Transcript Slide 1

Continuous Cohomology and Profinite
Higher
K-theory of Exact Categories,
Schemes and Twisted Laurent Series
Rings
By
Aderemi Kuku
University of Iowa
Iowa City
Abstract
This talk illustrates, among others, the benefits of methods of
category theory and homological algebra in gaining computational
and other insights into apparently different mathematical structures.
After a quick introduction to cohomology theory in Abelian
categories, we introduce continuous cohomology and associated exact
sequence useful for computations with examples from algebra
(groups) algebraic geometry (schemes) and algebraic topology (CWcomplexes)).
Next we introduce exact category and associated Higher Algebraic Ktheory with copious examples as well as mod-m and profinite Ktheory with associated computation-friendly exact sequences
analogous to those earlier obtained.
Finally, we discuss some of the results and computations obtained for
higher K-theory and profinite higher K-theory of schemes, groupschemes (e.g., twisted flag varieties) as well as orders and twisted
Laurent series rings over orders in semi-simple algebras over number
fields with consequences for group rings of virtually infinite cyclic
groups.
1.
AN INTRODUCTION TO COHOMOLOGY THEORY IN ABELIAN
CATEGORIES
1.1 Abelian Categories – Definitions and Examples
1.1.1 Examples of Abelian Categorie
(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.
1.1.2 Definitions:
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
exist
in A.
homA (M , N ) is an Abelian group such that
(c)
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).
1.1.3 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  : P  K such that h  g  k
g
f
K 
M 
N
k
h
P
Equivalently: given an object P in A , we have an exact sequence
0  homA (P, K ) homA (P, M ) homA (P, N )
is exact.
s
fp
Analogously, a morphism g : N  C is called a cokernel of
f : M  N if for any P  0 b A
f^
0  homA (C, P)  homA ( N , P) 
homA (M , P)
is exact.
f
g
B

C is said to be exact at B if
Note: A sequence A 
ker(g )  Im( f ) .
1.2.1 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 AB = {functors : B  A)
Morphisms are natural transformations of functors.
1.2.2 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
(1)
where
ob( E )  elementsof E
homE ( x, y )   unless x  y.
(2)
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).
Definition 1.2.4
A sheaf on X is a presheaf F satisfying:
If U i  is an open covering of subset U  X , then we have an exact
sequence:
0  F (U )  F Ui 
 F U i  U j 
i j
(i.e., if f i  F U i  are such that f i and f j agree on 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.
1.3
Cohomology Theory in an Abelian Category
1.3.1
Definition:
 
A cochain complex C * in an Abelian category A is a family C n of A
-objects together with maps d n : C n  C n1 such that d n1  d n  0 .
Call
Z n C* :  ker d n called n  cocyclesin A;
B n C *  Im d n1  C n - called
n-coboundaries
H n C*  Z n C* Bn C* is called the nth cohomology group
of C *
 
 
   
 
   
1.3.2
Coch (A) : = the collection of cochain complexes in A form
an Abelian category.
1.3.3
An object I in A is said to be injective if given any exact

 A 

B and a map
sequence 0 

A

I , thereexists,  : B  I making the day
0
 A 
 B

commute.
I

Let F : A  B be a left exact functor between Abelian categories A,
B.
(i.e., 0  A  A  A  0 exact in
A  0  F ( A)  F ( A)  F ( A) is exact in B)
If A has enough injectives (if for any A  A , there exist a map
A  I and hence an injective resolution A  I * ).
  
Define the right derived functor R i F of F by Ri F ( A)  H i F I * .
1.3.4
Example – Sheaf Cohomology
Let X be a topological space,  : Sh( X )  Z - Mod defined by
( F )  F ( X ) (global section functor)
Then H i ( X , F ) : Ri ( F ) .
2
CONTINUOUS COHOMOLOGY
2.1 General Considerations
1. Let ( N , ) be the natural numbers regarded as a poset (and hence
a category), A and Abelian category.
1
Then A N is an Abelian category where ob A N are inverse
systems  An , d n  .

2

A N has enough injectives iff A has enough injectives.
2. A left exact functor h : A  B induces a left exact functor.
hN : A N  B N
1
If A and hence A N has enough injectives and the inverse
limits over N exist in B, define the functor limh : A N  B by
n
 An , d n   limh An , hd n  i.e., limh :is given by
n
N
2
3
4
A B N B .
lim is left exact iff h is
Hence get right derived functors Ri (lim h) : A N  B .
hN
lim
Write lim n for the right derived functors of lim.
3.
Definition
An Abelian category is said to satisfy AB4 if for any collection {Ai}
of A-objects,  (Ai) exists (i.e., A is complete) and direct products of
epi’s is epi.
(4) Theorem:
If A has enough injectives, h : A  B is left exact and B satisfies AB4,
then there exists, an exact sequence
0  lim1 Ri 1hAn  Ri lim h Ai , di   limRi hAn  0


n
for i ≥ 0.
5. Note: In trying to compute the middle term, one sometimes
finds inverse systems (An, dn) for which left hand term of (I) vanishes.
One such situation is ‘Mittag-Leffler condition!
6. Definition An inverse system (An, dn) in an Abelian category A
is said to satisfy the Mittag-Leffler condition if for each k, there
exists, j ≥ k s.t.
image Ai  Ak  image of Aj  Ak  i  j e.g., when all Ai+1 
Ai is onto.
7. Theorem
Let A satisfy AB4 and let A ob( A ) satisfy Mittag-Lefler condition.
Then lim1 An  0.


2.2 Continuous Group Cohomology
(1) Let G be a profinite group (i.e., G  limGi , Gi finite). A discrete
G-module is a G-module A s.t. if A is given the discrete topology,
then the multiplication map G  A  A is continuous.

Note: A is a discrete G-module iff A =  AU where U runs
through open normal subgroups of G and AU={a  A | ga = a
 g  U}.
(2)
The category M(G) of discrete G-modules is an Abelian
category with enough injectives.
 The right derived functors of the left exact functor
M(G)  -mod : A  AG
are the cohomology groups of G with coefficients in A and denoted
by Hn(G,A)
(3)
We also have a left exact functor M(G)  -mod given by
M n , dn   H 0 G, M n , dn   lim H 0 G, M n 


with right derived functor
(Mn,dn)  Hi (G, (Mn,dn))
So by 2.1 (4), we have a short exact sequence
0  lim1 H i1 G, M n   H i G, M n , d n   lim H i G, M n   0


n


n
Call H i G, M n , dn  continuous cohomology of G1 in view of the
following theorem.
4.
Theorem
Let (An, dn) be an inverse system of discrete G-modules satisfying the
Mittag-Leffler condition and let A  limAn . Then, there exists an
isomorphism
i
Hcont
(G, A)  H i G,  An , dn 
i
for i ≥ 0 where H cont
(G, A) denotes the continuous cohomology
group defined by Tace.
2.3
Continuous etále cohomology
Let X be a scheme, Xet an etále site of X (i.e., Xet is a category with
Grothendieck topology s.t ob Xet :- all etále maps U X. Covering
families are morphisms
φi : Ui U over X s.t   Ui  = U
S(Xet) : = category of sheaves on Xet is an Abelian category. Let
N
S X et   Z-Mod be the functor, given
by
Fn , d n   lim H 0  X , Fn .


(I)
Define H i  X , Fn dn   derived functor of (I). Then we have an exact
sequence.
0  lim H i1  X , Fn   H i  X , Fn , dn   limH i  X , Fn   0
1
i
Call Hcont
X , F  : H i X , Fn , dn  the continuous cohomology of X.
2.4
CW-Complexes
(1)
Let {Ci, dn} be an inverse system of cochain complexes of
Abelian groups. Put C  lim Ci . Then we have the following


0  lim1 H n 1 Ci   H n (C )  lim H n Ci   0.




(2)
Let X be a CW-complex and Hn(X) the integral cohomology of
X, {Xi) an increasing sequence of sub-complexes with X   X i .
Then we have an exact sequence
0  lim1 H n 1  X i   H n ( X )  lim H n  X i   0.




Note: J. Milnor observed that analogous sequence hold for
generalized cohomology theory like K-theory (J. Mihor: On
Axiomatic homology theory. Pacific J. Math, 1962, 337–345)
3. HIGHER K-THEORY OF EXACT CATEGORY C
3.1 What is K-theory?
3.1.1
Kn
Roughly speaking, K-theory is the study of functors (bridges)
: ( Nice categories) 
 (categoryof Abelian groups)
nZ
Note:
C 
 K n (C ),
For n ≤ 0, we have Negative K-theory
For n ≤ 2, we have Classical K-theory
For n ≥ 3, Higher K-theory.
3.1.2
Brief Historical Remarks
K-theory was so christened by A. Grothendieck who first studied
K0(C) where for a scheme X, C is the category of coherent sheaves of
OX-modules, ‘K’ is the first letter in the German world ‘Klass’ which
means ‘class’. This is because K0(C) classifies the isomorphism
classes of objects of C.
Next M.F. Atiyah and F. Hirzebruch studied K0(C) when C is the
category of vector bundles on a compact space X-yielding what
became known as topological K-theory.
Later H. Bass, R.G. Swan, etc. studied K0(C) where (C) is thecategory
P(A) (resp M(A)) of finitely generated projective (resp. finitely
generated) modules over a ring A with identity marking the beginning
of Algebraic K-theory.
The algebraic K1 was defined by H. Bars and K2 by J. Milnori.
Negative K-theory was defined by H. Bars and M. Karoubi. Then D.
Quillen came forward in 1970 with the definition of all Kn for an n ≥
0 – leading to Higher k-theory.
One particularly important example of ‘nice’ category is an exact
category which we now define.
3.1.2 Definition
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)
(ii)
E is the class of sequences in C that are exact in A
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 brief construction of Kn (C) n ≥ 0, we give some
relevant examples of C and develop notations for Kn (C).
3.1.3
Examples
1.
Let A be any ring with identity C = P(A) (resp. M(A)) the
category of finitely generated projective (resp. finitely generated) Amodules. 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.
(iv)
Definition
Let R be a Dedekind domain with quotient field F (e.g., R = Z (resp.


Z p ), F  Q(resp Qp )

ˆ
p a rational prime or more generally R , F ( p a prime ideal of R).
p
p
An R-order  in semi-simple F-algebra  is a subring of  such that
R is contained in the centre of  ,  is a finitely generated R-module
and


F R   , (E.g.,   ZG, Z pG, RG, RpG G a finite group)
(v)
Let A be a ring (with 1), : A  A an automorphism of A,
A(T) = A (t, t –1) : 
-twisted Laurent series ring over A (i.e., Additively A[T] = A[T],
with multiplication given by ( at i)  (bt i) = a 1(b) t i + j for a, b  A).
Let A[t] be the subring of A(T) generated by A and t.
Note:
If  = RG,  [T] = R V where V  G  | T is a virtually

infinite cyclic group and G is a finite group,  an automorphism of G
and the action of the infinite cyclic group
T = t on G is given by (g) = tgt1 for all g  G.
(2)
X a compact topological space, F = or , VectF(X) : 
category of vector finite dimensional vector bundles on X. Write
KnF (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))
Gelford-Naimark theorem says that any commulative C*-algebra A
has the form A  C(X) for some locally compact space X. If A is a
non-commulative C^-algebra, then K-theory of A leads to “noncommulative geometry” in the sense that A could be conceived as
ring of functions on a “non-commutative or quantum” space.
(3)
Let X be a scheme. Let P(X) be the category of locally free
sheaves of OX-modules 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 comutative ring we recover Kn(A) and Gn(A).
(4)
Let G be an algebraic group over a field F, and X a G-scheme,
i.e., there exists an action  : G  X  X . Let M(G,X) be the
F
category of G-modules M over X. (i.e., M is a coherent OX-module
together with an isomorphism of OG  X -module *(M) = p2* (M),
F
where p2 : G  X  X ; satisfying some co-cycle conditions). Write
F
Gn (G, X) for Kn (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).
~
(5)
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 

 det a det c  1 a  GLn c  GLn k 
G  SLn P  


 0 c 

is the Grassmanian variety of k-dimensional linear subspaces of an ndimensional vector space.
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(X1B) for Kn(VBG(X,B)). In particular, in the notation of (5),
we write Kn(F, B) for Kn(VBG(F, B)).
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  : S  C. The [S,
C] is also an exact category where a sequence
0          0 is said to be exact in [S, C] if
0   (s)   (s)   (s)  0 is exact in C. Write KnG (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 AHmodules that are projective over A. (i.e., AH lattices)
K n G / H , P( A) : Gn ( A, H )
K n G / H , M( A) : Gn ( AH).
If A is regular, then Gn ( A, H )  Gn ( AH ).


3.1.3 Some features of K n (C )
(1)
Kn (C ) sometimes reflects the structure of objects of C.
For example,
(i) F a field K0 (PF ( FG))  G0 ( FG) classifies representations of
G in P(F ) .
(ii)
K 0 (ZG) contains topological / geometric invariants.
E.g., Swan-Well invariants.
(iii)
Ki (ZG) contains Whitehed torsion – a topological
invariants.
(2)
Each Kn (C ) yields a theory which could map or coincide with
other theories.
For example,
(i)
Galois, etale or Motivic cohomology theories
(ii)
Representation theory, e.g.,
K0 PF ( FG)  G0 ( FG) concides with Abelian group of
characters of G.
(3)
Kn (C ) satisfies various exact sequences connecting
K n , K n 1 , etc. For example, Localization sequences, Mayer-victories
sequence, etc. These sequences are useful for computations …
Brief Construction of K n ( C )  n  0
3.2
3.2.1 Let A be a small category. The nerve of A written NA is the
simplicial set whose n-simplices are diagrams
fi
f3
fn
f2
An  A0 
A1 
A2 

An Ai  ob A ,


and fi A f i are A-morphisms. The classifying space BA is the
geometric realization |(NA)| of NA.
Now let C be an exact category, QC a new
category s.t ob(QC )  ob C . A morphism from M to P is an
j
i

N

P where i.j are parts
isomorphism class of diagrams M 
of s.e.s in E. (see 3.1.2)
When n = 0, K0 (C ) coincides with the classical definition i.e.,
K0 (C ) is the Abelian group generated by isomorphism classes (C) of
C-objects subject to the relations C  C  (C) whenever
0  C   C  C   0 is an exact sequence in C.
It is such that every s.e.s 0  C   C  0 splits, e.g.,
C  P( A), Vect C ( X ) then (C , ) is a ‘symmetric monoidal
category’ with the property that the isomorphism classes of objects of
C form and Abelian monoidal and K 0 (C ) is then the ‘group
completion’ of such a monoid. In this case, we have a generalisation
of the elementary procedure of constructing integers from natural
numbers.
3.2.2
Examples 3.2.3
2
3
4
If R is a field or division ring or a local ring, then K 0 ( R) 
If X is a compact space, write KU (X ) for
K0 VBC ( KO( X ) and KO(X ) for K0VBR ( X ) .
K1 ( F )  F (units of F = non-zero elements of F). For a
D*
division ring D, K1 ( D)  *
. For any ring R, with
[ D , D
identity K1 ( R) :  GL( R) / E ( R) where GL(R) is the full linear
group and E(R) is the subgroup of GL(R) generated
elementary matrices.
K 2 ( ) is cyclic of order 2.
K2 of any finite field = 1.
K 2 ( F ( x))  K 2 ( F ).
Some Relevant Finiteness Results for K (n C )
3.3
3.3.1 Theorem
Let R be the ring of integers in a number field F,  any R-order in a
semi-simple F-algebra  . Then,
(i)
(ii)
(iii)
(iv)
For all n  1, Kn (), Gn () are finitely generated Abelian
group
(Kuku, J-algebra 1984, AMS contemp. Math, 1986).
For all n  1, K2n (), G2n () are finite Abelian groups. (Ktheory 2005).
If F is totally real, then G2m1 () is also finite for all odd
m 1
(Algebras and Rep. Theory - to appear)
For all n  1, G2n  (T ) is a finitely generated Abelian
group where  (T ) is the twisted Laurent series ring over
 . (Kuku: Algebras and Rep theory - to appear)
(i)
There exists isomorphism
Q  Kn  (T )  Q  Gn  (T )  Q  Kn  (T )  n  2
(Kuku – Algebras and Rep. theory - to appear)
3.3.2
Note: Above results (i), (ii), (iii) apply to
  RG (a finite group) while (iv) and (v) apply to
 (T )  ( RG) (T )  RV where V  G T is a virtually infinite

cyclic group.
3.3.3
Theorem Kuku (MPIM – Bonn preprint 2007
~
(a) Let G be a semi-simple, simply corrected and corrected F~
split algebraic group over a number field F, P a parabolic
~
subgroup of G ,  the 1-cocycle
~
~ ~
 , Gal Fsep F  G Fsep , F  G P the flag variety and  F the  -


 
twisted form of F, B a finite-dimensional separable F-algebra.
Then for all n  1
(i)
K2n 1  F, B is finitely generated Abelian group.
(ii)
K2n  F, Bis torson.
(b) If V is a severi-Brauer variety over a number field F,
then K 2 n1 (V ) is a finitely generated Abelian group.
4.
Mod-m and Profinite Higher K-theory of Exact
Category C
4.1
Mod-m K-theory
1.
Let X be an H-space, m,n positive integers, M mn an n-dim mod-
m Moore space, i.e., the space obtained from S n1 by attaching an ncell via a map of degree m.
2.
Another way to visualize M mn
Let G be a discrete Abelian group, M n (G) the space with only one
~
non-zero reduced integral cohomology group H n (M n (G)) . Suppose
~
that H n (M n (G))  G
Then
M mn  M n (Z / m); M n (Z )  S n
(G  Z / m
(G  Z ).


Write  n ( X , Z / m) for M mn , X the set of homotopy classes of
3.
maps from M mn to X .
 If C is an exact category, and X  BQC , write
Kn (C , Z / m) for  n 1 ( BQC , Z / m) n  1 and call this group
the mod-m higher K-theory of C.
 If C  P(A) write K n ( A, Z / m) for K n (P( A), Z / m)



If X a scheme, write K n ( X , Z / m) for K n (P( A), Z / m)
If A is a Northerian ring, write Gn ( A, Z / m ) for
K n (M( A), Z / m
If X is a Noetherian scheme, write Gn ( X , Z / m) for
K n (M( X ), Z / m )
4.2
Profinite Higher K-theory - Definitions and Examples
4.2.1 In 4.1, take m   s where  is a prime, s a positive integer. So
M ns1 is the (n  1) -dimensional mod-  s Moore space and
M n1;  lim M ns1.
s


 

Define profinite K-theory of C by Knpr C, Zˆ : M n1, BQC .
4.2.2 Problem
Study Knpn (C, Zˆ ) for various C.
Write Kn C, Zˆ  for lim Kn C, Z (s ) and also study Kn C , Zˆ 






4.2.3 Examples

 

1.
If A is a ring, write K npn A, Zˆ  for M nw1 , BQ(P( A)) and call
this the profinite K-theory of A.
 If A is Noetherian scheme, write
Gnp A, Zˆ : M n1 , BQ(M( A)) .


 

We are interested in A   [T ], the twisted Laurent series
ring over an order  .




If X is a scheme, write Knp X , Zˆ for M n1 , BQ(P( X ))
If X is a Noetherian scheme write Gnpr X , Zˆ for M n1 , BQ(M( X )) .
2.
3.



Let G be an algebraic group over a field F, X a G-scheme,
write K np G, X , Zˆ  for Knp P(G, X ); Zˆ and Gnp G, X , Zˆ for
K p MG, X , Zˆ .
n










4.
Let B be a finite dimensional separable F-algebra, X a smooth
projecture variety equipped with the action of an affine algebraic
group G over F. Write Knp ( X , B), Zˆ for Knp VBG  X , B, Zˆ .

4.3



Some General Results on Knp C , Zˆ


For proofs of these results, see the ‘Book’ A.O. Kuku (Representation
theory and Higher Algebraic K-theory, Chapman & Hall 2007
(Chapter 8)) or A.O. Kuku. Profinite and Continuous Higher K-theory
of Exact Categors, orders, and grouprings K-theory 22, (2001) 367 –
392 .
4.3.1 Theorem
Let C be an exact category,  for rational prime. Then for all n  1 ,
there exists an exact sequence
1
0  lim Kn C , Z / s  Knp C, Zˆ  Kn C, Zˆ  0.






Note: This sequence is analogous to that obtained in Section 2.
From above, we have
(i)
For any ring A,

0  lim Kn1 A, Z / s  Knp A, Zˆ  Kn A, Zˆ  0.
(ii)
For any scheme X,

0  lim Kn1 X , Z / s  Knp X , Zˆ  Kn X , Zˆ  0.






Similarly, for C  M( X ), M( A) , etc.






4.3.2
Theorem
Let C be an exact category  a prime s.t  n  0, Kn (C ) has no nontrivial divisible subgroups.
Then
(i) Kn (C ) s  Knp C, Zˆ s n  1
(ii)
Kn (C ) / s  Knp C, Zˆ / s  n  2.
 
4.3.3

 
 
Theorem
Let C be an exact category such that Kn (C ) is a finitely generated
Abelian group for all n  0 . Let  be a rational prime. Then
Knp C, Zˆ is an  -complete profinite Abelian group.


4.3.4
Note:
(a) An Abelian group A is  -complete if A  lim A /  s A .
s
(b) (4.3.2) and (4.3.3) above apply to
C  P(), and C  M () where  is any R-order in a semi(1)
single algebra over a number field F – and in particular to
  RG, G any finite group.
Let C  M [T ] ,  as in (i);  an automorphism of  and
 [T ] a twisted Laurent series ring over  . (See Kuku (2007)
Algebras and representation theory (to appear).
(i)
(iii) C  M [T )] where  [t ] is the  -twisted polynomial ring
over  .
(iv) C  P (X ) where X is a smooth projective curve over a finite
field of character p.
4.4
Some Further Results
4.4.1
Theorem (Kuku 2007) MPIM Bonn (Preprint).
~
Let G be a semi-simple, simply connected and connected F-split
~
~
algebraic group over a number F. P a parabolic subgroup of G ,  the
~ ~
1 cocycle  : Gal Fsep F   GFsep , F  G P the flag variety and
F the twisted form of F, B a finite-dimensional separable F-algebra.

Then for all n  1,
(a) K2n 1 F, B is a finitely generated Abelian group.
(b) K2n F, B is a torsion group.
(c)
(d)
K2prn F, B is an  -complete Abelian group.
K2prn (F, B), Zˆ  0


We next present the local situation.
4.4.2
Theorem (Kuku 2007) MPIM Bonn (Preprint)
Let p be a rational prime, F a p-adic field, G a semi-simple, connected
~ ~
and simply space connected split algebraic group over F, F  G P .
~
~

parabolic
subgroup
of
a
1-cocycle
Pa
G,
 : GalFsep F   GFsep , F the  -twisted form of the flag variety F ,
 a rational prime such that   p .
Then, for all n  2 .
(a)
(b)
(c)




Knp F, B, Zˆ is an  -complete profinite Abelian group.
Knp F, B, Zˆ  Kn F, B, Zˆ
The map  : K F, B  K p F, B Zˆ ) induces isomorphisms
n
(1)
(2)





n
 
 

Kn F, B s  Knp F, B Zˆ . s
K F, B s  K p F, B Zˆ s
n
n

(d)
(e)
kernel and cokernal of
Kn F, B  Knp F, B, Zˆ
are uniquely  -divisible.
div Knp F, B, Zˆ  0 for n  2 .




We close with the following result on profinite G-theory for the
twisted Laurent series ring  [T ] of an arbitrary R-order  in a
number field F with ring of integers R.
4.4.3
Theorem (Kuku 2007) (Algebras and Representation theory
(to appear))
Let R be the ring of integers in a number field F,  any R-order in a
semi-simple F-algebra ,  :    an R-automorphism and
 [T ] the  -twisted Laurent series ring over  .
Then for all n  2,
(a)
(b)
(c)


div Gnp  [T ], Zˆ  0
Gnp  [T ], Zˆ  Gn  [T ], Zˆ is an  -complete profinite
Abelian group.
The map Gn  [T ]  Gnp  [T ], Zˆ is injective with
uniquely  -divisible cokernel.






Note: 4.4.3 above applies to the group ring RV where V is a
virtually infinite cyclic group of the form V  G  | T where G is a

finite group,  an automorphism of G and the action of the infinite
cyclic group T  t on G is given by  ( g )  tgt1  g  G.
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