Algebraic K-theory

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

Kn(R)

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.

History[edit]

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 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.

Lower K-groups[edit]

The lower K-groups were discovered first, and given various ad hoc descriptions, which remain useful. Throughout, let A be a ring.

K0[edit]

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 AB gives a map K0(A) → K0(B) by mapping (the class of) a projective A-module M to MA B, making K0 a covariant functor.

If the ring A is commutative, we can define a subgroup of K0(A) as the set

\tilde{K}_0\left(A\right) = \bigcap\limits_{\mathfrak p\text{ prime ideal of }A}\mathrm{Ker}\dim_{\mathfrak p},

where :

\dim_{\mathfrak p}:K_0\left(A\right)\to \mathbf{Z}

is the map sending every (class of a) finitely generated projective A-module M to the rank of the free A_{\mathfrak p}-module M_{\mathfrak p} (this module is indeed free, as any finitely generated projective module over a local ring is free). This subgroup \tilde{K}_0\left(A\right) 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 = BZ be the extension of B to a ring with unity obtaining by adjoining an identity element (0,1). There is a short exact sequence BAZ and we define K0(B) to be the kernel of the corresponding map K0(A) → K0(Z) = Z.[2]

Examples[edit]

K0(A) = Pic(A) ⊕ Z,

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

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

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.[5]

Relative K0[edit]

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

D(A,I) = \{ (x,y) \in A \times A : x-y \in I \} \ .

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

K_0(A,I) = \ker \left({ K_0(D(A,I)) \rightarrow K_0(A) }\right) \ .

where the map is induced by projection along the first factor.

The relative K0(A,I) is isomorphic to K0(I), regarding I as a ring without identity. The independence from A is an analogue of the Excision theorem in homology.[2]

K0 as a ring[edit]

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.[3] The exterior product similarly induces a λ-ring structure. The Picard group embeds as a subgroup of the group of units K0(A).[8]

K1[edit]

Hyman Bass provided this definition, which generalizes the group of units of a ring: 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 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 K1[edit]

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

K_1(A,I) = \ker \left({ K_1(D(A,I)) \rightarrow K_1(A) }\right) \ .

There is a natural exact sequence[11]

 K_1(A,I) \rightarrow K_1(A) \rightarrow K_1(A/I) \rightarrow K_0(A,I) \rightarrow K_0(A) \rightarrow K_0(A/I) \ .

Commutative rings and fields[edit]

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:

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.

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.[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 SK1 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 Milnor (1971, corollary 16.3) shows that SK1(A) vanishes.[14]

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.[13] 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.[15]

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[edit]

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,[16] and this may be extended to square-free degree.[17] Wang also showed that SK1(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 SK1(A) is non-trivial.[17]

K2[edit]

John Milnor found the right definition of K2: it is the center of the Steinberg group St(A) of A.

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.

For a field, K2 is determined by Steinberg symbols: this leads to Matsumoto's theorem.

One can compute that K2 is zero for any finite field.[19][20] The computation of K2(Q) is complicated: Tate proved[20][21]

K_2(\mathbf{Q}) = (\mathbf{Z}/4)^* \times \prod_{p \text{ odd prime}} (\mathbf{Z}/p)^* \

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 K2(F) is the direct sum of a finite cyclic group of order m, say, and a divisible group K2(F)m.[24]

We have K2(Z) = Z/2,[25] and in general K2 is finite for the ring of integers of a number field.[26]

We further have K2(Z/n) = Z/2 if n is divisible by 4, and otherwise zero.[27]

Matsumoto's theorem[edit]

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

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

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[edit]

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

 K_2F \rightarrow \oplus_{\mathbf p} K_1 A/{\mathbf p} \rightarrow K_1 A \rightarrow K_1 F \rightarrow \oplus_{\mathbf p} K_0 A/{\mathbf p} \rightarrow K_0 A \rightarrow K_0 F \rightarrow 0 \

where p runs over all prime ideals of A.[30]

There is also an extension of the exact sequence for relative K1 and K0:[31]

K_2(A) \rightarrow K_2(A/I) \rightarrow K_1(A,I) \rightarrow K_1(A) \cdots \ .

Pairing[edit]

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 x y x^{-1} y^{-1} is an element of K2.[32] The map is not always surjective.[33]

Milnor K-theory[edit]

Main article: 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

\left \{a\otimes(1-a): \ a \neq 0,1 \right \}.

For n = 0,1,2 these coincide with those below, but for n ≧ 3 they differ in general.[34] For example, we have KM
n
(Fq) = 0 for n ≧ 2 but KnFq is nonzero for odd n (see below).

The tensor product on the tensor algebra induces a product  K_m \times K_n \rightarrow K_{m+n} making  K^M_*(F) a graded ring which is graded-commutative.[35]

The images of elements a_1 \otimes \cdots \otimes a_n in K^M_n(k) are termed symbols, denoted \{a_1,\ldots,a_n\}. For integer m invertible in k there is a map

\partial : k^* \rightarrow H^1(k,\mu_m)

where \mu_m denotes the group of m-th roots of unity in some separable extension of k. This extends to

\partial^n : k^* \times \cdots \times k^* \rightarrow H^n\left({k,\mu_m^{\otimes n}}\right) \

satisfying the defining relations of the Milnor K-group. Hence \partial^n may be regarded as a map on K^M_n(k), 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[edit]

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 RK(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[edit]

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

 K_n(R) = \pi_n(BGL(R)^+),

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

 K_n(R) = \pi_n(BGL(R)^+\times K_0(R))

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[edit]

Main article: 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 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

 M'\longleftarrow N\longrightarrow M'',

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

 K_i(P)=\pi_{i+1}(\mathrm{BQ}P,0)

with a fixed zero-object 0, where BQP 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 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.

The S-construction[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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 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) [42]

See also[edit]

Notes[edit]

  1. ^ Soulé, C.; Abramovich, Dan; Burnol, J.-F.; Kramer, Jürg (1992). Lectures on Arakelov geometry. Cambridge Studies in Advanced Mathematics 33. Joint work with H. Gillet. Cambridge: Cambridge University Press. p. 36. ISBN 0-521-47709-3. Zbl 0812.14015. 
  2. ^ a b Rosenberg (1994) p.30
  3. ^ a b Milnor (1971) p.5
  4. ^ Milnor (1971) p.14
  5. ^ Karoubi, Max (2008), K-Theory: an Introduction, Classics in mathematics, Berlin, New York: Springer-Verlag, ISBN 978-3-540-79889-7 , see Theorem I.6.18
  6. ^ Rosenberg (1994) 1.5.1, p.27
  7. ^ Rosenberg (1994) 1.5.3, p.27
  8. ^ Milnor (1971) p.15
  9. ^ J.H.C. Whitehead, Simple homotopy types Amer. J. Math. , 72 (1950) pp. 1–57
  10. ^ Rosenberg (1994) 2.5.1, p.92
  11. ^ Rosenberg (1994) 2.5.4, p.95
  12. ^ Rosenberg (1994) Theorem 2.3.2, p.74
  13. ^ a b Rosenberg (1994) p.75
  14. ^ Rosenberg (1994) p.81
  15. ^ Rosenberg (1994) p.78
  16. ^ Gille & Szamuely (2006) p.47
  17. ^ a b Gille & Szamuely (2006) p.48
  18. ^ Wang, Shianghaw (1950). "On the commutator group of a simple algebra". Am. J. Math. 72: 323–334. doi:10.2307/2372036. ISSN 0002-9327. Zbl 0040.30302. 
  19. ^ Lam (2005) p.139
  20. ^ a b Lemmermeyer (2000) p.66
  21. ^ Milnor (1971) p.101
  22. ^ Milnor (1971) p.102
  23. ^ Gras (2003) p.205
  24. ^ Milnor (1971) p.175
  25. ^ Milnor (1971) p.81
  26. ^ a b Lemmermeyer (2000) p.385
  27. ^ Silvester (1981) p.228
  28. ^ Matsumoto, Hideya (1969), "Sur les sous-groupes arithmétiques des groupes semi-simples déployés", Ann. Sci. École Norm. Sup. (4) (in French) (2): 1–62, ISSN 0012-9593, MR 0240214, Zbl 0261.20025 
  29. ^ Rosenberg (1994) Theorem 4.3.15, p.214
  30. ^ Milnor (1971) p.123
  31. ^ Rosenberg (1994) p.200
  32. ^ Milnor (1971) p.63
  33. ^ Milnor (1971) p.69
  34. ^ (Weibel 2005), cf. Lemma 1.8
  35. ^ Gille & Szamuely (2006) p.184
  36. ^ Gille & Szamuely (2006) p.108
  37. ^ Voevodsky, Vladimir (2003), "Motivic cohomology with Z/2-coefficients", Institut des Hautes Études Scientifiques. Publications Mathématiques 98 (98): 59–104, doi:10.1007/s10240-003-0010-6, ISSN 0073-8301, MR 2031199 
  38. ^ Rosenberg (1994) pp. 245–246
  39. ^ Rosenberg (1994) p.246
  40. ^ Rosenberg (1994) p.289
  41. ^ Waldhausen, Friedhelm (1985), "Algebraic K-theory of spaces", Algebraic K-theory of spaces, Lecture Notes in Mathematics 1126, Berlin, New York: Springer-Verlag, pp. 318–419, doi:10.1007/BFb0074449, ISBN 978-3-540-15235-4, MR 802796 . See also Lecture IV and the references in (Friedlander & Weibel 1999)
  42. ^ (Friedlander & Weibel 1999), Lecture VI

References[edit]

Further reading[edit]

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