 Algebraic Ktheory

In mathematics, algebraic Ktheory 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 integers n. For historical reasons, the lower Kgroups K_{0} and K_{1} are thought of in somewhat different terms from the higher algebraic Kgroups 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 Kgroups 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 QuillenSuslin 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 Ktheory has increasingly had applications to algebraic geometry. For example, motivic cohomology is closely related to algebraic Ktheory.
Contents
History
Alexander Grothendieck discovered Ktheory in the mid1950s as a framework to state his farreaching generalization of the RiemannRoch theorem. Within a few years, its topological counterpart was considered by Michael Atiyah and Hirzebruch and is now known as topological Ktheory.
Applications of Kgroups 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 Ktheory and KKtheory. It also became clear that Ktheory could play a role in algebraic cycle theory in algebraic geometry (Gersten's conjecture): here the higher Kgroups 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 work of Robert Steinberg 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 Quillen (1973, 1974), who gave several definitions of K_{n}(A) for arbitrary nonnegative n, via the +construction and the Qconstruction.
Lower Kgroups
The lower Kgroups 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 Amodule 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 Amodule 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 Ktheory of A.
Examples: (Projective) modules over a field k are vector spaces and K_{0}(k) is isomorphic to Z, by dimension. For A a Dedekind ring,
 K_{0}(A) = Pic(A) ⊕ Z,
where Pic(A) is the Picard group of A, and similarly the reduced Ktheory is given by
An algebrogeometric variant of this construction is applied to the category of algebraic varieties; it associates with a given algebraic variety X the Grothendieck's Kgroup of the category of locally free sheaves (or coherent sheaves) on X. Given a compact topological space X, the topological Ktheory K^{top}(X) of (real) vector bundles over X coincides with K_{0} of the ring of continuous realvalued functions on X. ^{[1]}
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 , by Whitehead's lemma. Indeed, the group was first defined and studied by Whitehead,^{[2]} and is called the Whitehead group of the ring A.
Commutative rings and fields
For A a commutative ring, one can define a determinant to the group of units of A, which vanishes on and thus descends to a map . As , one can also define the special Whitehead group . This map splits via the map (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
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: .
When A is a Euclidean domain (e.g. a field, or the integers) SK_{1}(A) vanishes, and the determinant map is an isomorphism. In particular, . 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 A such that SK_{1}(A) is nonzero was given by Grayson in 1981. If A is a Dedekind domain whose quotient field is a finite extension of the rationals then Milnor (1971, corollary 16.3) shows that SK_{1}(A) vanishes.
For a noncommutative ring, the determinant cannot be defined, but the map generalizes the determinant.
K_{2}
See also: Steinberg group (Ktheory)John Milnor found the right definition of K_{2} for fields: it is the center of the Steinberg group of A.
It can also be defined as the kernel of the map
or as the Schur multiplier of the group of elementary matrices.
Matsumoto's theorem says that for a field k, the second Kgroup is given by^{[3]}
Matsumoto's original theorem is even more general: For any root system, it gives a presentation for the unstable Ktheory. This presentation is different from the one given here only for symplectic root systems. For nonsymplectic root systems, the unstable second Kgroup with respect to the root system is exactly the stable Kgroup for GL(A). Unstable second Kgroups (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 Kgroups.
Milnor Ktheory
Main article: Milnor KtheoryThe above expression for K_{2} of a field k led Milnor to the following definition of "higher" Kgroups by
 ,
thus as graded parts of a quotient of the tensor algebra of the multiplicative group k^{×} by the twosided ideal, generated by the
for a ≠ 0,1. For n = 0,1,2 these coincide with those below, but for n≧3 they differ in general.^{[4]} For example, we have for n≧3. Milnor Ktheory modulo 2 is related to étale (or Galois) cohomology of the field by the Milnor conjecture, proven by Voevodsky.^{[5]} The analogous statement for odd primes is the BlochKato conjecture, proved by Voevodsky, Rost, and others.
Higher Ktheory
The definitive definitions of higher Kgroups were given by Quillen (1973), after a few years during which several incompatible definitions were suggested.
The +construction
One possible definition of higher algebraic Ktheory of rings was given by Quillen
 K_{n}(R) = π_{n}(BGL(R) ^{+} ),
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 Ktheory 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 Qconstruction
The Qconstruction 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 Kgroups, defined via the Qconstruction 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 diagrams
where the first arrow is an admissible epimorphism and the second arrow is an admissible monomorphism.
The ith Kgroup of P is then defined as
 K_{i}(P) = π_{i + 1}(BQP,0)
with a fixed zeroobject 0, where BQ is the classifying space of Q, which is defined to be the geometric realisation of the nerve of Q.
This definition coincides with the above definitions of K_{0}.
The Kgroups K_{i}(A) of the ring A are then the Kgroups K_{i}(P_{A}) where P_{A} is the category of finitely generated projective Amodules. More generally, for a scheme X, the higher Kgroups of X are by definition the Kgroups 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 Kgroups are usually called Ggroups, or higher Gtheory. When A is a noetherian regular ring, then G and Ktheory coincide. Indeed, the global dimension of regular local rings is finite, i.e. any finitely generated module has a finite projective resolution, so the canonical map K_{0} → G_{0} is surjective. It is also injective, as can be shown. This isomorphism extends to the higher Kgroups, too.
The Sconstruction
A third construction of Ktheory groups is the Sconstruction, due to Waldhausen.^{[6]} It applies to categories with cofibrations (also called Waldhausen categories). This is a more general concept than exact categories.
Examples
While the Quillen algebraic Ktheory has provided deep insight into various aspects of algebraic geometry and topology, the Kgroups have proved particularly difficult to compute except in a few isolated but interesting cases.
Algebraic Kgroups of finite fields
The first and one of the most important calculations of the higher algebraic Kgroups 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 , and
 K_{2i − 1}(F_{q}) = Z / (q^{i} − 1)Z for i≥1.
Algebraic Kgroups 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 Kgroups 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)
 for positive i unless i = 4k + 1 with k positive
and (modulo torsion)
 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 QuillenLichtenbaum conjecture for more details.
Applications and open questions
Algebraic Kgroups are used in conjectures on special values of Lfunctions and the formulation of an noncommutative main conjecture of Iwasawa theory and in construction of higher regulators.
Another fundamental conjecture due to Hyman Bass (Bass conjecture) says that all Ggroups G(A) (that is to say, Kgroups of the category of finitely generated Amodules) are finitely generated when A is a finitely generated Zalgebra.^{[7]}
References
 ^ Karoubi, Max (2008), KTheory: an Introduction, Classics in mathematics, Berlin, New York: SpringerVerlag, ISBN 9783540798897, see Theorem I.6.18
 ^ J.H.C. Whitehead, Simple homotopy types Amer. J. Math. , 72 (1950) pp. 1–57
 ^ Matsumoto, Hideya (1969), "Sur les sousgroupes arithmétiques des groupes semisimples déployés", Ann. Sci. École Norm. Sup. 4 (2): 1–62, MR0240214, http://www.numdam.org/item?id=ASENS_1969_4_2_1_1_0
 ^ (Weibel 2005), cf. Lemma 1.8
 ^ Voevodsky, Vladimir (2003), "Motivic cohomology with Z/2coefficients", Institut des Hautes Études Scientifiques. Publications Mathématiques 98 (98): 59–104, doi:10.1007/s1024000300106, ISSN 00738301, MR2031199
 ^ Waldhausen, Friedhelm (1985), "Algebraic Ktheory of spaces", Algebraic Ktheory of spaces, Lecture Notes in Mathematics, 1126, Berlin, New York: SpringerVerlag, pp. 318–419, doi:10.1007/BFb0074449, ISBN 9783540152354, MR802796. See also Lecture IV and the references in (Friedlander & Weibel 1999)
 ^ (Friedlander & Weibel 1999), Lecture VI
 Friedlander, Eric; Grayson, Daniel, eds. (2005), Handbook of KTheory, Berlin, New York: SpringerVerlag, ISBN 9783540304364, MR2182598, http://www.springerlink.com/content/9783540230199/
 Friedlander, Eric M.; Weibel, Charles W. (1999), An overview of algebraic Ktheory, World Sci. Publ., River Edge, NJ, pp. 1–119, MR1715873
 Milnor, John Willard (1969 1970), "Algebraic Ktheory and quadratic forms", Inventiones Mathematicae 9 (4): 318–344, doi:10.1007/BF01425486, ISSN 00209910, MR0260844
 Milnor, John Willard (1971), Introduction to algebraic Ktheory, Princeton, NJ: Princeton University Press, MR0349811 (lower Kgroups)
 Quillen, Daniel (1973), "Higher algebraic Ktheory. I", Algebraic Ktheory, I: Higher Ktheories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), Lecture Notes in Math, 341, Berlin, New York: SpringerVerlag, pp. 85–147, doi:10.1007/BFb0067053, ISBN 9783540064343, MR0338129
 Quillen, Daniel (1975), "Higher algebraic Ktheory", Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 1, Montreal, Quebec: Canad. Math. Congress, pp. 171–176, MR0422392 (Quillen's Qconstruction)
 Quillen, Daniel (1974), "Higher Ktheory for categories with exact sequences", New developments in topology (Proc. Sympos. Algebraic Topology, Oxford, 1972), London Math. Soc. Lecture Note Ser., 11, Cambridge University Press, pp. 95–103, MR0335604 (relation of Qconstruction to +construction)
 Rosenberg, Jonathan (1994), Algebraic Ktheory and its applications, Graduate Texts in Mathematics, 147, Berlin, New York: SpringerVerlag, ISBN 9780387942483, MR[http://wwwusers.math.umd.edu/~jmr/KThy_errata2.pdf Errata 1282290 [http://wwwusers.math.umd.edu/~jmr/KThy_errata2.pdf Errata]], http://books.google.com/books?id=TtMkTEZbYoYC
 Seiler, Wolfgang (1988), "λRings and Adams Operations in Algebraic KTheory", in Rapoport, M.; Schneider, P.; Schappacher, N., Beilinson's Conjectures on Special Values of LFunctions, Boston, MA: Academic Press, ISBN 9780125811200
 Weibel, Charles (2005), "Algebraic Ktheory of rings of integers in local and global fields", Handbook of Ktheory, Berlin, New York: SpringerVerlag, pp. 139–190, MR2181823, http://www.math.uiuc.edu/Ktheory/0691/KZsurvey.pdf (survey article)
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