Introduction To K-theory and Some

Applications*

By

Aderemi Kuku
Department of Mathematics
University of Iowa
Iowa City, USA
1. GENERAL INTRODUCTION AND OVERVIEW
1.1 What is K-theory?
1.1.1 Roughly speaking, K-theory is the study of functors
(bridges)
K n : (Nice categories) → (category of Abelian groups
n∈Z

C → K nC

(See 2.4 (ii) for a formal definition of a functor).

Note: For n ≤ 0 , we have Negative K-theory

For n ≤ 2 , we have Classical K-theory
For n ≥ 3 , Higher K-theory
1.1.2 Some Historical Remarks
K-theory was so christened in 1957 by A. Grotherdieck who first
studied K 0 (C) (then written K (C) ) where for a scheme X, C is the
category P(X) of locally free sheaves of OX-modules. Because
K 0 (C) classifies the isomorphism classes in C and he wanted the
name of the theory to reflect ‘class’, he used the first letter ‘K’ in
‘Klass’ the German word meaning ‘class’.

Next, M.F. Atiyah and F. Hirzebruch, in 1959 studied

K 0 (C) where C is the category VectC ( X ) of finite dimensional
complex vector bundles over a compact space X yielding what
became known as topological K-theory. It is usual to denote
K 0 ( VectC ( X ) ) by KU ( X ) or K top ( X ) .
0
In 1962, R.G. Swan proved that for a compact space X, the
category VectC ( X ) is equivalent to the category P (C ( X )) of
finitely generated projective modules over the ring C ( X ) of
complex valued functios on X.
i.e.,
VectC ( X ) ≈ P (C ( X )). So K 0 ( VectC ( X ) ) ≈ K 0 ( P (C ( X )) ) .

Thereafter, H. Bass, R.G. Swan, etc. started replacing C ( X ) by

arbitrary rings A and studied K 0 ( P ( A) ) for various rings A leading
to the birth of Algebraic K-theory. Here P ( A) denotes the
category of finitely generated projective modules over any ring A.
It is usual to denote K 0 ( P ( A) ) by K 0 ( A) for any ring A. K1 ( A) of a
ring A was defined by H. Bass and K 2 ( A) by J. Milnor. (see [3],
[58] and [79]).
In 1970, D. Quillen came up with the definitions of all
K n (C) for all n ≥ 0 in such a way that K 0 ( P ( A) ) coincides with
K n ( A) ∀ n ≥ 0 .

1.1.3 Some Features of K n (C)

(1) K n (C) sometimes reflects the structure of objects of C.

For example,
(i) Let F be a field, G a finite group, M(FG) the category
of finitely generated FG-modules. Then
K 0 (M ( FG )) : = G0 ( FG ) classifies representations of G
in P (F ) whose P (F ) is the category of finite-
dimensional vector spaces (see [42]),

(ii) K 0 ( ZG ) contains topological / geometric invariants.

E.g., Swan-Well
Invariants (see 2.7.1)
 (iii) K i (ZG ) contains Whitehead torsion – a topological

invariant (see
[3.2.3] or [57]).
(2) Each K n (C) yields a theory which could map or coincide
with other theories.
For example,
(i) Galois, etale or Motivic cohomology theories (see
[37]).
(ii) De Rham, cyclic cohomology (see [7] or [9, 10])
(iii) Representation theory, e.g.,
K 0 ( M ( FG ) ) ≈ G0 ( FG ) concides with Abelian group of
characters of G (see [8, 42] or 2.3 vii).

(3) K n (C) satisfiesvarious exact sequences connecting

K n , K n −1 , etc. For example, Localization sequences, Mayer-
victories sequence, etc. These sequences are useful for
computations (see [42] or [62]).
1.1.4 A Basic problem in this field is to understand and compute
the Abelian groups K n (C) for various categories ‘C’.

Two important examples of ‘nice’ categories are ‘Abelian

categories’ and ‘exact categories’. We now formally define these
categories with copious examples and also develop notations for
K n (C) for various C.
1.2 Abelian and Exact Categories – Definitions, Examples
and Notations

1.2.1 A category consists of a class C of objects together with a

set Hom C ( X , Y ) of morphisms from X to Y, for each ordered pair
(X,Y) of objects of C such that
(1) For each triple (X,Y,Z) of objects of C, we have
composition Hom C (Y , Z ) × Hom C ( X , Y ) → Hom C ( X , Z ) .

(2) Composition of morphisms is associative i.e., for

composable morphisms f.g.h g (hf ) = ( gh) f
(3) There exists identity 1 X ∈ Hom( X , X ) such that if
g ∈ Hom C ( X , Y ) and h ∈ Hom C ( Z , X ), g 1 X = g , and 1 X h = h .

Examples:
(i) Sp : = category of topological spaces, ob(Sp ) = topological

spaces, Hom Sp ( X , Y ) = {continuous maps X → Y }.

(ii) Gp : = category of groups. ob( G p ) are groups

Hom GP (G , H ) = groups homomorphisms G → G ′ .

For more examples (see [55]).

1.2.2 Examples of Abelian Categories (for motivation)
(1) A b or -Mod : = category of Abelian groups.
ob ( Ab) = Abelian groups
. Morphisms are Abelian group homomorphism.

(2) F a field; F-Mod : = category of vector spaces over

F.
ob ( F -Mod) : = vector spaces
Morphisms are linear transformation
(3) R a ring with identity.
(R- Mod) : = category of R-modules
Morphisms are R-module homomorphisms.
1.2.3 Definitions of an Abelian Category
A category A is called an Abelian category if
(1) it is an Addictive category, that is:
(a) There exists a zero object ‘0’ in A
(b) Direct sum (= direct product) of any two objects of
A exists in A.
(c) Hom A ( M , N ) is an Abelian group such that
composition distributes over addition.
(2) Every morphism in A has a kernel and a cokernel.
(3) For any morphism f, coker (ker f) = ker (coker f).
1.2.4 Note: A morphism g : K → M is called a kernel of a
morphism f : M → N if for any morphism h : P → M with
f ⋅ h = 0 , there exists a unique arrow κ : P → K such that h = g  k
K →
g
M →
f
N

k h
P
Equivalently: given an object P in A , we have an exact sequence
f
0 → hom A ( P, K ) →
sκ
hom A ( P, M ) →
p
hom A ( P, N )
is exact.
Analogously, a morphism g : N → C is called a cokernel of
f : M → N if for any P ∈ 0 b A
0 → hom A (C , P) → hom A ( N , P) →
f^
hom A ( M , P)
is exact.
Note: A sequence A →
f
B →
g
C is said to be exact at B if
ker( g ) = Im( f ) .
1.2.5 Definition of an Exact Category
An exact category is a small additive category C (embeddable in
an Abelian category A) together with a family E of short exact
sequences 0 → C′ → C → C″ → 0 (I) such that

(i) E is the class of sequences in C that are exact in A

(ii) C is closed under extensions i.e., for any exact sequence
0 → C′ → C → C″ → 0 in A with C′, C″ in C, we also
have C ∈ C.

Before giving a construction of Kn (C) n ≥ 0, we give some

relevant examples of C and develop notations for Kn (C).
1.2.6 Examples
1. An Abelian category is an exact category when it is
considered together with a family of short exact sequences.
2. Let A be any ring with identity C = P(A) (resp. M(A)) the
category of finitely generated projective (resp. finitely generated)
A-modules. Write Kn(A) for Kn(P(A) and Gn (A) for Kn (M(A))
For n ≥ 0, e.g.,
(i) A= , , , .
(ii) A = integral domain, R.
A = F (a field, - could be quotient field of R)
A = D (a division ring)
(iii) G any discrete group (could be finite)
A = G, RG, G, G, G (in the notation of (i) or (ii).
- These are group-rings.
(iv) G a finite group, ZG is an example of a Z-order in the
semi-simple algebra QG.
(v) Definition
Let R be a Dedekind domain with quotient field F (e.g., R = Z
 
(resp. Z p ), F = Q(resp Q p )

ˆ
p a rational prime or more generally R p , Fp ( p a prime ideal of
R). An R-order Λ in semi-simple F-algebra ∑ is a subring of ∑
such that R is contained in the centre of Λ , Λ is a finitely
generated R-module and
 
F ⊗ R Λ = Σ, (E.g., Λ = ZG, Z pG, RG, R pG G a finite group).
(vi) Let A be a ring (with 1), α: A → A an automorphism of A,
Aα(T) =
Aα (t, t –1) : = α-twisted Laurent series ring over A (i.e.,
Additively Aα[T] = A[T], with multiplication given by ( at i) ⋅ (bt i)
= a α−1(b) t i + j for a, b ∈ A). Let Aα[t] be the subring of Aα(T)
generated by A and t.
Note: If Λ = RG, Λ α[T] = R V where V = G × | T is a virtually
α

infinite cyclic group and G is a finite group, α an automorphism

of G and the action of the infinite cyclic group
T = 〈t〉 on G is given by α(g) = tgt −1 for all g ∈ G.
(3) X a compact topological space, F = or , VectF(X) : =
category of finite dimensional vector bundles on X. (See [2]).
Write K nF ( X ) for Kn (VectF (X).
Theorem (Swan): There exists an equivalence of categories
VectC(X) ≈ P ( X) where X is the ring of complex-valued
functions on X. Hence
K0(X): = Kn(VectF(X) ≈ Kn( ( X)) = Kn(C(X)) (I)
Note: (I) gives the first connection between topological and
Algebraic K-theory. (See [7])
Gelford-Naimark theorem says that any unital commutative C*-
algebra A has the form A ≈ C(X) for some compact space X. If A is
a non-commulative C^-algebra, then K-theory of A leads to “non-
commutative geometry” in the sense that A could be conceived as
ring of functions on a “non-commutative or quantum” space. Note
that any not necessarily unital commutative C-algebra A has the
form C 0 ( X ) where X is a locally compact space and
X + = X { p ∞ } , the one point compactification of X. When X is
compact C0 ( X ) = C ( X ) .
{ }
Note that C0 ( X ) = α : X + → C α continuous and α ( C0 ) = 0 .
(4) Let X be a scheme (e.g., an affine or projective algebraic
variety). (See [8] or below). Let P(X) be the category of
locally free sheaves of OX-modules. Write Kn(X) for
Kn(P(X). Let M(X) be the category of coherent sheaves of
OX-modules. Write Gn(X) for Kn(M(X). Note that if X =
Spec (A), A commutative ring we recover Kn(A) and Gn(A).

Recall (Definition of Affine and Projective Varieties)

(a) Let K be an algebraically closed field (e.g., or algebraic
closure of a finite field. Can regard polynomials in
A = An = K [ t1 ,  , t n ] as functions f : K n → K . An algebraic set
in K n = { x ∈ K n satisfying f i ( x) = 0 1 ≤ i ≤ r , f i ∈ A} .
 {
• If S ⊂ A, V ( S ) = x ∈ K n f ( x) = 0 ∀ f ∈ S } define closed
sets for a topology (Zariski topology) on the affine space
K n , also denoted An (K ) .
Note that ( V ( S1 ) V ( S 2 ) ) = V ( S1 S 2 )
V ( S i ) = V ( S j ), V ( A) = φ , V (φ ) = K n ) .
i⊂ I

• Also if E ⊂ K n , I ( E ) = { f ∈ A f ( x) = 0 ∀ x ∈ E} is an
ideal in A.
• Let X ⊂ K n be an algebraic set. A function ϕ : X → K is
said to be regular if ϕ = f X for some f ∈ A .
• The regular functions on A form a K-algebra K[X] and
K [ X ] ≅ A / a where a = I ( X ) .
• Call ( X 1 K [ X ]) an affine algebraic variety where
K[ X ] = OX ( X ) .
(b) Let V ∈ P ( K ), P(V ) = set of lines (i.e., 1-dim subspaces) of V.
Write Pn (K ) for P( K n ) . Elements of Pn (K ) are classes of (n +
1)-tangles [ x0 , x1 , , x n ] where [ x0 , , x n ] ≅ [ λ x0 ,  , λ x n ] if
λ ≠ 0 in K.
• If S ⊂ K [ t 0 , , t n ] is a set of homogeneous polynomials
V ( S ) = { x ∈ Pn ( K ) f ( x) = 0 ∀ f ∈ S } . The V(S) are closed
sets for Zariski topology on Pn (K ) .
• A projective algebraic variety X is a closed subspace of
Pn (K ) together with its induced structure sheaf O X = OP | X . n
(5) Let G be an algebraic group over a field F, (a closed subgroup
of GLn (F ) ) e.g., SLn ( F ), On ( F ) and X a G-scheme, i.e., there
exists an action θ : G × X → X . Let M(G,X) be the category of
F
G-modules M over X. (i.e., M is a coherent OX-module
together with an isomorphism of OG × X -module θ *(M) = p2*
F

(M), with p2 : G × X → X ; satisfying some co-cycle conditions)

F
(see [83]). Write Gn (G, X) for K n (M (G, X )) .
• Let P(G,X) be the full subcategory of M(G,X) consisting of
locally free OX-modules. Write Kn(G,X) for Kn(P(G,X)).
(see [43]).
 ~
(6) Let G be a semi-simple, connected, and simply connected
~
algebraic group over a field F . T ⊂ G a maximal G-split torus
~ ~ ~ ~
of G , P ⊂ G a parabolic subgroup of G containing the torus
~
T .
~ ~
The factor variety G F is smooth and projective. Call F
~ ~
= G P a flag variety.
E.g.,
~ ~   a b  

G = SLn P =   det a det c = 1 a ∈GLn c ∈GLn −k .

 0 c  

~ ~
Then F = G P is the Grassmanian variety of k-dimensional
linear subspaces of an n-dimensional vector space. Write
K n (G , F ) for K n ( P (G , F ) ) . (See [43])
6. Let F be a field and B a separable F-algebra, X a smooth
projective variety equipped with the action of an affine
algebraic group G over F. Let VBG (X1B) be the category of
vector bundles on X equipped with left B-module structure.
Write Kn(X, B) for Kn(VBG(X,B)). In particular, in the
notation of (5), we write Kn(F, B) for Kn(VBG(F, B)). (See
[43])
7. Let G be a finite group, S a G-set. Let S be a category
defined by ob S = {elements of S); S (s.,t) = {(g,s)| g ∈ G, g
s = t}. Let C be an exact category. [S, C] the category of
functors ξ : S → C Then [S, C] is also an exact category
where a sequence
0 → ξ ′ → ξ → ξ ′′ → 0 is said to be exact in [S, C] if
0 → ξ ′( s ) → ξ ( s) → ξ ′′( s ) → 0 is exact in C. Write
K nG ( S , C) for K n ([ S , C]) .
E.g., C = M ( A) , A a commutative ring,
S = G / H , then [G / H , M ( A))] = M ( AH ) .
• [G / H , P ( A)] = PA ( AH ) = category of finitely generated AH-
modules that are projective over A. (i.e., AH lattices)

K n ( G / H , M ( A) ) := Gn ( AH ).
If A is regular, then Gn ( A, H ) ≅ Gn ( AH ). (See [25])
2. K 0 (C), C AN EXACT CATEGORY: DEFINITIONS AND
EXAMPLES
2.1 Define the Grathendieck group K 0 (C) of an exact category
C as the Abelian group generated by isomorphism classes (C) of
C-objects subject to the relations ( C ′) + ( C ′′) = (C ) wherever
0 → C ′ → C → C ′′ → 0 is exact in C.

2.2 Remarks
(i) K 0 (C) ≅ F R where F is the free Abelian group on the
isomorphism classes (C) of C -objects and R is the
subgroup generated by all ( C ′) + ( C ′′) − (C ) for each short
exact sequence 0 → C ′ → C → C ′′ → 0 in C. Denote by [C]
the class of (C) in K 0 (C) .
(ii) The construction in 2.1 satisfies a universal property. If
χ C → A is a map from C to an Abelian group A, given
that χ (C ) depends only on the isomorphism class of C and
χ ( C ′′) + χ ( C ′) = χ (C ) for any exact
sequence 0 → C ′ → C → C ′′ → 0 , then there exists, a unique
homomorphism χ ′ : K 0 (C ) → A such that χ (C ) = χ ′(C ) for
any C-object C.

(i) Let F : C → D be an exact functor between two exact

categories C, D (i.e., F is additive and takes short exact
sequences in C to short exact sequences in D). Then F
induces a group homomorphism K 0 (C ) → K 0 ( D) .

(ii) Note that an Abelian category A is also an exact category

and the definition of K 0 ( A ) is the same as in definition 2.1.
(i) If C is an exact category in which every s.e.s
0 → C ′ → C → C ′′ → 0 splits. E.g., P ( A), VectC ( X ) , then
K 0 (C) is the Abelian group on isomorphism classes of C-
objects with relation (C ′) + ( C ′′) = ( C ′ ⊕ C ′) . In this case,
(C, ⊕) is an example of a “symmetric monoidal category”
with one property that the isomorphism classes of objects
of C form an Abelian monoid and K 0 (C) is then the ‘group
completion’ or ‘Grathendiuk group’ of such a monoid (see
[42], Chapter 1, 1.2, 1.3). In fact, this construction
generalizes standard procedure of constructing integers
from the natural numbers.
2.3 Examples
(i) If A is a field or division ring or a local ring or a principal
ideal domain, then K 0 ( A) ≅ Z . This follows from the fact
that every P ∈ P ( A) is free (i.e., P ≅ As for some s) and
moreover, A satisfies the invariant bases property i.e.,
Ar ≅ As ⇒ r = s .

(ii) Let A be a (left) Noetherian ring (i.e., every left ideal is

finitely generated). Then the category ( M (Λ ) of finitely
generated (left)-A-modules is an exact category and we
denote K 0 ( M (Λ ) ) by G0 (Λ ) . The inclusion functor
P (Λ) → M (Λ ) induces a map K 0 ( A) → G0 (Λ ) called the
Cartan map. For example, Λ = RG (R a Dedekind domain,
G a finite group) yields a Cartan map K 0 ( RG ) → G0 ( RG ) .
If Λ is left Artinian i.e., the left ideals of Λ satisfy
descending chain condition, then G0 (Λ) is free Abelian on
[ S1 ], , [ Sr ] where the [ Si ] are distinct classes of simple Λ -
modules, while K 0 (Λ) is free Abelian on [ I1 ], , [ I t ] and tho
I i are distinct classes of indecomposable projective Λ -
modules (see [8]). So, the map K 0 ( A) → G0 (Λ ) gives
matrix ( ai j ) where ai j = the number of times S j occurs in a
composition series for I i . This matrix is known as the
Cartan matrix.

If Λ is left regular (i.e., every finitely generated left Λ -

module has finite resolution by finitely generated
projective left Λ -modules), then it is well known that the
Cartan map is an isomorphism.
(iii) Recall also that a maximal R-order Γ in Σ is an order that is
not contained in any other R-order. Note that Γ is regular.
So, as in (ii) above, we have Cartan maps
K 0 (Γ) → G0 (Γ) and when Γ is a maximal order, we have
K 0 (Γ) ≅ G 0 (Γ ) .

(i) Let R be a commutative ring with identity. Λ an R-algebra.

Let PR (Λ) be the category of left Λ -modules that are
finitely generated and projective as R-modules (i.e., Λ -
lattices). Then PR (Λ ) is an exact category and we write
G0 ( R, Λ ) for K 0 ( PR ( A) ) . If Λ = RG , G a finite group, we
write PR (G ) for PR (RG ) and also write G0 ( R, G ) for
G0 ( R, RG ) . If M , N ∈ PR (Λ) , then, so is ( M ⊗ R N ) , and
hence the multiplication given in G0 ( R, G ) by
[ M ][ N ] = ( M ⊗ R N ) makes G0 ( R, G ) a commutative ring
with identity.
(v) If R is a commutative regular ring and Λ is an R-algebra
that is finitely generated and projective as an R-modules
(e.g., Λ = RG, G a finite group or R is a Dedekind domain
with quotient field F, and Λ is an R-order in a semi-simple
F-algebra), then G0 R, Λ) ≅ G0 (Λ)

(i) Let F be a field, G a finite group. A representation of G in

P(F) is a group homomorphism p : G → Aut(V ) V ∈ P ( F ) .
Call V a representation space for ρ . The dimension of V
over F is called the degree of ρ .

Note:
• ρ determines a G-action on V i.e.,
G × V → V ( g , v) → ρ ( g )v = gv and vice versa.
• Two representations (V1ρ ) and (V1′ ρ ′) are said to be
equivalent if there exists an F-isomomorphism
β : V ≅ V ′ such that ρ ′( g ) = βρ ( g )
 • There exists, 1 – 1 correspondence between
representations of P in P (F ) and FG-modules.
• Can write a representation of G in P (F ) as a pair
(V1ρ ) . V ∈ P (F ) and ρ : G → Aut(V ) .
• If C is any category and G a group. A representation
of G in C (or a G-object in C) is a pair
( X , ρ ) X ∈ ob C , ρ : G → Aut( X ) a group-
homomorphism.

The G-objects in C forms a category CG where for

( X , ρ ), ( X ′, ρ ′) , morC ( X , ρ ) , ( X ′, ρ ′) is the set of all C -
G

morphisms α : X → X ′ such that for each g ∈ G , the diagram

ρg
X X
α α commutes
ρ ′g
X′ X′
(vii) Let G be a finite group, S a G-set, S the category associated
to S, C an exact category, [ S , C] the category of covariant
functors ς : S → C . We write ς s for ς ( s ), s ∈ S . Then, [ S , C]
is an exact category where the sequence
0 → ς ′ → ς → ς ′′ → 0 in [ S , C] is defined to be exact if
0 → ς s′ → ς s → ς s′′ → 0 is exact in C for all s ∈ S . Denote by
K 0G ( S , C) the K 0 of [ S , C] . Then K 0G (−, C) : G Set → Ab is a
functor called ‘Mackey’ functor. We also note the fact that
K nG (−, C ), n ≥ 0 is also a ‘Mackey’ functor. (See [42])
If S = G / G , then [ G / G, C] ≅ CG analogous constructions to
the one above can be done for G, a profinite group, and
compact Lie groups (see [42], [28], [35]).
Now if R is a commutative Noetherian ring with identity,
we have [ G / G, P ( R)] ≅ P ( R) G ≅ PR ( RG ) , and so,
K 0G ( G / G, P ( R ) ) ≅ G0 ( R, G ) ≅ G0 ( RG ) . This provides an
initial connection between K-theory of the group ring RG
and Representation theory. As observed in (iv) above
G0 ( R, G ) is also a ring.
In particular, when R = C , P (C ) = M (C ) , and
K 0 (P (C ) G ) ≅ G0 (C,G ) = G0 (CG ) is the Abelian group of
characters, χ : G → C (see [30]), as already observed in this
paper.
 If the exact category C has a pairing C × C → C, which is
naturally associative and commutative, and there exists
E ∈ C such that ( E , M ) = ( M , E ) = M for all M ∈ C , then
K 0G (−, C) is a Green functor and moreover, for all n ≥ 0 ,
K nG (−, C ) is a module over K 0G (−, C ) . (See [42])

2.4 K 0 of Schemes

(i) More Examples of Abelian Categories:

Functor Categories and Sheaves
• Let B be a small category i.e., (ob B is a set), A an Abelian
category. Then the category of functors B → A is also an
Abelian category denoted by AB.
Note: ob A = {functors : B → A)
B

Morphisms are natural transformations of functors.

• Recall. Let C, D be two categories. A covariant (resp.
contravarient) functor from C to D is an assignment to each
object C ∈ ob(C) an object F(C) in D as well as an
assignment to each morphism f , C → C ′, a D-morphism
F ( f ) : F (C ) → F ( C ′) (resp. F ( C ′) → F (C ) ) such that
1. F (1C ) = 1F (C ) for any C ∈ C ;
2. F ( gf ) = ( F ( g ) F ( f ) (resp. F ( gf ) = F ( f ) F ( s) .

Example:
1. R a commutative ring, F : R -Mod → -Mod given by
F = Hom R (−, N )
N fixed in R-Mod. F is contravariant F ′ = Hom R ( M −)
is covariant.
 In fact Hom R (−,−) is a bifunctor
R- M od × R - M od → -Mod
( M , N ) → Hom R ( M , N )
covariant in N and contravarient in M.

2. F : (Groups) → -mod
G → G /[G, G ]

is covariant – called Abelianization functor.

• Let F , F ′ be two functors - from C to D. A natural

transformation from F to F ′ is an assignment to an object
C ∈ C a D-morphisms η C : F (C ) → F (C ) such that if
α : C → C ′ is a C-morphism, then the diagram
ηC
FC F ′C
F (α ) F ′(α ) commutes
η C′
FC ′ F ′C ′
• Note: A functor (roughly speaking) is a ‘bridge’ for
crossing from one category to another.

• Any partially ordered set ( E , ≤) has the structure of a

category where
ob( E ) = elements of E
hom E ( x, y ) = φ unless x ≤ y.

• Let X be a topological space, U the poset of open subsets of

X. A contravariant functor F : U → A (A an Abelian
category) is called a presheaf on X.

Note: The presheaves on X form an Abelian category

denoted by
Presh (X).
 A sheaf on X is a presheaf F satisfying:
If {U i } is an open covering of a subset U ⊂ X , then we
have an exact sequence:
0 → F (U ) → Π F (U i ) Π F (U i ∩ U j )
i< j

(i.e., if f i ∈ F (U i ) are such that f i and agree on

fj
F (U i ∩ U j ) , then there exists, a unique f ∈ F (U ) that maps
to every f i under F (U ) → F (U i ) .
Note: Sh(X) is also an Abelian category. (See [93] or
[18])
(i) A ringed space ( X , OX ) is a topological space X
together with a sheaf OX of rings on X.
(ii) An OX -module is a sheaf M together with a sheaf
morphism O X × M → M s.t for each U ⊂ X , M (U ) is a
unitary OX (U ) -module.
(ii) Let R be a commutative ring with identity Spec(R) =
{prime ideals of R}
A subset Y ⊂ Spec( R) is closed off
Y = V ( I ) = { p ∈ Spec( R ) p ⊃ I } , I an ideal of R.

One could view R as the ring of functions on Spec (R) and

V[I] as the set of points y ∈ Spec( R ) at which all the
functions in I vanish. If f ∈ R is viewed as a function on
Spec (R), its value at y ∈ Spec( R ) is its image in the residue
class field k ( y ) : = the field of fractions of R/y.

• If X = Spec( R) , there exists a sheaf of rings OX on X

where OX (U ) = S −1R and S = { f ∈ R ∀ y ∈ U , f ∉ y}
OX ( X ) = R . Call the ringed space Spec( X , OX ) an
affine scheme.
(iii) A scheme is a topological space X together with a sheaf of
rings on X such that X = U i , (U i open in X ) and
(U i , O X U i ) is an affine scheme.
A morphism of schemes f , X → Y is a continuous map of
the underlying topological space together with (for each
open set U ⊂Y ) a ring homomorphism
f U* : OY (U i ) → O X ( f −1U ) compatible with the restriction
maps for each V ⊂ U . In addition, we require hat for
x ∈ f −1 (U ) g ∈ OY (U ), if g vanishes on f(x), then
f * ( y ) ∈ O X ( f −1U ) vanishes at x.
Note: Say that f ∈ OY (U ) vanishes at a point y ∈ U if given any
affine neighbourhood W of y, the image of f in OW (U W ) lies in
the prime ideal corresponding to y.

Recall: k[ X ] = k [ t1 , , t n ] a X . View f ∈ k[ X ] as a function on the

set of points of X.

(iv) A scheme X over Z is a morphism of schemes X → Z

Let X 1 , Y , be schemes over Z
X ×Y → Y
Z
↓ ↓ g (I ) pull back
X 
→f
Z
X × Y is the fibre product in the category of schemes over Z
Z
given by the diagram (I).
 X × Y is the fibre product in the category of schemes over Z
Z
given by the diagram (I).

• If X = Spec( A), Y = Spec( B ) Z = Spec(C )

X × Y = Spec A ⊗ B
Z
( C
)
• Put AXn = Spec( Z [ t1 , , t n ] )
AXn = AZ × X
Spec ( Z )

(v) Let X be a scheme. Define an algebraic bundle on X as a

morphism of schemes π : E → X together with maps

s : E×E → E
X

µ : AZ1 × E→E
Spec ( Z )
(satisfying axioms similar to those of a topological vector
bundles) together with local triviality: i.e., there exists an open
covering X = U α of X together with isomorphism
E U ≅ π −1 (U α ) ≅ An
α

Recall that a topological vector bundle E over X consists of

continuous maps π : E → X and C × E → E (scalar multiplicator),
ρ : E × E → E (addition) satisfying
X
(1) for λ ∈ C , v ∈ E , π (λ ⋅ (v)) = π (v) ,
π ( ρ (v, w)) = π (v)
(2)
= π ( w)
(3) If E x = π ′( x), µ : C × Ex → Ex , σ x : Ex × Ex → E x makes
E x into a complex vector space.
(vi) It is usual to view a vector bundle π : E → X via its sheaf of
sections E (U ) = {maps s: X → E s.t π  s = id } i.e., E is required
to be a locally free sheaf of OX -modules i.e., there exists an open
cover X = U α such that
E U ≅ AUn for each nα ∈ N .
α
α
α

A morphism of bundles is just an OX -linear map f : E → F i.e.,

for each open set U ⊂ X we have an OX (U ) -linear map of
modules f U
E (U ) → F (U ) s.t for V ⊂ U , the map ρvu f U
= f V ρVU .

(i) If X is a scheme. Define K 0 ( X ) := K 0 (P ( X )) .

If E is a vector bundle, E a locally free sheaf with
[ E ] = [E ] ∈ K 0 ( X )
[
[E ] ⋅ [ F ] = E ⊗ O X F ] (product in K 0 ( X ) where
( E ⊗ F ) (U ) = E (U ) ⊗ O F(U) ).
X

So K 0 ( X ) is a commutative ring.

• If f : X → Y is a morphism of schemes, there exists an

exact functor f * : P (Y ) → P ( X ) : E → f * E

Note: If U ⊂ X , V ⊂ Y , are affine open sets with f (U ) ⊂ V , then

f * : K 0 (Y ) → K 0 ( X )
So K 0 is a contravarient functor (schemes) 
→ (commutative
rings)
(iii) If X is a smooth projective curve over a field k, (see [18])
then

K 0 ( X ) ≈ Z ⊕ Pic( X )
(
[ E ] → rk ( E ) ⊕ Λrk ( E ) E )
where Pic(X) = group of isomomorphism classes of line bundles
(i.e., variant bundles of rank 1) over X.

(iv) K 0 (Pkn ) ≅ Z n +1

(v) If X is a regular scheme (i.e., any coherent sheaf of O X -

modules has a finite global resolution by locally free sheaves)
then K 0 ( X ) ≅ G0 ( X ) .
2.5 Some Topological K-theory
2.5.1 Let X be a compact space.
Recall: KC0 ( X ) : = K 0 ( VectC ( X ) ) ≅ K 0 (CX ) . KC0 ( X ) is also
written K t0or ( X ) or KU ( X ) .
K R0 ( X ) : = K 0 ( Vect R ( X ) ).

Write KO( X ) for K 0 ( Vect R ( X ) ) .

Note: K t0or ( X ) as a generalized cohomology theory arises as

homotopy groups of spectra. We now introduce the notion of
spectra.
2.5.2 An Ω -spectrum E is a set of pointed spaces
{ E 0 , E1 ,} each of which has the homotopy type of a CW-complex
such that each map E i → Ω( E i +1 ) is a homotopy equivalence i.e.,
we have a ‘sequence of homotopy equivalences
E 0 ≅ Ω E1 ≅ Ω2 E 2 ≅ ⋅ ≅ Ωn E n .

2.5.3 Theorem (see [2]).

Let E be an Ω -spectrum. For any topological space A ⊂ X , put
hEn ( X , A) = [( X , A), E n ] n ≥ 0 .
Then ( X , A) → hE* ( X , A) is a generalized cohomology theory,
namely, it satisfies all of the Eulenberg-Steenrod axioms except
that its value at a point (∗, φ ) may not be that of ordinary
cohomology.
So,
(1) hE* (−) is a functor (Topological pairs) → (Graded Abelian
groups),
(2) For each n ≥ 0 , and each pair ( X , A) of spaces, there exists,
a functorial connecting homomorphism
α : hEn ( A) → hEn +1 ( X , A)

(3) The connecting homomorphisms in (2) determine long

exact sequence for every pair ( X , A) .

(4) hEn (−) satisfies excision i.e., for every pair ( X , A) and
every subspace U ⊂ A s.t. U ⊂ Int ( A)
hE* ( X , A) ≅ hEm ( X − U , A − U )

Note: Above, hE∞ ( X ) : = hE* ( X ,φ ) = hE* ( X + ,∗) where X + is the

disjoint union of X and a point *.
2.5.4 KOt∗on (−), K t∗on (−) = KU (−) are the generalized cohomology
theories associated to the Ω -spectrum given by BO × Z and
BU × Z
i.e.,
K t2onj ( X ) = [ X i BU × Z ]

K t2onj −1 ( X ) = [ X ,U ]; K .

2.5.5 Bott Periology

1. BO × Z ~ Ω 8 ( BO × Z )
Moreover, the homotopy groups π i ( BO × Z ) ≅ KO i are
given by Z , Z / 2, Z / 2, 0, Z , 0, 0, 0 respectively for
i ≡ 0,1,2,3,4,5,6,7 (mod 8)
 Z if i is even
2. BU × Z ~ Ω ( BU × Z ) and π i ( BU × Z ) = 
2
.
0 if i is odd

3. For any topological space X, and any i ≥ 0 , we have a

natural homomorphism
β : K top
−1 −i −2
( X ) → K top (X )

Note:
 K top
0
( X ) for i even

For i ∈ Z , K top
i
(X ) =  ,
−1
 K top ( X ) for i odd
Let S 0 = (∗,∗) = ∗.
Then
Z if n is even
K n
top (∗) = 
0 if n is odd

Z if n + n is even
K i
top (S ) = 
n

0 if n is odd

2.6 K-theory of C*-algebras

2.6.1 A C*-algebra is a Banach algebra satisfying a ∗ a = a for
2

all a ∈ A . Let A be a C*-algebra. Define

K iton ( A) : = π i ( BGL( A) ) = π i −1 ( GL( A) ) ⋅ (GLA) is a topological group).
Note: K 0 ( A) = K 0 ( P( A)) ≈ K 0ton ( A) = π 0 (GL( A)) .
K i ( A) : = GL( A) GL0 ( A) where GL0 ( A) of the connected
component of the identity in GL(A) …
Bott periodicity is also satisfied i.e., K nton ( A) = K n + 2 ( A) ∀ n ≥ 0
and so, the theory is Z 2 -graded having only K 0ton ( A) = K 0 ( A) and
K 1ton ( A) .

2.6.2 Example
1. Let G be a discrete groups, 2 (G ) the Hilbert space of
square summable complex-valued functions on G, i.e., any
element of f ∈ 2 (G ) can be written as

f = ∑ λg g , λg ∈C , ∑ (λg ) 2 < ∞ .
g∈G g∈G

The group algebra C G is a subspace of 2 (G ) . There exists a left

regular representation λG of G on the space 2 (G ) given by
  
λG ( g ) ∑ λh h  = ∑ λG gh
 h∈G  g∈G

where g ∈ G and
∑λ h h ∈ 2 G.
This unitary representation extends linearly to C G .

Now define reduced C*-algebra C r*G of G by the image of

λG (CG ) in the C*-algebra of bounded operators on 2 (G ) .

• If G is finite, the C rα (G ) = C G and K 0 (C G ) = R(G ) the

additive groups of representation ring of G.
(i) K 0 (C ) = Z , K1 (C ) = π G GL(C ) = 0 such that GL(C ) is
connected.
(ii) H G = Z / 2, K 0 ( C r* (G ) ) ≅ K 0 (C ) ⊕ K 0 (C ) ≅ Z ⊕ Z since
C rn G ≅ C G = C ⊕ C .
2.7 Some Applications of K 0 ( C )
2.7.1 Geometric and Topological Invarients
Let R = Zπ 1 ( X ) , the integral grouping of the fundamental group
of a space of the homotopy type of a CW-complex.

Theorem (Wall) [87]

1. Let C = (C* , d ) be a chain complex of projective R-modules
that is homotopic to a chain complex of finite type of projective
R-modules. Then C = (C* , d ) is chain homotopic to a chain
complex of finite type of free R-modules iff the Euler
characteristics χ (C ) = 0 in K 0 ( R) .
Note: A bounded chain complex C = ( C r , d ) of R-modules is of
finite type if all C i are finitely generated. The Euler character of
α
C = ( C r , d ) is given by χ (C ) = ∑ (−1) [ C ] ∈ K
i = −∞
r
i 0 ( R) .

2. Computation of the group (SSP)

The calculation of G0 ( RG ) , G Abelian is connected to the
calculation of the group (SSF) which houses obstructions
constructed by Shub and Franks in their study of Morse-Smele
diffeomorphisms.

3. Dynamical Systems
Dynamical systems can be classified by means of K 0 of C*-
algebras.
2.7.2 Some other Miscellaneous Applications
1. Several classical problems in topology were solved via K-
theory e.g., finding the number of independent vector fields on the
n-space.

2. Index of Elliptic Operators

Let M be a closed manifold and D : C ∞ ( E ) → C ∞ ( E ) be an elliptic
differential operator between two bundles E, F on M. Let
~
M → M be a normal covering of M with deck transformation
~
group G (see [7]). Then, we can lift D to M and obtain an elliptic
G-equivalent differential operators D : C ∞ ( E ) → C ∞ ( F ) where
~
~
E , F are bundles on M . Since the action is free, one can define
an analytic index ind G ( D ) in K 0 ( C rs G ) (see [7]).
~
3. THE FUNCTORS K 1 , K 2 - BRIEF REVIEW
We shall follow the historical development of the subject by
briefly discussing k1 , K 2 of rings and their classical formulations.

3.1 K1 of a Ring – Definition and Basic Properties

3.1.1 Let R be a ring with identity GLn (R) the group of invertible
 A 0
n × n matrices over R. Note that GLn ( R ) ⊂ GLn +1 ( R ) A →  .
 0 1
∞
Put GL( R) = lim GLn ( R) = GLn ( R ).
n =1

Let E n (R) be the subgroup of GLn (R) generated by the elementary

matrices, ei j (a ) where
ei j (a ) is the n × n matrix with 1’s along the diagonal, a in the
(i, j ) -position with i ≠ j and zeros elsewhere. Put
E ( R ) = lim E n ( R ) .
3.1.2 Note that the matrices ei j (a ) satisfy the following.
(i) ei j (a ) ei j (b) = ei j (a + b) ∀ a, b ∈ R
(ii) [eij (a), e jk (b)] = eik (ab) ∀ i ≠ k , a, b ∈ R
(iii) [eij (a), ekl (b)] = 1 ∀ i ≠ , j ≠ k .

3.1.3 Whitehead Lemma

(i) E ( R) = [ E ( R), E ( R)] i.e., E (R) is perfect
(ii) E ( R) = [GL( R)], GL( R)] .

3.1.4 Definition
K1 ( R) : = GL( R) / E ( R) = GL( R) /[GL( R), GL( R)]

= H 1 (GL( R))
3.1.5 Note that:
(i) K 1 is functorial in R i.e., R → R ′ is a ring homomorphism,
we have K 1 ( R) → K 1 ( R ′)
(ii) K 1 ( R ) ≅ K 1 ( M n ( R ) ) for any positive integer n and any ring
R
(iii) K 1 ( R ) ≅ K 1 (P ( R )) .

3.1.6 If R is a commutative ring with identity, the determinant

map det : GLn ( R ) → R ∗ commutes with GLn ( R ) → GLn +1 ( R ) and
hence defined a map det : GL( R) → R * which is surjective since
 a 0
given a ∈ R there exists A = 
*
 such that det( A) = a .
 0 1
• We also have an induced map
det : GL( R ) [GL( R ), GL( R )] → R *
i.e., det K1 ( R) → R * that is split by a map
 a 0
α : R → K 1 ( R) : a → 
/*

 0 1
i.e., det α = 1R . So K 1 ( R) ≅ R * ⊕ SK 1 ( R) where
SK 1 ( R) := ker(det : K 1 ( R) → R * ) ;
• Note that SK 1 ( R) = SL( R) E ( R) where
SL( R) = limSLn ( R ) and SLn ( R ) = { x ∈ GLn ( R) det x = 1} .

3.1.6 Examples
(i) If R is a field F , SK 1 ( F ) = 0 and K 1 ( F ) ≅ F *
(ii) If R is a divisin ring K1 ( R) ≅ R * [ R * , R * ] .
3.1.7 Stability for K1
Stability results are useful for reducing computations of K 1 ( R) to
computations of matrices of manageable size.

Definition: Let A be a ring with identity. An integer n is said to

satisfy stable range condition ( SRn ) for GL( A) if whenever r > n ,
and ( a1 , a 2 ,  ., a r ) generates the unit ideal ΣAai = A , then there
exists b1 , b2 , , br −1 ∈ A such that

( a1 + a r b, a 2 + a r bb ,, a r −1 + a r br −1 ) also
generates the unit ideal i.e.,
∑ A( a i + a r bi ) = A
E.g., a semi-local ring (i.e., a ring with a finite number of
maximal ideals satisfy SR2 ).

3.1.8 Theorem
If SRn is satisfied, then
(a) GLm ( A) E m ( A) → GL( A) E ( A) is onto for m ≥ n and
injective for all m > n .
(b) Em ( A)∆GLm ( A) for m ≥ nt 1
(c) GLm ( A) E m ( A) is Abelian for m > n .

3.2 K1, SK1 of Orders and Group-rings

3.2.1 Let R be a Dedekind domain with quoted field F, Λ an R-
order in a semi-simple F-algebra.
Put SK1 (Λ) := ker( K1 (Λ) → K1 (Σ) ) .
Hence understanding K1 (Λ ) reduces to understanding SK 1 (Λ ) and
K 1 (Σ) . Now Σ = ΠM n ( Di ) . Di a division ring.
i
 • So K i (Σ) ≅ ΠK1 ( Di ) .
• One way of understanding SK1 (Λ ) is via reduced norm
which generalizes the notion of determinant.

3.2.2 Let R be the ring of integers in a number field or p-adii

field F. then there exists an extension E of F since that E is a
splitting field of Σ i.e., E ⊗ F Σ is a direct sum of metric algebras
over E i.e.,
E ⊗ F Σ ≅ ⊕ M n (E ) .
i

Let C be the centre of Σ .

If a ∈ Σ, | ⊗ a ∈ E ⊗ F Σ can be represented as a direct sum of
matrices over E and so we have a map nr : GL(Σ) → C * .
If
Σ = ⊕Σ i = ⊕ M ni ( E ), and C = ⊕ C i .
i −1 i −1
We could compute n r (a) component-wise via GL( Σ i ) → C i . Since
C n is Abelian, we have
nr : K 1 (Σ) → C * .

• SK1 (Λ ) = { x ∈ K1 (Λ) n r ( x) = 1} .

Hence we have access to SK1 ( RG ) where G is any finite group.

3.2.3 Applications
1. Whitehead Torsion
J.H.C. Whitehead observed that if X is a topological space, with
fundamental group π , ( X ) = G , then the elementary row and
column transformation of matrices over ZG have some
topological meaning.
To enable him study homotopy between spaces, he introduce the
group Wh(G ) = K1 (ZG ) w(±G ) where w is the map
G → GL1 (ZG ) → GL(ZG ) → K 1 (ZG ) such that if f : X → Y is a
homotopy equivalence, then there exists an invariant
τ ( f ) ∈ Wh (G ) such that τ ( f ) = 0 iff f is induced by elementary
deformations transforming X to Y. The invariant τ ( f ) is called
Whitehead torsion. (see [57])

• K 1 (ZG ) ≅ (±1) × G ab × SK 1 (ZG ) and so rank

K 1 (ZG ) = rank Wh(G ) and SK1 (ZG ) is the full torsion
subgroup of Wh(G). So, computations of Tor ( K 1 (ZG ) )
reduces to computation of SK1 (ZG ) .
For information on computations of SK1 (Z G ) (see [8],
[60]).
3.3 K2 of Rings and Fields
3.3.1 Let A be a ring with identity. The Stenberg group of order
n (n ≥ 1) over A, denoted St n ( A) is the group generated by
xlij (a ) i ≠ j , 1 ≤ i, j ≤ n, a ∈ A , with relations

(i) x i j ( a ) x i j (b ) = x i j ( a + b )
(ii) [x ij ]
(a ), x kl (b) = 1, j ≠ k , i ≠ 
(iii) [x ij (a), x jk (b)] = x ik (ab), i, j , k distant
(iv) [x ij (a ), x jk (b)] = x ij (−ba ), j ≠ k .

Note: Since the generator eij (a ) of E n ( A) satisfies relations (i) to

(iv) above, we have a unique surjective homomorphism
ϕ n : St n ( A) → E n ( A) given by ϕ i ( xi j (a) ) = ei j (a) .
Moreover the relations for St n+l ( A) include those of St n ( A) and so,
there are maps St n ( A) → St n+l ( A) . Then we have a conical map

St ( A) → E ( A).

3.3.2 Define K 2M ( A) : = ker St ( A) → E ( A) .

3.3.3 Theorem: K 2M ( A) is an Abelian group and is the centre

of St ( A) . Hence St ( A) is a central extension of E(A).
i.e., we have a exact sequence
1 → K 2M ( A) → St ( A) → E ( A) → 1.

3.3.4 Definition: An exact sequence of groups of the form

ϕ
1→ A → E  → G → 1 is called a central extension of G by A if A
is central in E. Write the extension as ( E ,ϕ ) . A central extension
( E ,ϕ ) of G by A is said to be universal if for any other central
extension ( E ′,ϕ ′) of G, there is a unique morphism
( E , ϕ ) → ( E ′, ϕ ′) .
3.3.5 St ( A) is the universal central extension of E ( A) . Hence
there exists a natural isomorphism K 2M ( A) ≅ H 2 ( E ( A), Z ) .
Note: The last statement follows from the fact that G (in this
case, E ( A) , the kernel of the universal central extension ( E ,ϕ ) (in
this case ( St ( A),ϕ ) is isomorphism to H 2 (G, Z ) (in this case
H 2 ( E ( A), Z ) .

3.3.6 Examples
(i) K 2 Z is a cyclic group of order 2
(ii) ( )
K 2 ( Z (i ) ) = 1 , so is K 2 Z − 7
(iii) K 2 (Fq ) = 1 where Fq is a finite field with q elements
(i) If F is a field, K 2 ( F [t ]) ≅ K 2 ( F ) more generally
K 2 ( R[t ]) ≅ K 2 ( R) if R is a regular ring.
Note: K 2M ( A) ≅ K 2 ( P ( A) ) = K 2 ( A) .
3.3.7 Let A be a commutative ring with 1, a ∈ A* . Put
xi j (u ) x j i ( − u −1 ) xi j (u ) .
Define hi j (u ) = wi j (u ) wi j (−) .
For u, v ∈ A r , one can easily check hat ϕ ( [h12 (u ), h13 (u )]) = 1 and so,
[ h12 (u ), h13 (v)] ∈ K 2 ( A) . One can also show that [ h12 (u ), h13 (v)] is
independent of [ h12 (u ), h13 (v)] and call this the Stenberg symbol.

3.3.8 Theorem
Let A be a commutative ring with 1. The Stenberg symbol
{ , } : A* × A → K 2 ( A) is skew symmetric and bilinear i.e.,
{u, v} = {u , v}−1 ;{u, u 2 , v} = {u1 , v}{u 2 , v} .
3.3.9 Theorem (Matsumoto)
Let F be a field. Then K 2M ( F ) is generated by {u, v), u, v ∈ F * with
relations
(i) {u u1 , v} = {u, v}{u1 , v}
(ii) {u, v v1} = {u, v}{u, v1}
(iii) { u,1 − u} = 1
i.e., K 2M ( F ) is the quotient of F * ⊗ Z F * by the subgroup
generated by the elements x ⊗ (1 − x), x ∈ F * .

3.4 Connections of K2 with Brauer Groups of Fields and

Galois
Cohomology
3.4.1 Let F be a field and Br (F ) the Brauer group of F i.e., the
group of stable isomorphism classes of central simple F-algebras
with multiplication given by tensor product of algebras (see [7]).

A central simple F-algebra is said to be split by an extension E of

F of E ⊗ A is E-isomorphic to Mr (E ) for some positive integer r.
It is well known that such E can be taken as some finite Galois
extension of F.
Let Br ( F , E ) be the group of stable isomorphism classes of E-split
central simple F-algebras. Then Br ( F ) := Br ( F , Fs ) where Fs is the
separable closure of F.

3.4.3 For any m > 0 , let µm be a group of mth rods of 1,

G = Gil( Fs ) ( F ) . Then we have a Kummer sequence of G-modules
0 → µm → Fs* → 0 from which we obtain an exact sequence of
Galois cohomology groups

( )
F * → F * → H 1 ( F , µm ) → H 1 F , Fs* → 0

where H 1 ( F , Fs* ) = 0 by Hilbert theorem 90 so, we obtain

homomorphism
χ m : F * mF * ≅ F * ⊗ Z / m → H ′( F , µm ).
Now, the composite
( ) (
F * ⊗Z F * → F * ⊗Z F * ⊗ Z / m → H 1 ( F , µm ) ⊗ H 1 ( F , µm ) → H 2 F , µm⊗
2
)
is given by a ⊗ b → χ m (a) χ m (b) (where is a cup product)
which can be shown to be a Stanberg symbol inducing a
homomorphism
g 2,m : K 2 ( F ) ⊗ Z / m Z → H 2 F , µ m⊗( 2
) (I)

we then have the following result

Theorem 3.4.4: Let F be a field, m an integer > 0 such that the

characteristic of F is prime to m. Then the map

g 2 ,m : K 2 ( F ) m K 2 ( P ) → H F , µ
2
( ⊗2
m )
is an isomorphism where H F , µ 2
( ⊗2
m ) can be identified with m
torsion subgroup of Br (F ) .
Remark 3.4.5: J. Milnov defined ‘higher Milnov K-groups’
K nM ( F ) (n ≥ 1) fields as follows:
Definition
K nM ( F ) := F * ⊗ F * ⊗  ⊗ F * {a 1 ⊗  ⊗ a n ai + a j = 1 for some i ≠ j , ai ∈ F }
n times

i.e., K nM (F ) is the quotient of F * ⊗ F *  F * (n times) by the

subgroup generated by all a1 ⊗ a 2 ⊗  ⊗ a n , ai ∈ F such that
ai + a j = 1 .

Note: ⊕ ∞ K nM ( F ) is a ring.
n>0

Remarks 3.4.6: By generalizing the process outlined in 3.4.3,

we obtain a map,
(
g n,m : K nm ( F ) m K nm ( F ) → H n F , µ m⊗ ,
n
)
• It is a conjecture of Bloch-Kato hat g n.m is an isomorphism
for all F, m, n.
• Theorem 3.4.4 above due to A. Merkurjev and A. Suslin, is
the g z ,m case of Bloch-Kato conjecture when m is prime to
the characteristic of F.
• A Merkurjev proved that theorem 3.4.4 holds without any
restriction of F with respect to m.
• It is also a conjecture of Milnor that g n , z is an
isomomorphism. In 1996, V. Voevodsky proved that
g n , 2 is an isomorphism for any r, leading to his being
r

awarded a Fields medal.

• It is now believed that M. Rost and V. Voeodsky have now
proved the Bloch-Kato conjective.
3.5 Applications
1. K2 and Pseudo-isotopy
Let R = ZG , G a group. For u ∈ R * put
wi j (u ) : = xi j (u ) x j i ( − u −1 ) xi j (u ) . Let Wi j be the subgroup of
St (R) generated by all wij ( g ), g ∈G .

Now, let M be a smooth n-dimensional compact connected

manifold without boundary. Two diffeomorphisms h0 , h, of M are
said to be isotopic of they lie in the same path component of the
diffeomorphism group. Say that h0 , h1 are pseudo-isotopic if there
is a diffeormorphism of the cylinder M × [0,1] restricted to h0 on
M × (0) and to h1 on M × {1} . Let P(M) be the pseudo-isotopy
space of M, i.e., the group of diffeomorphism L of M × [0,1]
restricting to the identity on M × (0) . Computation of π 0 ( P ( M 2 ) )
helps to understand the differences between isotopes to and we
have the following result due to A. Hatcher and J. Wagover.
Theorem: Let M be an n-dimensional (n ≥ 3) smooth compact
manifold with boundary. Then there exists a subjective map

π 0 ( P( M ) → Wh2 ( π 1 ( M ) ) )

where π 1 ( M ) is the fundamental group of M.

4. HIGHER ALGEBRAIC K-THEORY

4.1 The Plus Construction for K n (A)
4.1.1 The plus construction of Kn of a ring A with identity makes
use of the following theorem of Quillen.

Theorem 4.1.2: Let X be a connected CW-complex N a perfect

normal subgroup of π 1 ( X ) . Then there exists a CW-complex
X + (depending on N) and a map X → X + such that

(i) ( )
i : π 1 ( X ) → π 1 X + is the quotient
map π 1 ( X ) → π 1 X N = π 1 ( X + )
(i) For any π 1 ( X ) N -module L, there is an isomorphism
ia : H a ( X , i * L ) → H i ( X + , L ) where i * L is L considered as a
π 1 ( X ) -module.
(ii) The space X + is universal in the sense that if Y is a CW-
complex and f : X → Y is a map such that
f * : π 1 ( X ) → π 1 (Y ) such that f α ( N ) = 0 then there exists a
unique map f + , X + → Y such that f + i = f .

Definition 4.1.3
Let A be a ring, X = BGL( A) the classifying space of the group
GL(A), (a CW-complex characterized by the property that
π 1 BGL( A) = GL( A) and π i BGL( A) = 0 for i ≠ 1 ). Then
π i BGL( A) = GL( A) contains E(A) as a perfect normal subgroup.
Hence, by theorem 4.1.2, there exists a BGL( A) + . Define
K n ( A) = π n ( BL( A) + ) .
Example/Remarks 4.1.4
(i) For n = 1,2, K n ( A) as defined above can be identified
with the classical definition.
(ii) π 1 BGL( A) H = GL( A) E ( A) = K1 ( A) .
(iii) BE ( A) + is the universal covering space of BGL( A) + and
so, we have
π 2 BGL( A) + ≅ π 2 ( BE ( A) + ) ≅ H 2 ( BE ( A) + ) ≅ H 2 ( BE ( A) )
≅ H 2 ( E ( A) ) ≅ K 2 ( A).

(iv) K 3 ( A) ≅ H 3 ( St ( A) ) (see [42])

(v) If A is a finite ring, K n ( A) is finite see [31] or [42]
(vi) For a finite field Fq with q elements
( )
K 2 n (Fq ) = 0 and K 2 n −1 (Fq ) = Z q n −1 .
4.2 Classifying Spaces and Simplical Objects
4.2.1 Definition
Let ∆ be a category defined as follows: ob(∆) := { n = {0 < 1 <  < n}}
Hom ∆ ( m, n ) = { monotone maps f , m → n i.e., f (i ≤ f ( j ) for i < j} .

4.2.2 For any category A, a simplical object in A is a

contravariant functor.
X : ∆ → A . Write X n for X (n)
A cosimplical object in A is a covariant functor X : ∆ → A .
• Equivalently, one could define a simplical object in a
category A as a set of objects X n (n ≥ 0) in A and a set of
morphisms δ i : X n → X n−1 (0 ≤ i ≤ n) called face maps as
well as a set morphisms si : X n → X n+1 (0 ≤ j ≤ n) called
degeneracies satisfying certain simplical identities (see
[93]).
 • The geometric n-simplex is the topological space
{
ˆ n = ( x , x , , x ) ∈ R n +1 0 ≤ x ≤| ∀ i and Σx = 1
∆ 0 1 n i i }
ˆ : Λ → spaces : n → ∆
A functor ∆ ˆ n is a co-simplical space..

4.2.4 Definition: Let X n be a simplical scl. The geometric

realization of X n , written X n is defined by
X n = X ×∆ =
∆
(X n
ˆ
×∆ n ) ≅
n ≥0
where the equivalence relations ≅ is generated by
( x,ϕn ( y ) ) ≅ (ϕ n ( x), y ) for any x ∈ X n y ∈Yn and ϕ : m → n ∈ ∆ and
where X n × ∆n is given the product topology and xn is considered
as a discrete space.

4.2.5 Definition
Now let A be a small category. The Nerve of A, written NA , is
the simplical set whose n-simplices are diagrams

{
An = A0 
f1
→ A1 
→ 
fn
→ An }
where the Ai ’s are A-objects and the f i are A-morphisms. The
classifying space of A is defined as NA and denoted by BA.

Remarks: BA is a CW-complex whose n-cells are in one-one

correspondence with the diagrams An above.

4.2.6 Definition
Now let C be an exact category. We form a new category QC
such that ob(QC) = ob C and morphisms from M to P, say is an
isomorphism class of diagrams M ← j
N →
i
P where i an
admissible monomorphism (or inflation) and j is an admissible
epi morphism or deflation) in C i.e., i and j are part of some exact
sequences 0→ N  →i
P → P ′ → 0 and
0 → N ′′ 
→
i
N →j
M → 0 , respectively.

Composition is also well defined (see [62]).

Definition 4.2.7: For n ≥ 0 , define
K n (C ) : = π n+1 ( BQC,0) n ≥ 0.

Examples: Recall earlier examples.

(A) (1) C = P ( A), K n (C) : = K n ( A) n ≥ 0
C = M ( A), K n (C) = Gn ( A) n ≥ 0
Note that K n ( P ( A) ) ≅ π n ( BGL( A+ ) ) for n ≥ 1
We shall be interested in various rings A.
(i) A = Z ,Q , R ,C
(ii) A = Integral domain R
(iii) A = F (field possibly quotient field of R)
(iv) A = D a dunsion ring
(v) A = Z G, RG,Q G, R G,C G (a finite group)
(vi) R = integers in a number field or p-adii field, A =
RG, G finite group or more generally A = r-order
Λ in a semi-simple F-algebra Σ
 (vii) A = Λ α (T ) where Λ isas in (vi) When
A = RG, A = Λ α (T ) = RV where V = G × T is virtually
α
cyclic group.

4.3 Some Sample Finiteness Results for K (n C ) -

( C = P(A), M(A))
4.3.1 Theorem
Let R be the ring of integers in a number field F, Λ any R-
order in a semi-simple F-algebra Σ . Then,

(i) For all n ≥ 1, K n (Λ), Gn (Λ) are finitely generated

Abelian group
(Kuku, J. algebra 1984, AMS contemp. Math,
1986).
(ii) For all n ≥ 1, K 2 n (Λ ), G2 n (Λ ) are finite Abelian
groups, Kuku (K-theory 2005).
(iii) If F is totally real, then G2 m+ 2 (Λ) is also finite for all odd
m ≥1
(Algebras and Rep. Theory - to appear)
(i) For all n ≥ 1, G2 n ( Λα (T ) ) is a finitely generated Abelian
group where Λα (T ) is the twisted Laurent series ring
over Λ . (Kuku (2007): Algebras and Rep theory - to
appear)
(ii) There exists isomorphism
Q ⊗ K n ( Λα (T ) ) ≈ Q ⊗ Gn ( Λα (T ) ) ≅ Q ⊗ K n ( ∑α (T ) ) ∀ n ≥ 2
(Kuku (2007): Algebras and Rep. theory - to appear)
(iii) If A is a finite ring, then K n ( A), Gn ( A) are finite for all
n ≥ 1 (Kuku AMS Cont. Mp. Math 1986).

Note: Above results (i), (ii), (iii) apply to Λ = RG (G a finite

group) while (iv) and (v) apply to
Λα (T ) = ( RG )α (T ) = RV where V = G ×T is a virtually infinite
α
cyclic group. (i) generalizes classical results known for n = 0,1
to higher dimensions.
4.3.2 Kn, SKn of Orders and Group rings
Let R be a Dederkind domain (i.e., an integral domain in which
every ideal is projective or equivalently R is Noetherian integrally
closed and every prime ideal is maximal or equivalently every
non-zero ideal a in R is invertible i.e., a a −1 = R where
−1
a = {x ∈ F | x a ⊂ R} . Let Λ be any R-order in a semi-simple F-
algebra Σ . For n ≥ 0 , let SK n (Λ) : = ker ( K n (Λ ) ) → K n (Σ) and
SGn (Λ ) = ker(Gn (Λ)) → Gn (Σ) ≅ K n (Σ) .
Note that for any regular ring R (e.g., Σ ), K n ( R) ≅ Gn ( R) .

As observed earlier, when Λ = RG (R integers in a number field,

G a finite group), SK 0 ( RG ) SK1 ( RG ) contain topological
invariants – respectively, e.g., Swan in variants and Whitehead
torsion). We have the following:
4.3.3 Theorem: (see Kuku Math. Zeit (1979) or Ku-Bk
(2007).
Let p be a rational prime. F a p-adii field with ring of integers R,
Γ a maximal R-order in a semi-simple F-algebra Σ , Then for all
n ≥ 1.

(a) SK 2 n (Γ) = 0
(b) SK 2n −1 (Γ) = 0 iff Σ is unified over its centre i.e., iff Σ is a
direct product of matrix algebras over fields.

Note: Above result applies to Γ = RG where ( | G |, p ) = 1 .

4.3.4 Theorem: See Ku-Bk (2007) or Kuku (1984) J-

algebra; Kuku (1986)
AMS Cont. Math; Kuku (2006) K-theory

Let R be the ring of integers in a number field F, Λ any R-order in

a semi-simple F-algebra Σ . Then

(a) SK n (Λ ), SGn (Λ) are finite groups and SG2 n (Λ) = 0 for all
n ≥1
(b) SK n Λ ( )
ˆ , SG Λ
p n p ( )
ˆ are finite groups and
(c) If Λ = ZG where G is a finite p-group, then SK 2 n−1 (ZG ) ,
( )
and SK 2 n−1 Ẑ P G are finite p-groups.

4.4 Higher Dimensional Class Groups of Orders and

Group rings
Let R be the ring of integers in a number field F, Λ any R-order in
a semi-simple F-algebra Σ . The higher class groups Cln (Λ) of
( )
Λ are defined for all n ≥ 0 by C ln(Λ) : = ker( SK n (Λ) ) → ⊕ SK n Λ
ˆ .
1

Note that Cln (Λ ) coincides with the usual class group Cl (Λ) of
Λ which in turn generalizes the notion of class groups of integers
in a number field. (see Ku-Bk (2007). For results on class groups
of Λ (see Curtis/Reiner (1987) [8]).
Note also that computations of Cl1 (Λ) which we already observed
reduces to computation of Whitehead torsion (see Oliver (1988)
[60]).
We now state known results for Cln (Λ) n ≥ 1 .

4.4.2 Theorem
Let R be the ring of integers in a number field F, Λ any R-order in
a semi-simple F-algebra Σ . Then

(i) For all n ≥ 1 , Cln (Λ) is a finite group (see Ku-Bk

(2007) or Kuku (1986) AMS Cont. Math.)
(ii) For all n ≥ 1 , p-torsion in Cl 2 n−1 (Λ ) can occur only for
primes p lying above prime ideals p at which Λ̂ p is not
maximal. Hence for any finite group G, for all n ≥ 1 , the
only p-torsion possible in Cl 2 n−1 ( RG ) is for those primes
p dividing the order of G. (see Kolster/Laubenbacher
(1988) Math. Zeit).
(iii) Let F be a number field with ring of integers R, Λ a
hereditary R-order in a semi-simple F-algebra or and
Eichler order in a quatermon algebra. Then the only p-
torsion possible is for those primes p lying below the
prime ideals p at which Λ p is not maximal. (see Ku-Bk
(2007) or Guo/Kuku (2005) Comm. in Alg.).
 (i) Let S n be a symmetric group of degree n. Then
Cl 2 n−1 ( Z S 2 ) is a finite z-torsion group (see Kolster
/Lauben bacher (1998) Math. Zeit).

4.5 Higher K-theory of Schemes

4.5.1 Recall: If X is a scheme, we write K n ( X ) for K n (P ( X )) and
when X is a Neotherian scheme, we write Gn ( X ) for K n (M ( X )) .

If G is an algebraic group over a field F, and X is a G-scheme, we

write K n (G , X ) for K n (P (G , X )) are Gn (G , X ) for K n (M (G , X )) .

Note:
(a) If G is trival group Gn (G, X ) = Gn ( X ) and
K n (G, X ) = K n ( X ) .
 (a) Gn (G ,−) is contravariant with respect to G-maps.
(b) Gn (G ,−) is covariant with respect to projective G-maps.
(c) K n (G,−) is contravariant with respect to any G-map.
(d) Gn (−, X ) is contravariant w.r.t. any group
homomorphism.
(e) K n (−, X ) is covariant w.r.t group homomorphisms. (see
Thomason (1987) K-theory Proc. Princeton.

4.5.2 Recall: Let B be a finite dimensional separate F-algebra. X

a smooth projective variety equipped with the action of an affine
algebraic group G over F, γ X the twisted form of X with respect
to a cocycle γ : Gal Fsep / F → G ( Fsep ) . Let VB G ( r , B ) be the
category of vector bundle on γ X equipped with left B-module
structure. We write K n ( γ X , B ) for K n (VBG ( γ X , B ) ) . (See Panin
(1994) K-theory; Merurjer (preprint).

We now have the following results.

4.5.3 Theorem: Kuku (2007) MPIM – Bonn, preprint
~
Let G be a semi-simple simply, connected and connected F-split
~
algebraic group over a field F, P a parabolic subgroup of G,
~ ~
F = G P the flag variety and γ F the twisted form of F, B a finite-
dimensional separable F-algebra.

(a) Let F be a number field, then for all n ≥ 1

(i) K 2 n+1 ( γ F , B ) is a finitely generated Abelian group;
(ii) K 2 n ( γ F , B ) is a torsion group and has no non-trivial
dunsible subgroups.

(b) Let F be a p-adii field,  a rational prime such that  ≠ p .

Then for all n ≥ 1 and any separate F-algebra B, K n ( γ F , B )  is a
finite group.
4.5.4 Theorem: (Kuku (2007) MPIM-Bonn (preprint))
Let V be a Brauer-Severi variety over a field F.

(a) If F is a number field, then K 2 n+1 (V ) is a finitely

generated Abelian group for all n ≥ 1 .
(b) If F is a p-adii field, then for all n ≥ 1, K n (V )  is a
finite group if  is a prime ≠ p .

4.6 Mod-m Higher K-theory of exact Categories, Schemes

and Orders
4.6.1 Let X be an H-space, m a positive integer
M mn an n-dimensional mod-m Moore space is the space obtained
from S n−1 by attaching an n-cell via a map of degree m, (See Ku-
Bk (2007) or Niesendorfer 1980/ AMS Memoir).
 • ). Write
π n ( X , Z / m ) for [ M mn , X ] n ≥ 2
π1 ( X , Z / m ) for π1 ( X ) ⊗ Z / m .
The cofibration sequence
β α
S n −1 →
m
S n −1 → M mn 
→ Sn →
m
Sn

yields an exact sequence

πn (X ) →
m
πn (X ) β
→ π n ( X , Z / m) α
→ π n −1 ( X ) →
m
π n−1 ( X )

and hence the following exact sequence

0 → π  ( X ) / m → π n ( X , Z / m ) → π n −1 ( X )[m] → 0

where
π n−1 ( X )[m] = { x ∈π n−1 ( X ) mx = 0} .
Example 4.6.2
(i) If C is an exact category, write K n ( C, Z / m ) for
π n+1 ( BQC, Z / m ); n ≥ 1 and write
K 0 ( C, Z / m ) for K 0 ( C ) ⊗ Z / m.
(ii) If C = P ( A), a ring with 1, write K n ( A, Z / m) for
K n ( P ( A), Z / m ) ;
(iii) If X is a scheme, and C = P ( X ), write K n ( X , Z / m ) for
K n ( P ( X ), Z / m ) . Note that if X = Spec( A) , A commutative,
we recover K n ( A, Z / m) .
(iv) Let A be a Noetherian ring. If C = M ( A) , we write
Gn ( A, Z / m ) for K n ( M ( A), Z / m ) .
(v) Let X be Noetherian scheme, C = M ( X ) . We write
Gn ( X , Z / m ) for K n ( M ( X ), Z / m ) . If X = Spec( A) , we
recover Gn ( A, Z / m ) .
(vi) Let G be an Abelian group over a field F, X a G-scheme,
C = M (G, X ) . Gn ( (G , X ), Z / m for K n ( M (G , X ), Z / m ) ) .
(vii) Let G be an algebraic group over a field F,X a G-scheme;
C = P (G, X ) . We write K n ( (G , X ), Z / m for K n ( P (G, X ), Z / m ) ) .
(viii) Let G be an algebraic group over a field F, X a G-scheme,
B a finite dimensional separable F-algebra, r X the twisted
form of X via a 1-cocycle r, C = VBG ( r X , B ) . We write
K n ( ( r X , B ) , Z / m for K n ( ( r X , B ) , Z / m ) ) .
4.6.2 Theorem: Kuku (2007) MPIM-Bonn Preprint
Let C, C′ be exact categories and f : C → C′ an exact factor which
induces Abelian group homomorphism f 0 : K n (C) → K n ( C′) for
each n ≥ 0 . Let  be a rational prime

(a) Suppose that f1 is injective (resp. surjective, resp.

bijective), then so is f1 : K n ( C, Z / m ) → K n ( C′, Z / m ) ;
(b) If fα is split surjective (resp. split injective), then so is
f : K n ( C, Z / m ) → K n ( C′, Z / m ) .

4.7 Profinite Higher K-theory of Exact Categories,

Schemes and Orders
4.7.1 Let C be an exact category,  a rational prime, s a positive
integer, put M n∞+1 = lim M ns+1 . We define the profinite K-theory of C by
( ) [ ] ( ) (
K npr C, Zˆ  = M n∞+1 , BQC . We also write K n C, Zˆ  for lim C, Z / s . )
Note: For all n ≥ 2 , we have an exact sequence

( ) ( ) ( )
0 → lim K 2 n+1 C, Z / s → K npr C, Zˆ  → K n C, Zˆ  → 0.
1
For more information on this construction, see Ku-Bk (2007),
chapter 8 or [42].

Example 4.7.2
(i) Let C = P ( A) , A a ring with 1. We write
( ) ( ) (
K npr A, Zˆ  for K n P ( A), Zˆ  and K n P ( A), Zˆ  for K n P ( A), Zˆ  . ) ( )
(ii) If X is a scheme and C = P ( X ) , we write
( ) ( ) (
K npr X , Zˆ  for K npr P ( X ), Zˆ  and K n ( X ), Zˆ  for K n P ( X ), Zˆ  . ) ( )
(iii) Let C = M (A) , write
( ) ( ) (
Gnpr A, Zˆ  for Gnpr M ( A), Zˆ  and Gn ( A), Zˆ  for K n M ( A), Zˆ  . ) ( )
(iv) If C = M ( X ) , X a scheme, write
( ) ( ) (
Gnpr X , Zˆ  for K npr M ( X ), Zˆ  and Gn X , Zˆ  for K n M ( X ), Zˆ  . If ) ( )
(
X = Spec( A) recover G pr A, Zˆ and G A, Zˆ .
n  ) n (  )
(v) Let G be an algebraic group over a field F, X a G-scheme,
(
C = M (G, X ) . We write Gnpr ( G , X ), Zˆ  for Gnpr M (G, X ), Zˆ  . ) ( )
(vi) Let G be an algebraic group over a field F, X a G-scheme,
( )
C = P (G, X ) , we write K npr ( G, X ) , Zˆ  for K npr P (G , X ), Zˆ  . ( )
(vii) Let G be an algebraic group over a field F, X a G-scheme,
γ X the twisted form of X and B a finite-dimensional separable
algebraic over F. If C = VBG (( r )
X , B ) , Ẑ  , we write
K npr (( r ) (
X , B ) , Zˆ  for K npr VBG , ( r X , B ) , Zˆ )
Theorem 4.7.3: Kuku (2007) MPIM –Bonn preprint
Let C, C′ be exact categories and f : C → C′ an exact factor
which induces an Abelian group homomorphism
f n , K n (C) → K n ( C′) for n ≥ 0 . Let  be a rational prime, s a
positive integer. Suppose that f α is injective (resp. surjective;
resp. bjective), then so is
( )
f α : K npr C, Zˆ  → K npr C′, Zˆ  . ( )
Theorem 4.7.4: Kuku (2007) MPIM-Bonn Preprint
~
Let F be a number field, G a semi-simple connected, simply
~
connected split algebraic group over F, P a parabolic subgroup of
G , F = G P , γ a 1-cocycle : Gal( Fsep F ) → G ( Fsep ) , γ F the γ -
~ ~ ~ ~

twisted form of F, B a finite-dimensional separable F-algebra.

Then for all n ≥ 1 ,
(i) ( )
K 2prn ( γ F , B ) , Ẑ  is an  -complete Abelian group;
(ii) ( )
div K npr ( F , B ) , Zˆ  = 0.

Theorem 4.7.5: Kuku (2007 – MPIM-Bonn Preprint

~
Let p be a rational prime, F a p-adii field, G a semi-simple
~
connected and simply connected split algebraic group over F, P a
~ ~ ~
parabolic subgroup of G , F = G / P the flag variety, γ a 1-cocycle
Gal( Fsep F ) → G ( Fsep ) , γ F the γ -twisted form of F, B a finite-
dimensional separable F-algebra,  a rational prime such that
 ≠ p . Then for all n ≥ 2 .
(i) ( )
K npr ( γ F , B ) Ẑ  is an  -complete profinite Abelian group.
(ii) ( ) (
K npr ( γ F , B ) Zˆ  = K n ( γ F , B ) Zˆ  ’ )
(
(iii) The map ϕ : K n ( γ F , B ) → K npr ( γ F , B ) , Ẑ  induces )
isomomorphiss
( )
- K n ( γ F , B ) , [] ≅ K npr ( γ F , B ) , Zˆ  , [s ]
- (
K n ( γ F , B ) ,  ≅ K npr ( γ F , B ) , Ẑ  s .
s
)
(iv) Kernel and cokernel of K n ( r F , B ) → K n
pr
(( r )
F , B ) , Ẑ  are
uniquely  -divisible.

K (( )
F , B ) , Zˆ  = 0 for n ≥ 2 .
pr
(v) div n r
5. Equivariant Higher K-theory Together with
Relative Generalizations
In this section, we exploit representation theoretic techniques
(especially induction theory) to define and study equivarient
higher K-theory and their relative generalizatins. Induction theory
has always aimed at computing various invariants of a group G in
terms of corresponding invariants of subgroups of G. For lack of
time and space, we discuss here finite group actions and note that
analogous results exist for pro-finite group and compact lie group
actions (see Ku-Bk (2007) chapter 9 –13).

5.1 Equivariant Higher K-theory for Exact Categories

for Finite Group Actions
5.1.1 Definition
Let B be a category with finite sums final object and finite
pullbacks (and hence finite products) e.g., category G-set of
(finite) G-Sets, where G is a finite groups, D an Abelian category
(e.g., R-Mod)
A pair of functors ( M α , M α ) : B → D is called a Marchey functor if

(i) M α : B → D is covariant, M * : B → D contravariant and

Mα (X ) = M α (X ) = M (X ) ∀ X ∈ ob B .
(ii) For any pull-back diagram
p2 f2 * M ( A2 )
A′ A2 M (A′)
p1 p2 in B, the diagram p1α p2+ commutes
f1 f1α
A1 A M ( A1 ) M ( A)

(iii) M α transforms finite coproducts in B into finite products in

D i.e., the embeddings X i →  X i induces an isomomorphism
i =1

M ( X i  X 2  X n ) ≅ M ( X1 ) × × M ( X n ) .
5.1.2 Note that (ii) above is an axiomatization of the Mackey
subgroup theorem in classical representation theory (Put B = G-
Set, A1 = G H ; A2 = G / H ′ G H × G H ′ can be identified with the
set D( H , H ′) = { HgH ′ g ∈ G} of double cosets of H and H ′ in G.
(see [8] for a statement of Mackey subgroup theorem).

5.1.3 We shall concentrate on exact categories in this section but

observe that analogous theories exist for symmetric monoidal and
Wildhanser category (see Ku-Bk (2007) chapters 9, 10, 13).

So, let C be an exact category, S a G-set, G a finite group, S the

translation category of S. Recall that the category [ S , C] of
covariant functors from S to C is also an exact category where a
sequence 0 → S ′ → S → S ′′ → 0 in [ S , C] is said to be exact if
0 → S ′( S ) → S ( S ) → S ′′( S ) → 0 is exact in C .
5.1.3 Definition
Let K nG ( S , C) be the nth algebraic K-group associated with the
exact category [ S , C] with respect to fibre-wise exact sequences.

Theorem 5.1.4
K nG (−, C) : GSet → Z - M od is a Mackey functor.
(For proof see Ku-Bk (2007) or Dress/Kuku Comm. in Alg.
(1981).

5.1.5 Note: We want to turn K nG (−, C) into a ‘Green’ functor and

see that for suitable category C, K nG (−, C) is a module over
K nG (−, C) . We first define these notions of ‘Green’ functor and
modules over ‘Green’ functors.
5.1.6 Definition
A Green functor G : B → R - M od is a Mackey functor together
with a pairing G× G → G such that for any B-object X, the R-
bilinear map G( X ) → G( X ) makes G(X ) into an R-algebra with a
unit 1∈ G( X ) such that for any morphism f : X → Y , we have
f * (1G (Y ) ) = 1G ( X ) .
A left (resp. right) G-module is a Mackey functor
M : B → R - M od together with a pairing
G× M → M (resp. M × G → M ) such that for any B-object X,
M(X) becomes a left (resp. right) unitary G(X)-module we shall
refer to left G-modules just as G-modules.
5.1.7 Definition
Let C1 , C2 , C3 be exact categories. An exact pairing ( , ).
C1 × C2 → C3 given by ( X 1 , X 2 ) → ( X 1  X 2 ) is a covariant functor
such that

Hom[ ( X 1 , X 2 ) , ( X 1′, X 2′ ) ]

= Hom( X 1 , X 1′ ) × Hom( X 2 , X 2′ ) → Hom( X 1  X 2 ) , ( X 1′  X 2′ )

is bi-additive and bi-exact (see Ku-Bk (2007) or [87]).

5.1.8 Theorem
(for Proof see Ku-Bk (200) or Dress/Kuku. Comm. in Alg.
(1981)
Let C1 , C2 , C3 be exact categories and C1 × C2 → C3 an exact
pairing of exact categories, S a G-Set. Then the pairing induces a
pairing [ S , C1 ] × [ S , C2 ] → [ S , C3 ] and hence a pairing
K nG ( S , C1 ) × K nG ( S , C2 ) → K nG ( S , C3 ) .
Suppose that C is an exact category such that the pairing
C × C → C is naturually associative and commutative and there
exists E ∈ C such that [ E  N ] = [ N  E ] = [ N ] ∀ N ∈ C . Then
K nG (−, C) is a Green functor and K nG ( −, C) is a unitary K nG (−, C) -
module.

5.1.9 Definition/Remarks
If M : GSet → Z -Mod is any Mackey functor, X a G-set, define a
Mackey functor M X : GSet → Z - M od by M X (Y ) = M ( X × Y ) . The
projection map pr : X × Y → Y defines a natural transformation
θ X : M X → M where θ X (Y ) = pr1 M ( X × Y → M (Y )) . M is said to
be X-projective if θ X is split surjective i.e., there exists a national
transformation ϕ : M → M X such that O X ϕ = id M .

Now define a defect base DM of M by DM = { H ≤ G X H ≠ φ } where

X is a G-set (called defect set of M) such that M is Y-projective iff
there is a G-map f, X → Y (See Ku-Bk (2007) Prop. 9.1.1).
If M is a module over a Green functor G, then M is X-projective iff
G is X-projective iff the induction map G( X ) → G(G / G ) is
surjective (see Ku-Bk. Theorem 9.3.1).

• In general, proving induction results reduce to determining

G-sets X for where G( X ) → G(G / G ) is surjective and this
in turn reduces to computing DG (see Ku-Bk 9.6.1).
Hence one could apply induction techniques to obtain results on
higher K-groups K nG (−, C) which are modules over Green functors
K nG (−, C) .
5.2 Relative Equivalent Higher Algebraic k-theory
Definition 5.2.1 Let S, T be G-Sets. Then the projection
ϕ ϕ
S × T → S gives rise to a functor S × T → S . Suppose that C
is an exact category. If ς ∈ [ S , C] , we write ς ′ for
ϕ ς
ς  ϕ : S × T → S → C . Then a sequence ς 1 → ς 2 → ς 3 of
functors in [ S , C] is said to be T-exact if the sequence
ς 1′ → ς 2′ → ς 3′ of restricted functors S × T → ϕ
S ς
→ C is split
exact. If ϕ : S 2 → S1 is a G-map, and ς 1 → ς 2 → ς 3 is a T-exact
sequence in [ S , C] , and we put ςˆi = ϕ  ς i , then ςˆ1 → ςˆ2 → ςˆ3 is T-
exact in [ S 1 , C] . Let K nG ( S ,C, T ) be the nth algebraic K-group
associated to the exact category [ S , C] with respect to T-exact
sequence.

Remarks: The use of the restriction functors ς ′, ςˆ in 5.2.1

constitute a special case of the following general situation. Let ς
be an exact category and B, B′ any small categories. We define
exactress in [B, C] relative to some covariant functor δ : B′ → B .
Thus a sequence ς 1 → ς 2 → ς 3 of functors in [B, C] is said to be
exact relative to δ : B′ → B if it is exact fibrewise and if the
sequence ς 1′ → ς 2′ → ς 3′ of restricted functors
ς 1′ := ς i  δ ′ : B′ →δ
B ς
→ C is split exact. Let K nG ( S , C, T ) be
the nth algebraic K-group associated to the exact category [S , C]
w.r.t exact sequences.

5.2.3 Definition
Let S, T be G-Sets. A functor ς ∈ [ S , C] is said to be T-projective
if any T-exact sequence ς 1 → ς 2 → ς is exact. Let [ S , C]T be the
additive category of T-projective functors in [ S , C] considered as
an exact category with respect to split exact sequences. Note that
ψ
the restriction functor associated to S1 → S 2 carries T-
projective functors ς ∈ [ S 2 , C] into T-projective functors
ς ψ ∈ [ S 2 , C] . Define PnG ( S , C , T ) as the nth algebraic K-group
associated to the exact category [ S , C]T , with respect to split exact
sequences.
5.2.3 Theorem
K nG (−, C , T ) and PnG (−, C , T ) are Mackey functors from GSet to Ab
for all n ≥ 0 . If the pairing C × C → C is naturally associative and
commutative and contains a natural unit, then
K nG (−, C , T ) : GSet → Ab is a Green functor, and K nG (−, C , T ) and
PnG (−, C , T ) are K 0G (−, C , T ) -modules.

Also, the induction functor ψ * : [ S 1 , C] → [ S 2 , C] associated to

ψ : S1 → S 2 preserves T-exact sequences and T-projective functors
and hence induces homomorphism
K nG (ψ , C , T ) * : K nG ( S1 , C , T ) → K nG ( S 2 , C , T ) and
PnG (ψ , C , T ) * : PnG ( S1 , C , T ) → PnG ( S 2 , C , T ) , thus making K nG (−, C , T )
and PnG ( S1 , C , T ) covariant functors. Other properties of Mackey
functors can be easily verified.
Observe that for any GSet T , the pairing [ S 1 , C] × [ S 2 , C] → [ S 3 , C]
takes T-exact sequences into T-exact sequences, and so, if
[ S i , C], i = 1,2 are considered as exact categories with respect to T-
exact sequences, then we have a pairing
K 0G ( S , C1 , T ) × K nG ( S , C2 , T ) → K nG ( S , C3 , T ) . Also if ς 3 is T-
projective, so is ς 1 , ς 2 . Hence, if [ S , C1] is considered as an exact
category with respect to T-exact sequences, we have an induced
pairing K 0G ( S , C1 , T ) × PnG ( S , C2 , T ) → PnG ( S , C3 , T ) . Now, if we put
C1 = C 2 = C3 = C such that the pairing C × C → C is naturally
associative and commutative and C has a natural unit, then, as in
theorem 5.1.8 K 0G (−, C , T ) is a Green functor and it is clear from
the above that K nG (−, C , T ) and PnG (−, C , T ) are K 0G (−, C , T ) -
modules.
5.2.4 Remarks
(i) In the notation of theorem 5.2.3, we have the following
natural transformation of functors:
PnG (−, C, T ) → K nG (−, C, T ) → K nG (−, C) , where T is any
G-set, G a finite group, and C an exact category. Note
that the first map is the ‘Cartan’ map.

(ii) If there exists a G-map T2 → T1 , we also have the

following natural transformations
PnG (−, C, T2 ) → PnG (−, C, T1 ) and
K nG (−, C, T1 ) → K nG (−, C, T2 ) since, in this case, any T1 -
exact sequence is T2 -exact.

5.3 Interpretation in Terms of Group-rings

In this subsection, we discuss how to interpret the theories in
previous sections in terms of group-rings.
5.3.1 Recall that any G-set S can be written as a finite as a finite
disjoint union of transitive G-sets, each of which is isomorphic to
a quotient set G/H for some subgroup H of G. Since Mackey
functors, by definition, take finite disjoint unions into finite direct
sums, it will be enough to consider exact categories [ G / H , C]
where C is an exact category.

For any ring A, let M ( A) be the category of finitely generated A-

modules and P ( A) the category of finitely generated projective A-
modules. Recall from … that if G is a finite group, H a subgroup
of G, A a commutative ring, then there exists and equivalence of
exact categories [ G / H , M ( A)] → M ( AH ) . Under this experience,
[ G / H , P ( A)] is identified with the category of finitely generated
A-projective left AH-modules, i.e., [ G / H , P ( A)] ≅ PA ( AH ) .
We now observe that a sequence of functors
ς 1 → ς 2 → ς 3 ∈ [ G / H , M ( A)] or [ G / H , P ( A)] is exact if the
corresponding sequence ς 1 ( H ) → ς 2 ( H ) → ς 3 ( H ) of AH-modules
is exact.

Remarks 5.3.2
(i) It follows that for every n ≥ 0, K nG [ G / H , P ( A)] can be
identified with the nth algebraic K-group of the
category of finitely generated A-projective AH-modules
while K nG [ G / H , P ( A)] = Gn ( AH ) if A is Noetherian. It is
well known that K nG [ G / H , P ( A)] = K nG [ G / H , M ( A)] is an
isomorphism when A is regular.
(i) Let ϕ : G H 1 → G H 2 be a G-map for H 1 ≤ H 2 ≤ G . We
may restrict ourselves to the case H 2 = G , and so, we
have ϕ * [G G, M ( A)] → [G H , M ( A)] corresponding to
the restriction functor M ( AG) → M ( AH ) , while
ϕ* : [G H , M ( A)] → [G G, M ( A)] corresponds to the
induction functor M ( AH ) → M ( AG ) given by
N → AG ⊗ AN N . Similar situations hold for functor
categories involving P ( A) . So, we have corresponding
restriction and induction homomorphisms for the
respective K-groups.

(ii) If C = P ( A) and A is commutative, then the tensor

product defines a naturally associative and commutative
pairing P ( A) × P ( A) → P ( A) with a natural unit, and so,
K nG ( −, P ( A) ) are K 0G ( −, P ( A) ) -modules.
5.3.3 We now interpret the relative situation. So let T be a G-set.
Note that a sequence ς 1 → ς 2 → ς 3 of functors in [G H , M ( A)] or
[G H , P ( A)] is said to be T-exact if ς 1 ( H ) → ς 2 ( H ) → ς 3 ( H ) is
AH ′ -split exac for all H ′ ≤ H such that T H ′ ≠ ∅ where
T H ′ → {t ∈ T ′ gt = t ∀ g ∈ H ′} . In particular, the sequence of G/H-
exact (resp. G/G-exact) if an only if the corresponding sequence
of AH-modules (resp. A/G-modules) is split exact. If ε is the
trivial subgroup of G, it is G / ε -exact if it is split exact as a
sequence of A-modules.

So, K nG ( G H ,P ( A), T ) (resp. K nG ( G H , M ( A), T ) is the nth

algebraic K-group of the category of finitely generated A-
projective AH-modules (resp. category of finitely generated AH-
modules) with respect to exact sequences that split when restricted
to the various subgroups H ′ of H such that T H ′ ≠ ∅ with respect
to exact sequences. In particular, K nG ( G H , P ( A), G ε ) = K n ( AH ) .
If A is commutative, then K nG ( − ,P ( A), T ) is a Green functor, and
K nG ( − ,P ( A), T ) and PnG ( − ,P ( A), T ) are K 0G ( − ,P ( A), T ) -modules.
Now, let us interpret the map, associated to G-maps S1 → S 2 . We
may specialize to maps ϕ : G H 1 → G H 2 for H 1 ≤ H 2 ≤ G , and for
convenience we may restrict ourselves to the case H 2 = G , which
we write H 1 = H . In this case, ϕ * : [G G, M ( A)] → [G H , M ( A)]
corresponds to the restriction of AG-modules to AH-modules, and
ϕ* : [G H , M ( A)] corresponds to the induction of AH-modules to
AG-modules.

Since any G-set S can be written as a disjoint union of transitive

G-sets isomorphic to some coset-set G/H, and since all the above
K-functors satisfy the additiveity condition, the above
identification extend to K-groups, defined on an arbitrary G-set S.
5.4 Some Applications
5.4.1 We are now in position to draw various conclusions just by
quoting well-established induction theorems concerning
K 0G ( − ,P ( A) ) and K 0G ( − ,P ( A), T ) , and more generally
R ⊗ Z K 0G ( −, P ( A) ) and R ⊗ Z K 0G ( −, P ( A), T ) for R, a subring of Q,
or just any commutative ring (see …) Since any exact sequence in
P ( A) is split exact, we have a canonical identification
K 0G ( −, P ( A), T ) = K 0G ( −, P ( A), G ε ) (ε the trivial subgroup of G) and
thus may direct our attention to the relative case only.

So, let T be a G-set. For p a prime and q a prime or 0, let

D( p, T , q ) denote the set of subgroups H ≤ G such that the
smallest normal subgroup H 1 of H with a q-factor group has a
normal Sylow-subgroup H 2 with T H ≠ ∅ and a cyclic factor
2

group H 1 H 2 . Let Hq denote the set of subgroups H ≤ G , which

are q-hyperelementary, i.e., have a cyclic normal subgroup with a
q-factor group (or are cyclic for q = 0 ).
For A and R being commutative rings, let D( A, T , R ) denote the
union of all D( p, T , q) with pA ≠ A and qR ≠ R , and let HR denote
the set of all Hq with qR ≠ R . Then , it has been proved (see [11],
[44]) R ⊗ Z K 0G ( −, P ( A), T ) is S-projective for some G-set S if
S H ≠ ∅ ∀ H ∈ D( A, T , R ) H R . Moreover, if A is a field of
characteristic p ≠ 0 , then K 0G ( −, P ( A), T ) is S-projective already if
S H ≠ ∅ ∀ H ∈ D( A, T , R ) . (Also see Ku-Bk).
5.4.2 Among the many possible applications of these results, we
discuss just one special case. Let A = k be a field of characteristic
p ≠ 0 , let R = Z ( 1p ) , and let S = ∪ H ∈D( k ,T , R ) G / H . Then,
R ⊗ Z K nG ( −,P ( k ), T ) are S-projective. Moreover, the Cartan map
K nG ( −, P (k ), T ) → K nG ( −, P ( k ), T ) is an isomorphism for any G-set S
for which the Sylow-p-subgroups H of the stabilizers of the
elements in X have a non-empty fixed point set T H ∈ T , since in
this case T-exact sequences over X are split exact and thus all
functors ς : X → P (k ) are T-projective, i.e., [ X , P (k )]τ [ X , P (k )]
is an isomomorphism if [ X , P ( A)] is taken to be exact with respect
to T-exact and thus split exact sequences. This implies in
particular that for G-sets X, the Cartan map
PnG ( X × S , P (k ), T ) → K nG ( X × S , P ( k ), T )
is an isomorphism since any stabilizer group of an element in
X × S is a subgroup of a stabilizer group of an element in S, and
thus, by the very definition of S and D( k , T , Z ( 1p ) ) , has a Sylow-p-
subgroup H with T H ≠ ∅ . This finally implies that
PnG ( −, P (k ), T ) s → K nG ( −, P (k ), T ) s is an isomorphism. So, by the
general theory of Mackey funcors,
1 1
Z   ⊗ PnG ( −, P (k )T ) → Z   ⊗ K 0G ( −, P (k )T )
 p  p
is an isomorphism. The special case (T = G / ε ) PnG (−, P (k ), G / ε ) ,
just the K-theory of finitely generated projective kG-modules and
K nG (−, P (k ), G / ε ) the K-theory of finitely generated kG-modules
with respect to exact sequences. Thus we have proved the
following.
Theorem 5.4.3
Let k be a filed of characteristics p, G a finite group. Then, for all
n ≥ 0 , the Cartan map K n (kG ) → Gn (kG ) induces isomorphisms
1 1
Z   ⊗ K n (kG ) → Z   ⊗ Gn (kG ).
 p  p
Here are some applications of theorem 5.4.3. These applications
are due to A.O. Kuku (see [42]).

Theorem 5.4.4
Let p be a rational prime, k a field of characteristic p, G a finite
group. Then for all n ≥ 1 .

(i) K 2 n (kG ) is a finite p-group.

(ii) The Cartan homomorphism ϕ 2 n−1 : K 2 n−1 (kG ) → G2 n−1 (kG ) is
surjective, and ker ϕ 2 n−1 is the Sylow-p-subgroup of K 2 n−1 (kG ).
Corollary 5.4.5
Let k be a field of characteristic p, C a finite E1 category. Then,
for all n ≥ 0 , the Cartan homomorphism K n (kC) → Gn (kC)
induces isomorphism

1 1
Z   ⊗ K n (kC) ≅ Z   ⊗ Gn (kC).
 
 p  p
Corollary 5.4.6
Let R be the ring of integers in a number field F, m a prime ideal
of R lying over a rational prime p. then for all, n ≥ 1 ,

(a) the Cartan map K n ( ( R / m ) C ) → Gn ( ( R / m ) C ) is

surjective;
(b) K 2 n ( ( R / m ) C ) is a finite p-group
Finally, with the identification of Mackey functors:
GSet → Ab with Green’s G-functors δ G → Ab as in [42] and
above interpretations of our equivariant theory in terms of
grouprings, we now have, from the forgoing, the following result,
which says that higher algebraic K-groups are hyperelementary
computable. First, we define this concept.

Definition 5.4.7
Let G be a finite group, U a collection of subgroups of G closed
under subgroups and isomorphic images, A a commutative ring
with identity. Then a Mackey functor M : δ G → A - M od is said to
be U-compatible if the restriction maps M (G ) → ∏H ∈U M(H)
induces an isomorphism M (G ) ≅ lim H ∈U M ( H ) where lim H∈U is the
subgroup of all ( x) ∈ ∏H ∈U M ( H ) such that for any H , H ′ ∈ U and
g ∈ G with gH ′g − ⊆ H , ϕ : H ′ → H given by h → ghg −1 , then
M (ϕ )( x H ) = x H .
Now, if A is a commutative ring with identity, M : δ G → Z - M od
a Mackey functor, then A ⊗ M (H ) . Now, let P be a set of rational
[ ]
primes, Z P = Z 1q q ∉ P , C (G ) the collection of all cyclic
subgroups of G, hP C (G ) the collection of all P-hyperelementary
subgroups of G, i.e.,

hP C (G ) = { H ≤ G ∃ H ′ ≤ H , H ′ ∈ (G ), H / H ′ a p - group for some p ∈ P }.

Then we have the following theorem,

Theorem 5.4.7
Let R be a Dedekind ring, G a finite group, M any of the Green
modules K n (k − 1), Gn (k − 1) SK n (k − 1), SGn ( R − 1) , Cln ( R − 1) over
G0 ( R − 1) then Z P ⊗ M is hP ( C (G ) ) -computable.
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