|
|
| Line 1: |
Line 1: |
| In [[mathematics]], '''algebraic K-theory''' is an important part of [[homological algebra]] concerned with defining and applying a sequence
| | When holiday seasons like Xas comes, lots of persons decide on to invest in discounted items. Typically they like to acquire up-graded goods. 1 of them are the UGG boots. As we all know, the high-quality and the fame of the UGG boot are equally incredibly fantastic in the world wide marketplace. For quite a few persons, UGG boots can preserve them warm in winter season and UGG is also the image of vogue. So in accordance to the investigation, UGG normally offered really effectively in holiday seasons. Ugg Boots,jordan shoes,adidas sneakers,MBT Footwear, <br><br> |
| :''K<sub>n</sub>(R)''
| |
| of [[functor]]s from [[ring (mathematics)|rings]] to [[abelian group]]s, for all nonnegative integers ''n.'' For historical reasons, the '''[[#Lower K-groups|lower K-groups]]''' ''K''<sub>0</sub> and ''K''<sub>1</sub> are thought of in somewhat different terms from the '''[[#Higher K-theory|higher algebraic K-groups]]''' ''K<sub>n</sub>'' 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 [[integer]]s).
| |
|
| |
|
| The group ''K<sub>0</sub>(R)'' generalises the construction of the [[ideal class group]] of a ring, using [[projective module]]s. Its development in the 1960s and 1970s was linked to attempts to solve a conjecture of [[Jean-Pierre Serre|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<sub>1</sub>(R)'' is a modification of the group of [[unit (ring theory)|units]] in a ring, using [[elementary matrix]] theory. The group ''K<sub>1</sub>(R)'' is important in [[topology]], especially when ''R'' is a [[group ring]], because its quotient the [[Whitehead torsion#The Whitehead group of a group|Whitehead group]] contains the [[Whitehead torsion]] used to study problems in [[CW complex|simple homotopy theory]] and [[surgery theory]]; the group ''K<sub>0</sub>(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. | | The check report of Ugg boots -Australia correlates employing the prolonged custom within the Australian men and women currently centered on significant responsibility sheepskin boots for their farming and rugged outside the house pursuits. For 200 a long time, Australians referred to their a hundred% merino sheepskin boots as ggs? quick for gly.?in your before 1970's, several several sheepskin boot manufacturing crops seasoned been developed in close proximity to to towards Australian community of Perth. <br><br><br>Located around to this neighborhood seasoned been well-known surfing shorelines, by which surfers tailored the Ugg bootsfor yr-spherical use. The insulating aspects of fleece held ft cozy in wintertime, and in summer time, it absorbed perspiration. Possessing a cozy genuinely sense and type-fitting fashion, Uggs equipped sock-like ease and comfort while going for walks greater than really hard or slippery terrain. Little by little, Uggs attained within the West Coastline within your path in the surfing neighborhoods of Australia East Coastline, fantastic soon after which professional been carried into snowboarding neighborhoods by surfers who appreciated the two sports activities. When Brian Smith, a indigenous Australian surfer frequented the U.S inside the past credited 1970's, he brought Uggs with him. Following realizing the acceptance of Uggs within the California browsing neighborhood, he formulated a decision to construct and company the sheepskin footwear. His conceptual notion arrived to fruition in 1978. Mr. Smith obtained a trademark for that phrase Ugg-and it is thought that he also obtained variants for the spelling-these types of as Ugg boots and Ugh. <br><br>Right after 1995, Mr. Smith marketed all legal rights of Ugg Holding to some Californian establishment in your producer of Decker outdoors Corporation. Upon the acquire, Dickers received the Ugg trademark producer for 24 nations and started out making use of the producer of Ugg-Australia. However, the generating inside of the sheepskin Ugg boots was transferred to China.<br><br>Here is more info in regards to [http://tinyurl.com/k7shbtq uggs on sale] look at our own internet site. |
| | |
| == 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 [[manifold]]s, in particular; and numerous other connections with classical algebraic problems were brought out.
| |
| | |
| A little later a branch of the theory for [[operator algebra]]s 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 ''K<sub>2</sub>(A)'' of a ring ''A'' as the center, isomorphic to H<sub>2</sub>(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<sub>1</sub>(A) × K<sub>1</sub>(A)'' to ''K<sub>2</sub>(A)''. In the special case of a field k, with ''K<sub>1</sub>(k)'' isomorphic to the multiplicative group GL(1,''k''), computations of Hideya Matsumoto showed that ''K<sub>2</sub>(k)'' is isomorphic to the group generated by ''K<sub>1</sub>(A) × K<sub>1</sub>(A)'' modulo an easily described set of relations.
| |
| | |
| Eventually the foundational difficulties were resolved (leaving a deep and difficult theory) by {{harvs|txt|authorlink=Daniel Quillen|last=Quillen|year1=1973|year2=1974}}, who gave several definitions of ''K<sub>n</sub>(A)'' for arbitrary non-negative ''n'', via the [[plus construction|+-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 (mathematics)|ring]]. | |
| | |
| === ''K<sub>0</sub>'' ===
| |
| The functor ''K<sub>0</sub>'' takes a ring ''A'' to the [[Grothendieck group]] of the set of isomorphism classes of its [[finitely generated module|finitely generated]] [[projective module]]s, regarded as a monoid under direct sum. Any ring homomorphism ''A'' → ''B'' gives a map ''K<sub>0</sub>(A)'' → ''K<sub>0</sub>(B)'' by mapping (the class of) a projective ''A''-module ''M'' to ''M'' ⊗<sub>''A''</sub> ''B'', making ''K<sub>0</sub>'' a covariant functor.
| |
| | |
| If the ring ''A'' is commutative, we can define a subgroup of ''K<sub>0</sub>(A)'' as the set
| |
| | |
| :<math>\tilde{K}_0\left(A\right) = \bigcap\limits_{\mathfrak p\text{ prime ideal of }A}\mathrm{Ker}\dim_{\mathfrak p},</math>
| |
| | |
| where :
| |
| | |
| :<math>\dim_{\mathfrak p}:K_0\left(A\right)\to \mathbf{Z}</math>
| |
| | |
| is the map sending every (class of a) finitely generated projective ''A''-module ''M'' to the rank of the free <math>A_{\mathfrak p}</math>-module <math>M_{\mathfrak p}</math> (this module is indeed free, as any finitely generated projective module over a local ring is free). This subgroup <math>\tilde{K}_0\left(A\right)</math> is known as the ''reduced zeroth K-theory'' of ''A''.
| |
| | |
| If ''B'' is a [[pseudo-ring|ring without an identity element]], we can extend the definition of K<sub>0</sub> 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<sub>0</sub>(''B'') to be the kernel of the corresponding map K<sub>0</sub>(''A'') → K<sub>0</sub>('''Z''') = '''Z'''.<ref name=Ros30>Rosenberg (1994) p.30</ref>
| |
| | |
| ====Examples====
| |
| * (Projective) modules over a [[field (mathematics)|field]] ''k'' are [[vector space]]s and K<sub>0</sub>(''k'') is isomorphic to '''Z''', by [[Dimension (vector space)|dimension]].
| |
| * Finitely generated projective modules over a [[local ring]] ''A'' are free and so in this case again K<sub>0</sub>(''A'') is isomorphic to '''Z''', by [[Rank of a free module|rank]].<ref name=Mil5>Milnor (1971) p.5</ref>
| |
| * For ''A'' a [[Dedekind domain]],
| |
| | |
| :K<sub>0</sub>(''A'') = Pic(''A'') ⊕ '''Z''',
| |
| | |
| where Pic(''A'') is the [[Picard group]] of ''A'',<ref name=Mil14>Milnor (1971) p.14</ref> and similarly the reduced K-theory is given by
| |
| | |
| :<math>\tilde K_0(A)=\operatorname{Pic} A.</math>
| |
| | |
| An algebro-geometric variant of this construction is applied to the category of [[algebraic variety|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<sup>top</sup>(''X'') of (real) [[vector bundle]]s over ''X'' coincides with ''K<sub>0</sub>'' of the ring of [[continuous function|continuous]] real-valued functions on ''X''.<ref>{{Citation | last1=Karoubi | first1=Max | title=K-Theory: an Introduction | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Classics in mathematics | isbn=978-3-540-79889-7 | year=2008}}, see Theorem I.6.18</ref>
| |
| | |
| ====Relative K<sub>0</sub>====
| |
| Let ''I'' be an ideal of ''A'' and define the "double" to be a subring of the [[Cartesian product]] ''A''×''A'':<ref name=Ros27>Rosenberg (1994) 1.5.1, p.27</ref>
| |
| | |
| :<math>D(A,I) = \{ (x,y) \in A \times A : x-y \in I \} \ . </math>
| |
| | |
| The ''relative K-group'' is defined in terms of the "double"<ref name=Ros27a>Rosenberg (1994) 1.5.3, p.27</ref>
| |
| | |
| :<math>K_0(A,I) = \ker \left({ K_0(D(A,I)) \rightarrow K_0(A) }\right) \ . </math>
| |
| | |
| where the map is induced by projection along the first factor.
| |
| | |
| The relative K<sub>0</sub>(''A'',''I'') is isomorphic to K<sub>0</sub>(''I''), regarding ''I'' as a ring without identity. The independence from ''A'' is an analogue of the [[Excision theorem]] in homology.<ref name=Ros30/>
| |
| | |
| ====''K''<sub>0</sub> 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<sub>0</sub> into a commutative ring with the class [''A''] as identity.<ref name=Mil5/> The [[exterior product]] similarly induces a [[λ-ring]] structure.
| |
| The [[Picard group]] embeds as a subgroup of the group of units K<sub>0</sub>(''A'')<sup>∗</sup>.<ref name=Mil15>Milnor (1971) p.15</ref>
| |
| | |
| === ''K''<sub>1</sub> ===
| |
| [[Hyman Bass]] provided this definition, which generalizes the group of units of a ring: ''K<sub>1</sub>(A)'' is the [[abelianization]] of the [[infinite general linear group]]:
| |
| | |
| :<math>K_1(A) = \operatorname{GL}(A)^{\mbox{ab}} = \operatorname{GL}(A) / [\operatorname{GL}(A),\operatorname{GL}(A)]</math>
| |
| | |
| Here
| |
| | |
| :<math>\operatorname{GL}(A) = \operatorname{colim} \operatorname{GL}(n, A)</math>
| |
| | |
| 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,<ref>J.H.C. Whitehead, ''Simple homotopy types'' Amer. J. Math. , 72 (1950) pp. 1–57</ref> and is called the '''Whitehead group''' of the ring ''A''.
| |
| | |
| ==== Relative ''K''<sub>1</sub> ====
| |
| The ''relative K-group'' is defined in terms of the "double"<ref name=Ros92>Rosenberg (1994) 2.5.1, p.92</ref>
| |
| | |
| :<math>K_1(A,I) = \ker \left({ K_1(D(A,I)) \rightarrow K_1(A) }\right) \ . </math>
| |
| | |
| There is a natural [[exact sequence]]<ref name=Ros95>Rosenberg (1994) 2.5.4, p.95</ref>
| |
| | |
| :<math> 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) \ . </math>
| |
| | |
| ==== 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<sub>1</sub>(A)'' → ''A*''. As E(''A'') ◅ SL(''A''), one can also define the '''special Whitehead group''' SK<sub>1</sub>(''A'') := SL(''A'')/E(''A''). This map splits via the map ''A*'' → GL(1, ''A'') → ''K<sub>1</sub>(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]]:
| |
| | |
| :<math>1 \to SK_1(A) \to K_1(A) \to A^* \to 1,</math>
| |
| | |
| which is a quotient of the usual split short exact sequence defining the [[special linear group]], namely
| |
| | |
| :<math>1 \to \operatorname{SL}(A) \to \operatorname{GL}(A) \to A^* \to 1.</math>
| |
| | |
| The determinant is split by including the group of units ''A*'' = GL<sub>1</sub>''(A)'' into the general linear group GL''(A)'', so ''K<sub>1</sub>(A)'' splits as the direct sum of the group of units and the special Whitehead group: ''K<sub>1</sub>(A)'' ≅ ''A*'' ⊕ SK<sub>1</sub> (''A'').
| |
| | |
| When ''A'' is a Euclidean domain (e.g. a field, or the integers) SK<sub>1</sub>(''A'') vanishes, and the determinant map is an isomorphism from K<sub>1</sub>(''A'') to ''A''<sup>∗</sup>.<ref name=Ros74>Rosenberg (1994) Theorem 2.3.2, p.74</ref> 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<sub>1</sub> is nonzero was given by Ischebeck in 1980 and by Grayson in 1981.<ref name=Ros75>Rosenberg (1994) p.75</ref> If ''A'' is a Dedekind domain whose quotient field is an [[algebraic number field]] (a finite extension of the rationals) then {{harvtxt|Milnor|1971|loc=corollary 16.3}} shows that SK<sub>1</sub>(''A'') vanishes.<ref name=Ros81>Rosenberg (1994) p.81</ref>
| |
| | |
| The vanishing of SK<sub>1</sub> can be interpreted as saying that K<sub>1</sub> is generated by the image of GL<sub>1</sub> in GL. When this fails, one can ask whether K<sub>1</sub> is generated by the image of GL<sub>2</sub>. For a Dedekind domain, this is the case: indeed, K<sub>1</sub> is generated by the images of GL<sub>1</sub> and SL<sub>2</sub> in GL.<ref name=Ros75/> The subgroup of SK<sub>1</sub> generated by SL<sub>2</sub> may be studied by [[Mennicke symbol]]s. For Dedekind domains with all quotients by maximal ideals finite, SK<sub>1</sub> is a torsion group.<ref name=Ros78>Rosenberg (1994) p.78</ref>
| |
| | |
| For a non-commutative ring, the determinant cannot in general be defined, but the map GL(''A'') → ''K<sub>1</sub>(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<sub>1</sub>(''A'') → ''F''<sup>∗</sup> and SK<sub>1</sub>(''A'') may be defined as the kernel. '''Wang's theorem''' states that if ''A'' has prime degree then SK<sub>1</sub>(''A'') is trivial,<ref name=GS47>Gille & Szamuely (2006) p.47</ref> and this may be extended to square-free degree.<ref name=GS48>Gille & Szamuely (2006) p.48</ref> [[Shianghao Wang|Wang]] also showed that SK<sub>1</sub>(''A'') is trivial for any central simple algebra over a number field,<ref name=Wang1950>{{cite journal | zbl=0040.30302 | last=Wang | first=Shianghaw | authorlink=Shianghao Wang | title=On the commutator group of a simple algebra | journal=Am. J. Math. | volume=72 | pages=323–334 | year=1950 | issn=0002-9327 }}</ref> but Platonov has given examples of algebras of degree prime squared for which SK<sub>1</sub>(''A'') is non-trivial.<ref name=GS48/>
| |
| | |
| === ''K''<sub>2</sub> ===
| |
| {{See also|Steinberg group (K-theory)}}
| |
| <!--Matsumoto's theorem (K-theory) links here-->
| |
| [[John Milnor]] found the right definition of ''K<sub>2</sub>'': it is the [[centre of a group|center]] of the [[Steinberg group (K-theory)|Steinberg group]] St(''A'') of ''A''.
| |
| | |
| It can also be defined as the [[kernel (algebra)|kernel]] of the map
| |
| | |
| :<math>\varphi\colon\operatorname{St}(A)\to\mathrm{GL}(A),</math>
| |
| | |
| or as the [[Schur multiplier]] of the group of [[elementary matrices]].
| |
| | |
| For a field, K<sub>2</sub> is determined by [[Steinberg symbol]]s: this leads to Matsumoto's theorem.
| |
| | |
| One can compute that K<sub>2</sub> is zero for any finite field.<ref name=Lam139>Lam (2005) p.139</ref><ref name=Lem66>Lemmermeyer (2000) p.66</ref> The computation of K<sub>2</sub>('''Q''') is complicated: Tate proved<ref name=Lem66/><ref name=Mil101>Milnor (1971) p.101</ref>
| |
| | |
| :<math>K_2(\mathbf{Q}) = (\mathbf{Z}/4)^* \times \prod_{p\ge 3} (\mathbf{Z}/p)^* \ </math>
| |
| | |
| and remarked that the proof followed [[Gauss]]'s first proof of the [[Law of Quadratic Reciprocity]].<ref name=Mil102>Milnor (1971) p.102</ref><ref name=Gras205>Gras (2003) p.205</ref> | |
| | |
| For non-Archimedean local fields, the group K<sub>2</sub>(''F'') is the direct sum of a finite [[cyclic group]] of order ''m'', say, and a [[divisible group]] K<sub>2</sub>(''F'')<sup>''m''</sup>.<ref name=Mil175>Milnor (1971) p.175</ref>
| |
| | |
| We have K<sub>2</sub>('''Z''') = '''Z'''/2,<ref name=Mil81>Milnor (1971) p.81</ref> and in general K<sub>2</sub> is finite for the ring of integers of a number field.<ref name=Lem385>Lemmermeyer (2000) p.385</ref>
| |
| | |
| We further have K<sub>2</sub>('''Z'''/''n'') = '''Z'''/2 if ''n'' is divisible by 4, and otherwise zero.<ref name=Sil228>Silvester (1981) p.228</ref>
| |
| | |
| ====Matsumoto's theorem====
| |
| '''Matsumoto's theorem''' states that for a field ''k'', the second ''K''-group is given by<ref>{{citation | mr=0240214 | last=Matsumoto | first= Hideya | title=Sur les sous-groupes arithmétiques des groupes semi-simples déployés | journal=Ann. Sci. École Norm. Sup. (4) | issue= 2 | year=1969 | pages= 1–62
| |
| | url=http://www.numdam.org/item?id=ASENS_1969_4_2_1_1_0 | zbl=0261.20025 | language=French | issn=0012-9593 }}</ref><ref name=Ros214>Rosenberg (1994) Theorem 4.3.15, p.214</ref>
| |
| | |
| :<math>K_2(k) = k^\times\otimes_{\mathbf Z} k^\times/\langle a\otimes(1-a)\mid a\not=0,1\rangle.</math>
| |
| | |
| 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<sub>n</sub>'' (''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]]
| |
| | |
| :<math> 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 \ </math>
| |
| | |
| where '''''p''''' runs over all prime ideals of ''A''.<ref name=Mil123>Milnor (1971) p.123</ref>
| |
| | |
| There is also an extension of the exact sequence for relative K<sub>1</sub> and K<sub>0</sub>:<ref name=Ros200>Rosenberg (1994) p.200</ref>
| |
| | |
| :<math>K_2(A) \rightarrow K_2(A/I) \rightarrow K_1(A,I) \rightarrow K_1(A) \cdots \ . </math>
| |
| | |
| ====Pairing====
| |
| There is a pairing on K<sub>1</sub> with values in K<sub>2</sub>. Given commuting matrices ''X'' and ''Y'' over ''A'', take elements ''x'' and ''y'' in the [[Steinberg group]] with ''X'',''Y'' as images. The commutator <math>x y x^{-1} y^{-1}</math> is an element of K<sub>2</sub>.<ref name=Mil63>Milnor (1971) p.63</ref> The map is not always surjective.<ref name=Mil6>Milnor (1971) p.69</ref>
| |
| | |
| == Milnor ''K''-theory ==
| |
| {{main|Milnor K-theory}}
| |
| | |
| The above expression for ''K<sub>2</sub>'' of a field ''k'' led Milnor to the following definition of "higher" ''K''-groups by
| |
| | |
| :<math> K^M_*(k) := T^*(k^\times)/(a\otimes (1-a)) </math>,
| |
| | |
| thus as graded parts of a quotient of the [[tensor algebra]] of the [[multiplicative group]] ''k''<sup>×</sup> by the [[two-sided ideal]], generated by the
| |
| | |
| :<math>\left \{a\otimes(1-a): \ a \neq 0,1 \right \}.</math>
| |
| | |
| For ''n'' = 0,1,2 these coincide with those below, but for ''n''≧3 they differ in general.<ref>{{Harvard citations|last=Weibel|year=2005}}, cf. Lemma 1.8</ref> For example, we have ''K''{{su|b=''n''|p=''M''}}''(F<sub>q</sub>) = 0'' for ''n'' ≧2
| |
| but ''K<sub>n</sub>F<sub>q</sub>'' is nonzero for odd ''n'' (see below).
| |
| | |
| The tensor product on the tensor algebra induces a product <math> K_m \times K_n \rightarrow K_{m+n}</math> making <math> K^M_*(F)</math> a [[graded ring]] which is [[graded-commutative]].<ref name=GS184>Gille & Szamuely (2006) p.184</ref>
| |
| | |
| The images of elements <math>a_1 \otimes \cdots \otimes a_n</math> in <math>K^M_n(k)</math> are termed ''symbols'', denoted <math>\{a_1,\ldots,a_n\}</math>. For integer ''m'' invertible in ''k'' there is a map
| |
| | |
| :<math>\partial : k^* \rightarrow H^1(k,\mu_m) </math>
| |
| | |
| where <math>\mu_m</math> denotes the group of ''m''-th roots of unity in some separable extension of ''k''. This extends to
| |
| | |
| :<math>\partial^n : k^* \times \cdots \times k^* \rightarrow H^n\left({k,\mu_m^{\otimes n}}\right) \ </math>
| |
| | |
| satisfying the defining relations of the Milnor K-group. Hence <math>\partial^n</math> may be regarded as a map on <math>K^M_n(k)</math>, called the ''Galois symbol'' map.<ref name=GS108>Gille & Szamuely (2006) p.108</ref>
| |
| | |
| The relation between [[étale cohomology|étale]] (or [[Galois cohomology|Galois]]) cohomology of the field and Milnor K-theory modulo 2 is the [[Milnor conjecture]], proven by Voevodsky.<ref>{{Citation | last1=Voevodsky | first1=Vladimir | author1-link=Vladimir Voevodsky | title=Motivic cohomology with '''Z'''/2-coefficients | doi=10.1007/s10240-003-0010-6 | mr=2031199 | year=2003 | journal=Institut des Hautes Études Scientifiques. Publications Mathématiques | issn=0073-8301 | issue=98 | pages=59–104 | volume=98}}</ref> 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 {{harvtxt|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 space]]s 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'').<ref name=Ros2456>Rosenberg (1994) pp.245-246</ref>
| |
| | |
| Quillen gave two constructions, the "+-construction" and the "''Q''-construction", the latter subsequently modified in different ways.<ref name=Ros246>Rosenberg (1994) p.246</ref> The two constructions yield the same K-groups.<ref name=Ros289>Rosenberg (1994) p.289</ref>
| |
| | |
| === The +-construction ===
| |
| One possible definition of higher algebraic ''K''-theory of rings was given by Quillen
| |
| | |
| :<math> K_n(R) = \pi_n(BGL(R)^+),</math>
| |
| | |
| Here π<sub>''n''</sub> is a [[homotopy group]], GL(''R'') is the [[direct limit]] of the [[general linear group]]s over ''R'' for the size of the matrix tending to infinity, ''B'' is the classifying space construction of [[homotopy theory]], and the <sup>+</sup> is Quillen's [[plus construction]].
| |
| | |
| This definition only holds for ''n>0'' so one often defines the higher algebraic ''K''-theory via
| |
| | |
| :<math> K_n(R) = \pi_n(BGL(R)^+\times K_0(R)) </math>
| |
| | |
| Since ''BGL''(''R'')<sup>+</sup> is path connected and ''K<sub>0</sub>(R)'' discrete, this definition doesn't differ in higher degrees and also holds for ''n=0''.
| |
| | |
| === The Q-construction ===
| |
| {{main|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 Q''P'' is defined, objects of which are those of ''P'' and morphisms from ''M''′ to ''M''″ are isomorphism classes of diagrams
| |
| | |
| :<math> M'\longleftarrow N\longrightarrow M'',</math>
| |
| | |
| 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
| |
| | |
| :<math> K_i(P)=\pi_{i+1}(\mathrm{BQ}P,0)</math>
| |
| | |
| 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 (category theory)|nerve]]'' of ''QP''.
| |
| | |
| This definition coincides with the above definition of ''K<sub>0</sub>(P)''. If ''P'' is the category of finitely generated [[projective module|projective ''R''-modules]], this definition agrees with the above ''BGL<sup>+</sup>''
| |
| definition of ''K<sub>n</sub>(R)'' for all ''n''.
| |
| <!--
| |
| The ''K''-groups ''K''<sub>i</sub>(''R'') of the ring ''R'' are then the ''K''-groups ''K''<sub>i</sub>(''P<sub>R</sub>'') where ''P<sub>R</sub>'' is the category of finitely generated [[projective module|projective ''R''-modules]].
| |
| -->
| |
| More generally, for a [[scheme (mathematics)|scheme]] ''X'', the higher ''K''-groups of ''X'' are defined to be the ''K''-groups of (the exact category of) locally free [[Coherent sheaf|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<sub>n</sub>(R)''. When ''R'' is a [[noetherian ring|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<sub>*</sub> → M'', and a simple argument shows that the canonical map ''K''<sub>0</sub>(R) → ''G''<sub>0</sub>(R) is an [[isomorphism]], with ''[M]=Σ ±[P<sub>n</sub>]''. 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 [[Friedhelm Waldhausen|Waldhausen]].<ref>{{Citation | last1=Waldhausen | first1=Friedhelm | author1-link=Friedhelm Waldhausen | title=Algebraic ''K''-theory of spaces | doi=10.1007/BFb0074449 | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Lecture Notes in Mathematics | mr=802796 | year=1985 | volume=1126 | pages=318–419 | chapter=Algebraic K-theory of spaces | isbn=978-3-540-15235-4}}. See also Lecture IV and the references in {{Harvard citations|last1=Friedlander|last2=Weibel|year=1999}}</ref> It applies to categories with cofibrations (also called [[Waldhausen category|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 field]]s:
| |
| | |
| If '''F'''<sub>''q''</sub> is the finite field with ''q'' elements, then:
| |
| | |
| * ''K''<sub>0</sub>('''F'''<sub>''q''</sub>) = '''Z''',
| |
| * ''K''<sub>2i</sub>('''F'''<sub>''q''</sub>)=0 for ''i'' ≥1,
| |
| * ''K''<sub>2i-1</sub>('''F'''<sub>''q''</sub>)= '''Z'''/(''q<sup> i</sup>''-1)'''Z''' for ''i'' ≥1.
| |
| | |
| === Algebraic K-groups of rings of integers ===
| |
| Quillen proved that if ''A'' is the [[ring of integers|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. [[Armand Borel|Borel]] used this to calculate K<sub>''i''</sub>(''A'') and K<sub>''i''</sub>(''F'') modulo torsion. For example, for the integers '''Z''', Borel proved that (modulo torsion)
| |
| | |
| * ''K''<sub>i</sub> ('''Z''')/tors.=0 for positive ''i'' unless ''i=4k+1'' with ''k'' positive
| |
| * ''K''<sub>4''k''+1</sub> ('''Z''')/tors.= '''Z''' for positive ''k''.
| |
| | |
| The torsion subgroups of K<sub>2''i''+1</sub>('''Z'''), and the orders of the finite groups K<sub>4''k''+2</sub>('''Z''') have recently been determined, but whether the latter groups are cyclic, and whether the groups K<sub>4''k''</sub>('''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 regulator]]s.<ref name=Lem385>Lemmermeyer (2000) p.385</ref>
| |
| | |
| [[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<sub>n</sub>(A)'' are finitely generated when ''A'' is a finitely generated '''Z'''-algebra. (The groups
| |
| ''G<sub>n</sub>(A)'' are the ''K''-groups of the category of finitely generated ''A''-modules) <ref>{{Harvard citations|last1=Friedlander| last2=Weibel | year=1999}}, Lecture VI</ref>
| |
| | |
| ==Notes==
| |
| {{reflist}}
| |
| | |
| == References ==
| |
| *{{citation | last=Bass | first=Hyman | authorlink=Hyman Bass | title=Algebraic ''K''-theory | series=Mathematics Lecture Note Series | location=New York-Amsterdam | publisher=W.A. Benjamin, Inc. | year=1968 | zbl=0174.30302 }}
| |
| *{{Citation | editor1-last=Friedlander |editorlink1=Eric Friedlander | editor1-first=Eric | editor2-last=Grayson | editor2-first=Daniel | title=Handbook of K-Theory | url=http://www.springerlink.com/content/978-3-540-23019-9/ | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-3-540-30436-4 | mr=2182598 | year=2005}}
| |
| * {{Citation | last1=Friedlander | first1=Eric M. | last2=Weibel | first2=Charles W. | title=An overview of algebraic ''K''-theory | publisher=World Sci. Publ., River Edge, NJ | mr=1715873 | year=1999 | pages=1–119}}
| |
| * {{citation | last1=Gille | first1=Philippe | last2=Szamuely | first2=Tamás | title=Central simple algebras and Galois cohomology | series=Cambridge Studies in Advanced Mathematics | volume=101 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2006 | isbn=0-521-86103-9 | zbl=1137.12001 }}
| |
| * {{citation | last=Gras | first=Georges | title=Class field theory. From theory to practice | series=Springer Monographs in Mathematics | location=Berlin | publisher=[[Springer-Verlag]] | year=2003 | isbn=3-540-44133-6 | zbl=1019.11032 }}
| |
| * {{citation | first=Tsit-Yuen | last=Lam | authorlink=Tsit Yuen Lam | title=Introduction to Quadratic Forms over Fields | volume=67 | series=Graduate Studies in Mathematics | publisher=[[American Mathematical Society]] | year=2005 | isbn=0-8218-1095-2 | zbl=1068.11023 | mr = 2104929 }}
| |
| * {{citation | last=Lemmermeyer | first=Franz | title=Reciprocity laws. From Euler to Eisenstein | series=Springer Monographs in Mathematics | location=Berlin | publisher=[[Springer-Verlag]] | year=2000 | isbn=3-540-66957-4 | zbl=0949.11002 | mr=1761696 | doi=10.1007/978-3-662-12893-0 }}
| |
| * {{Citation | last1=Milnor | first1=John Willard | author1-link= John Milnor | title=Algebraic ''K''-theory and quadratic forms | mr=0260844 | year=1970 | month=1969 | journal=Inventiones Mathematicae | issn=0020-9910 | volume=9 | pages=318–344 | doi=10.1007/BF01425486 | issue=4}}
| |
| * {{Citation | last1=Milnor | first1=John Willard | author1-link= John Milnor | title=Introduction to algebraic K-theory | publisher=[[Princeton University Press]] | location=Princeton, NJ | mr=0349811 | year=1971 | zbl=0237.18005 | series=Annals of Mathematics Studies | volume=72 }} (lower K-groups)
| |
| *{{Citation | last1=Quillen | first1=Daniel | author1-link=Daniel Quillen | title=Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Lecture Notes in Math | doi=10.1007/BFb0067053 | mr=0338129 | year=1973 | volume=341 | chapter=Higher algebraic K-theory. I | pages=85–147 | isbn=978-3-540-06434-3}}
| |
| * {{Citation | last1=Quillen | first1=Daniel | author1-link= Daniel Quillen | title=Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 1 | publisher=Canad. Math. Congress | location=Montreal, Quebec | mr=0422392 | year=1975 | chapter=Higher algebraic K-theory | pages=171–176}} (Quillen's Q-construction)
| |
| * {{Citation | last1=Quillen | first1=Daniel | title=New developments in topology (Proc. Sympos. Algebraic Topology, Oxford, 1972) | publisher=[[Cambridge University Press]] | series=London Math. Soc. Lecture Note Ser. | mr=0335604 | year=1974 | volume=11 | chapter=Higher K-theory for categories with exact sequences | pages=95–103}} (relation of Q-construction to +-construction)
| |
| *{{Citation | last1=Rosenberg | first1=Jonathan | authorlink=Jonathan Rosenberg (mathematician) | title=Algebraic K-theory and its applications | url=http://books.google.com/books?id=TtMkTEZbYoYC | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=[[Graduate Texts in Mathematics]] | isbn=978-0-387-94248-3 | mr=1282290 | zbl=0801.19001 | year=1994 | volume=147}}. [http://www-users.math.umd.edu/~jmr/KThy_errata2.pdf Errata]
| |
| * {{Citation | last1=Seiler | first1=Wolfgang | editor1-last=Rapoport | editor1-first=M. | editor2-last=Schneider | editor2-first=P. | editor3-last=Schappacher | editor3-first=N. | title=Beilinson's Conjectures on Special Values of L-Functions | publisher=Academic Press | location=Boston, MA | isbn=978-0-12-581120-0 | chapter=λ-Rings and Adams Operations in Algebraic K-Theory | year=1988}}
| |
| * {{citation | last=Silvester | first=John R. | title=Introduction to algebraic K-theory | series=Chapman and Hall Mathematics Series | location=London, New York | publisher=[[Chapman and Hall]] | year=1981 | isbn=0-412-22700-2 | zbl=0468.18006 }}
| |
| * {{Citation | last1=Weibel | first1=Charles | author1-link=Charles Weibel | title=Handbook of K-theory | url=http://www.math.uiuc.edu/K-theory/0691/KZsurvey.pdf | publisher=[[Springer-Verlag]] | location=Berlin, New York | mr=2181823 | year=2005 | chapter=Algebraic K-theory of rings of integers in local and global fields | pages=139–190}} (survey article)
| |
| | |
| ==Further reading==
| |
| *{{citation | last=Srinivas | first=V. | title=Algebraic ''K''-theory | edition=Paperback reprint of the 1996 2nd | series=Modern Birkhäuser Classics | location=Boston, MA | publisher=[[Birkhäuser]] | year=2008 | isbn=978-0-8176-4736-0 | zbl=1125.19300 }}
| |
| *C. Weibel "[http://www.math.rutgers.edu/~weibel/Kbook.html The K-book: An introduction to algebraic K-theory]"
| |
| | |
| == See also ==
| |
| *[[Bloch's formula]]
| |
| *[[Redshift conjecture]]
| |
| *[[K-theory spectrum]]
| |
| *[[Fundamental theorem of algebraic K-theory]]
| |
| | |
| == External links ==
| |
| | |
| * [http://www.math.uiuc.edu/K-theory/ K theory preprint archive]
| |
| | |
| {{DEFAULTSORT:Algebraic K-Theory}}
| |
| [[Category:Algebraic K-theory| ]]
| |
| [[Category:Algebraic geometry]]
| |