Algebraic K-theory over the infinite dihedral group: an algebraic approach

0.33 MB·English

by  James F. Davis

ALGEBRAIC K-THEORY
OVER THE INFINITE DIHEDRAL GROUP:
AN ALGEBRAIC APPROACH
0
1
0
JAMESF.DAVIS,QAYUMKHAN,ANDANDREWRANICKI
2

b
e Abstract. WeprovethattheWaldhausennilpotentclassgroupofaninjective
F index2amalgamatedfreeproductisisomorphictotheFarrell–Bassnilpotent
classgroupofatwistedpolynomialextension. Asanapplication,weshowthat
2 theFarrell–JonesConjectureinalgebraicK-theorycanbesharpenedfromthe
2 familyofvirtuallycyclicsubgroupstothefamilyoffinite-by-cyclicsubgroups.

]
T
K
h. Introduction
t
a Theinfinitedihedralgroupisbothafreeproductandanextensionoftheinfinite
m cyclic group Z by the cyclic group Z of order 2
2
[
D = Z ∗Z = Z⋊Z
∞ 2 2 2
3
v withZ acting onZby −1. AgroupG is saidto be over D if itis equipped with
2 ∞
9 an epimorphism p : G → D . We study the algebraic K-theory of R[G], for any
∞
3
ring R and any group G over D . Such a group G inherits from D an injective
6 ∞ ∞
amalgamated free product structure G = G ∗ G with F an index 2 subgroup
1 1 F 2
. of G1 and G2. Furthermore, there is a canonical index 2 subgroup G ⊂ G with
3 an injective HNN structure G = F ⋊ Z for an automorphism α : F → F. The
0 α
various groups fit into a commutative braid of short exact sequences:
8
0
: Z
rXiv ssssssssss9999 M&& MMMMMMMMMM&&
a t99 tttttGttt=99 F ⋊αK%% KZKKKKθKKKK%% qqqqpqqDqq∞qq88=88 Z2C∗CCZCCπ2CCC!!!!
F =G ∩G G=G ∗ G Z
1 2 1 F 2 2
%% 99 ::::
π◦p
The algebraic K-theory decomposition theorems of Waldhausen for injective
amalgamated free products and HNN extensions give
K (R[G]) = K (R[F]→R[G ]×R[G ])⊕Nil (R[F];R[G −F],R[G −F]) (∗)
∗ ∗ 1 2 ∗−1 1 2
f
2000 Mathematics Subject Classification. Primary: 19D35;Secondary: 57R19.
1
2 J.DAVIS,Q.KHAN,ANDA.RANICKI
and
K (R[G]) = K (1−α:R[F]→R[F])⊕Nil (R[F],α)⊕Nil (R[F],α−1) .
∗ ∗ ∗−1 ∗−1
(∗∗)
f f
We establish isomorphisms
Nil (R[F];R[G −F],R[G −F]) ∼= Nil (R[F],α) ∼= Nil (R[F],α−1)
∗ 1 2 ∗ ∗
whichfwe use to prove that the Farrell–Jonefs isomorphism cofnjecture in algebraic
K-theory can be reduced to the family of finite-by-cyclic groups, so that virtually
cyclic groups of infinite dihedral type need not be considered.
0.1. Algebraic semi-splitting. Ahomotopyequivalencef :M →X =X ∪ X
1 Y 2
offiniteCW complexesissplitalongY ⊂X ifitisacellularmapandtherestriction
g = f| : N = f−1(Y)→Y
is also a homotopy equivalence. The Nil -groups arise as the obstruction groups
∗
to splitting homotopy equivalences of finite CW complexes in the case of injective
f
π (Y) → π (X) (Farrell–Hsiang, Waldhausen). In this paper we introduce the
1 1
considerably weaker notion of a homotopy equivalence being semi-split, as defined
in§0.2. The following is a specialcase ofour main algebraicresult(1.1, 2.7) which
shows that there is no obstruction to semi-splitting.
Theorem 0.1. Let G be a group over D , with
∞
F = G ∩G = G = F ⋊ Z⊂G = G ∗ G .
1 2 α 1 F 2
(i) For any ring R and n ∈ Z the corresponding reduced Nil-groups are naturally
isomorphic:
Niln(R[F];R[G1−F],R[G2−F]) ∼= Niln(R[F],α) ∼= Niln(R[F],α−1)
(ii) Tfhe inclusion θ :R[G]→R[G] determinfes induction and tfransfer maps
θ : K (R[G])→K (R[G]) , θ! : K (R[G])→K (R[G]).
! n n n n
Foralln61,themapsθ andθ! restricttoisomorphismsontheNil-Nil-components
!
in the decompositions (∗) and (∗∗).
f f
Proof. Part (i) is a special case of Theorem 0.5.
Part(ii)followsfromProposition3.20(ii)(induction)andProposition3.22(trans-
fer). (cid:3)
The n=0 case will be discussed in more detail in §0.3 and §3.1.
Remark 0.2. We do not seriously doubt that a more assiduous application of
higher K-theory would extend Theorem 0.5 (ii) to all n∈Z (see also [DQR]).
As an application of Theorem 0.1, we shall prove the following theorem.
Theorem 0.3. Let Γ be any group, and let R be any ring. Then the following map
of equivariant homology groups with coefficients in the algebraic K-theory functor
K is an isomorphism:
R
H∗Γ(EfbcΓ;KR)−→H∗Γ(EvcΓ;KR) .
ALGEBRAIC K-THEORY OVER D∞: AN ALGEBRAIC APPROACH 3
Infact,this is aspecialcaseofourmoregeneralfiberedversion(Theorem3.28).
The original reduced Nil-groups Nil (R) = Nil (R,1) feature in the decomposi-
∗ ∗
tions of Bass [Bas68] and Quillen [Gra76]:
f f
K (R[t]) = K (R)⊕Nil (R) ,
∗ ∗ ∗−1
K∗(R[Z]) = K∗(R)⊕Kf∗−1(R)⊕Nil∗−1(R)⊕Nil∗−1(R) .
In §3 we shall compute severalexamples which refquire Theorfem 0.1:
K (R[Z ])⊕K (R[Z ])
K (R[Z ∗Z ]) = ∗ 2 ∗ 2 ⊕Nil (R)
∗ 2 2 ∗−1
K (R)
∗
K (R[Z ])⊕K (R[Z ]) f
K (R[Z ∗Z ]) = ∗ 2 ∗ 3 ⊕Nil (R)∞
∗ 2 3 ∗−1
K (R)
∗
Wh(G ×Z )⊕Wh(G ×Zf)
Wh(G ×Z ∗ G ×Z ) = 0 2 0 2 ⊕Nil (Z[G ])
0 2 G0 0 2 Wh(G ) 0 0
0
f
where G = Z ×Z ×Z. The point here is that Nil (Z[G ]) is an infinite torsion
0 2 2 0 0
abelian group. This provides the first example (Example 3.27) of a non-zero Nil
f
group in the amalgamated product case and hence the first example of a non-zero
obstruction to splitting a homotopy equivalence in the two-sided case (A).
0.2. Topological semi-splitting. Let (X,Y) be a separating, codimension 1, fi-
nite CW pair, with X = X ∪ X a union of connected CW complexes such
1 Y 2
that π (Y) → π (X) is injective. Let X¯ (resp. X¯ ,X¯ ) be the connected cover
1 1 1 2
of X (resp. X , X ) classified by π (Y) ⊂ π (X) (resp. π (Y) ⊂ π (X ),
1 2 1 1 1 1 1
π (Y) ⊂ π (X )). Then X¯ = X¯− ∪ X¯+ with π (X¯−) = π (X¯+) = π (Y).
1 1 2 Y 1 1 1
Note X¯ ⊂X¯− and X¯ ⊂X¯+ with X¯ ∩X¯ =Y ⊂X¯.
1 2 1 2
A homotopy equivalence f : M → X from a finite CW complex is semi-split
along Y ⊂ X if N = f−1(Y) ⊂ M is a subcomplex and the restriction (f,g) :
(M,N)→(X,Y)isamapofpairssuchthattherelativehomologykernelZ[π (Y)]-
1
modules K (M¯ ,N) vanish. Equivalently, f is semi-split along Y if the following
∗ 2
induced Z[π (Y)]-module morphisms are isomorphisms:
1
ρ2 : K∗(M¯+,N)−→K∗(M¯+,M¯2)=Z[π1(X2)−π1(Y)]⊗Z[π1(Y)]K∗(M¯−,N) .
The notation of CW-splitting is explained more in §0.4.
We referto §3.23forthe definitionofanalmost-normalsubgroup. Inparticular,
finite-index subgroups and normal subgroups are almost-normal.
Theorem 0.4. Let (X,Y) be a separating, codimension 1, finite CW pair, with
X = X ∪ X a union of connected CW complexes such that π (Y) → π (X) is
1 Y 2 1 1
injective. Suppose π (Y) is an almost-normal subgroup of π (X ). Then, for any
1 1 2
finite CW complex M, any homotopy equivalence h:M →X is simple homotopic
to a semi-split homotopy equivalence h′ :M →X along Y.
0.3. Algebraic exposition. For any ring R, we establish isomorphisms between
two types of codimension 1 splitting obstruction nilpotent class groups. The first
type,forseparatedsplitting,arisesinthedecompositionsofthealgebraicK-theory
oftheR-coefficientgroupringR[G]ofagroupGoverD ,withanepimorphismp:
∞
G→D onto the infinite dihedralgroupD . The secondtype, fornon-separated
∞ ∞
splitting, arises from the α-twisted polynomial ring R[F] [t], with F =ker(p) and
α
4 J.DAVIS,Q.KHAN,ANDA.RANICKI
α:F →F an automorphism such that
G = ker(π◦p:G→Z ) = F ⋊ Z
2 α
where π :D →Z is the unique epimorphism with infinite cyclic kernel. Note:
∞ 2
(A) D = Z ∗Z is the free product of two cyclic groups of order 2, whose
∞ 2 2
generators will be denoted t ,t .
1 2
(B) D =h t ,t | t2 =1=t2 i contains the infinite cyclic group Z=hti as a
∞ 1 2 1 2
subgroup of index 2 with t = t t . In fact there is a short exact sequence
1 2
with a split epimorphism
π
{1} //Z //D //Z // {1} .
∞ 2
More generally, if G is a group over D , with an epimorphism p:G→D , then:
∞ ∞
(A) G=G ∗ G is a free product with amalgamation of two groups
1 F 2
G = ker(p :G→Z ) , G = ker(p :G→Z )⊂G
1 1 2 2 2 2
amalgamated over their common subgroup F =ker(p)=G ∩G of index
1 2
2 in both G and G .
1 2
(B) G has a subgroup G = ker(π◦p : G → Z ) of index 2 which is an HNN
2
extensionG=F⋊ Zwhereα:F →F isconjugationbyanelementt∈G
α
with p(t)=t t ∈D .
1 2 ∞
The K-theoryof type (A). ForanyringRandR-bimodulesB ,B ,theNil-groups
1 2
Nil (R;B ,B )aredefinedtobethealgebraicK-groupsK (Nil(R;B ,B ))ofthe
∗ 1 2 ∗ 1 2
exact category Nil(R;B ,B ) with objects quadruples (P ,P ,ρ ,ρ ) with P ,P
1 2 1 2 1 2 1 2
f.g. projective R-modules and
ρ : P →B ⊗ P , ρ : P →B ⊗ P
1 1 1 R 2 2 2 2 R 1
R-module morphisms such that ρ ρ : P → B ⊗ B ⊗ P is nilpotent (or
2 1 1 1 R 2 R 1
equivalently such that ρ ρ : P → B ⊗ B ⊗ P is nilpotent). The reduced
1 2 2 2 R 1 R 2
Nil-groups Nil are such that
∗
fNil (R;B ,B ) = K (R)⊕K (R)⊕Nil (R;B ,B ) .
∗ 1 2 ∗ ∗ ∗ 1 2
As already noted above, Waldhausen [Wal78] decomfposed the algebraic K-theory
of R[G] for an injective amalgamated free product G=G ∗ G as
1 F 2
K (R[G]) = K (R[F]→R[G ]×R[G ])⊕Nil (R[F];R[G −F],R[G −F]) .
∗ ∗ 1 2 ∗−1 1 2
In particular, there is defined a split monomofrphism
σ : Nil (R[F];R[G −F],R[G −F]) // // K (R[G]) ,
A ∗−1 1 2 ∗
which for ∗=1 isfgiven by
σ : Nil (R[F];R[G −F],R[G −F]) // // K (R[G]) ;
A 0 1 2 1
[Pf,P ,ρ ,ρ ]7→ R[G]⊗ (P ⊕P ),( 1 ρ2 .
1 2 1 2 (cid:20) R[F] 1 2 (cid:18)ρ1 1(cid:19)(cid:21)
ALGEBRAIC K-THEORY OVER D∞: AN ALGEBRAIC APPROACH 5
The K-theory of type (B). Given a ring R and an R-bimodule B, let
T (B) = R ⊕ B ⊕ B⊗ B ⊕ ···
R R
bethetensoralgebraofB overR. TheNil-groupsNil (R;B)aredefinedtobethe
∗
algebraic K-groups K (Nil(R;B)) of the exact category Nil(R;B) with objects
∗
pairs (P,ρ) with P a f.g. (finitely generated) projective R-module and ρ : P →
B⊗ P a R-module morphism, nilpotent in the sense that for some k, we have
R
ρk =0:P →B⊗ P →···→B⊗ ···⊗ B⊗ P.
R R R R
The reduced Nil-groups Nil are such that
∗
Nifl∗(R;B) = K∗(R)⊕Nil∗(R;B) .
Waldhausen [Wal78] provedthat if B is f.g. profjective as a left R-module and free
as a right R-module, then
K (T (B)) = K (R)⊕Nil (R;B)
∗ R ∗ ∗−1
with a split monomorphism f
σ : Nil (R;B) // // K (T (B)) ,
B ∗−1 ∗ R
which for ∗=1 is given by f
σ : Nil (R;B) // // K (T (B)) ; [P,ρ]7→[T (B)⊗ P,1−ρ] .
B 0 1 R R R
In particular,ffor B =R
Nil (R;R) = Nil (R) , Nil (R;R) = Nil (R) ,
∗ ∗ ∗ ∗
T (B) = R[t] , K (R[t]f) = K (R)⊕Nfil (R) .
R ∗ ∗ ∗−1
Relating the K-theory of types (A) and (B). Recall thatfa category I is filtered if:
• for any pair of objects α,α′ in I, there exist an object β and morphisms
α→β and α′ →β in I, and
• for any pair of morphisms u,v : α → α′ in I, there exists an object β and
morphism w :α′ →β such that w◦u=w◦v.
Note that any directed poset I is a filtered category.
Theorem 0.5 (General Algebraic Semi-splitting). Let R be a ring. Let B ,B be
1 2
R-bimodules. Suppose that I is a small, filtered category and B =colim Bα is
2 α∈I 2
a direct limit of R-bimodules such that each Bα is a f.g. projective left R-module.
2
Then, for all n ∈ Z, the Nil-groups of the triple (R;B ,B ) are related to the
1 2
Nil-groups of the pair (R;B ⊗ B ) by isomorphisms
1 R 2
Niln(R;B1,B2) ∼= Niln(R;B1⊗RB2)⊕Kn(R) ,
Niln(R;B1,B2) ∼= Niln(R;B1⊗RB2) .
In particular, fofr n=0 there are definfed inverse isomorphisms
∼
=
i : Nil (R;B ⊗ B )⊕K (R) // Nil (R;B ,B ) ;
∗ 0 1 R 2 0 0 1 2
ρ
([P ,ρ :P →B ⊗ B ⊗ P ],[P ])7→[P ,B ⊗ P ⊕P , 12 ,(1 0)] ,
1 12 1 1 R 2 R 1 2 1 2 R 1 2 (cid:18) 0 (cid:19)
∼
=
j : Nil (R;B ,B ) // Nil (R;B ⊗ B )⊕K (R) ;
∗ 0 1 2 0 1 R 2 0
[P ,P ,ρ :P →B ⊗ P ,ρ :P →B ⊗ P ]7→([P ,ρ ◦ρ ],[P ]−[B ⊗ P ]) .
1 2 1 1 1 R 2 2 2 2 R 1 1 2 1 2 2 R 1
6 J.DAVIS,Q.KHAN,ANDA.RANICKI
The reduced versions are the inverse isomorphisms
∼
=
i : Nil (R;B ⊗ B ) // Nil (R;B ,B ) ; [P ,ρ ]7→[P ,B ⊗ P ,ρ ,1] ,
∗ 0 1 R 2 0 1 2 1 12 1 2 R 1 12
∼
j : Nfil (R;B ,B ) = // Nilf(R;B ⊗ B ) ; [P ,P ,ρ ,ρ ]7→[P ,ρ ◦ρ ] .
∗ 0 1 2 0 1 R 2 1 2 1 2 1 2 1
Proof.fThis follows immediatefly from Theorems 1.1 and 2.7. (cid:3)
Remark0.6. Theorem0.5wasalreadyknowntoPierreVogelin1990-see[Vog90].
0.4. Topological exposition. The proof of Theorem 0.5 is motivated by the ob-
structiontheoryofWaldhausen[Wal69]forsplittinghomotopyequivalencesoffinite
CW complexes X alongcodimension 1 subcomplexes Y ⊂X with π (Y)→π (X)
1 1
injective,andthe subsequentalgebraicK-theorydecompositiontheoremsofWald-
hausen [Wal78].
A codimension 1 pair (X,Y ⊂ X) is a pair of spaces such that the inclusion
Y =Y×{0}⊂X extendstoanopenembeddingY×R⊂X. Amapofcodimension
1 pairs (f,g) : (M,N) → (X,Y) is a cellular map f : M → X with g = f| : N =
f−1(Y)→Y.
Let(X,Y)beacodimension1finiteCW pair. Ahomotopyequivalencef :M →
X fromafinite CW complexsplits along Y ⊂X ifthereisamapofcodimension1
pairs (f,g) :(M,N) → (X,Y) so that g : N → Y is also a homotopy equivalence.
A map f : M → X between finite CW complexes is simple homotopic to a map
f′ : M′ → X if M′ is a finite CW complex, s : M′ → M is a simple homotopy
equivalenceandf◦sishomotopictof′. Ahomotopyequivalencef :M →X from
a finite CW complex is splittable along Y ⊂ X if f is simple homotopic to a map
which splits along Y ⊂X.
A codimension 1 pair (X,Y) is injective if X,Y are connected and π (Y) →
1
π (X)isinjective. LetX betheuniversalcoverofX. Asin§0.2letX¯ =X/π (Y),
1 1
sothat(X¯,Y)isacodimension1pairwithX¯ =X¯−∪ X¯+forconnectedsubspaces
Y
X¯−,X¯+ ⊂ X¯ with π (Xe¯) = π (X¯−) = π (X¯+) = π (Y). As usual, thereeare two
1 1 1 1
cases, according as to whether Y separates X or not:
(A) X −Y is disconnected, so
X = X ∪ X
1 Y 2
with X ,X connected. By the Seifert-van Kampen theorem
1 2
π (X) = π (X )∗ π (X )
1 1 1 π1(Y) 1 2
is the amalgamated free product, with π (Y) → π (X ), π (Y) → π (X )
1 1 1 1 1 2
injective. The labeling is chosen such that
X¯ = X /π (Y)⊂X¯− , X¯ = X /π (Y)⊂X¯+ , X¯ ∩X¯ =Y .
1 1 1 2 2 1 1 2
(B) X −Y eis connected, so e
X = X /{y ∼ty|y ∈Y}
1
for a connected space X (a deformation retract of X−Y) which contains
1
two disjoint copies Y,tY ⊂X of Y. We shall only consider the case when
1
π (Y)→π (X ), π (tY)→π (X ) are isomorphisms, so that
1 1 1 1 1 1
π (X) = π (Y)⋊ Z
1 1 α
ALGEBRAIC K-THEORY OVER D∞: AN ALGEBRAIC APPROACH 7
for an automorphism α : π (Y) → π (Y) and X¯ is an infinite cyclic cover
1 1
of X with a generating covering translation t : X¯ → X¯. The labeling is
chosen such that
X¯ = X /π (Y)⊂X¯− , tX¯ ⊂X¯+ , X¯ ∩tX¯ =Y .
1 1 1 1 1 1
In both cases (X¯,Ye) is an injective codimension 1 pair of type (A).
The kernel Z[π (X)]-modules of a map f : M → X are the relative homology
1
Z[π (X)]-modules
1
K (M) = H (f :M →X)
r r+1
withX theuniversalcoverofX,M =f∗X theepufllbackecoverofM,andf :M →X
a π (X)-equivariant lift of f. For a map of injective codimension 1 CW pairs
1 e f e e f e
(f,g):(M,N)→(X,Y) the kernel Z[π (Y)]-modules fit into an exact sequence
1
···→K (N)→K (M¯)→K (M¯+,N)⊕K (M¯−,N)→K (N)→... .
r r r r r−1
If f is a homotopy equivalence and g : π (N) → π (Y) is an isomorphism, then
∗ 1 1
g is a homotopy equivalence if and only if K (N) = 0, which occurs if and only if
∗
K (M¯+,N)=K (M¯−,N)=0. In particular, if f is of type (A) and split, then f
∗ ∗
is semi-split.
Theorem 0.7 (Waldhausen [Wal69] for type (A), Farrell–Hsiang [FH73] for type
(B)). Let(X,Y)beaninjective, codimension 1,finiteCW pair. Supposef′ :M′ →
X is a homotopy equivalence from a finite CW complex.
(i) The homotopy equivalence f′ is simple homotopic to a map of pairs (f,g) :
(M,N) → (X,Y) such that g : π (N) → π (Y) is an isomorphism and for some
∗ 1 1
n>2 we have
K (N) = 0 for r 6=n .
r
Moreover, the Z[π (Y)]-modules K (M¯±,N) are f.g. projective, and the direct
1 n+1
sum
K (N) = K (M¯−,N)⊕K (M¯+,N)
n n+1 n+1
is stably f.g. free. Hence the projective classes are complementary:
[K (M¯−,N)] = −[K (M¯+,N)]∈K (Z[π (Y)]) .
n+1 n+1 0 1
(ii) In the separating case (A) there is defined an exactesequence
···−→Wh(π (X ))⊕Wh(π (X ))−→Wh(π (X))
1 1 1 2 1
−→K (Z[π (Y)])⊕Nil (Z[π (Y)];B ,B )−→···
0 1 0 1 1 2
where e f
B = Z[π (X )−π (Y)] , B = Z[π (X )−π (Y)] .
1 1 1 1 2 1 2 1
The Whitehead torsion τ(f)∈Wh(π (X)) has image
1
[τ(f)] = ([K (M¯−,N)],[K (M¯−,N),K (M¯+,N),ρ ,ρ ])
n+1 n+1 n+1 1 2
where
ρ1 : Kn+1(M¯−,N)−→Kn+1(M¯−,M¯1) = B1⊗Z[π1(Y)]Kn+1(M¯+,N) ,
ρ2 : Kn+1(M¯+,N)−→Kn+1(M¯+,M¯2) = B2⊗Z[π1(Y)]Kn+1(M¯−,N) .
8 J.DAVIS,Q.KHAN,ANDA.RANICKI
The homotopy equivalence f is splittable along Y if and only if [τ(f)]=0.
(iii) In the non-separating case (B) there is defined an exact sequence
1−α
···−→Wh(π (Y)) // Wh(π (Y))−→Wh(π (X))
1 1 1
−→K (Z[π (Y)])⊕Nil (Z[π (Y)],α)⊕Nil (Z[π (Y)],α−1)−→··· .
0 1 0 1 0 1
The Whiteheadetorsion τ(f)∈Wfh(π1(X)) has imagfe
[τ(f)] = ([K (M¯+,N)],[K (M¯+,N),ρ ],[K (M¯−,N),ρ ])
n+1 n+1 1 n+1 2
where
ρ : K (M¯−,N)−→K (M¯−,M¯ ) = t−1K (M¯−,N) ,
1 n+1 n+1 1 n+1
ρ : K (M¯+,N)−→K (M¯+,tM¯ ) = tK (M¯+,N) .
2 n+1 n+1 1 n+1
The homotopy equivalence f is splittable along Y if and only if [τ(f)]=0.
Proof of Theorem 0.4 (outline). The proof of [Wal69, Theorem 0.7(i)] was based
on a one-one correspondence between the elementary operations in the algebraic
K-theory of the nilpotent categories and the elementary operations (‘surgeries’
or cell-exchanges) for maps of injective codimension 1 pairs. The proof of our
Theorem 0.5 shows that there is no algebraic obstruction to making a homotopy
equivalence semi-split by elementary operations, and hence there is no geometric
obstruction. (cid:3)
1. Higher Nil-groups
In this section, we shall prove Theorem 0.5 for non-negative degrees.
Quillen[Qui73]definedtheK-theoryspaceKE :=ΩBQ(E)ofanexactcategory
E. The space BQ(E) is the geometric realization of the simplicial set N Q(E),
•
which is the nerve of a certain category Q(E) associated to E. The algebraic K-
groups of E are defined for ∗∈Z
K (E) := π (KE)
∗ ∗
using a nonconnective delooping for ∗6−1. In particular, the algebraic K-groups
of a ring R are the algebraic K-groups
K (R) := K (Proj(R))
∗ ∗
of the exact category Proj(R) of f.g. projective R-modules. The Nil-categories
defined in the Introduction all have the structure of exact categories.
Theorem 1.1. Let R be a ring. Let B ,B be R-bimodules. Suppose that I is a
1 2
filtered category and B = colim Bα is a direct limit of R-bimodules Bα, each
2 α∈I 2 2
of which is a f. g. projective left R-module. Let i be the exact functor of exact
categories of projective nil-objects:
i:Nil(R;B ⊗ B )−→Nil(R;B ,B ); (Q,σ :Q→B BαQ)7−→(Q,BαQ,σ,1).
1 R 2 1 2 1 2 2
Then the induced map of K-theory spaces is a homotopy equivalence:
K¯i:KNil(R;B ⊗ B )−→KNil(R;B ,B )/(0×K(R)).
1 R 2 1 2
In particular, for all n∈N, there is an induced isomorphism of abelian groups:
i :Nil (R;B ⊗ B )⊕K (R)−→Nil (R;B ,B ).
∗ n 1 R 2 n n 1 2
ALGEBRAIC K-THEORY OVER D∞: AN ALGEBRAIC APPROACH 9
The exact functor
j :Nil(R;B ,B )−→Nil(R;B ⊗ B ) ; (P ,P ,ρ ,ρ )7−→(P ,ρ ◦ρ )
1 2 1 R 2 1 2 1 2 1 2 1
satisfies j◦i=1.
Proof. It is straightforward to show that tensor product commutes with colimits
over a category. Moreover, for any object x = (P ,P ,ρ : P → B P ,ρ : P →
1 2 1 1 1 2 2 2
B P ),sinceP isfinitelygenerated,thereexistsα∈I suchthatρ factorsthrough
2 1 2 2
a map P →BαP , and similarly for short exactsequences of nil-objects. We thus
2 2 1
obtain induced isomorphisms of exact categories:
colimNil(R;B ⊗ Bα) −→ Nil(R;B ⊗ B )
1 R 2 1 R 2
α∈I
colimNil(R;B ,Bα) −→ Nil(R;B ,B ).
1 2 1 2
α∈I
So, by Quillen’s colimit observation [Qui73, §2, Equation (9), page 20], we obtain
induced weak homotopy equivalences of K-theory spaces:
colimKNil(R;B ⊗ Bα) −→ KNil(R;B ⊗ B )
1 R 2 1 R 2
α∈I
colimKNil(R;B ,Bα) −→ KNil(R;B ,B ).
1 2 1 2
α∈I
Therefore,foreachα∈I,itsufficestoshowthattherestrictionK¯iα isahomotopy
equivalence.
Our setting is the exact category Nil(R;B ,Bα). By assumption, we may con-
1 2
sider objects
x := (P ,P ,ρ ,ρ )
1 2 1 2
0
x′ := (P ,BαP ⊕P , , 1 ρ )
1 2 1 2 (cid:18)ρ1(cid:19) 2
(cid:0) (cid:1)
x′′ := (P ,BαP ,ρ ◦ρ ,1)
1 2 1 2 1
a := (0,P ,0,0)
2
a′ := (0,BαP ,0,0).
2 1
Define morphisms
0
f := (1, ):x−→x′
(cid:18)1(cid:19)
f′ := (1, 1 ρ ):x′ −→x′′
2
(cid:0)−ρ (cid:1)
g := (0, 2 ):a−→x′
(cid:18) 1 (cid:19)
g′ := (0, 1 0 ):x′ −→a′
h := (0,ρ(cid:0) ):a(cid:1)−→a′.
2
There are exact sequences
f g
 
0 1 g′ h
0 −−−−→ x⊕a −−−−−−→ x′⊕a −(cid:16)−−−−−→(cid:17) a′ −−−−→ 0
g f′
0 −−−−→ a −−−−→ x′ −−−−→ x′′ −−−−→ 0.
Define functors F′,F′′,G,G′ :Nil(R;B ,Bα)−→Nil(R;B ,Bα) by
1 2 1 2
F′(x)=x′, F′′(x)=x′′, G(x)=a, G′(x)=a′.
10 J.DAVIS,Q.KHAN,ANDA.RANICKI
Thus we have two exact sequences of exact functors
0 −−−−→ 1⊕G −−−−→ F′⊕G −−−−→ G′ −−−−→ 0
0 −−−−→ G −−−−→ F′ −−−−→ F′′ −−−−→ 0.
Recall j◦i = 1, and note i◦j = F′′. By Quillen’s Additivity Theorem [Qui73, p.
98, Cor. 1], we obtain homotopies KF′ ≃1+KG′ and KF′ ≃KG+KF′′. Then
Ki◦Kj = KF′′ ≃1+(KG′−KG) ,
wherethesubtractionusestheloopspacestructure. ObserveG,G′ :Nil(R;B ,Bα)→
1 2
0×Proj(R). Therefore the functor iα induces a homotopy equivalence
K¯iα :KNil(R;B ⊗ Bα)−→KNil(R;B ,Bα)/(0×K(R)).
1 R 2 1 2
(cid:3)
Remark 1.2. TheproofofTheorem1.1isbestunderstoodintermsoffinitechain
complexes x=(P ,P ,ρ ,ρ ) in the category Nil(R;B ,B ), assuming that B is
1 2 1 2 1 2 2
a f. g. projective left R-module. Any such x represents a class
∞
[x] = (−1)r[(P ) ,(P ) ,ρ ,ρ ]∈Nil (R;B ,B ) .
1 r 2 r 1 2 0 1 2
Xr=0
Thekeyobservationisthatxdeterminesafinitechaincomplexx′ =(P′,P′,ρ′,ρ′)
1 2 1 2
in Nil(R;B ,B ) such that ρ′ :P′ →B ⊗ P is a chain equivalence and
1 2 2 2 2 R 1
[x] = [x′]∈Nil (R;B ,B ) . (∗)
0 1 2
Specifically, let P′ =P , P′ =M(ρ ), tfhe algebraic mapping cylinder of the chain
1 1 2 2
map ρ :P →B ⊗ P , and let
2 2 2 R 1
0
ρ′1 = 0 : P1′ =P1 −→ B1⊗RP2′ =M(1B1 ⊗ρ2) ,
ρ
1
ρ′ = 10 ρ : P′ = M(ρ ) −→ B ⊗ P ,
2 2 2 2 2 R 1
(cid:0) (cid:1)
so that P′/P = C(ρ ) is the algebraic mapping cone of ρ . Moreover, the proof
2 2 2 2
of (∗) is sufficiently functorial to establish not only that the following maps of the
reduced nilpotent class groups are inverse isomorphisms:
∼
=
i : Nil (R;B ⊗ B ) // Nil (R;B ,B ) ; (P,ρ)7→(P,B ⊗ P,ρ,1) ,
0 1 R 2 0 1 2 2 R
∼
j : Nfil (R;B ,B ) = // Nilf(R;B ⊗ B ) ; [x]7→[x′] ,
0 1 2 0 1 R 2
f f
but also that there exist isomorphisms of Nil for all higher dimensions n > 0, as
n
shown above. In order to prove equation (∗), note that x fits into the sequence
f
(1,u) (0,v)
0 // x //x′ //y // 0 (∗∗)