Z/2 if n is divisible by 4, and otherwise zero.
Matsumoto's original theorem is even more general: For any root system, 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 An (n > 1) and, in the limit, stable second K-groups.
Long exact sequences
where p runs over all prime ideals of A.
There is also an extension of the exact sequence for relative K1 and K0:
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 is an element of K2. The map is not always surjective.
The above expression for K2 of a field k led Milnor to the following definition of "higher" K-groups by
For n = 0,1,2 these coincide with those below, but for n ≧ 3 they differ in general. For example, we have KM
n(Fq) = 0 for n ≧ 2 but KnFq is nonzero for odd n (see below).
The images of elements in are termed symbols, denoted . For integer m invertible in k there is a map
where denotes the group of m-th roots of unity in some separable extension of k. This extends to
satisfying the defining relations of the Milnor K-group. Hence may be regarded as a map on , called the Galois symbol map.
The relation between étale (or Galois) cohomology of the field and Milnor K-theory modulo 2 is the Milnor conjecture, proven by Voevodsky. The analogous statement for odd primes is the Bloch-Kato conjecture, proved by Voevodsky, Rost, and others.
The accepted definitions of higher K-groups were given by Quillen (1973), 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 spaces 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 of a fibration K(R,I) → K(R) → K(R/I).
One possible definition of higher algebraic K-theory of rings was given by Quillen
Here πn is a homotopy group, 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, and the + is Quillen's plus construction.
This definition only holds for n > 0 so one often defines the higher algebraic K-theory via
Since BGL(R)+ is path connected and K0(R) discrete, this definition doesn't differ in higher degrees and also holds for n = 0.
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
The i-th K-group of the exact category P is then defined as
This definition coincides with the above definition of K0(P). If P is the category of finitely generated projective R-modules, this definition agrees with the above BGL+ definition of Kn(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 Gn(R). When R is a noetherian regular ring, 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 K0(R) → G0(R) is an isomorphism, with [M]=Σ ± [Pn]. This isomorphism extends to the higher K-groups, too.
A third construction of K-theory groups is the S-construction, due to Waldhausen. It applies to categories with cofibrations (also called Waldhausen categories). This is a more general concept than exact categories.
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.
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 fields:
If Fq is the finite field with q elements, then:
- K0(Fq) = Z,
- K2i(Fq) = 0 for i ≥1,
- K2i–1(Fq) = Z/(q i − 1)Z for i ≥ 1.
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. Borel used this to calculate Ki(A) and Ki(F) modulo torsion. For example, for the integers Z, Borel proved that (modulo torsion)
- Ki (Z)/tors.=0 for positive i unless i=4k+1 with k positive
- K4k+1 (Z)/tors.= Z for positive k.
The torsion subgroups of K2i+1(Z), and the orders of the finite groups K4k+2(Z) have recently been determined, but whether the latter groups are cyclic, and whether the groups K4k(Z) vanish depends upon Vandiver's conjecture about the class groups of cyclotomic integers. See Quillen–Lichtenbaum conjecture for more details.
Applications and open questions
Parshin's conjecture 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) 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) 
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