Algebraic K-theory - Citizendia

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

K_n(R)

of functors from rings to abelian groups, for all integers n. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Homological algebra is the branch of Mathematics which studies homology in a general algebraic setting In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the For historical reasons, the lower K-groups K₀ and K₁ 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 integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French

The group K₀(R) generalises the construction of the ideal class group of a ring, using projective modules. In Mathematics, the extent to which Unique factorization fails in the ring of integers of an Algebraic number field (or more generally any Dedekind domain In Mathematics, particularly in Abstract algebra and Homological algebra, the concept of projective module over a ring R is a more flexible generalisation 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. The Quillen–Suslin theorem, also known as Serre's problem or Serre's conjecture, is a Theorem in Commutative algebra about the relationship Similarly, K₁(R) is a modification of the group of units in a ring, using elementary matrix theory. In Mathematics, a unit in a ( Unital) ring R is an invertible element of R, i In Mathematics, an elementary matrix is a simple matrix which differs from the Identity matrix in a minimal way 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. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of In Mathematics, a group ring is a ring R constructed from a ring R and a group G (written multiplicatively In Mathematics, Whitehead torsion is an Invariant of an h- Cobordism in a Whitehead group that is important in Simple homotopy theory and In Mathematics, Whitehead torsion is an Invariant of an h- Cobordism in a Whitehead group that is important in Simple homotopy theory and In Topology, a CW complex is a type of Topological space introduced by J In Mathematics, specifically in Topology, surgery theory is the name given to a collection of techniques used to produce one Manifold from another in a Since the 1980s, algebraic K-theory has increasingly had applications to algebraic geometry. Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with For example, motivic cohomology is closely related to algebraic K-theory. Motivic cohomology is a cohomological theory in Mathematics, the existence of which was first conjectured by Alexander Grothendieck during the 1960s

1 History
2 Lower K-groups
3 Milnor K-theory
4 Higher K-theory
- 4.1 The +-construction
- 4.2 The Q-construction
5 Examples
- 5.1 Algebraic K-groups of finite fields
- 5.2 Algebraic K-groups of rings of integers
6 References
7 External links

History

Alexander Grothendieck invented K-theory in the mid-1950s as a framework to state his far-reaching generalization of the Riemann-Roch theorem. Experimental infobox see Wikipedia talkPersondata before changing --> Alexander Grothendieck (born March 28, 1928 in Berlin, Germany In Mathematics, specifically in Complex analysis and Algebraic geometry, the Riemann–Roch theorem is an important tool in the computation of the dimension Within a few years, its topological counterpart was considered by Atiyah and Hirzebruch and is now known as topological K-theory. Friedrich EP Hirzebruch (born 17 October 1927) is a German mathematician working in the fields of Topology, Complex In Mathematics, topological K-theory is a branch of Algebraic topology.

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. In Mathematics, specifically in Topology, surgery theory is the name given to a collection of techniques used to produce one Manifold from another in a A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be

A little later a branch of the theory for operator algebras was fruitfully developed, resulting in operator K-theory and KK-theory. In Functional analysis, an operator algebra is an algebra of continuous Linear operators on a Topological vector space with the multiplication In Mathematics, operator K-theory is a variant of K-theory on the Category of Banach algebras (In most applications these Banach algebras are This article is on the generalization of operator K-theory and K-homology 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. In Mathematics, an algebraic cycle on an Algebraic variety V is roughly speaking a Homology class on V that is represented by a Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with The problem was that the definitions were lacking (or, too many and not obviously consistent). A definition of K₂ for fields by John Milnor, for example, gave an attractive theory that was too limited in scope, constructed as a quotient of the multiplicative group of the field tensored with itself, with some explicit relations imposed; and closely connected with central extensions. John Willard Milnor (b February 20, 1931 in Orange New Jersey) is an American Mathematician known for his work in Differential

Eventually the foundational difficulties were resolved (leaving a deep and difficult theory) by Daniel Quillen, who gave several definitions of higher algebraic K-theory, via the +-construction and the Q-construction. Daniel Gray ("Dan" Quillen (born June 22, 1940) is an American Mathematician and a Fields Medalist From 1984 to 2006 In Mathematics, the plus construction is a method for simplifying the Fundamental group of a space without changing its homology and Cohomology

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. In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real

K₀

The 0th K-group is related to dimension and the Picard group.

The (covariant) functor K₀ goes from the category of rings to the category of groups, taking A to the Grothendieck group of the set of isomorphism classes of its finitely generated projective modules, regarded as a monoid under direct sum. In Mathematics, the category of rings, denoted by Ring, is the category whose objects are rings (with identity and whose Morphisms In Mathematics, the category Grp has the class of all groups for objects and Group homomorphisms for Morphisms As such In Mathematics, the Grothendieck group construction in Abstract algebra constructs an Abelian group from a Commutative Monoid in the In Mathematics, particularly in Abstract algebra and Homological algebra, the concept of projective module over a ring R is a more flexible generalisation

(Projective) modules over a field k are vector spaces and K₀(k) is isomorphic to $\mathbf{Z}$ , by dimension. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added For A a Dedekind ring,

$K_0(A)=\operatorname{Pic} A\times\mathbf Z,$

where Pic(A) is the Picard group of A, and similarly the reduced K-theory is given by

$\tilde K_0(A)=\operatorname{Pic} A.$

K₁

Hyman Bass provided this definition, which generalizes the group of units of a field: K₁(A) is the abelianization of the infinite general linear group:

$K_1(A) = \operatorname{GL}(A)^{\mbox{ab}} = \operatorname{GL}(A) / [\operatorname{GL}(A),\operatorname{GL}(A)]$

Here

$\operatorname{GL}(A) = \operatorname{colim} \operatorname{GL}_n(A)$

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 $\operatorname{E}(A)=[\operatorname{GL}(A),\operatorname{GL}(A)]$ , by Whitehead's lemma. In Abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an Integral domain in which every nonzero Proper In Mathematics, the Picard group of a Ringed space X is the group of Isomorphism classes of Invertible sheaves on X, with Hyman Bass (born 1932 is an American Mathematician, known for work in algebra In Mathematics, more specifically in Abstract algebra, the commutator subgroup or derived subgroup of a group is the Subgroup In Mathematics, the general linear group of degree n is the set of n × n invertible matrices, together with the operation In Mathematics, the direct limit (also called the inductive limit) is a general method of taking colimits of "directed families of objects" In the mathematical discipline of Matrix theory, a block matrix or a partitioned matrix is a partition of a matrix into rectangular smaller matrices In Mathematics, more specifically in Abstract algebra, the commutator subgroup or derived subgroup of a group is the Subgroup Whitehead's lemma is a technical result in Abstract algebra, used in Algebraic K-theory, It states that a matrix of the form Indeed, the group $\operatorname{GL}(A) / \operatorname{E}(A)$ was first defined and studied by Whitehead,^[1] and is called the Whitehead group of the ring^[2] A.

As $\operatorname{E}(A) \triangleleft \operatorname{SL}(A)$ , one can also define the special Whitehead group $SK_1(A) := \operatorname{SL}(A)/\operatorname{E}(A)$ .

Commutative rings and fields

For A a commutative ring, one can define a determinant $\det\colon \operatorname{GL}(A) \to A^*$ to the group of units of A, which vanishes on $\operatorname{E}(A)$ and thus descends to a map $\det\colon K_1(A) \to A^*$ . In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property In Mathematics, a unit in a ( Unital) ring R is an invertible element of R, i This map splits via the map $A^* \stackrel{\sim}{\to} GL_1(A) \to K_1(A)$ (unit in the upper left corner), and hence is onto, and has the special Whitehead group as kernel, yielding the split short exact sequence:

$1 \to SK_1(A) \to K_1(A) \to A^* \to 1,$

which is a quotient of the usual split short exact sequence defining the special linear group, namely

$1 \to \operatorname{SL}(A) \to \operatorname{GL}(A) \to A^* \to 1.$

Thus, since the groups in question are abelian, $K 1 (A)$ splits as the direct sum of the group of units and the special Whitehead group: $K_1(A) \approx A^* \oplus SK_1(A)$ . In Mathematics, especially in Homological algebra and other applications of Abelian category theory as well as in Differential geometry and Group In Mathematics, the special linear group of degree n over a field F is the set of n × n matrices with

When A is a Dedekind domain (e. g. a field, or the ring of algebraic integers in an algebraic number field), SK₁(A) vanishes, and the determinant map is an isomorphism. In particular, $\det\colon K_1(F) \stackrel{\sim}{\to} F^*$ .

For a non-commutative ring, the determinant cannot be defined, but the map $\operatorname{GL}(A) \to K_1(A)$ generalizes the determinant.

K₂

See also: Steinberg group (K-theory)

John Milnor found the right definition of K₂: it is the center of the Steinberg group $\operatorname{St}(A)$ of A. In Algebraic K-theory, a field of Mathematics, the Steinberg group \operatorname{St}(A of a ring A, is the Universal central extension John Willard Milnor (b February 20, 1931 in Orange New Jersey) is an American Mathematician known for his work in Differential In Abstract algebra, the center of a group G is the set Z ( G) of all elements in G which commute with all the In Algebraic K-theory, a field of Mathematics, the Steinberg group \operatorname{St}(A of a ring A, is the Universal central extension

It can also be defined as the kernel of the map

$\varphi\colon\operatorname{St}(A)\to\mathrm{GL}(A),$

or as the Schur multiplier of the group of elementary matrices. In Mathematics, the word kernel has several meanings Kernel may mean a subset associated with a mapping The kernel of a mapping is the set of elements that In Mathematics, more specifically in Group theory, the Schur multiplier is an important invariant of a group that has applications in many areas of mathematics In Mathematics, an elementary matrix is a simple matrix which differs from the Identity matrix in a minimal way

For a field k one has

$K_2(k) = k^\times\otimes_{\mathbf Z} k^\times/\langle a\otimes(1-a)\mid a\not=0,1\rangle.$

Milnor K-theory

The above expression for K₂ 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 of the multiplicative group k^× by the two-sided ideal, generated by the

$a\otimes(1-a)$

for a ≠ 0,1. In Mathematics, the tensor algebra of a Vector space V, denoted T ( V) or T &bull( V) is the algebra In Mathematics and Group theory the term multiplicative group refers to one of the following concepts depending on the context any group \scriptstyle\mathfrak In Ring theory, a branch of Abstract algebra, an ideal is a special Subset of a ring. For n = 0,1,2 these coincide with those above, but for n≧3 they differ in general. For example, we have $K^M_n(\mathbb{F}_q) = 0$ for n≧3.

Higher K-theory

The master, definitive definitions of K-theory were given by Daniel Quillen, after an extended period in which uncertainty had reigned. Daniel Gray ("Dan" Quillen (born June 22, 1940) is an American Mathematician and a Fields Medalist From 1984 to 2006

The +-construction

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

K_n(R) = π_n(BGL(R)⁺),

a very compressed piece of abstract mathematics. Here π_k 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. In Mathematics, homotopy groups are used in Algebraic topology to classify Topological spaces The base point preserving maps from an n -dimensional In Mathematics, the direct limit (also called the inductive limit) is a general method of taking colimits of "directed families of objects" In Mathematics, the general linear group of degree n is the set of n × n invertible matrices, together with the operation In Mathematics, a classifying space BG in Homotopy theory of a Topological group G is the quotient of a Weakly contractible In Topology, two continuous functions from one Topological space to another are called homotopic ( Greek homos = identical In Mathematics, the plus construction is a method for simplifying the Fundamental group of a space without changing its homology and Cohomology

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 +-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 exact diagrams

$M'\longleftarrow N\longrightarrow M''.$

where the first arrow is an admissible epimorphism and the second arrow is an admissible monomorphism. In Mathematics, an exact category is a concept of Category theory due to Daniel Quillen which is designed to encapsulate the properties of Short exact In Category theory an epimorphism (also called an epic morphism or an epi) is a Morphism f: X &rarr Y which In the context of Abstract algebra or Universal algebra, a monomorphism is simply an Injective Homomorphism.

The i-th K-group of P is then defined as

K i (P) = π i + 1 (BQ P,0)

with a fixed zero-object 0, where BQ is the classifying space of Q, which is defined to be the geometric realisation of the nerve of Q. In Mathematics, a simplicial set is a construction in categorical Homotopy theory which is a purely algebraic model of the notion of a " Well-behaved In Category theory, the nerve N ( C) of a Small category C is a Simplicial set constructed from the objects and morphisms

This definition coincides with the above definitions of K₀, K₁ and K₂.

The K-groups $K i (A)$ of the ring A are then the K-groups $K i (P A)$ where $P A$ is the category of finitely generated projective A-modules. In Mathematics, particularly in Abstract algebra and Homological algebra, the concept of projective module over a ring R is a more flexible generalisation More generally, for a scheme X, the higher K-groups of X are by definition the K-groups of (the exact category of) locally free coherent sheaves on X. In Mathematics, a scheme is an important concept connecting the fields of Algebraic geometry, Commutative algebra and Number theory. In Mathematics, especially in Algebraic geometry and the theory of Complex manifolds coherent sheaves are specific class of sheaves having

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 called G-groups, or higher G-theory. When A is a noetherian regular ring, then G- and K-theory coincide. In Commutative algebra, a regular ring is a commutative Noetherian ring, such that the localization at every Maximal ideal is a Regular Indeed, the global dimension of regular local rings is finite, i. In Ring theory and Homological algebra, the global dimension (or global homological dimension; sometimes just called homological dimension) of e. any finitely generated module has a finite projective resolution, so the canonical map K₀ → G₀ is surjective. In Mathematics, a function f is said to be surjective or onto, if its values span its whole Codomain; that is for every It is also injective, as as can be shown. This isomorphism extends to the higher K-groups, too.

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:

Theorem. In Abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements Let F be a finite field with q elements. Then

K 0 (F) = Z

K 2 i (F) = 0

for $i\neq 0$ , and

$K_{2i-1}(F)= \mu_{q^i-1}$ for $i=1,2,\dots$

where $μ r$ denotes the cyclic group with r elements.

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. In Mathematics, the ring of integers is the set of Integers made an Algebraic structure Z with the operations of integer addition In Mathematics, an algebraic number field (or simply number field) F is a finite (and hence algebraic) Field extension of the Borel used this to calculate K_i(A) and K_i(F) modulo torsion. Armand Borel ( 21 May 1923 &ndash 11 August 2003) was a Swiss Mathematician, born in La Chaux-de-Fonds, and was For example, for the integers Z, Borel proved that (modulo torsion)

K i (Z) = 0

for positive i unless

i = 4 k + 1

with k positive

and (modulo torsion)

K 4 k + 1 (Z) = 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. In Mathematics, Vandiver's conjecture concerns a property of Algebraic number fields Although attributed to American mathematician Harry Vandiver, the

References

^ J. H. C. Whitehead, Simple homotopy types Amer. J. Math. , 72 (1950) pp. 1–57
^ Not to be confused with the Whitehead group of a group. In Mathematics, Whitehead torsion is an Invariant of an h- Cobordism in a Whitehead group that is important in Simple homotopy theory and

Milnor, John Willard (1970), “Algebraic K-theory and quadratic forms”, Inventiones Mathematicae 9: 318–344, MR 0260844, ISSN 0020-9910
Milnor, John Willard (1971), Introduction to algebraic K-theory, Princeton, NJ: Princeton University Press, MR 0349811 (lower K-groups)
Quillen, Daniel (1975), “Higher algebraic K-theory”, Proceedings of the International Congress of Mathematicians (Vancouver, B. John Willard Milnor (b February 20, 1931 in Orange New Jersey) is an American Mathematician known for his work in Differential Mathematical Reviews is a journal and online database published by the American Mathematical Society that contains brief synopses (and occasionally evaluations of many An International Standard Serial Number ( ISSN) is a unique eight-digit number used to identify a print or electronic Periodical publication. The Princeton University Press is an independent publisher with close connections to Princeton University. Mathematical Reviews is a journal and online database published by the American Mathematical Society that contains brief synopses (and occasionally evaluations of many Daniel Gray ("Dan" Quillen (born June 22, 1940) is an American Mathematician and a Fields Medalist From 1984 to 2006 C. , 1974), Vol. 1, Montreal, Quebec: Canad. Math. Congress, pp. 171–176, MR 0422392 (Quillen's Q-construction)
Quillen, Daniel (1974), “Higher K-theory for categories with exact sequences”, New developments in topology (Proc. Mathematical Reviews is a journal and online database published by the American Mathematical Society that contains brief synopses (and occasionally evaluations of many Sympos. Algebraic Topology, Oxford, 1972), vol. 11, London Math. Soc. Lecture Note Ser. , Cambridge University Press, pp. Cambridge University Press (known colloquially as CUP is a Publisher given a Royal Charter by Henry VIII in 1534 95–103, MR 0335604 (relation of Q-construction to +-construction)
Seiler, Wolfgang, “λ-Rings and Adams Operations in Algebraic K-Theory”, in Rapoport, M. Mathematical Reviews is a journal and online database published by the American Mathematical Society that contains brief synopses (and occasionally evaluations of many ; Schneider, P. & Schappacher, N. , Beilinson's Conjectures on Special Values of L-Functions, Boston, MA: Academic Press, ISBN 978-0-12-581120-0
Weibel, Charles (2005), “Algebraic K-theory of rings of integers in local and global fields”, Handbook of K-theory, Berlin, New York: Springer-Verlag, pp. Springer Science+Business Media or Springer (ˈʃpʁɪŋɐ is a worldwide Publishing company based in Germany, which publishes textbooks academic 139–190, MR 2181823, <http://www.math.uiuc.edu/K-theory/0691/KZsurvey.pdf> (survey article)

External links

C. Mathematical Reviews is a journal and online database published by the American Mathematical Society that contains brief synopses (and occasionally evaluations of many Weibel "The K-book: An introduction to algebraic K-theory"

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