# Algebraic K-theory

In mathematics, **algebraic K-theory** is an important part of homological algebra concerned with defining and applying a sequence

*K*_{n}(*R*)

of functors from rings to abelian groups, for all nonnegative integers *n*. For historical reasons, the **lower K-groups** *K*_{0} and *K*_{1} are thought of in somewhat different terms from the **higher algebraic K-groups** *K*_{n} 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 *K*_{0}(*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, *K*_{1}(*R*) is a modification of the group of units in a ring, using elementary matrix theory. The group *K*_{1}(*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 *K*_{0}(*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.

## Contents

## History

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.

Applications of *K*-groups were found from 1960 onwards in surgery theory for manifolds, in particular; and numerous other connections with classical algebraic problems were brought out.

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):^{[1]} 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 *K*_{2}(*A*) of a ring *A* as the center, isomorphic to *H*_{2}(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 *K*_{1}(*A*) × *K*_{1}(*A*) to *K*_{2}(*A*). In the special case of a field k, with *K*_{1}(*k*) isomorphic to the multiplicative group GL(1,*k*), computations of Hideya Matsumoto showed that *K*_{2}(*k*) is isomorphic to the group generated by *K*_{1}(*A*) × *K*_{1}(*A*) modulo an easily described set of relations.

Eventually the foundational difficulties were resolved (leaving a deep and difficult theory) by Template:Harvs, who gave several definitions of *K*_{n}(*A*) for arbitrary non-negative *n*, via the plus construction and the *Q -construction.*

## 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 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

where :

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 K_{0} 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 K_{0}(*B*) to be the kernel of the corresponding map K_{0}(*A*) → K_{0}(**Z**) = **Z**.^{[2]}

#### Examples

- (Projective) modules over a field
*k*are vector spaces and K_{0}(*k*) is isomorphic to**Z**, by dimension. - Finitely generated projective modules over a local ring
*A*are free and so in this case again K_{0}(*A*) is isomorphic to**Z**, by rank.^{[3]} - For
*A*a Dedekind domain,

- K
_{0}(*A*) = Pic(*A*) ⊕**Z**,

where Pic(*A*) is the Picard group of *A*,^{[4]} and similarly the reduced K-theory is given by

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 K^{top}(*X*) of (real) vector bundles over *X* coincides with *K _{0}* of the ring of continuous real-valued functions on

*X*.

^{[5]}

#### Relative K_{0}

Let *I* be an ideal of *A* and define the "double" to be a subring of the Cartesian product *A*×*A*:^{[6]}

The *relative K-group* is defined in terms of the "double"^{[7]}

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.^{[2]}

*K*_{0} 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 K_{0} into a commutative ring with the class [*A*] as identity.^{[3]} The exterior product similarly induces a λ-ring structure.
The Picard group embeds as a subgroup of the group of units K_{0}(*A*)^{∗}.^{[8]}

*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:

Here

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,^{[9]} and is called the **Whitehead group** of the ring *A*.

#### Relative *K*_{1}

The *relative K-group* is defined in terms of the "double"^{[10]}

There is a natural exact sequence^{[11]}

#### 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**SK

_{1}(

*A*) := SL(

*A*)/E(

*A*). This map splits via the map

*A**→ GL(1,

*A*) →

*K*(unit in the upper left corner), and hence is onto, and has the special Whitehead group as kernel, yielding the split short exact sequence:

_{1}(A)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** = GL_{1}(*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**⊕ SK

_{1}(

*A*).

When *A* is a Euclidean domain (e.g. a field, or the integers) SK_{1}(*A*) vanishes, and the determinant map is an isomorphism from K_{1}(*A*) to *A*^{∗}.^{[12]} 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 SK_{1} is nonzero was given by Ischebeck in 1980 and by Grayson in 1981.^{[13]} If *A* is a Dedekind domain whose quotient field is an algebraic number field (a finite extension of the rationals) then Template:Harvtxt shows that SK_{1}(*A*) vanishes.^{[14]}

The vanishing of SK_{1} can be interpreted as saying that K_{1} is generated by the image of GL_{1} in GL. When this fails, one can ask whether K_{1} is generated by the image of GL_{2}. For a Dedekind domain, this is the case: indeed, K_{1} is generated by the images of GL_{1} and SL_{2} in GL.^{[13]} The subgroup of SK_{1} generated by SL_{2} may be studied by Mennicke symbols. For Dedekind domains with all quotients by maximal ideals finite, SK_{1} is a torsion group.^{[15]}

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 SK_{1}(*A*) may be defined as the kernel. **Wang's theorem** states that if *A* has prime degree then SK_{1}(*A*) is trivial,^{[16]} and this may be extended to square-free degree.^{[17]} Wang also showed that SK_{1}(*A*) is trivial for any central simple algebra over a number field,^{[18]} but Platonov has given examples of algebras of degree prime squared for which SK_{1}(*A*) is non-trivial.^{[17]}

*K*_{2}

{{#invoke:see also|seealso}}
John Milnor 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 of the map

or as the Schur multiplier of the group of elementary matrices.

For a field, K_{2} is determined by Steinberg symbols: this leads to Matsumoto's theorem.

One can compute that K_{2} is zero for any finite field.^{[19]}^{[20]} The computation of K_{2}(**Q**) is complicated: Tate proved^{[20]}^{[21]}

and remarked that the proof followed Gauss's first proof of the Law of Quadratic Reciprocity.^{[22]}^{[23]}

For non-Archimedean local fields, the group K_{2}(*F*) is the direct sum of a finite cyclic group of order *m*, say, and a divisible group K_{2}(*F*)^{m}.^{[24]}

We have K_{2}(**Z**) = **Z**/2,^{[25]} and in general K_{2} is finite for the ring of integers of a number field.^{[26]}

We further have K_{2}(**Z**/*n*) = **Z**/2 if *n* is divisible by 4, and otherwise zero.^{[27]}

#### Matsumoto's theorem

**Matsumoto's theorem** states that for a field *k*, the second *K*-group is given by^{[28]}^{[29]}

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 *A*_{n} (*n* > 1) and, in the limit, stable second *K*-groups.

#### Long exact sequences

If *A* is a Dedekind domain with field of fractions *F* then there is a long exact sequence

where * p* runs over all prime ideals of

*A*.

^{[30]}

There is also an extension of the exact sequence for relative K_{1} and K_{0}:^{[31]}

#### Pairing

There is a pairing on K_{1} with values in K_{2}. 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 K_{2}.^{[32]} The map is not always surjective.^{[33]}

## Milnor *K*-theory

{{#invoke:main|main}}

The above expression for *K*_{2} of a field *k* led Milnor to the following definition of "higher" *K*-groups by

thus as graded parts of a quotient of the tensor algebra of the multiplicative group *k*^{×} by the two-sided ideal, generated by the

For *n* = 0,1,2 these coincide with those below, but for *n* ≧ 3 they differ in general.^{[34]} For example, we have *K*Template:Su*(F _{q}) = 0* for

*n*≧ 2 but

*K*is nonzero for odd

_{n}F_{q}*n*(see below).

The tensor product on the tensor algebra induces a product making a graded ring which is graded-commutative.^{[35]}

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.^{[36]}

The relation between étale (or Galois) cohomology of the field and Milnor K-theory modulo 2 is the Milnor conjecture, proven by Vladimir Voevodsky.^{[37]} The analogous statement for odd primes is the Bloch-Kato conjecture, proved by Voevodsky, Rost, and others.

## Higher *K*-theory

The accepted definitions of higher *K*-groups were given by Template:Harvtxt, 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*).^{[38]}

Quillen gave two constructions, the "plus-construction" and the "*Q*-construction", the latter subsequently modified in different ways.^{[39]} The two constructions yield the same K-groups.^{[40]}

### The +-construction

One possible definition of higher algebraic *K*-theory of rings was given by Quillen

Here π_{n} 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 *K*_{0}(*R*) discrete, this definition doesn't differ in higher degrees and also holds for *n* = 0.

### The Q-construction

{{#invoke:main|main}}

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 Q*P* is defined, objects of which are those of *P* and morphisms from *M*′ to *M*″ are isomorphism classes of diagrams

where the first arrow is an admissible epimorphism and the second arrow is an admissible monomorphism.

The *i*-th ** K-group** of the exact category

*P*is then defined as

with a fixed zero-object 0, where B*QP* is the *classifying space* of *QP*, which is defined to be the geometric realisation of the *nerve* of *QP*.

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, 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, 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.^{[41]} 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.

### 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 **F**_{q} is the finite field with *q* elements, then:

*K*_{0}(**F**_{q}) =**Z**,*K*_{2i}(**F**_{q}) = 0 for*i*≥1,*K*_{2i–1}(**F**_{q}) =**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 *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*_{4k+1}(**Z**)/tors.=**Z**for positive*k*.

The torsion subgroups of K_{2i+1}(**Z**), and the orders of the finite groups K_{4k+2}(**Z**) have recently been determined, but whether the latter groups are cyclic, and whether the groups *K*_{4k}(**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

Algebraic *K*-groups are used in conjectures on special values of L-functions and the formulation of an non-commutative main conjecture of Iwasawa theory and in construction of higher regulators.^{[26]}

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 *G _{n}(A)* are finitely generated when

*A*is a finitely generated

**Z**-algebra. (The groups

*G*are the

_{n}(A)*K*-groups of the category of finitely generated

*A*-modules)

^{[42]}

## See also

## Notes

- ↑ {{#invoke:citation/CS1|citation |CitationClass=book }}
- ↑
^{2.0}^{2.1}Rosenberg (1994) p.30 - ↑
^{3.0}^{3.1}Milnor (1971) p.5 - ↑ Milnor (1971) p.14
- ↑ {{#invoke:citation/CS1|citation |CitationClass=citation }}, see Theorem I.6.18
- ↑ Rosenberg (1994) 1.5.1, p.27
- ↑ Rosenberg (1994) 1.5.3, p.27
- ↑ Milnor (1971) p.15
- ↑ J.H.C. Whitehead,
*Simple homotopy types*Amer. J. Math. , 72 (1950) pp. 1–57 - ↑ Rosenberg (1994) 2.5.1, p.92
- ↑ Rosenberg (1994) 2.5.4, p.95
- ↑ Rosenberg (1994) Theorem 2.3.2, p.74
- ↑
^{13.0}^{13.1}Rosenberg (1994) p.75 - ↑ Rosenberg (1994) p.81
- ↑ Rosenberg (1994) p.78
- ↑ Gille & Szamuely (2006) p.47
- ↑
^{17.0}^{17.1}Gille & Szamuely (2006) p.48 - ↑ {{#invoke:Citation/CS1|citation |CitationClass=journal }}
- ↑ Lam (2005) p.139
- ↑
^{20.0}^{20.1}Lemmermeyer (2000) p.66 - ↑ Milnor (1971) p.101
- ↑ Milnor (1971) p.102
- ↑ Gras (2003) p.205
- ↑ Milnor (1971) p.175
- ↑ Milnor (1971) p.81
- ↑
^{26.0}^{26.1}Lemmermeyer (2000) p.385 - ↑ Silvester (1981) p.228
- ↑ {{#invoke:citation/CS1|citation |CitationClass=citation }}
- ↑ Rosenberg (1994) Theorem 4.3.15, p.214
- ↑ Milnor (1971) p.123
- ↑ Rosenberg (1994) p.200
- ↑ Milnor (1971) p.63
- ↑ Milnor (1971) p.69
- ↑ Template:Harvard citations, cf. Lemma 1.8
- ↑ Gille & Szamuely (2006) p.184
- ↑ Gille & Szamuely (2006) p.108
- ↑ {{#invoke:citation/CS1|citation |CitationClass=citation }}
- ↑ Rosenberg (1994) pp. 245–246
- ↑ Rosenberg (1994) p.246
- ↑ Rosenberg (1994) p.289
- ↑ {{#invoke:citation/CS1|citation |CitationClass=citation }}. See also Lecture IV and the references in Template:Harvard citations
- ↑ Template:Harvard citations, Lecture VI

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