Algebraic K-theory of generalized free products and functors Nil
aa r X i v : . [ m a t h . K T ] J u l Algebraic K-theory of generalized free products and functors Nil.
Pierre Vogel Abstract.
In this paper, we extend Waldhausen’s results on algebraic K-theory ofgeneralized free products in a more general setting and we give some properties of theNil functors. As a consequence, we get new groups with trivial Whitehead groups.
Keywords:
Algebraic K-theory, functor Nil, Whitehead groups.
Mathematics Subject Classification (2020):
Introduction.
Quillen’s construction associates to any essentially small exact category A itsalgebraic K-theory which is an infinite loop space K ( A ) and this correspondence isa functor from the category of essentially small exact categories to the category Ω sp of infinite loop spaces.If A is the category P A of finitely generated projective right modules over somering A , one gets a functor A K ( A ) = K ( P A ) and this functor can be enriched intoa new functor K containing also the negative part of the algebraic K-theory. That is K is a functor from the category of rings to the category Ω sp of Ω-spectra and thenatural transformation K ( A ) → K ( A ) induces a homotopy equivalence from K ( A )to the 0-th term of K ( A ).By a left-flat bimodule we mean a pair ( A, S ) where A is a ring and S is an A -bimodule flat on the left. The left-flat bimodules form a category where a morphism( A, S ) −→ ( B, T ) is a ring homomorphism f : A −→ B together with a morphism of A -bimodules ϕ : S −→ f ∗ ( T ).For each left-flat bimodule ( A, S ) one has an exact category N il ( A, S ) where theobjects are the pairs (
M, f ) where M is an object in P A and f : M → M ⊗ A S is anilpotent morphism of right A -modules.The correspondence M ( M,
0) induces a morphism K ( A ) → K ( N il ( A, S )) andthis morphism has a retraction coming from the functor (
M, f ) M . Thus there isa functor N il from the category of left-flat bimodules to Ω sp which is unique up tohomotopy such that: K ( N il ( A, S )) ≃ K ( A ) × N il ( A, S ) Theorem 1.
There is a functor
N il from the category of left-flat bimodules to thecategory Ω sp of Ω -spectra and a natural transformation N il → N il such that thefollowing holds for every left-flat bimodule ( A, S ) : Universit´e de Paris, Institut de Math´ematiques de Jussieu-Paris Rive Gauche, Bˆatiment SophieGermain, Case 7012, 75205–Paris Cedex 13 France, Email: [email protected] the map N il ( A, S ) → N il ( A, S ) induces a homotopy equivalence from N il ( A, S ) to the 0-th term of N il ( A, S ) • if R is the tensor algebra of S , then there is a natural homotopy equivalence in Ω sp : K ( R ) ∼ −→ K ( A ) × Ω − ( N il ( A, S )) Moreover if A is regular coherent on the right, every spectrum N il ( A, S ) is con-tractible. Following a terminology of Waldhausen, a ring homomorphism α : A → B willbe called pure if it is split injective as an A -bimodule homomorphism. Theorem 2.
Let α : C −→ A and β : C −→ B be pure ring homomorphisms. Let R be the ring defined by the push-out diagram: C α −−−−→ A β y y B −−−−→ R and K ′ ( R ) be the Ω -spectrum defined by the homotopy fibration in Ω sp : K ( C ) f −→ K ( A ) × K ( B ) −→ K ′ ( R ) where f is the map K ( α ) × − K ( β ) .Suppose A and B are C -flat on the left. Then there exist a left-flat bimodule ( C × C, S ) and a homotopy equivalence in Ω sp : K ( R ) ∼ −→ K ′ ( R ) × Ω − ( N il ( C × C, S )) Theorem 3.
Let C and A be two rings and α and β be two pure ring homomorphismsfrom C to A . Let R be the ring generated by A and an invertible element t with theonly relations: ∀ c ∈ C, α ( c ) t = tβ ( c ) and K ′ ( R ) be the Ω -spectrum defined by the homotopy fibration: K ( C ) f −→ K ( A ) −→ K ′ ( R ) where f is the map K ( α ) − K ( β ) . uppose A is C -flat on the left via both α and β . Then there exist a left-flatbimodule ( C × C, S ) and a homotopy equivalence in Ω sp : K ( R ) ∼ −→ K ′ ( R ) × Ω − ( N il ( C × C, S ))The connective part of these theorems are generalization of the Waldhausen’sresults in [Wa1], in the sense that the condition free on the left in [Wa1] may bereplaced by the condition flat on the left and (in the polynomial extension case) thecondition on the right may be removed.Because of these results, the functor
N il detects in some sense the default ofexcision in algebraic K-theory. So it would be useful to know when the spectrum
N il ( A, S ) is contractible, especially if A is not regular coherent. Actually we havethe following result: Theorem 4.
Let A and B be two rings, S be an ( A, B ) -bimodule and T be a ( B, A ) -bimodule. Suppose S and T are flat on both sides. Using projections A × B −→ A and A × B −→ B , S and T may be considered as A × B -bimodules. Then we havenatural homotopy equivalences of spectra: N il ( A × B, S ⊕ T ) ∼ −→ N il ( A, S ⊗ B T ) N il ( A × B, S ⊕ T ) ∼ −→ N il ( B, T ⊗ A S )We have other results concerning the spectrum N il ( A, S ) when the bimodule S is a direct sum of bimodules: S = ⊕ i ∈ I S i .Let W ( I ) be the set of words in the set I . This set is the unitary monoid freelygenerated by I . Let CW ( I ) the set of cyclic words in I . The set CW ( I ) is thequotient of W ( I ) by the equivalence relation uv ∼ vu in W ( I ). A word u ∈ W ( I ) issaid to be reduced if we have the following: ∀ v ∈ W ( I ) , ∀ p > , u = v p The set of reduced words is denoted by W ( I ) and its image in CW ( I ) is denotedby CW ( I ). A subset X ⊂ W ( I ) is said to be admissible if the projection W ( I ) −→ CW ( I ) induces a bijection X ∼ −→ CW ( I ).For every u ∈ W ( I ), we have a bimodule S u defined by: u = 1 = ⇒ S u = Au = vi, with i ∈ I, = ⇒ S u = S v ⊗ A S i Theorem 5:
Let A be a ring and S i , i ∈ I , be a family of A -bimodules. Supposeeach bimodule S i is flat on both sides. Let X be an admissible set in W ( I ) . Thenwe have a homotopy equivalence of spectra: N il ( A, ⊕ i S i ) ∼ −→ ⊕ u ∈ X N il ( A, S u ) where ⊕ is the coproduct in the category of spectra. Moreover, using theorems 4 and 5, we can deduce this last result:
Theorem 6:
Let A be a ring, S be an A -bimodule and I be a set. For each i ∈ I , let A i be a right regular coherent ring, E i be an ( A, A i ) -bimodule and F i be an ( A i , A ) -bimodule. Suppose all these bimodules are flat on both sides. Then the inclusion: S ⊂ S ⊕ ⊕ i E i ⊗ A i F i induces a homotopy equivalence: N il ( A, S ) ∼ −→ N il ( A, S ⊕ ⊕ i E i ⊗ A i F i )The paper is organized as follows:In section 1, we construct many categories and functors in such a way they aredefined in each of the three cases: the general polynomial extension case, the gener-alized free product case and the generalized Laurent extension case. We prove alsomany algebraic properties of these categories and functors.In section 2, we apply these properties to algebraic K-theory and prove theorems1, 2 and 3.The section 3 is devoted to the proof of theorems 4, 5 and 6 about Nil functors.In the last section we apply all these theorems and get new results about White-head spectra. In particular we construct a class Cl bigger that Waldhausen’s classCl such that every group in Cl has trivial Whitehead groups.
1. The categories V and M V and their algebraic properties.
In order to simplify the notations, the following writing conventions will often beused: • Convention 1: if Φ : A → B is a functor, then for every morphism α in A , itsimage under Φ will be still denoted by α . So, if α : X → Y is a morphism, we havea morphism α : Φ( X ) → Φ( Y ). • Convention 2: if E is a right module over some ring A and F is a left moduleover the same ring, then the module E ⊗ A F will be simply denoted be EF . In the4ame spirit, if E is an A -bimodule, the tensor product E ⊗ A . . . ⊗ A E of n copies of E will be denoted by E n . Let A be a ring and S be an A -bimodule. Let M be a right A -module and f : M → M S be an A -linear map. So by iteration, we get for eachinteger n > a morphism f n : M → M S n . We say that f is nilpotent if everyelement in M is killed by some power f n of f . Let ( A, S ) be a left-flat bimodule. Let M be a right A -module and f : M → M S be an A -linear map. Then f is nilpotent if and only if there is afiltration of M : M ⊂ M ⊂ M ⊂ . . . ⊂ M by right A -submodules, such that: • M is the union of the M i ’s • for every i > , one has: f ( M i ) ⊂ M i − S . Proof:
If such a filtration exists, then f is clearly nilpotent.Suppose f is nilpotent. For every integer n ≥
0, denote by M n the kernel of f n : M −→ M S n . By construction we have an increasing sequence0 = M ⊂ M ⊂ M ⊂ M . . . and M is the union of the M i ’s.Since S is flat on the left, we have, for every n ≥
0, an isomorphism:Ker( f n : M S −→ M S n +1 ) ≃ Ker( f n : M −→ M S n ) S and then an equality: M n +1 = f − ( M n S ). The result follows. N il ( A, S ) and the space N il ( A, S ) . Let (
A, S ) be a left-flat bimodule. The pairs (
M, f ) where M is a right A -moduleand f : M → M S is a nilpotent morphism are the objects of a category denoted by N il ( A, S ) ∨ . Let A be the category of finitely generated projective right A -modules.The full subcategory of N il ( A, S ) ∨ generated by pairs ( M, f ) with M ∈ A will bedenoted by N il ( A, S ).If 0 −→ ( M, f ) −→ ( M ′ , f ′ ) −→ ( M ′′ , f ′′ ) −→ N il ( A, S ), wesay that this sequence is exact if the following diagram is commutative with exactlines: 0 −−−−→ M −−−−→ M ′ −−−−→ M ′′ −−−−→ f y f ′ y f ′′ y −−−−→ M S −−−−→ M ′ S −−−−→ M ′′ S −−−−→ N il ( A, S ) becomes an exact category in thesense of Quillen and its algebraic K-theory K ( N il ( A, S )) is a well defined infiniteloop space (see [Q]).Actually N il is a functor from the category of left-flat bimodules to the categoryof essentially small exact categories and exact functors.We have two exact functors M ( M,
0) from A to N il ( A, S ) and (
M, f ) M from N il ( A, S ) to A inducing two maps: K ( A ) F −→ K ( N il ( A, S )) G −→ K ( A )where G is a retraction of F . Denote by N il ( A, S ) the homotopy fiber of G . Then N il ( A, S ) is an infinite loop space and we have a decomposition: K ( N il ( A, S )) ≃ K ( A ) × N il ( A, S )Throughout this paper, we’ll consider many categories and functors and, in par-ticular, many exact categories and their abelianizations, where an abelianization ofan exact category is defined as follows:
Definition:
Let E be an exact category. We say that a category E ∨ is an abelian-ization of E if the following holds: • E ∨ is an abelian category • E is a fully exact subcategory of E ∨ i.e. E is a full subcategory of E ∨ and, forevery sequence S = (0 −→ X −→ Y −→ Z −→ in E , S is exact in E if and onlyif S is exact un E ∨ • E is stable in E ∨ under extension. Notice that the Gabriel-Quillen embedding theorem produces an abelianizationfor every essentially small exact category (see [TT], thm A.7.1 or [K], prop A.2).Following Waldhausen we consider three situations: the generalized free productcase, the generalized Laurent extension case and the generalized polynomial extensioncase.In case 1 (i.e. the generalized polynomial extension case), we have a ring C anda C -bimodule S which is flat on the left. In this case the ring R is the tensor algebraof S : R = C ⊕ S ⊕ S ⊕ S ⊕ . . . In case 2 (i.e. the generalized free product case) we have two pure morphisms ofrings α : C → A and β : C → B and we suppose that A and B are C -flat on the left.6e denote by R the ring defined by the cocartesian diagram: C α −−−−→ A β y y B −−−−→ R In case 3 (i.e the generalized Laurent extension case) we have two rings C and A and two pure morphisms α and β from C to A . We suppose that A is C -flat on theleft via both α and β and we denote by R the ring generated by A and an invertibleelement t with the only relations: ∀ c ∈ C, α ( c ) t = tβ ( c )So we have a morphism γ : A → R and γ ◦ α and γ ◦ β are conjugate.From now on we will consider α and γ as inclusions. So in each case A , B and R are C -bimodules.We denote by A , B , C and R the categories of finitely generated projective rightmodules over the rings A , B , C and R respectively. We set also: D = C in case 1, D = A × B in case 2 and D = A in case 3. This category is the category of finitelygenerated projective right modules over the ring C or A × B or A .The categories A , B , C , D and R are contained in the abelian categories A ∨ , B ∨ , C ∨ , D ∨ and R ∨ of right-modules over the corresponding rings and these cate-gories are abelianizations of A , B , C , D and R respectively. Notice that N il ( A, S ) ∨ is also an abelianization of N il ( A, S ).We denote by C the ring C in case 1 and C × C in the other cases and also by C ∨ the category of right C -modules and by C the subcategory of C ∨ generated byfinitely generated projective modules. The category C ∨ is also an abelianization of C . We will define the C -bimodule S and many categories and functors in order togive a common proof of theorems 1, 2 and 3 (at least for the connective part of it). s and σ and the bimodule S . Consider the case C = C × C . Let π and π be the two projections C × C → C .Consider a right C -module M and an integer i ∈ { , } . The ring C × C acts on M via π i and becomes a right C × C -module M i . This functor M M i from C ∨ to C ∨ has an adjoint functor (on both sides) from C ∨ to C ∨ denoted by E E i .The two functors M M and M M induce an equivalence of categories from C ∨ × C ∨ to C ∨ and the functors E E and E E induce an inverse of it.The two functors s i : M M i from C to C (and also from C ∨ to C ∨ ) are exactand it is the same for they adjoint functors σ i : E E i from C to C (and from C ∨ C ∨ ). So s , s and s = s ⊕ s are exact functors and their adjoint functors σ , σ and σ = σ ⊕ σ are also exact.In case 1, s and σ are defined to be identities. Therefore s and σ are well definedin all cases: s is an exact functor from C to C (and also from C ∨ to C ∨ ) and σ isan exact functor from C to C (and also from C ∨ to C ∨ ).Moreover, for every module E in C (or C ∨ ) the module σ ( E ) is nothing else butthe module E equipped with the C -action induced by the identity or the diagonalmap from C to C .We can do the same for left modules and we have functors M i M and E i E .In case of bimodules, we gets functors M i M j and E i E j . Using these notationswe have the following, for every right C × C -module E and left C × C -module F : E ⊗ C × C F = EF = E F ⊕ E F = ⊕ i E i i F In case 1, the C -bimodule S is already defined.Consider the case 2. Since α : C → A and β : C → B are pure morphisms, A and B have two complements A ′ and B ′ . These objects are C -bimodules and we have twodecompositions of C -bimodules: A = α ( C ) ⊕ A ′ and B = β ( C ) ⊕ B ′ . Then we definethe bimodule S by: S = A ′ S = B ′ S = S = 0Consider the case 3. Ring homomorphisms α and β induce two left C -actions on A and we get two ( C, A )-bimodules α A and β A . By doing the same on the right, weget four C -bimodules α A α , α A β , β A α and β A β . Moreover, since α and β are puremorphisms, we have two decompositions of C -bimodules: α A α = α ( C ) ⊕ A ′ β A β = β ( C ) ⊕ A ′′ Then we define the bimodule S by: S = A ′ S = A ′′ S = β A α S = α A β Then in the three cases S is a well defined C -bimodule. It is easy to see that S is flat on the left. D , M V and V and the functors T , F and b F . We have a functor T : D ∨ −→ R ∨ defined as follows: • in case 1: T ( E ) = E ⊗ C R = ER • in case 2: T ( E A , E B ) = E A ⊗ A R ⊕ E B ⊗ B R = E A R ⊕ E B R • in case 3: T ( E ) = E ⊗ A R = ER It is easy to check that T is an exact functor sending D to R .Let E be an object in D ∨ , M be an object in C ∨ and ϕ : T ( E ) −→ M R be amorphism in R ∨ . We say that ϕ is admissible if the following holds:8 ϕ ( E ) ⊂ M ⊕ M S in case 1 • ϕ ( E A ) ⊂ M A and ϕ ( E B ) ⊂ M B in case 2 (with: E = ( E A , E B )) • ϕ ( E ) ⊂ M A ⊕ M tA ⊂ M R in case 3.The set of admissible morphisms ϕ : T ( E ) −→ M R will be denote by F ( E, M ).Following Waldhausen, we define a splitting diagram as a triple X = ( E, M, ϕ )with: E ∈ D ∨ , M ∈ C ∨ and ϕ ∈ F ( E, M ).The splitting diagram (
E, M, ϕ ) is called a Mayer Vietoris presentation (resp.a splitting module) if ϕ is surjective (resp. bijective). The splitting modules, theMayer Vietoris presentations and the splitting diagrams define three categories V ∨ ⊂ M V ⊂ S ∨ . Moreover categories V ∨ and S ∨ are abelian.If we replace D ∨ and C ∨ by D and C , we get three subcategories V ⊂ M V ⊂ S .The correspondences ( E, M, ϕ ) E and ( E, M, ϕ ) M define two functorsΦ : S −→ D and Φ : S −→ C (and also from S ∨ to D ∨ and from S ∨ to C ∨ ).We have an extra functor Φ sending ( E, M, ϕ ) to the kernel of ϕ .Consider a sequence S = (0 −→ X −→ Y −→ Z −→
0) in V or in S . We saythat this sequence is exact if it is sent to an exact sequence under Φ and Φ . If S is a sequence in M V , we say that S is exact if it is sent to an exact sequence underΦ , Φ and Φ .With these exact sequences, V , M V and S become exact categories and theinclusions V ⊂ M V ⊂ S are exact functors. Moreover Φ : M V −→ R is an exactfunctor. In some sense V is the kernel of the functor Φ : M V −→ R .Since M V is not an abelian category, it will be useful to construct an abeliancategory
M V ∨ containing M V . The category
M V is equivalent to the category oftuples (
U, E, M, µ, ϕ ) where (
U, E, M ) is an object of R × D × C and µ : U −→ T ( E )and ϕ : T ( E ) −→ M R are morphisms in R such that ϕ is admissible and thesequence: 0 −→ U µ −→ T ( E ) ϕ −→ M R −→ M V ∨ is defined as the category of tuples ( U, E, M, µ, ϕ )where (
U, E, M ) is an object of R ∨ × D ∨ × C ∨ and µ : U −→ T ( E ) and ϕ : T ( E ) −→ M R are morphisms in R ∨ such that ϕ is admissible and ϕµ = 0. It is easy to seethat M V ∨ is an abelian category and the inclusions of exact categories V ⊂ M V and
M V ⊂ S extend to functors V ∨ −→ M V ∨ and M V ∨ −→ S ∨ . Moreover thecategories V ∨ , M V ∨ and S ∨ are abelianizations of V , M V and S respectively.We have a functor F from C ∨ to D ∨ defined as follows: • in case 1, F is the identity • in case 2, F ( M ) = ( σ ( M ) A, σ ( M ) B ) ∈ D ∨ • in case 3, F ( M ) = σ ( M ) α A ⊕ σ ( M ) β A ∈ D ∨ where α A and β A are the module A equipped with the ( C, A )-bimodule structureinduced by α and β respectively.This functor F is exact and sends C to D . It has a right adjoint functor b F from D ∨ to C ∨ and we have: • in case 1, b F is the identity • in case 2, b F ( E A , E B ) = s ( E A ) ⊕ s ( E B )9 in case 3, b F ( E ) = s ( E α ) ⊕ s ( E β )where E α and E β are the module E equipped wihe the right C -module structureinduced by α and β respectively.In case: C = C × C , we have another functor M f M from C ∨ (or C ) to itselfdefined by: M = ( M ′ , M ′′ ) = ⇒ f M = ( M ′′ , M ′ ) Suppose C = C × C . Then for every right C × C -module M , we havenatural isomorphisms: sσ ( M ) ≃ M ⊕ f M b F F ( M ) ≃ M ⊕ f M S
Moreover the induced projection sσ ( M ) −→ M and the induced injections M −→ sσ ( M ) and M −→ b F F ( M ) are adjoint to identities. Proof:
Let M = ( M ′ , M ′′ ) be a module in C = C × C . We have: s ( σ ( M )) = s ( M ′ ⊕ M ′′ ) = s ( M ′ ⊕ M ′′ ) ⊕ s ( M ′ ⊕ M ′′ ) ≃ M ⊕ ( s ( M ′′ ) ⊕ s ( M ′ )) ≃ M ⊕ f M In case 2, we have: b F F ( M ) = b F ( M ′ A, M ′′ B ) = ( M ′ ( C ⊕ A ′ ) , M ′′ ( C ⊕ B ′ )) ≃ ( M ′ , M ′′ ) ⊕ ( M ′ S , M ′′ S ) ≃ M ⊕ f M S
In the case 3 we have: b F F ( M ) = b F ( M ′ α A ⊕ M ′′ β A ) = s ( M ′ α A α ⊕ M ′′ β A α ) ⊕ s ( M ′ α A β ⊕ M ′′ β A β )= s ( M ′ ⊕ M ′ S ⊕ M ′′ S ) ⊕ s ( M ′ S ⊕ M ′′ ⊕ M ′′ S ) ≃ M ⊕ s ( f M S ) ⊕ s ( f M S ) ≃ M ⊕ f M S and the result follows. M [ S ] and the transformations e , ε and τ . For every M ∈ C ∨ , we set: M [ S ] = M ⊕ M S ⊕ M S ⊕ . . . and M [ S ] is a right C -module. In case 1, M [ S ] is isomorphic to M R .We have a stabilization map
M S [ S ] −→ M [ S ] induced by the identities M SS i −→ M S i +1 . There exist natural transformations: e : T F ( P ) ∼ −→ σ ( P ) R : σ ( b F ( E )[ S ]) ∼ −→ T ( E ) τ : σ ( s ( M )[ S ]) −→ M R for all P ∈ C ∨ , E ∈ D ∨ and M ∈ C ∨ such that: • e is an isomorphism of R -modules • ε is an isomorphism of C -modules • τ is an epimorphism of C -modules and the following diagram is exact (i.e.cartesian and cocartesian): σs ( M ) −−−−→ σ ( s ( M )[ S ]) τ y y τ M −−−−→ M R where the horizontal maps are the canonical inclusions and τ is adjoint to the identityof s ( M ) . Proof:
In case 1, it is easy to see that e , ε , τ and τ can be chosen to be identities.In the other cases consider a module P ∈ C ∨ . So we have two C -modules M = P and N = P .In case 2, we have: T ( F ( P )) = T ( M A, N B ) =
M A ⊗ A R ⊕ N B ⊗ B R ≃ M R ⊕ N R ≃ ( M ⊕ N ) R = σ ( P ) R and we get the isomorphism e .In case 3, we have: T ( F ( P )) = T ( M α A ⊕ N β A ) ≃ M α R ⊕ N β R but the multiplication on the left by t induces a isomorphism of ( C, R )-bimodulesfrom β R to α R . Then we have: T ( F ( P )) ≃ M α R ⊕ N α R = σ ( P ) α R = σ ( P ) R and we get the isomorphism e .In order to construct the morphism τ , we need to give an explicit description of R as a C -bimodule.We set: α U α = A ′ β U β = B ′ α U β = β U α = 0in case 2 and: α U α = A ′ β U β = tA ′′ t − α U β = At − β U α = tA in case 3. 11or each sequence Σ = ( i , j , i , j , . . . , i n , j n ) in the set { α, β } , the ring structureon R induce a well defined morphism of C -bimodules:Φ(Σ) : i U j i U j . . . i n U j n −→ R and the sum of these morphisms is an epimorphism. On the other hand it iseasy to see that, for each ( i, j, k ) in { α, β } the image of Φ( i, j, j, k ) is containedin C ⊕ i U k . Therefore, if there is some k < n such that j k = i k +1 in the sequence Σ =( i , j , i , j , . . . , i n , j n ), every element of the image of Φ(Σ) is reducible. As a conse-quence the sum of the morphisms Φ(Σ), for each sequence Σ = ( i , j , i , j , . . . , i n , j n )such that j k = i k +1 for every k < n is still an epimorphism.Actually this sum is an isomorphism and we have a description of R as C -bimodule: R = C ⊕ M(cid:16) i U j i U j . . . i n U j n (cid:17) the sum being taken over all non empty sequences ( i , j , i , j , . . . , i n , j n ) in { α, β } such that j k = i k +1 for all k < n .This fact was proven in [Wa1], p. 140 (or in [C]) for the case 2 and in [Wa1], p.150 (with a suggestion of S. Cappell) for the case 3.Denote by f (resp. g ) the unique bijection from { , } to { α, β } such that f (1) = β (resp. g (1) = α ). Then, because of the definition of S , we have in case 2: ∀ i, j ∈ { , } , i S j = f ( i ) U g ( j ) In case 3, we check that the multiplication by t or 1 on the left and by t − or 1 onthe right induce for each i, j in { , } an isomorphism of C -bimodules i S j ∼ −→ f ( i ) U g ( j ) .Then in cases 2 and 3 we have an isomorphism of C -bimodules: R ≃ C ⊕ M(cid:16) i S j i S j . . . i n S j n (cid:17) the sum being taken over all non empty sequences ( i , j , i , j , . . . , i n , j n ) in { , } such that j k = i k +1 for all k < n . Hence we get an isomorphism of C -bimodules: R ≃ C ⊕ ⊕ i,j ( i S j ⊕ i ( S ) j ⊕ i ( S ) j ⊕ . . . ) ≃ C ⊕ ⊕ i,j,n> i ( S n ) j So we are able to define the morphism τ . If M is a right C -module we have: M R ≃ M ⊕ ⊕ i,j,n> M i ( S n ) j ≃ M ⊕ ⊕ j,n> ( s ( M ) S n ) j ≃ M ⊕ ⊕ n> σ ( s ( M ) S n )But we have: σ ( s ( M )[ S ]) = σs ( M ) ⊕ ⊕ n> σ ( s ( M ) S n )Then we define the morphism τ to be the identity on the direct sum of the σ ( s ( M ) S n )and the morphism τ : σs ( M ) → M induced by the adjunction on the first term σs ( M ).The last thing to do is to construct the isomorphism ε in cases 2 and 3.12n case 2 with E = ( E A , E B ), we have: T ( E ) = E A ⊗ A R ⊕ E B ⊗ B R ≃ E A ( C ⊕ B ′ ⊕ B ′ A ′ ⊕ B ′ A ′ B ′ ⊕ . . . ) ⊕ E B ( C ⊕ A ′ ⊕ A ′ B ′ ⊕ A ′ B ′ A ′ ⊕ . . . ) ≃ E A ⊕ E B ⊕ E A ( ⊕ j,n> ( S n ) j ) ⊕ E B ( ⊕ j,n> ( S n ) j ) ≃ σ ( b F ( E )) ⊕ ( ⊕ n> σ ( b F ( E ) S n )) ≃ σ ( b F ( E )[ S ])which give the isomorphism ε .In the last case we have: R ≃ C ⊕ M i,j,n> i ( S n ) j ≃ ( C ⊕ S )( C ⊕ M j,n> ( S n ) j ) ⊕ S ( C ⊕ M j,n> ( S n ) j ) ≃ A ( C ⊕ M j,n> ( S n ) j ) ⊕ At − ( C ⊕ M j,n> ( S n ) j ) ≃ A α ( C ⊕ M j,n> ( S n ) j ) ⊕ A β ( C ⊕ M j,n> ( S n ) j and we check that the isomorphism from R to this last module is an isomorphism of( A, C )-bimodules. Since E belongs to A ∨ , we have; T ( E ) ≃ E α ( C ⊕ M j,n> ( S n ) j ) ⊕ E β ( C ⊕ M j,n> ( S n ) j ) ≃ σ ( b F ( E )) ⊕ M n> σ ( b F ( E ) S n ) ≃ σ ( b F ( E )[ S ])which give the isomorphism ε in this last case. We have an explicit description of the morphism τ and the isomor-phism ε .Consider two modules M ∈ C ∨ and E ∈ D ∨ .In case 1, set: u = u for each u ∈ E .In case 2, the maps E A −→ E A R and E B −→ E B R induce a map σ ( E ) −→ T ( E )and in case 3, the maps E α −→ ER and E β −→ EtR −→ ER induce also a map σ ( E ) −→ T ( E ). Denote by v v this map.So we have a morphism v v from σ ( E ) to T ( E ) in the three cases.Denote also by s s the isomorphism i S j ∼ −→ f ( i ) U g ( j ) ⊂ R in cases 2 or 3 andthe identity S −→ S in case 1.With these notations, we have the following description of τ and ε :In case 1, for each integer n ≥
0, each u ∈ M , each v ∈ E and each sequence( s , s , . . . , s n ) in S we have: τ ( us s . . . s n ) = u s s . . . s n ∈ M Rε ( vs s . . . s n ) = v s s . . . s n ∈ T ( E )13n case 2 or 3, for each integer n ≥
0, each sequence ( i , i , . . . , i n ) in { , } , each u ∈ M , each v ∈ E i and each sequence ( s , s , . . . s n ), with s k ∈ i k − S i k , we have: τ ( us s . . . s n ) = u s s . . . s n ∈ M Rε ( vs s . . . s n ) = v s s . . . s n ∈ T ( E ) There is a natural transformation:
Λ : F ( E, M ) −→ Hom( b F ( E ) , s ( M ) ⊕ s ( M ) S ) for each ( E, M ) ∈ × D ∨ × C ∨ with the following properties: • Λ is injective in case 2 or 3 and bijective in case 1 • for each ( P, M ) ∈ C ∨ × C ∨ , if f is the map P −→ b F ( F ( P )) induced byadjunction, the composite morphism: F ( F ( P ) , M ) Λ −→ Hom( b F ( F ( P )) , s ( M ) ⊕ s ( M ) S ) f ∗ −→ Hom(
P, s ( M ) ⊕ s ( M ) S ) is bijective • for each ϕ ∈ F ( E, M ) , we have a commutative diagram: σ ( b F ( E )[ S ]) g −−−−→ σ ( s ( M )[ S ]) ε y y τ T ( E ) ϕ −−−−→ M R where g is the composite morphism: σ ( b F ( E )[ S ]) Λ( ϕ ) −→ σ (( s ( M ) ⊕ s ( M ) S )[ S ]) h −→ σ ( s ( M )[ S ]) and h the morphism induced by the identity s ( M )[ S ] −→ s ( M )[ S ] and the stabiliza-tion map s ( M ) S [ S ] −→ s ( M )[ S ] . Proof:
Consider the case 1. We have an isomorphism from Hom R ( ER, M R ) toHom C ( E, M R ) inducing an isomorphism F ( E, M ) ≃ Hom(
E, M ⊕ M S ) and we getthe isomorphism Λ : F ( E, M ) ∼ −→ Hom(
E, M ⊕ M S ) = Hom( b F ( E ) , s ( M ) ⊕ s ( M ) S ).Consider the case 2. We have E = ( E A , E B ) ∈ A × B and we get isomorphisms:Hom R ( T ( E ) , M R ) ≃ Hom R ( E A R ⊕ E B R, M R ) ≃ Hom A ( E A , M R ) ⊕ Hom B ( E B , M R )and then isomorphisms: F ( E, M ) ≃ Hom( E A , M A ) ⊕ Hom( E B , M B ) ≃ Hom( E, ( M A, M B )) = Hom(
E, F ( s ( M )))14onsider now the last case. We have:Hom R ( T ( E ) , M R ) ≃ Hom R ( ER, M R ) ≃ Hom A ( E, M R )and then: F ( E, M ) ≃ Hom(
E, M A ⊕ M tA ) ≃ Hom(
E, M α A ⊕ M β A ) ≃ Hom(
E, F ( s ( M )))Therefore, in case 2 and 3, we have an isomorphism:( ∗ ) F ( E, M ) ∼ −→ Hom(
E, F ( s ( M )))On the other hand, the morphism b F induces an injection:Hom( E, F ( s ( M ))) −→ Hom( b F ( E ) , b F ( F ( s ( M )))) ≃ Hom( b F ( E ) , s ( M ) ⊕ s ( M ) S )(see lemma 1.6) and we get the desired injectionΛ : F ( E, M ) −→ Hom( b F ( E ) , s ( M ) ⊕ s ( M ) S )Let ( P, M ) be an object in C ∨ × C ∨ . In case 1, the morphism P −→ b F ( F ( P ))is the identity and the composite map: F ( F ( P ) , M ) Λ −→ Hom( b F ( F ( P )) , s ( M ) ⊕ s ( M ) S ) f ∗ −→ Hom(
P, s ( M ) ⊕ s ( M ) S )is bijective.Consider the other cases. Let ϕ ∈ F ( F ( P ) , M ) be an admissible morphismand α : F ( P ) −→ F ( s ( M )) be the corresponding morphism (via the isomorphism( ∗ )). Let f : P −→ b F F ( P ) be the morphism adjoint to the identity of F ( P ). Byadjunction, the composite morphism P f −→ b F F ( P )) α −→ b F F ( s ( M )) is the morphismobtained from α by adjunction and we have a bijection Hom( F ( P ) , F ( s ( M ))) ≃ Hom( P, b F F ( s ( M )). Hence we have a bijection: F ( F ( P ) , M ) ≃ Hom( P, b F F ( s ( M )) ≃ Hom(
P, s ( M ) ⊕ s ( M ) S )which is nothing else but the map f ∗ ◦ Λ.Denote by ( D ) the diagram of the lemma.In case 1, ε and τ are identities and we have: g = ϕ . Hence ( D ) is commutative.Consider the other cases. Via the bijection F ( E, M ) ≃ Hom(
E, F ( s ( M ))), themorphism ϕ ∈ F ( E, M ) corresponds to a morphism e ϕ : E −→ F ( s ( M )) and wehave a diagram: σ ( b F ( E )[ S ]) e ϕ −−−−→ σ ( b F F ( s ( M ))[ S ]) g −−−−→ σ ( s ( M )[ S ]) ε y ε y τ y T ( E ) e ϕ −−−−→ T ( F ( s ( M ))) ϕ −−−−→ M R
15n this diagram, the square on the left is commutative by naturality and thesquare on the right ( D ) is the diagram ( D ) in the case: E = F ( s ( M )). Moreoverthe total square is the diagram ( D ). Hence, to prove the commutativity of ( D ) it isenough to prove that ( D ) is commutative.In ( D ), the morphism g is induced by the isomorphism b F F ( s ( M )) ∼ −→ s ( M ) ⊕ s ( M ) S , the identity s ( M )[ S ] −→ s ( M )[ S ] and the stabilization map s ( M ) S [ S ] −→ s ( M )[ S ].The morphism ϕ is the composite map: T ( F ( s ( M ))) ∼ −→ ( M A ⊕ M B ) R ≃ M R ⊕ M R + −→ M R in case 2 and the composite map: T ( F ( s ( M ))) ∼ −→ ( M A ⊕ M tA ) R ≃ M R ⊕ M R + −→ M R in case 3. Hence ϕ is the composite map: T ( F ( s ( M ))) e −→ ∼ σs ( M ) R a −→ M R where a : σs ( M ) −→ M is adjoint to the identity of s ( M ).Consider an element u ∈ b F F ( s ( M )), an integer n ≥
0, a sequence ( i , i , . . . , i n )in { , } and a sequence ( s , s , . . . , s n ) with s k ∈ i k − S i k . Denote by v the image of u under the isomorphism b F F ( s ( M )) ≃ s ( M ) ⊕ s ( M ) S . If v belongs to s ( M ) or to s ( M ) S i , we have, because of remark 1.9, the following: ϕ ε ( us s . . . s n ) = v s s . . . s n = τ g ( us s . . . s n )and ( D ) is therefore commutative. Φ : N il ( C , S ) ∨ −→ V ∨ and Ψ : S ∨ −→ N il ( C , S ) ∨ . Let H be a module in C ∨ . Because of lemma 1.10, we have an isomorphism: ζ : F ( F ( H ) , σ ( H )) ∼ −→ Hom(
H, sσ ( H ) ⊕ sσ ( H ) S )Therefore, for every morphism θ : H −→ HS , we have a unique morphism ϕ θ in F ( F ( H ) , σ ( H ) such that ζ ( ϕ θ ) is the composite morphism: H − θ −→ H ⊕ HS i −→ sσH ⊕ sσHS where i : H −→ sσH is adjoint to the identity. Thus Φ( H, θ ) = ( F ( H ) , σ ( H ) , ϕ θ ) isa well defined split diagram in S ∨ . The correspondence above induces two equivalences of categories
Φ : N il ( C , S ) ∼ −→ V and Φ : N il ( C , S ) ∨ ∼ −→ V ∨ . oreover there is a functor Ψ : S ∨ −→ N il ( C , S ) ∨ and a morphism of functors π from I ΦΨ to the identity, where I : V ∨ −→ S ∨ is the inclusion, such that thefollowing holds for every X ∈ S ∨ (with: Ψ( X ) = ( H, θ ) ): • we have a natural exact sequence in C ∨ : −→ Φ ( X ) −→ σ ( H ) π −→ Φ ( X ) • a splitting diagram X belongs to M V (resp. to V ∨ ) if and only if the morphism π : ΦΨ( X ) −→ X is an epimorphism (resp. an isomorphism). Remark:
Actually, everything works without any flatness condition from subsection1.4 to 1.11. But this condition is strongly needed for lemma 1.12, essentially forconstructing the functor Ψ.
Proof of lemma 1.12:
Let (
H, θ ) be an object of N il ( C , S ) ∨ . If θ = 0, themorphism ϕ : T ( F ( H ) −→ σ ( H ) R is nothing else but the isomorphism e defined inlemma 1.8. Therefore Φ( H, θ θ ) belongs to V ∨ in this case.Suppose θ is nilpotent. Then, because of lemma 1.2, there is a filtration 0 = H ⊂ H ⊂ H ⊂ . . . of H such that H is the union of the H i ’s and, for all i > θ ( H i ) iscontained in H i − S .On the other hand it is easy to see that the functor Φ is exact. So we get afiltration: 0 = Φ( H , θ ) ⊂ Φ( H , θ ) ⊂ Φ( H , θ ) ⊂ . . . ⊂ Φ( H, θ )Because each Φ( H i /H i − , θ ) = Φ( H i /H i − ,
0) belongs to V ∨ , each Φ( H i , θ ) is also in V ∨ and then Φ( H, θ ) is a splitting module.Therefore Φ is a functor from N il ( C , S ) ∨ to V ∨ and also from N il ( C , S ) to V .Let X = ( E, M, ϕ ) be a splitting diagram with E ∈ D ∨ and M ∈ C ∨ . Denote by P the module b F ( E ) ∈ C ∨ . By composing the morphism ϕε : σ ( P [ S ]) −→ M R withthe identity P [ S ] = σ ( P [ S ]), we get a Z -linear map f : P [ S ] −→ M R . Denote by H the Z -submodule f − ( M ) and by i the inclusion map H −→ P [ S ].Because of lemma 1.10, ϕ is determined by a morphism ψ = Λ( ϕ ) from P = b F ( E )to s ( M ) ⊕ s ( M ) S . Then we have two morphisms λ : P −→ s ( M ) and γ : P −→ s ( M ) S such that: ψ = λ − γ . Moreover the composite map P [ S ] = σ ( P [ S ]) ϕε −→ M R is equal to : τ ( λ − γ ).We have: u ∈ H ⇐⇒ ϕε ( u ) ∈ M ⇐⇒ τ ( λ − γ )( u ) ∈ M ⇐⇒ ∀ k ≥ , τ ( λ ( u k +1 ) − γu k )) = 0But the morphism τ : σ ( s ( M ) S k ) −→ M R is injective for all k > u ∈ H ⇐⇒ ∀ k ≥ , λ ( u k +1 ) = γ ( u k )17ince λ and γ are morphisms in C ∨ , H is a C -module and we have an exactsequence in C ∨ : 0 −→ H i −→ P [ S ] δ −→ N [ S ]where N is the module s ( M ) S and δ is the morphism sending u = u + u + u + . . . (with u k ∈ P S k for every k ≥
0) to: δ ( u + u + u + . . . ) = X k ≥ ( λ ( u k +1 ) − γ ( u k ))If U is a module in C ∨ , we have a morphism θ U : U [ S ] −→ U [ S ] S sending u + u + u + . . . ∈ U [ S ] (with u k ∈ P S k for every k ≥
0) to: θ U ( u + u + u + . . . ) = u + u + u + . . . Notice that θ kU sends u + u + u + . . . ∈ U [ S ] to u k + u k +1 + . . . and θ U is nilpotent.Hence ( U [ S ] , θ U ) belongs to N il ( C , S ) ∨ .Consider the following diagram:( D ) P [ S ] δ −−−−→ N [ S ] θ P y θ N y P [ S ] S δ −−−−→ N [ S ] S Let n ≥ u be an element in P and s , s , . . . , s n be elements in S .We have the following: n = 0 = ⇒ δθ P ( u ) = θ N δ ( u ) = 0 n = 1 = ⇒ δθ P ( us ) = θ N δ ( us ) = − γ ( u ) s n > ⇒ δθ P ( us . . . s n ) = θ N δ ( us . . . s n ) = λ ( u ) s . . . s n − γ ( u ) s . . . s n and the diagram ( D ) is commutative.Since S is flat on the left, N il ( C , S ) ∨ is an abelian category and there is aunique nilpotent morphism θ : H −→ HS such that the following sequence is exactin N il ( C , S ) ∨ : 0 −→ ( H, θ ) i −→ ( P [ S ] , θ P ) δ −→ ( N [ S ] , θ N )Therefore Ψ( X ) = ( H, θ ) is a well defined object in N il ( C , S ) ∨ and we get thedesired functor Ψ : S ∨ −→ N il ( C , S ) ∨ .Consider the splitting diagram ΦΨ( X ) = ( F ( H ) , σ ( H ) , ϕ θ ). The morphism ϕ θ corresponds to the composite morphism H − θ −→ H ⊕ HS −→ sσ ( H ) ⊕ sσ ( H ) S . We18ave to construct a morphism π : ΦΨ( X ) −→ X in S ∨ . This morphism is given bytwo morphisms π : F ( H ) −→ E and π : σ ( H ) −→ M .For each u ∈ H we set: π ′ ( u ) = u , with i ( u ) = u + u + . . . and u k ∈ P S k forall k . So we have two morphisms π ′ : H −→ b F ( E ) and λπ ′ : H −→ s ( M ) and, byadjunction, two morphisms π : F ( H ) −→ E and π : σ ( H ) −→ M .For u ∈ H , with: i ( u ) = u + u + . . . , we have: ψπ ′ ( u ) = ψ ( u ) = λ ( u ) − γ ( u ) = λ ( u ) − λ ( u ) = λπ ′ ( u )) − λπ ′ θ ( u ) = λπ ′ (1 − θ )( u )and the following diagram is commutative: H ψ −−−−→ sσ ( H ) ⊕ sσ ( H ) S π ′ y π y P ψ −−−−→ s ( M ) ⊕ s ( M ) S Therefore the two morphisms π : F ( H ) −→ E and π : σ ( H ) −→ M induce a welldefined morphism π : ΦΨ( X ) −→ X and we get a commutative diagram:( D X ) T ( F ( H )) ϕ θ −−−−→ ∼ σ ( H ) R π y π y T ( E ) ϕ −−−−→ M R
We have an exact sequence in R ∨ :0 −→ Φ ( X ) µ −→ T ( E ) ϕ −→ M R and then an exact sequence in C ∨ :0 −→ Φ ( X ) ε − µ −→ σ ( P [ S ]) ϕε −→ M R
Therefore we have a commutative diagram in C ∨ with exact lines:0 −−−−→ Φ ( X ) −−−−→ σ ( H ) π −−−−→ M = y εi y j y −−−−→ Φ ( X ) µ −−−−→ T ( E ) ϕ −−−−→ M R j : M −→ M R is the inclusion, and the top line of this diagram is the desiredexact sequence.We have the following equivalences: X ∈ M V ⇐⇒ the morphism ϕ : T ( E ) −→ M R is surjective ⇐⇒ the image of ϕ contains M ⇐⇒ the image of σ ( P [ S ]) −→ M R contains M ⇐⇒ σ ( H ) π ′ −→ σ ( P ) λ ′ −→ M is surjectivewhere λ ′ is adjoint to λ . Therefore X belongs to M V if and only if the morphism π : σ ( H ) −→ M is surjective.Suppose X belongs to M V . Let E ′ be the image of π : F ( H ) −→ E . Becauseof the diagram above, T ( E ′ ) contains the image of µ : Φ ( X ) −→ T ( E ) and theimage of εi : σ ( H ) −→ T ( E ). But T ( E ′ ) is a R -submodule of T ( E ) and ϕ ( T ( E ′ ))is a R -submodule of M R containing M . Therefore ϕ : T ( E ′ ) −→ M R is surjectiveand T ( E ′ ) contains the kernel Φ ( X ) of ϕ . Hence we have: T ( E ′ ) = T ( E ) and then: E ′ = E .Consequently, if X belongs to M V , π : σ ( H ) −→ M and π : F ( H ) −→ E are surjective and π : ΦΨ( X ) −→ X is an epimorphism (in S ∨ ). Conversely, if π : ΦΨ( X ) −→ X is an epimorphism, π : σ ( H ) −→ M is surjective and X belongsto M V .Suppose X is in V ∨ . Then π and π are epimorphisms. Because of the exactsequence: 0 −→ Φ ( X ) −→ σ ( H ) π −→ M the morphism π : σ ( H ) −→ M is an isomorphism. Therefore in the diagram ( D X ), π and ϕ are isomorphisms and π : ΦΨ( X ) −→ X is an isomorphism too. As aconsequence, the functor Φ induces two equivalences of categories N il ( C , S ) −→ V and N il ( C , S ) ∨ −→ V ∨ .Conversely, if π is an isomorphism, π and π are isomorphisms and ϕ is anisomorphism too. Therefore X belongs to V ∨ . Let ( A, S ) be a left-flat bimodule and X = ( M, θ ) be an object in N il ( A, S ) ∨ . Let V be a finitely generated projective right A -module and f : V −→ M be a morphism. Then there exist an object Y = ( M ′ , θ ′ ) ∈ N il ( A, S ) , a split injectivemorphism f ′ : V −→ M ′ and a morphism g : Y −→ X making the following diagram ommutative: V f ′ −−−−→ M ′ = y g y V f −−−−→ M Proof:
Denote by A ∨ the category of right A -modules and A ⊂ A ∨ the categoryof finitely generated projective modules in A ∨ . Because of lemme 1.2 there is afiltration: 0 = M ⊂ M ⊂ M ⊂ . . . of M by A -modules such that M is the union of the M i ’s and θ ( M i ) ⊂ M i − S forevery i >
0. Since V is finitely generated, there is an integer n > M n contains the image of f : V −→ M .Then we’ll construct modules F i ∈ A , morphisms h i : F i −→ M i for i = 0 , , . . . , n and morphisms θ ′ : F i −→ F i − S for i = 1 , , . . . , n such that: • F n = V , F = 0 and h n = f • for each i = 1 , , . . . , n , the following diagram is commutative:( D i ) F i h i −−−−→ M iθ ′ y θ y F i − S h i − −−−−→ M i − S Let p be an integer with 0 ≤ p ≤ n . Denote by H ( p ) the following property: • There exist modules F i ∈ A and morphisms h i : F i −→ E i for i = p, . . . , n andalso morphisms θ ′ : F i −→ F i − S for i = p + 1 , . . . , n such that F n = V , h n = f , thediagram ( D i ) is commutative for i = p + 1 , . . . , n and F = 0 (if p = 0).This property is clearly true if p = n . Suppose H ( p ) is true with p > λ : F p −→ M p θ −→ M p − S . If p = 1, this morphismis trivial and we set: F = 0. Therefore the property H ( p −
1) = H (0) is true.Consider the case p >
1. Since F p is finitely generated, M p − contains a finitelygenerated submodule M ′ such that: λ ( F p ) ⊂ M ′ S . Let F p − be a module in A and µ : F p − −→ M ′ be an epimorphism. Since F p is projective the morphism F p −→ M ′ S F p − S and we have a commutative diagram: F p θ ′ −−−−→ F p − S = y µ y F p λ −−−−→ M ′ S So we define the morphism h p − as the composite map: F p − µ −→ M ′ ⊂ M p − and wehave the property H ( p − H (0) and all the data are constructed.Then we set: M ′ = F ⊕ F ⊕ . . . ⊕ F n . The morphisms h i induce a morphism g : M ′ −→ M and the morphisms θ ′ : F i −→ F i − S induce a morphism θ ′ : M ′ −→ M ′ S . The lemma is now easy to check.Denote by R ′ the full subcategory of R generated by modules on the form U = V R with V ∈ C . This category is exact and cofinal in R , that is, for each module M ∈ R ,there is a module M ′ ∈ R such that M ⊕ M ′ belongs to R ′ . Let X be a split diagram in S ∨ , V be a module in R ′ and f : V −→ Φ ( X ) be a morphism in R ∨ . Then there exist an object Y ∈ M V , a morphism g : Y −→ X in S ∨ and an isomorphism ε : V ∼ −→ Φ ( Y ) such that the followingdiagram is commutative: V ε −−−−→ ∼ Φ ( Y ) = y g y V f −−−−→ Φ ( X ) Moreover, if X belongs to M V , the morphism g can be chosen to be an epimor-phism in S ∨ . Proof:
The split diagram X is a triple ( E, M, ϕ ) with E ∈ D ∨ and M ∈ C ∨ . Since V belongs to R ′ , there is a module W ∈ C such that: V = W R and we get a morphism f ′ : W −→ Φ ( X ) in C ∨ .Denote by K = ( H, θ ) the object Ψ( X ) ∈ N il ( C , S ) ∨ . Because of lemma 1.12,we have an exact sequence:0 −→ Φ ( X ) µ −→ σ ( H ) π −→ M µf ′ : W −→ σ ( H ) has an adjoint λ : s ( W ) −→ H . Because oflemma 1.13, there are an object K ′ = ( H ′ , θ ′ ) ∈ N il ( C , S ), a morphism h : K ′ −→ K and a split injective morphism λ ′ : s ( W ) −→ H ′ making the following diagramcommutative: s ( W ) λ ′ −−−−→ H ′ = y h y s ( W ) λ −−−−→ H The morphism e λ ′ : W −→ σ ( H ′ ) adjoint to λ ′ is still split injective and we get acommutative diagram with exact lines:0 −−−−→ W e λ ′ −−−−→ σ ( H ′ ) π ′ −−−−→ M ′ −−−−→ f ′ y h y y −−−−→ Φ ( X ) µ −−−−→ σ ( H ) π −−−−→ M where M ′ is the cokernel of e λ ′ .We have now an object Y ′ = Φ( K ′ ) = ( F ( H ′ ) , σ ( H ′ ) , ϕ θ ” ) ∈ V , an object Y =( F ( H ′ ) , M ′ , π ′ ϕ θ ′ ) ∈ M V and an epimorphism u : Y ′ −→ Y . But the morphismΦ( K ′ ) −→ Φ( K ) π −→ X vanishes on the kernel of u and factorizes by a morphism g : Y −→ X . So we have morphisms Y ′ −→ Y −→ X inducing a commutativediagram with exact lines:0 −−−−→ −−−−→ T ( F ( H ′ )) −−−−→ σ ( H ′ ) R −−−−→ y y π ′ y −−−−→ W R −−−−→ T ( F ( H ′ )) −−−−→ M ′ R −−−−→ f y y y −−−−→ Φ ( X ) −−−−→ T ( E ) −−−−→ M R
Hence, we get an isomorphism ε : V = W R −→ Φ ( Y ) and a commutative23iagram: V ε −−−−→ ∼ Φ ( Y ) = y g y V f −−−−→ Φ ( X )Suppose X belongs to M V . Because of lemma 1.12, the morphism π : ΦΨ( X ) −→ X is an epimorphism in S ∨ inducing two epimorphisms F ( H ) −→ E and σ ( H ) −→ M . Since E is finitely generated, H contains a finitely generated C -submodule H ⊂ H such that the composite morphism F ( H ) −→ F ( H ) −→ E is an epimorphism.Therefore there exist a module P ∈ C and a morphism u : s ( P ) −→ H such that theimage of u contains the submodule H . Because of lemma 1.13, there are an object K ′ = ( H ′ , θ ′ ) ∈ N il ( C , S ), a morphism h : K ′ −→ K and a split injective morphism λ ′ ⊕ u ′ : s ( W ) ⊕ s ( P ) −→ H ′ making the following diagram commutative: s ( W ) ⊕ s ( P ) λ ′ ⊕ u ′ −−−−→ H ′ = y h y s ( W ) ⊕ s ( P ) λ ⊕ u −−−−→ H Therefore the composite morphism F ( H ′ ) h −→ F ( H ) −→ E is an epimorphism.If we continue the construction above by using this morphism h : K ′ −→ K , weget a morphism g : Y −→ X such that the morphism Φ ( Y ) −→ Φ ( X ) is isomorphicto the morphism F ( H ′ ) −→ E which is an epimorphism. Hence Φ ( Y ) −→ Φ ( X ) isalso an epimorphism. On the other hand, we have a commmutative diagram: T (Φ ( Y )) −−−−→ Φ ( Y ) R y y T ( E ) ϕ −−−−→ M R where ϕ : T ( E ) −→ M R is surjective. Then Φ ( Y ) R −→ M R is surjective andΦ ( Y ) −→ M is surjective too. The result follows.24 . Algebraic K-theory of categories V and M V .2.1 About Waldhausen K-theory.
A Waldhausen category is a category with a zero object and two subcategories:the category of cofibrations and the category of (weak-)equivalences. These cate-gories have to satisfy certain conditions (see [Wa2]). Waldhausen associates to anyessentially small Waldhausen category C an infinite loop space K ( C ) and K (calledthe Waldhausen K-theory functor) is a functor from the category of essentially smallWaldhausen categories to the category of infinite loop spaces.An exact category E may be considered as a Waldhausen category, where a cofi-bration is an admissible monomorphism of E (i.e. a morphism f appearing is an exactsequence 0 −→ X f −→ Y −→ Z −→ E ) and an equivalence is an isomorphism in E . Moreover, if E is essentially small, we have a natural homotopy equivalence fromthe Quillen K-theory of E to the Waldhausen K-theory of E .To every exact category E we can associate the following category E ∗ :The objects of E ∗ (called the E -complexes) are the complexes: C = (cid:16) . . . d −→ C n d −→ C n − d −→ C n − d −→ . . . (cid:17) where each C n is an object of E and each d is a morphism of E such that the sum ⊕ n C n exists in E and each morphism d is zero.The morphisms in this category are morphisms respecting degrees and differen-tials. A sequence 0 −→ X −→ Y −→ Z −→ E ∗ is said to be exact if it inducesan exact sequence in E on each degree. With these exact sequences, E ∗ becomes anexact category.If E is the category of right modules (resp. the category of finitely generatedprojective right modules) over a ring A , the E -complexes are called A -complexes(resp. finite A -complexes).Suppose E is an exact subcategory of an abelian category E ∨ . Then E ∗ is aWaldhausen category, where cofibrations are admissible monomorphisms and equiv-alences are morphisms inducing an isomorphism in homology (where homologies arecomputed in E ∨ ). Moreover E ∗ is saturated and has a cylinder functor satisfying thecylinder axiom (in the sense of Waldhausen [Wa2]). We have the following result([Wa2], [We]): Let E be an essentially small exact categorycontained in an abelian category E ∨ . Suppose E is stable in E ∨ by kernel of epimor-phisms. Then the inclusion E ⊂ E ∗ of Waldhausen categories induces a homotopyequivalence in K-theory. Let E be an essentially small exact category and C and C ′ be two E -complexes.For each integer n , we set:Hom( C, C ′ ) n = Y p Hom E ( C p , C ′ n + p )25nd Hom( C, C ′ ) is a graded Z -module. We have on Hom( C, C ′ ) a natural differential d of degree − ∀ f ∈ Hom(
C, C ′ ) n , d ( f ) = d ◦ f − ( − n f ◦ d An element of Hom(
C, C ′ ) n is called a linear map of degree n , a cycle in Hom( C, C ′ ) n is called a morphism of degree n and a boundary of Hom( C, C ′ ) n is called a homotopyof degree n . The morphisms of degree 0 are the morphisms in the category E ∗ .In this category, we have also a notion of n -cone:Consider a morphism f : X −→ Y in E ∗ and an integer n ∈ Z . We set: C = X ⊕ Y .So we have four linear maps: i : Y −→ C , p : C −→ X , r : C −→ Y and s : X −→ C .The map i is an injection, p is a projection, r is a retraction of i and s is a sectionof p . There is a unique way to modify degrees and differentials on C such that thefollowing holds: ∂ ◦ i = n ∂ ◦ r = − n ∂ ◦ p = − − n ∂ ◦ s = 1 + nd ( i ) = 0 d ( p ) = 0 d ( r ) = − ( − n f p d ( s ) = if With these new degrees and differentials, C is an E -complex called the n -cone of f .If n = 0, C is the classical mapping cone, the map i : Y −→ C is a cofibration in E ∗ and we have an exact sequence in E ∗ :0 −→ X −→ T ( f ) −→ C −→ T ( f ) is the cylinder of f .If n = −
1, the map p : C −→ X is also a morphism in E ∗ .Let F : A −→ B be an exact functor between two Waldhausen categories. Wesay that F has the approximation property if the following holds: • (App1) a morphism in A is an equivalence if and only if its image under F isan equivalence • (App2) for every ( X, Y ) ∈ A × B and every morphism f : F ( X ) −→ Y , thereexist a morphism α : X −→ X ′ in A and a commutative diagram in B : F ( X ) f −−−−→ Y α y = y F ( X ′ ) f ′ −−−−→ ∼ Y where f ′ is an equivalence. We have the following theorem of Waldhausen ([Wa2]): Let F : A −→ B be an exact functorbetween two essentially small saturated Waldhausen categories. Suppose A has a ylinder functor satisfying the cylinder axiom and F has the approximation property.Then F induces a homotopy equivalence in K-theory. The functor Φ × Φ : M V −→ D × C induces a homotopy equivalencein K-theory: K ( M V ) ∼ −→ K ( D × C ) = K ( D ) × K ( C ) Proof:
The proof will be done by using Waldhausen K-theory.The three exact categories
M V , D and C are contained in the abelian categories M V ∨ , D ∨ and C ∨ respectively. Moreover each of these exact categories are closed bykernel of epimorphisms in the corresponding abelian categories. Therefore, because ofGillet-Waldhausen theorem, it’s enough to prove that the functor Φ × Φ : M V ∗ −→ D ∗ × C ∗ induces a homotopy equivalence in K-theory.By replacing the category of equivalences of M V ∗ by the category of morphisms f : X −→ Y inducing a homology equivalence Φ ( X ) ∼ −→ Φ ( Y ) we get a newWaldhausen category denoted by M V ′∗ .Denote also by M V ∗ the Waldhausen subcategory of M V ∗ of objects X withacyclic Φ ( X ). Because of the fibration theorem (see [Wa2]), the sequence: M V ∗ −→ M V ∗ −→ M V ′∗ induces a fibration in K-theory.Denote by D ∗ the Waldhausen subcategory of D ∗ of acyclic complexes in D ∗ .Since each morphism in D ∗ is an equivalence, D ∗ has trivial K-theory. We have acommutative diagram: M V ∗ −−−−→ M V ∗ −−−−→ M V ′∗ Φ × Φ y Φ × Φ y Φ y D ∗ × C ∗ −−−−→ D ∗ × C ∗ −−−−→ D ∗ where each line induces a fibration in K-theory. Therefore it will be enough toprove that functors Φ : M V ′∗ −→ D ∗ and Φ × Φ : M V ∗ −→ D ∗ × C ∗ inducehomotopy equivalences in K-theory or, equivalently, that Φ : M V ′∗ −→ D ∗ andΦ : M V ∗ −→ C ∗ induce homotopy equivalences in K-theory.Because of the approximation theorem of Waldhausen, in order to prove thatΦ : M V ′∗ −→ D ∗ and Φ : M V ∗ −→ C ∗ induces a homotopy equivalence in K-theory, it’s enough to show that these two functors have the approximation property.The property (App1) it easy to check. Then the last thing to do is to show thatΦ : M V ′∗ −→ D ∗ and Φ : M V ∗ −→ C ∗ have the property (App2).27onsider an object X ∈ M V ′∗ , an object F ∈ D ∗ and a morphism f : Φ ( X ) −→ F in D ∗ . The object X is a triple X = ( E, M, ϕ ) where E is a D -complex, M is a C -complex and ϕ is an element in F ( E, M ) inducing a surjective morphism T ( E ) −→ M R . So f is a morphism from E to F in D ∗ .Denote by E the − f . So we have linear maps i : F −→ E , p : E −→ E , r : E −→ F and s : E −→ E . The map p is an epimorphism in D ∗ , i is amorphism of degree − d ( s ) = if d ( r ) = f p Consider the triple ( E , M, ϕp ). Since p is an epimorphism, this triple is anobject X ∈ M V ∗ and we have a morphism g : X −→ X . Denote by Y the 0-coneof g . We have a cofibration j : X −→ Y and four linear maps: j : E −→ Φ ( Y ), q : Φ ( Y ) −→ E , ρ : Φ ( Y ) −→ E and σ : E −→ Φ ( Y ). Moreover we have; d ( σ ) = jp d ( ρ ) = − pq Consider the linear map g = f ρ + rq . The differential d ( g ) vanishes and g is amorphism from Φ ( Y ) to F . It is easy to see that g is surjective and its kernel isisomorphic to the 0-cone of the identity of E . Moreover we have: gj = f . Therefore g : Φ ( Y ) −→ F is a homology equivalence and the following diagram is commutative:Φ ( X ) f −−−−→ F j y = y Φ ( Y ) g −−−−→ ∼ F Then the functor Φ : M V ′∗ −→ D ∗ has the property (App2) and Φ : M V ′∗ −→ D ∗ induces a homotopy equivalence in K-theory.Consider an object X ∈ M V ∗ , an object N ∈ C ∗ and a morphism f : Φ ( X ) −→ N in C ∗ . The object X is a triple ( E, M, ϕ ) where E is an acyclic D -complex, M is a C -complex and ϕ is an element in F ( E, M ) inducing a surjective morphism T ( E ) −→ M R . So f is a morphism from M to N .Let U be the − N . Then U is acyclic and we have anepimorphism p : U −→ N . Consider the composite morphism: ϕ ′ : T ( F s ( U )) p −→ T ( F s ( N )) e −→ ∼ σs ( N ) R −→ N R
This morphism is surjective and the triple ( E ⊕ F s ( U ) , N, f ϕ ⊕ ϕ ′ ) is an object Y in M V ∗ . Moreover we have a morphism g : X −→ Y inducing the inclusion E ⊂ E ⊕ F s ( U ) and the morphism f : M −→ N , making the following diagram28ommutative: Φ ( X ) f −−−−→ N g y = y Φ ( Y ) = −−−−→ N Hence Φ : M V ∗ −→ C ∗ has the property (App2) and this functor induces a homo-topy equivalence in K-theory.Denote by M V ′ the full subcategory of M V consisting of objects X ∈ M V suchthat Φ ( X ) belongs to R ′ . Then an object X ∈ M V belongs to
M V ′ if and only ifthere is an isomorphism Φ ( X ) ≃ V R for some V ∈ C . This category is exact andcofinal in M V . Moreover the functor Φ sends M V ′ to R ′ .We define the following categories: • The category E of modules in R ′ and isomorphisms • The category E of objects in M V ′ and morphisms inducing isomorphismsunder Φ • The category E of objects in M V ′ and epimorphisms inducing isomorphismsunder Φ • The category E of objects in M V ′ × V , where a morphism in E from ( X, V )to (
Y, W ) is an morphism X ⊕ V −→ Y ⊕ W in E sending X to Y .We have four functors f : E −→ E , f : E −→ E , g : E −→ E and g : E −→ E where f is induced by Φ , f is the inclusion and g (resp. g ) is thecorrespondence ( X, V ) X (resp. ( X, V ) X ⊕ V ).In order to prove a connective version of theorems 1, 2 and 3, we’ll need to provethat the sequence V ⊂ M V ′ −→ R ′ induces a fibration in K-theory and, for that,it will be useful to prove that f f is a homotopy equivalence. The functor f is a homotopy equivalence. Proof:
It is enough to prove that the fiber category f /U is contractible for eachobject U ∈ E = R ′ .Consider an object U in R ′ and denote by F the fiber category f /U . By applyinglemma 1.14 with X = 0 and V = U , we get an object Y ∈ M V and an isomorphism U ≃ Φ ( Y ). Hence the category F is nonemptyConsider two objects ( X , ε ) and ( X , ε ) in F . For i = 1 , ε i is an isomorphismfrom Φ ( X i ) to U .By applying lemma 1.14 with X = X ⊕ X and f = ε − ⊕ ε − : U −→ Φ ( X ),we get an object Y ∈ M V , a morphism g : Y −→ X ⊕ X and an isomorphism ε : U ∼ −→ Φ ( Y ) such that ε − ⊕ ε − is the composite morphism U ε −→ Φ ( Y ) g −→ Φ ( X ).Therefore ( Y, ε − ) is an object in F and we have two morphisms ( Y, ε − ) −→ ( X , ε )and ( Y, ε − ) −→ ( X , ε ). 29onsider two objects ( X , ε ) and ( X , ε ) in F and two morphisms h and h from ( X , ε ) to ( X , ε ). Denote by X the kernel of h − h (in S ∨ ).For i = 1 ,
2, we have an exact sequence:0 −→ Φ ( X i ) µ i −→ T Φ ( X i ) ϕ i −→ Φ ( X i ) R −→ −→ U µ i ε − i −→ T Φ ( X i ) ϕ i −→ Φ ( X i ) R −→ i = 1 , −−−−→ U µ ε − −−−−→ T Φ ( X ) ϕ −−−−→ Φ ( X ) R −−−−→ = y h i y h i y −−−−→ U µ ε − −−−−→ T Φ ( X ) ϕ −−−−→ Φ ( X ) R −−−−→ −→ U −→ T Φ ( X ) −→ Φ ( X ) R inducing an isomorphism ε : U −→ Φ ( X ). By applying lemma 1.14 with ε , we get anobject ( Y, u − ) ∈ F and a morphism k : ( Y, u − ) −→ ( X , ε ) such that: h k = h k .Because of these properties, each fiber category F is cofiltered and then con-tractible. Hence the functor f is a homotopy equivalence. The functor g is a homotopy equivalence. Proof:
Let X be an object of M V ′ and F be the fiber category g /X . Applyinglemma 1.14 to the morphism 0 −→ Φ ( X ), we get an object E ∈ M V , a morphism α : E −→ X such that: Φ ( E ) = 0 and Φ ( E ) −→ Φ ( X ) and Φ ( E ) −→ Φ ( X )are epimorphisms. Then E belongs to V and the morphism α : E −→ X inducesepimorphisms on Φ and Φ . Therefore, for each morphism f : Y −→ X in E , themorphism Y ⊕ E f ⊕ α −→ X belongs to E .An object in F is a triple ( Y, V, f ) where (
Y, V ) belongs to
M V ′ × V and f : Y −→ X is a morphism in E . A morphism ϕ : ( Y, V, f ) −→ ( Y ′ , V ′ , f ′ ) is a morphism ϕ : Y ⊕ V −→ Y ′ ⊕ V ′ in E sending Y to Y ′ such that: f = f ′ ϕ .The category F is nonempty because it contains the object ( X, , Id).We have three functors G , G , G from F to F sending each ( Y, V, f ) ∈ F to G ( Y, V, f ) = (
Y, V, f ), G ( Y, V, f ) = (
Y, V ⊕ E, f ) and G ( Y, V, f ) = ( X, , Id)respectively.The inclusion 0 ⊂ E induces a morphism G −→ G . The morphism f ⊕ ⊕ α : Y ⊕ V ⊕ E −→ X induces a morphism ( Y, V ⊕ E, f ) −→ ( X, , Id) and we get a30orphism G −→ G . Therefore the identity of F is homotopic to G and then to G which is constant. Hence F is contractible and, since each fiber category of g iscontractible, g is a homotopy equivalence. The functor g is a homotopy equivalence. Proof:
Let X be an object in E and F be the fiber category g /X . An objectin F is a triple ( Y, V, f ) where (
Y, V ) belongs to E and f : Y ⊕ V −→ X is amorphism in E . A morphism ϕ : ( Y, V, f ) −→ ( Y ′ , V ′ , f ′ ) in F is a morphism ϕ : Y ⊕ V −→ Y ′ ⊕ V ′ in E such that ϕ ( Y ) ⊂ Y ′ and f = f ′ ϕ . Therefore it is easy tosee that, for every ( Y, V, f ) ∈ F , we have a unique morphism in F from ( Y, V, f ) to( X, , Id). Hence F has a final object and is contractible. Since each fiber categoryof g is contractible, g is a homotopy equivalence. The functor f is a homotopy equivalence. Proof:
The inclusion X ⊂ X ⊕ V for all ( X, V ) ∈ E induces a morphism from g to f g and g is homotopic to f g . But g and g are homotopy equivalences.Therefore f is a homotopy equivalence too. The following diagram of exact categories: V ⊂ M V ′ Φ −→ R ′ induces a fibration in K-theory. Proof:
In [Wa1] (lemma 10.2 p. 206), Waldhausen proved that Q V −→ Q M V ′ −→ Q F R is a homotopic fibration in a situation similar to ours. In our situation we’llprove, essentially in the same way, that the sequence Q V −→ Q M V ′ −→ Q R ′ is ahomotopic fibration.Following Waldhausen’s notations, if F is an exact subcategory of an exact cate-gory E , we have a bicategory Q ep ( E , F ) where the horizontal maps form the Quillen’scategory Q ( E ) and the vertical morphisms form the category of epimorphisms in E with kernel in F . In particular we have an equivalence between Q ( E ) and Q ep ( E , Q ( E ) may be considered as a bicategory.We have a commutative diagram of bicategories: Q ( V ) −→ Q ep ( V , V ) y y Q ( M V ′ ) −→ Q ep ( M V ′ , V )which is homotopically cartesian. Moreover Q ep ( V , V ) is contractible. Therefore the31iagram of bicategories: Q ( V ) −→ Q ( M V ′ ) −→ Q ep ( M V ′ , V )is a homotopic fibration.On the other hand the morphism f = f f : E −→ R ′ induces a morphism f ∗ : Q ep ( M V ′ , V ) −→ Q ep ( R ′ ,
0) and we want to prove that f ∗ is a homotopyequivalence. Actually, the proof of lemma 10.2 in [Wa1] works exactly the same inour situation except maybe for the sublemma (p. 209). In this sublemma, we have afiltered object M ⊂ M ⊂ . . . M n − ⊂ M n in M V ′ where each quotient M i /M i − is in M V ′ . Since Φ is exact and each modulein R ′ is projective, the morphism Φ ( M n ) −→ Φ ( M n /M n − ) is surjective and has asection s from U = Φ ( M n /M n − ) to Φ ( M n ).Because of lemma 1.14, there are an object N ∈ M V , a morphism g : N −→ M n inducing epimorphisms on Φ and Φ , and an isomorphism U ∼ −→ Φ ( N ) making thefollowing diagram commutative: U ∼ −−−−→ Φ ( N ) = y g y U s −−−−→ Φ ( M n )Therefore the morphism N −→ M n induces epimorphisms on Φ and Φ and thecomposite morphism N −→ M n −→ M n /M n − is an epimorphism with kernel in V . Hence the sublemma can be proven in our situation and, since f is a homotopyequivalence, the proof of lemma 10.2 applies completely here. The lemma follows. Let R ′′ be the full subcategory of R generated by the image of Φ : M V −→ R . Then the diagram: V ⊂ M V Φ −→ R ′′ induces a fibration in K-theory. Proof:
Let X = ( E, M, ϕ ) be an object in
M V . Since E is projective in D ,there is a module E ∈ D such that E ⊕ E is free in case 1 or 3 and on the form E ⊕ E = ( F A , F B ) ∈ A × B with F A and F B free. Therefore T ( E ⊕ E ) is free inall cases. Let X be the object ( E , , ∈ M V . We have an exact sequence in R :0 −→ Φ ( X ⊕ X ) −→ T ( E ⊕ E ) −→ M R −→ ( X ⊕ X ) is stably in R ′ . Hence there is another object X ∈ M V such thatΦ ( X ⊕ X ⊕ X ) belongs to R ′ and M V ′ is cofinal in M V .Consider the following commutative diagram:( D ) M V ′ Φ −−−−→ R ′ y y M V Φ −−−−→ R ′′ where the vertical maps are the canonical cofinal inclusions.Let K be the fiber product of K ( M V ) and K ( R ′ ) over K ( R ′′ ). The commu-tativity of the diagram K ( D ) induce a map λ : K ( M V ′ ) −→ K . Since the map K ( M V ′ ) −→ K ( M V ) is injective (by cofinality) and factorizes through K , themap λ is injective.Let w = ( u, v ) be an element in K . Then we have: u ∈ K ( M V ), v ∈ K ( R ′ )and Φ ( u ) and v are the same in K ( R ′′ ).For every object X in some exact category A , the class of X in the Grothendieckgroup K ( A ) will be denoted by [ X ].So there are two objects X , Y in M V such that: u = [ X ] − [ Y ]. Since M V ′ iscofinal in M V , there is an object Y in M V such that Y ⊕ Y is in M V ′ . Let usset: w ′ = w + λ [ Y ⊕ Y ] and X ′ = X ⊕ Y . We have: w ′ = ( u ′ , v ′ ) with u ′ = [ X ′ ]and [Φ ( X ′ )] belongs to K ( R ′ ). Then Φ ( X ′ ) is stably isomorphic to a module in R ′ . Up to adding to X (and then to X ′ ) an object in M V ′ on the form ( E, , ( X ′ ) belongs to R ′ and then that X ′ belongs to M V ′ .Therefore we have: w ′ − λ [ X ′ ] = (0 , v ′′ ) where v ′′ is an element in K ( R ′ ) killedin K ( R ′′ ). But the morphism K ( R ′ ) −→ K ( R ′′ ) is injective. So we have: w ′ − λ [ X ′ ] = 0 = ⇒ w ′ = λ ( X ′ )and w ′ and then w are in the image of λ . Therefore λ is surjective and then bijective.Consequently the diagram K ( D ) is exact (cartesian and cocartesian) and, bycofinality, the diagram K ( D ) is homotopically cartesian. The lemma follows.As a consequence we get the following result, which is, in some sense, a connectiveversion of theorems 1, 2 and 3: Let X be the homotopy fiber of the map K ( C ) −→ K ( D ) × F ( C ) induced by the functor F × σ : C −→ D × C . Then there is a natural homotopyequivalence Ω K ( R ) ∼ −→ N il ( C , S ) × X . Proof:
Because of lemma 1.12 the functor Φ : N il ( C , S ) −→ V is an equivalenceof categories and the lemma 2.10 implies that the following diagram: N il ( C , S ) Φ −→ M V Φ −→ R ′′ K ( R ′′ ) ≃ Ω K ( R ) is homotopically equivalentto the homotopy fiber of the map induced by Φ : N il ( C , S ) −→ M V in K-theory.Because of lemma 2.4, we have a commutative diagram N il ( C , S ) Φ −−−−→ M V = y y Φ × Φ N il ( C , S ) Φ ′ −−−−→ D × C where the functor Φ × Φ induces a homotopy equivalence in K-theory. There-fore Ω K ( R ) is homotopically equivalent to the homotopy fiber of the map Φ ′ : K ( N il ( C , S )) −→ K ( D × C ).The functor Φ ′ sends an object ( H, θ ) ∈ N il ( C , S ) to the pair ( F ( H ) , σ ( H )) ∈ D × C and Φ ′ factorizes by the forgetful map N il ( C , S ) −→ C . Therefore Ω K ( R )is homotopically equivalent to the homotopy fiber of Φ ′′ : K ( C ) × N il ( C , S ) −→ K ( D ) × K ( C ) where Φ ′′ is trivial on N il ( C , S ) and induced by the functor F × σ on K ( C ). The lemma follows. K and N il . The K-theory of Quillen is a functor K from the category of rings to the categoryof infinite loop spaces. There are different methods to construct a so called negativeK-theory: that is a functor K ′ from the category of rings to the category Ω sp ofΩ-spectra (see [B] and [KV]) and a natural homotopy equivalence from K ( A ) to the0-th term of K ′ ( A ) such that the following sequence is exact for every ring A andevery integer i ∈ Z :0 −→ K i ( A ) −→ K i ( A [ t ]) ⊕ K i ( A [ t − ]) −→ K i ( A [ t, t − ]) −→ K i − ( A ) −→ K i ( A ) is the i -th homotopy group of K ′ ( A ). This exact sequence was provenby Bass [B] for i = 1 and generalized by Quillen for i > A ⊂ A [ t ] ⊂ A [ t, t − ], A ⊂ A [ t − ] ⊂ A [ t, t − ] except for the map ∂ : K i ( A [ t, t − ]) −→ K i − ( A ). But this map ∂ hasa section induced by the multiplication by t ∈ K ( Z [ t, t − ]). Therefore the exactsequence above is natural in A .Inspired by the Karoubi-Villamayor method [KV], we’ll define our version K ( A )of negative K-theory as follows:Denote by E the set of infinite square matrices with entries in Z having onlyfinitely many nonzero entries in each row and each column. This ring has a two-sided ideal M ( Z ) of matrices having only finitely many nonzero entries. So we set:Σ = E/M ( Z ).For every ring A , EA = E ⊗ Z A and Σ A = Σ ⊗ Z A are rings and the morphism f : EA −→ Σ A is a surjective ring homomorphism. It is easy to see that the kernel of34 is isomorphism, as a pseudo ring, to M ( A ) and that EA is a flasque ring. Therefore(see [KV]) we have a natural homotopy equivalence: K ( A ) ∼ −→ Ω K (Σ A )and the sequence K (Σ n A ) is an Ω-spectrum. This spectrum will be denoted by K ( A )and K is a negative K-theory. For each integer i ∈ Z we set also: K i ( A ) = π i ( K ( A )). Let ( A, S ) be a left-flat bimodule. Then for each integer n ≥ , (Σ n A, Σ n S ) is a left-flat bimodule and the sequence N il (Σ n A, Σ n S ) is an Ω -spectrumdenoted by N il ( A, S ) . Moreover we have a natural homotopy equivalence from N il ( A, S ) to the 0-th term of N il ( A, S ) and, for each integer i ∈ Z , we have anexact sequence which is natural on the left-flat bimodule ( A, S ) : −→ N il i ( A, S ) −→ N il i ( A [ t ] , S [ t ]) ⊕ N il ( A [ t − ] , S [ t − ]) −→ N il i ( A [ t, t − ] , S [ t, t − ]) −→ N il i − ( A, S ) −→ where N il i (?) is the i -th homotopy group of N il (?) . Proof:
Let (
A, S ) be a left-flat bimodule. For every ring B , BS = B ⊗ Z S is a BA -bimodule. Since S is flat on the left S , as a left A -module, is isomorphic to a filteredcolimit of free left A -modules E i and BS is also isomorphic to a filtered colimit offree BA -modules BE i . Therefore BS is flat on the left and ( BA, BS ) is a left-flatbimodule. In particular each (Σ n A, Σ n S ) is a left-flat bimodule. Moreover we have anatural isomorphism of rings: Σ( A [ S ]) ≃ (Σ A )[Σ S ].Consider the case 1 (with C = A ). Because of proposition 2.11, we have a naturalhomotopy equivalence: Ω K ( A [ S ]) ∼ −→ N il ( A, S ) × X where X is the homotopy fiber of the map: K ( C ) F × σ −→ K ( D × C ). But in case 1, C , D and C are equal to the category A of finitely generated projective right A -modulesand functors F and σ are the identity. Therefore X is nothing else but the loop spaceof K ( A ). So we get a natural homotopy equivalence:Ω K ( A [ S ]) ∼ −→ N il ( A, S ) × Ω K ( A )By naturality, we get a homotopy equivalence from N il ( A, S ) to the homotopy fiberof the map Ω K ( A [ S ]) −→ Ω K ( A ) induced by the canonical ring homomorphism A [ S ] −→ A .We have a commutative diagram: K ( A [ S ]) −−−−→ K ( A ) ∼ y ∼ y Ω K (Σ( A [ S ])) −−−−→ Ω K (Σ A )35here the horizontal maps are induced by the canonical morphism A [ S ] −→ A andthe vertical maps are homotopy equivalences.But Σ( A [ S ]) is isomorphic to (Σ A )[Σ S ]. So we get a commutative diagram:Ω K ( A [ S ]) −−−−→ Ω K ( A ) ∼ y ∼ y Ω K ((Σ A )[Σ S ])) −−−−→ Ω K (Σ A )inducing a homotopy equivalence N il ( A, S ) ∼ −→ Ω N il (Σ A, Σ S ). Then the sequenceof spaces N il (Σ n A, Σ n , S ) is a well defined Ω-spectrum N il ( A, S ) and we have anatural decomposition: Ω K ( A [ S ]) ∼ −→ N il ( A, S ) × Ω K ( A )Let us set: N il i ( A, S ) = π i ( N il ( A, S )). Then we have, for every left-flat bimodule(
A, S ) and every integer i ∈ Z an isomorphism K i ( A [ S ]) ≃ N il i − ( A, S ) ⊕ K i ( A ).For the ring A we have, for every integer i ∈ Z , the following exact sequence S i ( A ):0 −→ K i ( A ) −→ K i ( A [ t ]) ⊕ K i ( A [ t − ]) −→ K i ( A [ t, t − ]) −→ K i − ( A ) −→ B = Z [ t ] or B = Z [ t − ] or B = Z [ t, t − ], the ring B ( A [ S ]) is isomorphic to( BA )[ BS ]. Then the sequence S i +1 ( A [ S ]) decomposes into S i +1 ( A ) and the followingsequence:0 −→ N il i ( A, S ) −→ N il i ( A [ t ] , S [ t ]) ⊕ N il i ( A [ t − ] , S [ t − ]) −→ N il i ( A [ t, t − ] , S [ t, t − ]) −→ N il i − ( A, S ) −→ In the proof of lemma 2.13, we have constructed the Ω-spectrum
N il (?) and thefirst two properties of theorem 1 have already been proven.Suppose A is regular coherent on the right, i.e. every finitely presented right A -module has a finite resolution by finitely generated projective modules. The category A of finitely generated projective right A -modules is contained in the category A ′ of finitely presented right A -modules. The category N il ( A, S ) is also contained inthe category N il ′ ( A, S ) of pairs (
H, θ ) with H ∈ A ′ and θ : H −→ HS nilpotent.Moreover A is stable in A ′ under extension and kernel of admissible epimorphismand the inclusion N il ( A, S ) ⊂ N il ′ ( A, S ) have the same properties. Therefore, by36he resolution theorem [Q], the inclusions A ⊂ A ′ and N il ( A, S ) ⊂ N il ′ ( A, S )induce homotopy equivalences in K-theory.On the other hand, we have an inclusion A ′ −→ N il ′ ( A, S ) sending H to thepair ( H,
0) and every object (
H, θ ) in N il ′ ( A, S ) has a finite filtration with subquo-tients in A ′ . Moreover A ′ and N il ′ ( A, S ) are abelian categories and A ′ is closedin N il ′ ( A, S ) under subobjects and quotients. Therefore, by the devissage theorem[Q], the inclusion A ′ ⊂ N il ′ ( A, S ) induces a homotopy equivalence in K-theory.As a consequence the inclusion A ⊂ N il ( A, S ) induce a homotopy equivalence inK-theory and the space
N il ( A, S ) is contractible.Hence for every left-flat bimodule (
A, S ), with A regular coherent, the space N il ( A, S ) is contractible and
N il i ( A, S ) is trivial for every i ≥ E ( n ) where n is any integer: • For every left-flat bimodule (
A, S ) with A regular coherent on the right, themodule N il i ( A, S ) is trivial for every i ≥ n .The property E (0) is then satisfied. Suppose E ( n ) is true and take a left-flatbimodule ( A, S ) with A regular coherent on the right. Then A [ t ], A [ t − ] and A [ t, t − ]are also regular coherent on the right and, in the exact sequence:0 −→ N il n ( A, S ) −→ N il n ( A [ t ] , S [ t ]) ⊕ N il n ( A [ t − ] , S [ t − ]) −→ N il n ( A [ t, t − ] , S [ t, t − ]) −→ N il n − ( A, S ) −→ N il n − ( A, S ). Hence the module
N il n − ( A, S ) is alsotrivial and the property E ( n −
1) is true.By induction E ( n ) is satisfied for all n . Hence N il ( A, S ) is contractible for everyleft-flat bimodule (
A, S ) with A regular coherent on the right and theorem 1 is proven.Consider the case 2. The ring R is defined by the cocartesian diagram:( D ) C α −−−−→ A β y y B −−−−→ R By tensoring by Σ n , we get a cocartesian diagram:(Σ n D ) Σ n C α −−−−→ Σ n A β y y Σ n B −−−−→ Σ n R
37e remark that the morphism α : Σ n C −→ Σ n A (resp. β : Σ n C −→ Σ n B ) ispure with complement Σ n A ′ (resp. Σ n B ′ ). Therefore in this new situation, we getnew rings: Σ n C , Σ n A , Σ n B , Σ n R , Σ n C × Σ n C and a new bimodule Σ n S .Because of proposition 2.11, we have a homotopy equivalence between Ω K (Σ n R )and N il (Σ n C × Σ n C, Σ n S ) × X n where X n is the homotopy fiber of f : K (Σ n C × Σ n C ) −→ K (Σ n A ) × K (Σ n B ) × K (Σ n C ).The morphism f is induced by the functor C × C −→ A × B × C sending( M, M ′ ) ∈ C × C to ( M A, M ′ B, M ⊕ M ′ ). Denote by f n ( α ) (resp. f n ( β )) the map K (Σ n C ) −→ K (Σ n A ) (resp. K (Σ n C ) −→ K (Σ n B )) induced by α (resp. β ). Then X n is homotopy equivalent to the homotopy fiber of f n ( α ) − f n ( β ) : K (Σ n C ) −→ K (Σ n A ) × K (Σ n B ) and we have a homotopy cartesian diagram of spaces:Ω H (Σ n C ) α −−−−→ Ω K (Σ n A )) β y y Ω K (Σ n B ) −−−−→ X n By naturality of the homotopy equivalence: K (?) ≃ Ω K (Σ?), we get a homotopyequivalence X n −→ Ω X n +1 and the sequence of X n defines an Ω-spectrum X togetherwith a homotopy cartesian diagram of spectra: K ( C ) α −−−−→ K ( A ) β y y K ( B ) −−−−→ Ω − X and that finishes the proof of theorem 2.Consider now the case 3. We proceed as before and we get a homotopy equivalencebetween Ω K (Σ n R ) and N il (Σ n C × Σ n C, Σ n S ) × X n where X n is the homotopy fiberof the map f : K (Σ n C × Σ n C ) −→ K (Σ n A ) × K (Σ n C ) induced by the functor F × σ : C × C −→ A × C sending ( M, M ′ ) ∈ C × C to ( M α A ⊕ M ′ β A, M ⊕ M ′ ).Denote by f n ( α ) (resp. f n ( β )) the map K (Σ n C ) −→ K (Σ n A ) induced by α (resp. β ). Then X n is homotopy equivalent to the homotopy fiber of f n ( α ) − f n ( β ).As before, we get a homotopy equivalence X n −→ Ω X n +1 and the sequence of X n defines an Ω-spectrum X which is the homotopy fiber of the map f ( α ) − f ( β ) : K ( C ) −→ K ( A ). Therefore we have a homotopy fibration of spectra: K ( C ) f ( α ) − f ( β ) −→ K ( A ) −→ Ω − X
3. Properties of the functor
N il . Consider two rings A and B . Every right A × B -module M is determined by tworight modules M a and M b , where M a is an A -module and M b is a B -module. Bysetting: R a = A and R b = B , we see that M i is a right R i -module for each i ∈ { a, b } .If E is an A × B -bimodule, E is determined by four bimodules a E a , a E b , b E a and b E b , and for each i, j in { a, b } , i E j is a ( R i , R j )-bimodule.Suppose f : M −→ M E is a morphism of right A × B -modules. Then f isdetermined by four morphisms i f j : M j −→ M i i E j and i f j is a morphism of right R j -modules. Let A and B be two rings and E be an A × B -bimodule. Suppose E is flat on the left. Then the correspondences ( M, f ) ( M b , b f b )( M, f ) ( M a , a f a + X k ≥ a f b ( b f b ) kb f a ) induce two well defined functors: Φ : N il ( A × B, E ) −→ N il ( B, b E b )Φ : N il ( A × B, E ) −→ N il ( A, a E a ⊕ ⊕ k ≥ a E b ( b E b ) kb E a ) Moreover, if E b is flat on the right, these functors induce a homotopy equivalenceof spectra: N il ( A × B, E ) ∼ −→ N il ( B, b E b ) × N il ( A, a E a ⊕ k ≥ a E b ( b E b ) kb E a )This lemma will be proven in 3.7.Using this result, we are able to prove theorem 4. Consider two rings A and B ,an ( A, B )-bimodule S and a ( B, A )-bimodule T . Suppose S and T are flat on bothsides. Define the A × B -bimodule E by: a E b = S b E a = T a E a = b E b = 0Actually, this bimodule is the bimodule S ⊕ T , where S and T are considered as A × B -bimodules (via the projections A × B −→ A and A × B −→ B ). Moreover E = S ⊕ T is flat on both sides.Applying lemma 3.1, we get two functors:Φ : N il ( A × B, E ) −→ N il ( B, : N il ( A × B, E ) −→ N il ( A, ST )and a homotopy equivalence of spectra:Φ(
S, T ) :
N il ( A × B, E ) ∼ −→ N il ( A, ST )By exchanging the roles of A and B , and S and T we get also a homotopy equivalenceof spectra: Φ( T, S ) =
N il ( A × B, E ) ∼ −→ N il ( B, T S ) Let A be a ring, I be a set and S i , i ∈ I be a family of A -bimodules flat on bothsides. The direct sum of the S i ’s will be denoted by S .Let ( M, f ) be an object of the category N il ( A, S ). The morphism f : M −→ M S decomposes into a finite sum: f = P i ∈ I f i , where f i is a morphism from M to M S i .If u = i i . . . i p is a word in W ( I ), we set: f u = f i f i . . . f i p and S u = S i S i . . . S i p If J is a subset of W ( I ), we set also: f J = X u ∈ J f u and S J = ⊕ u ∈ J S u We check that f u is a morphism from M to M S u and f J is a morphism from M to M S J .Let u be a non empty word in W ( I ). Then the correspondence ( M, f ) ( M, f u )induces an exact functor ϕ u : N il ( A, S ) −→ N il ( A, S u ). If J is a subset of W ( I ) thatdoes not contains the empty word, the correspondence ( M, f ) ( M, f J ) induces alsoan exact functor ϕ J : N il ( A, S ) −→ N il ( A, S J ). These functors are compatible withtensor product with any power of Σ and induce morphisms of spectra on K-theory.Consider two words u and v in W ( I ). We have four A -bimodules: S u , S v , S uv and S vu . Using notations in 1.4, for each A -bimodule T and each ( i, j ) ∈ { , } weget an A × A -bimodule i T j where A × A acts on the left of T via the i -th projectionand on the right via the j -th projection. We have a commutative diagram of exactcategories: N il ( A, S ) F −−−−→ N il ( A × A, S u ⊕ S v ) ψ ( u,v ) −−−−→ N il ( A, S uv ) = y G y ∼ N il ( A, S ) F ′ −−−−→ N il ( A × A, S u ⊕ S v ) ψ ( v,u ) −−−−→ N il ( A, S vu )40here F and F ′ are defined by: F ( M, f ) = ( M ⊕ M , f u + f v ) F ′ ( M, f ) = ( M ⊕ M , f v + f u )for each ( M, f ) ∈ N il ( A, S ) and the morphisms G , ψ ( u, v ) and ψ ( v, u ) are definedby sending each ( M, M ′ , f + f ′ ) (with M, M ′ ∈ A , f : M −→ M ′ S v and f ′ : M ′ −→ M S u ) to G ( M, M ′ , f + f ′ ) = ( M ′ , M, f + f ′ ) ψ ( u, v )( M, M ′ , f + f ′ ) = ( M, f ′ f ) ψ ( v, u )( M, M ′ , f + f ′ ) = ( M ′ , f f ′ )Because of theorem 4, this diagram induces a homotopy commutative diagram ofΩ-spectra: N il ( A, S ) F −−−−→ N il ( A × A, S u ⊕ S v ) ψ ( u,v ) −−−−→ ∼ N il ( A, S uv ) = y G y ∼ H y ∼ N il ( A, S ) F ′ −−−−→ N il ( A × A, S u ⊕ S v ) ψ ( v,u ) −−−−→ ∼ N il ( A, S vu )where ψ ( u, v ) and ψ ( v, u ) are homotopy equivalences and H is the map ψ ( u, v ) − Gψ ( v, u )(up to homotopy).On the other hand we have: ψ ( u, v ) F = ϕ uv and ψ ( v, u ) F ′ = ϕ vu . Then we havea homotopy commutative diagram: N il ( A, S ) ϕ uv −−−−→ N il ( A, S uv ) = y H y ∼ N il ( A, S ) ϕ vu −−−−→ N il ( A, S vu )and the homotopy class of the map ϕ u depends only on the class of u in CW ( I ).Therefore, to prove the theorem, its is enough to prove it for some admissible set X . Suppose I has exactly two elements i and j . Let J ⊂ W ( I ) be the setof elements i k j , for k ≥ . Then the functors: ϕ i : N il ( A, S ) −→ N il ( A, S i ) and ϕ J : N il ( A, S ) −→ N il ( A, S J ) induce a homotopy equivalence of spectra: N il ( A, S ) ∼ −→ N il ( A, S i ) × N il ( A, S J )41 roof: We apply lemma 3.1 in the following case: B = A a E a = S i b E b = 0 b E a = A a E b = S j and we get a homotopy equivalence of spectra: N il ( A × A, E ) ∼ −→ N il ( A, S i ⊕ S j ) = N il ( A, S )induced by the functor (
P, Q, f ) ( P, a f a + a f b b f a ).By exchanging the role of a and b , we have also a homotopy equivalence of spectra: N il ( A × A, E ) ∼ −→ N il ( A, S i ) × N il ( A, ⊕ k ≥ S ki S j ) = N il ( A, S i ) × N il ( A, S J )and this map is induced by the two functors:( P, Q, f ) ( P, a f a )( P, Q, f ) ( Q, X k ≥ b f a ( a f a ) ka f b )Because of theorem 4, this last functor is, after applying the functor N il , equivalentto the functor: (
P, Q, f ) ( P, X k ≥ ( a f a ) ka f b b f a )Therefore we get a homotopy equivalence of spectra: N il ( A, S ) ∼ −→ N il ( A, S i ) × N il ( A, S J )induced by the two functors: ( P, f ) ( P, f i )( P, f ) ( P, X k ≥ f ki f j )and the lemma follows.From now one, the coproduct in the category of spectra will be denoted by ⊕ andthe trivial spectrum will be denoted by 0. Actually, is E j is a family of spectra, thespectrum ⊕ j E j is nothing else but the filtered colimit of finite products of the E j ’s.Let X be an admissible set in W ( I ). Then the map: X ⊂ W ( I ) π −→ CW ( I )induces a bijection X ∼ −→ CW ( I ).For every object ( M, f ) ∈ N il ( A, S ) there are only finitely many non zero mor-phisms f u and the product of the ϕ u , for u ∈ X , induces a map: N il ( A, S ) −→ Y u ∈ X N il ( A, S u )42ith values in ⊕ u ∈ X N il ( A, S u ). Hence we have a morphism of spectra: F : N il ( A, S ) −→ ⊕ u ∈ X N il ( A, S u )and the last thing to do is to prove that F is a homotopy equivalence.On the other hand, we have a homotopy commutative diagram:lim −→ N il ( A, ⊕ j ∈ J S j ) −−−−→ lim −→ (cid:16) ⊕ u ∈ X ∩ W ( J ) N il ( A, S u ) (cid:17) ∼ y ∼ y N il ( A, S ) −−−−→ ⊕ u ∈ X N il ( A, S u )where the limit is taken over all finite subset J of I . Moreover vertical arrows of thisdiagram are homotopy equivalences and, in order to prove the theorem, it is enoughto consider the case where I is finite.If I has at most 1 elements there is nothing to prove. So we may suppose that I is a finite set with at least 2 elements.If J is a subset of W ( I ) and j an element of J , we denote by Z ( J, j ) the set ofwords in W ( I ) on the form j p j ′ , where j ′ is any element in J distinct from j and p is any non negative integer. Suppose I is finite with at least elements. Then there exists asequence ( J n , j n ) , for n ≥ , with the following properties: • for every integer n ≥ , J n is a subset of W ( I ) , j n is an element of J n and J n +1 is the set Z ( J n , j n ) • the set Y = { j , j , j , j , . . . } ⊂ W ( I ) is admissible • for every integer p > , there is an integer m ≥ such that: ∀ n > m, ∀ u ∈ J n , | u | > p where | u | is the length of u . Proof:
In order to define the sequence ( J n , j n ), it’s enough to define J and to chooseeach j n in J n . So we set: J = I and, for each n , j n is chosen to be an element of J n of minimal length in W ( I ).Let n ≥ J = J n , j = j n and denote by J ′ the complement of j in J . The inclusion J n +1 = Z ( J, j ) ⊂ W ( I ) factorizes through W ( J ) and J n +1 canbe considered as a subset of W ( J ). Every word u in W ( J ) is written uniquely in thefollowing form: u = j n j j n j j n . . . j p j n p p ≥ n ∗ ≥ j ∗ ∈ J ′ .Suppose u is reduced.If p = 0, then u is a power of j and, because u is reduced, we have: u = j .If p >
0, then u is, up to conjugation, on the form: u = j n j j n j . . . j n p − j p and u is conjugate to an element in W ( Z ( J, j )) = W ( J n +1 ). Moreover u is reducedin W ( J ) if and only if u is reduced in W ( I ).Therefore every element in CW ( J ) = CW ( J n ) belongs to the image of the map CW ( J n +1 ) −→ CW ( J n ) except the element j = j n and the inclusion J n +1 ⊂ J n induces a bijection { j n } ` CW ( J n +1 ) ∼ −→ CW ( J n ). As a consequence, we have, foreach n ≥
0, a bijection: { j , j , . . . , j n − } a CW ( J n ) ∼ −→ CW ( I )For each integer n ≥
0, denote by p n the minimal length of the words in J n . Since J n +1 is contained in W ( J n ), we have: p n +1 ≥ p n and the sequence ( p n ) is increasing.We’ll prove that this sequence is unbounded.Let n ≥ u ∈ J n , we have: | u | ≥ p n . For every integer m , denote by H m the set of elements u ∈ J m with | u | = p n and by CH m its image in CW ( I ). Since I is finite and every element of H m is reduced in W ( J m ), CH m is afinite set contained in CW ( I ).Denote by q the cardinal of CH n . Since j n is an element in J n of minimallength j n belongs to H n . Then we have: q > { j n } ` CW ( J n +1 ) ∼ −→ CW ( J n ), the cardinal of CH n +1 is q −
1. For the samereason, we have the following: ∀ i ∈ { , , . . . , q } , card( CH n + i ) = q − i and then: card( CH n + q ) = 0 and p n + q > p n Therefore the sequence ( p n ) is unbounded and the last property of the lemma isproven.Since the map { j , j , . . . , j n − } ` CW ( J n ) ∼ −→ CW ( I ) is bijective, the map f : Y −→ CW ( I ) is injective. Let u be an element in CW ( I ) with length p . Sincethe sequence p n is unbounded, there is an integer n such that p n > p and u doesn’tbelong to the image of CW ( J n ) −→ CW ( I ). Hence u belongs to the image of { j , j , . . . , j n − } −→ CW ( I ) and f is surjective. Therefore Y is admissible and thelemma is proven.For every n > Y n the set { j , j , . . . , j n − } . For all integer n ≥ , the maps ϕ u : N il ( A, S ) −→ N il ( A, S u ) , for u ∈ Y n and the map ϕ J n : N il ( A, S ) −→ N il ( A, S J n ) induce a homotopy equivalenceof spectra: G n : N il ( A, S ) ∼ −→ ⊕ u ∈ Y n N il ( A, S u ) ⊕ N il ( A, S J n )44 roof: We’ll prove the lemma by induction on n . The map G is the identity (andthen a homotopy equivalence).Suppose n ≥ G n is a homotopy equivalence. Consider the bimodule S ′ = S ′ i ⊕ S ′ j with: S ′ i = S j n S ′ j = S J n \{ j n } By applying lemma 3.3 with this bimodule, we get a homotopy equivalence of spectra:
N il ( A, S ′ ) = N il ( A, S J n ) ∼ −→ N il ( A, S j n ) ⊕ N il ( A, b S )where b S is the following bimodule: b S = ⊕ u,k (cid:16) S j n (cid:17) k S u the sum being taken over all integer k ≥ u = j n in J n . So we haveisomorphisms: b S ≃ ⊕ u,k S j kn u ≃ ⊕ v ∈ J n +1 S v = S J n +1 and then a homotopy equivalence: N il ( A, S J n ) ∼ −→ N il ( A, S j n ) ⊕ N il ( A, S J n +1 )Therefore we have homotopy equivalences: N il ( A, S ) ∼ −→ ⊕ u ∈ Y n N il ( A, S u ) ⊕ N il ( A, S J n ) ∼ −→ ⊕ u ∈ Y n N il ( A, S u ) ⊕ N il ( A, S j n ) ⊕ N il ( A, S J n +1 ) ∼ −→ ⊕ u ∈ Y n +1 N il ( A, S u ) ⊕ N il ( A, S J n +1 )and G n +1 is a homotopy equivalence. The lemma follows by induction.We are now able to finish the proof of theorem 5.As we have said before it is enough to consider the case where I is finite with atleast two elements.Consider the map Φ : N il ( A, S ) −→ ⊕ u ∈ Y N il ( A, S u ) induced by the functors ϕ u : N il ( A, S ) −→ N il ( A, S u ). We have to prove that G is a homotopy equivalence and,for doing that, it will be enough to prove that G induces an isomorphism:Φ i : N il i ( A, S ) −→ ⊕ u ∈ Y N il i ( A, S u )for every integer i ∈ Z . 45et i ∈ Z be an integer and y be an element in ⊕ u ∈ Y N il i ( A, S u ). There is aninteger n ≥ y is in the direct sum of N il i ( A, S u ), for u ∈ Y n . Because oflemma 3.5, there is an element x ∈ N il i ( A, S ) such that: G n ( x ) = y ⊕ ∈ ⊕ u ∈ Y n N il i ( A, S u ) ⊕ N il i ( A, S J n )Hence x is sent to y by Φ i and Φ i is surjective.Suppose i ≥
0. Let x ∈ N il i ( A, S ) be an element killed by Φ i . This elementcan be lifted in an element y ∈ K i ( N il ( A, S )) = π i +1 ( B ( Q N il ( A, S ))) and there isa finite subcategory E of the Quillen category Q N il ( A, S ) such that y can be liftedin an element z ∈ π i +1 ( B E ). This category E involves only finitely many objects( M, f ) ∈ N il ( A, S ). Denote by F the set of these morphisms f .Since each f ∈ F is nilpotent, there is an integer p such that f u is trivial for each f ∈ F and each element u ∈ W ( I ) of length ≥ p . Because of the last property oflemma 3.4, the set of integers n such that there is some u ∈ J n and some f ∈ F with f u = 0 is finite. Hence, for n big enough, the composite functor E −→ Q N il ( A, S ) ϕ Jn −→ Q N il ( A, S J n )factorizes through the category Q P A and the composite mapΩ B E −→ Ω BQ N il ( A, S J n ) −→ N il ( A, S J n )is trivial. Hence z is killed in N il i ( A, S J n ) and x ∈ N il i ( A, S ) is killed by ϕ J n : N il i ( A, S ) −→ N il ( A, S J n ).But x is killed by Φ i and x is also killed by the map N il i ( A, S ) −→ ⊕ u ∈ Y n N il i ( A, S u ).Therefore x is killed by G n which is a homotopy equivalence and x is zero. Hencethe morphism Φ i is bijective for every integer i ≥ i is a negative integer, we may replace A and S by Σ p A and Σ p S for some p > − i and the bijectivity of Φ i + p for (Σ p A, Σ p S ) implies that Φ i (for ( A, S )) isbijective. Hence Φ is a homotopy equivalence of spectra. Moreover this is true forthe set Y and then for every admissible set in W ( I ). Then we get the desired resultand the theorem follows. Denote by S ′ the following bimodule: S ′ = S ⊕ ⊕ i E i ⊗ A i F i Let J be the disjoint union of I and { } . We set: ∀ i ∈ I, S i = E i ⊗ A i F i = E i F i S = S S ′ = ⊕ j ∈ J S j Because of theorem 5, there exist a family of A -bimodules U k , k ∈ K such that thefollowing holds: • each U k is flat on both sides • N il ( A, S ′ ) ≃ N il ( A, S ) ⊕ ⊕ k N il ( A, U k ) • each U k has the form: U k = S j S j . . . S j n , with j , j , . . . , j n in J and j q ∈ I forsome q .Then, because of theorem 4, there exist a family of A -bimodules V k , k ∈ K andelements i k ∈ I such that • each V k is flat on both sides • N il ( A, S ′ ) ≃ N il ( A, S ) ⊕ ⊕ k N il ( A, S i k V k )and we have, for each k : N il ( A, S i k V k ) = N il ( A, E i k F i k V k ) ≃ N il ( A i k , F i k V k E i k )But A i k is regular coherent and each spectrum N il ( A i k , F i k V k E i k ) is contractible.The result follows. With the notations of lemma 3.1, we denote by F = B ⊕ b E b ⊕ ( b E b ) ⊕ . . . thetensor algebra of b E b and by b E the bimodule b E = a E a ⊕ a E b F b E a .For every object ( M, f ) ∈ N il ( A × B, E ), we set: g = a f a , g = a f b , g = b f a , g = b f b Since f is nilpotent, there is an integer n > f n = 0. Then, for everyintegers i , i , . . . , i n in { , , , } the morphism g i g i . . . g i n is a part of f n and wehave: g i g i . . . g i n = 0. Hence the two morphisms b f b and b f = a f a + P k ≥ a f b ( b f b ) kb f a are nilpotent and we have a functorΦ( A, B, E ) : N il ( A × B, E ) −→ N il ( B, b E b ) × N il ( A, b E )sending each object ( M, f ) ∈ N il ( A × B, E ) to:Φ(
A, B, E )( M, f ) = (cid:16) ( M b , b f b ) , ( M a , b f ) (cid:17) Moreover this functor is exact.It is easy to see that lemma 3.1 is equivalent to the fact that Φ(Σ n A, Σ n , B, Σ n E )induces, for all n ≥
0, a homotopy equivalence in K-theory. Therefore it is enough toprove that Φ(
A, B, E ) induces a homotopy equivalence in K-theory for every left-flatbimodule ( A × B, E ) such that E b is flat on the right and that will be done by usingK-theory of Waldhausen categories. 47n our situation, we have three exact categories: the categories A , B and C of finitely generated projective right modules over the rings A , B and C = A × B respectively. These categories are contained in the corresponding abelian categories A ∨ , B ∨ and C ∨ of right modules over the corresponding rings.If ( A, S ) is a left-flat bimodule, we have also an exact category N il ( A, S ) and aWaldhausen category N il ( A, S ) ∗ . Moreover N il ( A, S ) is contained in the abeliancategory N il ( A, S ) ∨ (see part 1.3) and, as a subcategory of N il ( A, S ) ∨ , N il ( A, S )is stable under taking kernel of epimorphisms. Hence the Gillet-Waldhausen theoremapplies to the categories N il (? , ?) and we have a commutative diagram: N il ( A × B, E ) Φ( A,B,E ) −−−−→ N il ( B, b E b ) × N il ( A, b E ) y y N il ( A × B, E ) ∗ Φ( A,B,E ) −−−−→ N il ( B, b E b ) ∗ × N il ( A, b E ) ∗ of Waldhausen categories where the vertical functors induce homotopy equivalencesin K-theory.Therefore, in order to prove the lemma, it is enough to prove that the functor:Φ( A, B, E ) ∗ : N il ( A × B, E ) ∗ −→ N il ( B, b E b ) ∗ × N il ( A, b E ) ∗ induces a homotopy equivalence in K-theory.If ( A, S ) is a left-flat bimodule, it is easy to see that an object of N il ( A, S ) ∗ isnothing else but a pair ( M, f ), where M is in A ∗ and f : M −→ M S is a nilpotentmorphism in A ∨∗ . Then the functor Φ( A, B, E ) ∗ is given by:Φ( A, B, E ) ∗ ( M, f ) = (cid:16) ( M b , b f b ) , ( M a , b f ) (cid:17) for every M ∈ C ∗ = A ∗ × B ∗ .Consider the Waldhausen category E = N il ( A × B, E ) ∗ . We may define a newsubcategory of equivalences by saying that ϕ : ( M, f ) −→ ( M ′ , f ′ ) is an equivalenceif the induced morphism M a −→ M ′ a is an isomorphism in homology. With this newequivalences, we get a new Waldhausen category E ′ . We have also a Waldhausensubcategory E of E generated by the objects ( M, f ) ∈ E such that M a is acyclic.Denote also by N il the Waldhausen subcategory of N il ( A, b E ) ∗ generated by pairs( M, f ) with M acyclic. Hence we have a commutative diagram of essentially small48aldhausen categories: E −−−−→ E −−−−→ E ′ Φ y Φ( A,B,E ) y Φ ′ y N il ( B, b E b ) ∗ × N il −−−−→ N il ( B, b E b ) ∗ × N il ( A, b E ) ∗ −−−−→ N il ( A, b E ) ∗ Since every morphism in N il is an equivalence, the category N il has trivialK-theory and, because of the fibration theorem, the two lines of the diagram inducefibrations in K-theory. Hence in order to prove the lemma, it’s enough to prove thatΦ and Φ ′ induce homotopy equivalences on K-theory and that’s equivalent to showthat Φ ′ and the functor:Φ ′ : E −→ N il ( B, b E b ) ∗ × N il pr −→ N il ( B, b E b ) ∗ induce homotopy equivalences on K-theory. Moreover these two functors have theapproximation properties (App1). The functor Φ ′ induces a homotopy equivalence in K-theory. Proof:
Because of the approximation theorem 2.3, we just have to prove that Φ ′ hasthe property (App2).Let X = ( M, f ) ∈ E and Y = ( Q, g ) ∈ N il ( B, b E b ) ∗ be two objects in thecorresponding categories. A morphism ϕ : Φ ′ ( X ) −→ Y is represented by a morphism ϕ : M b −→ Q in B ∗ making the following diagram commutative: M b ϕ −−−−→ Q b f b y g y M b b E b ϕ −−−−→ Q b E b Denote by C the cylinder of ϕ and by C ′ its mapping cone (or its 0-cone as definedin 2.1). The morphisms b f b and g induce two nilpotent morphisms λ : C −→ C b E b λ ′ : C ′ −→ C ′ b E b . By naturality, we have a commutative diagram in C ∨∗ : M b i −−−−→ C p −−−−→ Q b f b y λ y g y M b b E b i −−−−→ C b E b p −−−−→ Q b E b where i : M b −→ C is a cofibration and p : C −→ Q a homotopy equivalence.Moreover we have: ϕ = pi .Define the object M ′ ∈ C ∗ by: M ′ a = M a and M ′ b = C . In order to define thedesired object X ′ ∈ E , we have to construct the morphism f ′ : M ′ −→ M ′ E .We set: a f ′ a = a f a , b f ′ b = λ and b f ′ a = i b f a . Since i : M b −→ C is a cofibrationand M a a E b is acyclic, there is no obstruction to extend a f b : M b −→ M a a E b to amorphism: a f ′ b : C −→ M a a E b . Hence we get a morphism f ′ : M ′ −→ M ′ E .Let C be the finite complex in C ∗ defined by: C a = 0 and C b = C ′ and λ : C −→ CE be the morphism in C ∨∗ associated with λ ′ under the canonical bijection:Hom C ∨∗ ( C, CE ) ≃ Hom B ∨∗ ( C ′ , C ′ b E b ). Since λ ′ is nilpotent, λ is also nilpotent.We have a commutative diagram in C ∨∗ with exact rows:0 −−−−→ M −−−−→ M ′ −−−−→ C −−−−→ f y f ′ y λ y −−−−→ M E −−−−→ M ′ E −−−−→ CE −−−−→ f and λ are nilpotent, f ′ is nilpotent also. More precisely, if we have: f p = 0 and λ q = 0, then we have f ′ p + q = 0.Since M ′ a is acyclic, X ′ = ( M ′ , f ′ ) is an object in E and the morphism i inducesa morphism α : X −→ X ′ . Moreover we have the following commutative diagram:Φ ′ ( X ) ϕ −−−−→ Y α y = y Φ ′ ( X ′ ) p −−−−→ Y and the property (App2) is satisfied. The lemma follows.50o the last thing to do is to prove that the functor Φ ′ : E ′ −→ N il ( A, b E ) ∗ )induces a homotopy equivalence in K-theory. As before, it’s enough to prove that Φ ′ has the property (App2). In order to do that we’ll need two technical results: Let A and B be two rings and S be an ( A, B ) -bimodule. Suppose S isflat on the left. Let X be a finite B -complex, Y be an A -complex and f : X −→ Y S be a morphism of B -complexes. Then there exist a finite A -complex Y ′ , a morphism g : Y ′ −→ Y and a commutative diagram: X −−−−→ Y ′ S = y g y X f −−−−→ Y S
Let A be a ring, X and Y two A -complexes and f : X −→ Y be amorphism. Suppose the module ⊕ n X n is finitely presented and each Y n is flat. Then f factorizes through a finite A -complex. These two lemmas will be proven at the end of the section.As a consequence of these two lemmas we have the following result:
Let A , B and C be three rings, S be a ( C, B ) -bimodule and T be a ( B, A ) -bimodule. Suppose S is flat on the right and T is flat on the left. Let X , Y and Z be finite complexes over the rings A , B and C respectively and f : X −→ Y T and g : Y −→ ZS be two morphisms. We suppose that the composite morphism: X f −→ Y T g −→ ZST is zero. Then the morphism g : Y −→ ZS factorizes through a finite complex Y ′ bya morphism λ : Y −→ Y ′ in such a way that the composite morphism: X f −→ Y T λ −→ Y ′ T is zero. Proof:
Let K be the kernel of g . Since T is flat on the left we have two exactsequences: 0 −→ K i −→ Y g −→ ZS −→ KT i −→ Y T g −→ ZST f factorizes through KT . Because of lemma 3.9 there exist a finite B -complex K ′ , a morphism j : K ′ −→ K and a commutative diagram: X f ′ −−−−→ K ′ T = y ij y X f −−−−→ Y T
Denote by Y the cokernel of ij : K ′ −→ Y . Since gi is the zero morphism, thereis a morphism g : Y −→ ZS making the following diagram commutative: Y g −−−−→ ZS y = y Y g −−−−→ ZS Since S is flat on the right, ZS is flat and, because of lemma 3.10, g factorizesthrough a finite complex Y ′ . The lemma follows. The functor Φ ′ : E ′ −→ N il ( A, b E ) ∗ ) has the property (App2). Proof:
Let’s set: H = a E a K = b E b S = a E b T = b E a b K = F = B ⊕ K ⊕ K ⊕ . . . Let X and Y be two objects in N il ( A × B, E ) ∗ and N il ( A, b E ) ∗ respectively and ϕ be a morphism from Φ ′ ( X ) to Y . By setting: ( U, f ) = X , ( M, g ) = Y , P = U a , Q = U b , we see that P and M are finite A -complexes, Q is a finite B -complex, ϕ isa morphism from P to M and g is written as a finite sum: g = λ + X k ≥ µ k where λ is a morphism from M to M H and each µ k is a morphism from M to M SK k T . Moreover the following diagrams are commutative: P ϕ −−−−→ M a f a y λ y P H ϕ −−−−→ M H P ϕ −−−−→ M a f b ( b f b ) kb f a y µ k y P SK k T ϕ −−−−→ M SK k T
52e want to construct an object X ′ ∈ N il ( A × B, E ) ∗ and a morphism α : X −→ X ′ such that Φ ′ ( α ) : Φ ′ ( X ) −→ Φ ′ ( X ′ ) is isomorphic to ϕ : Φ ′ ( X ) −→ Y . So we wantto have: X ′ = ( U ′ , f ′ ) and U ′ a = M and, to determine X ′ , we need to define Q ′ = U ′ b ,the morphism f ′ and the morphism α : Q −→ Q ′ . Actually, Q ′ will be constructedas a direct sum: Q ′ = ⊕ i ≥ Q i such that b f ′ b ( Q i +1 ) is contained in Q i K for all i ≥ i = − a f ′ b vanishes on each Q i , for i > B -complexes Q i , a morphism e : Q −→ M S andmorphisms α i : Q −→ Q i , θ i : Q i +1 −→ Q i K and β i : M −→ Q i T (with Q i = 0 for i big enough) and the morphism f ′ is defined by: a f ′ a = λ b f ′ b = X i ≥ θ i b f ′ a = X i ≥ β i a f ′ b = e pr where pr : Q ′ −→ Q is the projection. The morphism α is equal to ϕ : P −→ M on P and to P i ≥ α i : Q −→ Q ′ on Q .But we have two conditions: the fact that α is a morphism and the equality:Φ ′ ( X ′ ) = Y . These conditions are equivalent to: ϕ a f b = eα α i b f a = β i ϕθ i α i +1 = α i b f b µ i = eθ θ . . . θ i − β i for all i ≥ e i = eθ θ . . . θ i − from Q i to M SK i . Then we have to construct, for each i ≥
0, the complex Q i and morphisms e i : Q i −→ M SK i , α i : Q −→ Q i , β i : M −→ Q i T and θ i : Q i +1 −→ Q i K , with thefollowing properties: A ( i ): ϕ a f b ( b f b ) i = e i α i B ( i ): α i b f a = β i ϕC ( i ): θ i α i +1 = α i b f b D ( i ): µ i = e i β i E ( i ): e i +1 = e i θ i for all i ≥ µ k ’s is finite, there is an integer n > µ i = 0 forall i > n .So we’ll construct ( Q i , e i , α i , β i , θ i ) by induction. Let i ≥ Q j , e j , α j , β j , θ j ) is defined for all j > i such that the properties A ( j ), B ( j ), C ( j ), D ( j ), E ( j ), are satisfied for all j > i . We begin this induction with i = n by setting: Q j = 0 for all j > n .We have to construct Q i and morphisms e i , α i , β i and θ i .Consider the following morphisms: ϕ a f b ( b f b ) i : Q −→ M SK i µ i : M −→ M SK i T i +1 : Q i +1 −→ M SK i +1 These morphisms induce a morphism h : Q ⊕ M ⊕ Q i +1 −→ M SK i ( B ⊕ T ⊕ K ) and,because of lemma 3.9, there are a finite complex Q i , a morphism e i : Q i −→ M SK i and a commutative diagram: Q ⊕ M ⊕ Q i +1 h ′ −−−−→ Q i ( B ⊕ T ⊕ K ) = y e i y Q ⊕ M ⊕ Q i +1 h −−−−→ M SK i ( B ⊕ T ⊕ K )The morphism h ′ induces morphisms: α i : Q −→ Q i β i : M −→ Q i Tθ i : Q i +1 −→ Q i K and properties A ( i ), D ( i ), ( E ( i ) are satisfied. Denote by u and v the defaults ofproperties B ( i ) and C ( i ): u = β i ϕ − α i b f a v = α i b f b − θ i α i +1 Because of properties A ( i ) and D ( i ), we have: e i u = e i β i ϕ − e i α i b f a = µ i ϕ − ϕ a f b ( b f b ) ib f a = 0and, because of properties A ( i ), E ( i ) and A ( i + 1), we have: e i v = e i α i b f b − e i θ i α i +1 = ϕ a f b ( b f b ) i +1 − e i +1 α i +1 = 0Since a tensor product of bimodules which are flat on the right is flat on the right,we can apply the lemma 3.11 to morphisms u ⊕ v : P −→ Q i ( T ⊕ K ) and e i : Q i −→ M SK i . Thus e i factorizes through a finite complex Q ′ i by a morphism ε : Q i −→ Q ′ i such that: ε ( u ⊕ v ) = 0.Hence, up to replacing Q i by Q ′ i , we may as well suppose that A ( i ), B ( i ), C ( i ), D ( i ), E ( i ) are satisfied. Therefore Q i , e i , α i , β i , θ i are defined and A ( i ), B ( i ), C ( i ), D ( i ), E ( i ) are satisfied for all i ≥ Q ′ = ⊕ i ≥ Q i is constructed and the morphism f ′ is definedby: a f ′ a = λ b f ′ b = ⊕ i ≥ θ i f ′ b = e pr b f ′ a = ⊕ i ≥ β i Hence the desired object X ′ is constructed and Φ ′ has the approximation property(App2). The lemma follows and then follow lemma 3.1 and theorems 4, 5 and 6. The situation is the following: (
A, S ) is a left-flat bimodule, X is a finite A -complex, Y is an A -complex and f : X −→ Y S is a morphism of A -complexes.We want to construct a finite A -complex Y ′ and a morphism Y ′ −→ Y such that f : X −→ Y S factorizes through Y ′ S .For each integer n , denote by X ( n ) the n -skeleton of X . Let n be an integer.Suppose the n -skeleton Y ′ ( n ) of Y ′ is constructed in such a way that we have amorphism g n : Y ′ ( n ) −→ Y and a commutative diagram:( D n ) X ( n ) h n −−−−→ Y ′ ( n ) S = y g n y X ( n ) f −−−−→ Y S If n is small enough, X ( n ) is null and Y ′ ( n ) can be chosen to be zero.Denote by Z n the kernel of the morphism d : Y ′ n −→ Y ′ n − and by U n +1 the moduledefined by the cartesian square: U n +1 α −−−−→ Y n +1 β y d y Z n g n −−−−→ Y b The composite morphism X n +1 d −→ X n h n −→ Y ′ n S takes values in Z n S and induces,together with the morphism f : X n +1 −→ Y n +1 S , a well defined morphism λ :55 n +1 −→ U n +1 . So we get a commutative diagram: X n +1 λ −−−−→ U n +1 S α −−−−→ Y n +1 S d y γ y d y X n h n −−−−→ Y ′ n S g n −−−−→ Y n S where γ is the composite morphism U n +1 β −→ Z n ⊂ Y ′ n .Since X n +1 is finitely generated, there is a finitely generated submodule M in U n +1 such that λ ( X n +1 ) is contained in M S . Let Y ′ n +1 be a finitely generated projective A -module and µ : Y ′ n +1 −→ M be an epimorphism. Since X n +1 is projective, themorphism λ : X n +1 −→ Y ′ S can be lifted in a morphism X n +1 −→ Y ′ n +1 S and we geta commutative diagram: X n +1 h −−−−→ Y ′ n +1 S g −−−−→ Y n +1 d y d y d y X n −−−−→ Y ′ n S −−−−→ Y n S where d : Y ′ n +1 −→ Y ′ n is the morphism γµ and g : Y ′ n +1 −→ Y n +1 is the morphism αµ .Thus we have constructed the complex Y ′ ( n + 1) and the commutative diagram( D n +1 ). By induction we have Y ′ ( n ) and the commutative diagram ( D n ) for everyinteger n and, for n big enough, Y ′ ( n ) and the diagram ( D n ) is a solution of theproblem. In the lemma, f : X −→ Y is a morphism between two A -complexes, the directsum of the X ′ n s if finitely presented and each Y n is flat.For every A -complex E , denote by E n its n -coskeleton i.e. the quotient of E byits ( n − n be an integer. Suppose that the n -coskeleton F n of a finite complex F isconstructed in such a way that the morphism f : X n −→ Y n induced by f : X −→ Y factorizes through F n via two morphisms α : X n −→ F n and β : F n −→ Y n . If n isbig enough X n is trivial and we may set: F n = 0.56e have a commutative diagram: X n +1 α −−−−→ F n +1 β −−−−→ Y n +1 d y d y d y X n α −−−−→ F n β −−−−→ Y n Let E be the A -module defined by the cocartesian square: X n −−−−→ F ′ nd y y X n − −−−−→ E where F ′ n is the cokernel of the morphism d : F n +1 −→ F n . Since X n , X n − and F ′ n are finitely presented, the A -module E is also finitely presented.We have a commutative diagram: X n +1 α −−−−→ F n +1 β −−−−→ Y n +1 d y d y d y X n α −−−−→ F n β −−−−→ Y nd y δ y d y X n − −−−−→ E −−−−→ Y n − where the composite morphism F n +1 d −→ F n −→ E is trivial.But E is finitely presented and Y n − is flat. Therefore the morphism E −→ Y n − factorizes through a finitely generated projective A -module F n − and, together withthe composite morphism F n δ −→ E −→ F n − , we get the desired finite complex F n − X n − α −−−−→ F n − β −−−−→ Y n − y y y X n α −−−−→ F n β −−−−→ Y n So we construct the complexes F n inductively and, for n small enough, the mor-phism f : X −→ Y factorizes through the finite complex F = F n . Remark:
It is not clear that the right flatness condition is necessary in theorems4, 5 and 6. Actually this condition is only used in order to prove that the functorΦ ′ : E ′ −→ N il ( A, b E ) ∗ (in the proof of lemma 3.1) is a homotopy equivalence. Theproof given here needs the right flatness condition (in the lemma 3 .10) but anotherproof without this condition is still possible. If E is an Ω-spectrum and X is a space, we denote by H ( X, E ) the Ω-spectrumassociated to the smash product X ∧ E . For every i ∈ Z , we have: π i ( H ( X, E )) ≃ H i ( X, E )In [Wa1], section 15, Waldhausen associates to any ring R and any group G anassembly map: H ( BG, K ( R )) −→ K ( R [ G ]) which is a map of infinite loop spaces.This assembly map induces assembly maps H ( BG, Σ n K ( R )) −→ K (Σ n R [ G ]) andthen an assembly map h : H ( BG, K ( R )) −→ K ( R [ G ]) which is a map of spectra. Sowe get a fibration of spectra: H ( BG, K ( R )) h −→ K ( R [ G ]) −→ W h R ( G )The spectrum W h R ( G ) is called the Whitehead spectrum of G relative to R .For every integer i , we set: W h i ( G ) = π i ( W h Z ( G ))For i <
0, the group
W h i ( G ) is isomorphic to K i ( Z [ G ]) and we have exact sequences:0 −→ Z −→ K ( Z [ G ]) −→ W h ( G ) −→ −→ Z / ⊕ H ( G, Z ) −→ K ( Z [ G ]) −→ W h ( G ) −→ W h ( G ) is the reduced K -group of Z [ G ], W h ( G ) is the classicalWhitehead group of G and W h ( G ) is the second Whitehead group of G as definedin [HW]. 58ollowing Waldhausen, a group G is said to be regular noetherian (resp. regularcoherent) if, for every ring R which is regular noetherian on the right, R [ G ] is regularnoetherian (resp. regular coherent) on the right. Since R op [ G ] is isomorphic to( R [ G ]) op , this condition is equivalent to the condition obtained by replacing right byleft. We denote also by G the category of groups and monomorphisms of groups. We have the following properties: • If G is the amalgamated free product of a diagram in G : H −−−−→ G y G where G and G are regular coherent and H regular noetherian, then G is regularcoherent. • If G is the HNN extension of a diagram in G : H α −−−→−−−→ β G where G is regular coherent and H is regular noetherian, then G is regular coherent. • If G is the colimit of a filtered system G i in G , where each G i is regular coherent,then G is regular coherent. • A subgroup of a regular coherent group is regular coherent.
Proof:
All these properties are proven in [Wa1] (in theorem 19.1) except the thirdone.Let G be the colimit of a filtered system G i in G and R be a ring which is regularnoetherian on the right. Suppose each G i is regular coherent. Set: A = R [ G ] and A i = R [ G i ].Each ring A i is regular coherent on the right and for each i ∈ I , the ring A is freeon the left over A i .Let M be a finitely presented right A -module. We have an exact sequence of right A -modules: F f −→ F −→ M −→ F and F are finitely generated free A -modules. The morphism f is representedby a finite matrix with entries in A . Since A is the colimit of the A i ’s, there is anelement i ∈ I such that A i contains all the entries of f . Therefore f comes from afinite matrix with entries in A i and there exist a finitely presented right A i -module M ′ and an isomorphism M ′ ⊗ A i A ≃ M .Since A i is regular coherent on the right we have an exact sequence of right A i -modules: 0 −→ C n −→ C n − −→ . . . −→ C −→ M ′ −→ C k is finitely generated projective. Since A is free on the left over A i , wehave an exact sequence of right A -modules:0 −→ C n ⊗ A i A −→ C n − ⊗ A i A −→ . . . −→ C ⊗ A i A −→ M −→ C k ⊗ A i A is finitely generated projective right A -module. Therefore everyfinitely presented A -module has a finite resolution by finitely generated projective A -modules and A is regular coherent. The result follows.Let Cl be the class of groups defined by Waldhausen in [Wa1]. This class is thesmallest class of groups satisfying the following: • The trivial group belongs to Cl. • If G is the amalgamated free product of a diagram in G : H −−−−→ G y G where G and G are in Cl and H regular coherent, then G belongs to Cl. • If G is the HNN extension of a diagram in G : H α −−−→−−−→ β G where G is in Cl and H is regular coherent, then G belongs to Cl. • If G is the colimit of a filtered system G i in G , where each G i is in Cl, then G belongs to Cl.This class contains free groups, torsion free abelian groups, poly- Z -groups, torsionfree one-relator groups and fundamental groups of many low-dimensional manifolds.It is also closed under taking subgroups. See theorem 19.5 in [Wa1]. For every group G in Cl and every ring R which is regular noetherianon the right, the Whitehead spectrum W h R ( G ) is contractible. Proof:
This is essentially theorem 19.4 in [Wa1]. We just have to replace spaces
N il ( A, S ) by spectra
N il ( A, S ). Since all these spectra are contractible, the resultfollows.We’ll construct a class of groups Cl obtained by replacing the condition ” H isregular coherent” (in the definition of Cl) by a weaker condition in such a way thattheorem 4.2 is still true for groups in Cl .60onsider a diagram of groups: H α −−−−→ G β y G We say that this diagram is regular coherent if the following holds: • α and β are monomorphisms • for every x ∈ G \ α ( H ) and every y ∈ G \ β ( H ), the intersection of the twogroups α − ( xα ( H ) x − ) and β − ( yβ ( H ) y − ) is regular coherent.Consider a diagram of groups: H α −−−→−−−→ β G We say that this diagram is regular coherent if the following holds: • α and β are monomorphisms • for every x ∈ G \ α ( H ) and every y ∈ G \ β ( H ), the intersection of the twogroups α − ( xα ( H ) x − ) and β − ( yβ ( H ) y − ) is regular coherent. • for every x ∈ G , the group β − ( xα ( H ) x − ) is regular coherent.Since the condition ”regular coherent” is stable under taking subgroups, it is easyto see that diagrams above are regular coherent if the subgroup H is regular coherent.So we define the class Cl as the smallest class of groups satisfying the following: • The trivial group belongs to Cl . • If G is the amalgamated free product of a diagram D : H −−−−→ G y G where G and G are in Cl and D is regular coherent, then G belongs to Cl . • If G is the HNN extension of a diagram D ′ : H α −−−→−−−→ β G where G is in Cl and D ′ is regular coherent, then G belongs to Cl . • If G is the colimit of a filtered system G i in G , where each G i is in Cl , then G belongs to Cl . 61 .3 Theorem: Let R be a ring which is regular noetherian on the right and G bethe amalgamated free product of a regular coherent diagram of groups: H α −−−−→ G β y G Then this diagram induces a homotopically cartesian diagram of spectra:
W h R ( H ) α −−−−→ W h R ( G ) β y y W h R ( G ) −−−−→ W h R ( G ) Let R be a ring which is regular noetherian on the right and G bethe HNN extension of a regular coherent diagram of groups: H α −−−→−−−→ β G Then this diagram induces a homotopy fibration of spectra:
W h R ( H ) f −→ W h R ( G ) −→ W h R ( G ) where f is the difference (in Ω sp) of maps induced by α and β . Proofs of theorems 4.3 and 4.4:
In the amalgamated case, we have a commutativediagram of spectra: H ( BH, K ( R )) α ⊕− β −−−−→ H ( BG , K ( R )) ⊕ H ( BG , K ( R )) −−−−→ H ( BG, K ( R )) y y y K ( R [ H ]) α ⊕− β −−−−→ K ( R [ G ]) ⊕ K ( R [ G ]) −−−−→ K ( R [ G ]) ′ K ( R [ G ]) ≃ K ( R [ G ]) ′ ⊕ Ω − N il ( R [ H ] × R [ H ] , S )for some R [ H ] × R [ H ] bimodule S . Therefore we have a fibration: W h R ( H ) −→ W h R ( G ) ⊕ W h R ( G ) −→ W h R ( G ) ′ and a homotopy equivalence: W h R ( G ) ≃ W h R ( G ) ′ ⊕ Ω − N il ( R [ H ] × R [ H ] , S )We can do the same for the HNN extension and we get a fibration: W h R ( H ) f −→ W h R ( G ) −→ W h R ( G ) ′ and a homotopy equivalence: W h R ( G ) ≃ W h R ( G ) ′ ⊕ Ω − N il ( R [ H ] × R [ H ] , S )for some R [ H ] × R [ H ] bimodule S .Hence the only thing to do is to prove that N il ( R [ H ] × R [ H ] , S ) is contractible.Let’s denote by C the ring R [ H ]. With the notations of 1.4, the bimodule S isdetermined by four C -bimodules i S j (with i, j ∈ { , } ). In order to describe thesebimodules we’ll introduce the following terminology:Let G be a group. A G -biset is a set X equipped with two compatible actions of G , one on the left and the other one on the right. We say that a G -biset X is free ifboth actions on X are free. If R is a ring, R [ X ] is naturally a R [ G ]-bimodule and, if X is free, R [ X ] is free on both sides. For any free G -biset X and any ring R , we set: N il R ( G, X ) =
N il ( R [ G ] , R [ X ])If G and G are two groups, we can also define a ( G , G )-biset as a set equippedwith two compatible actions: a left action of G and a right action of G . Then, forevery ring R and any ( G , G )-biset X , R [ X ] is a ( R [ G ] , R [ G ])-bimodule.Consider the amalgamated case. We have two monomorphisms α : H −→ G and β : H −→ G . Denote by X the complement of α ( H ) in G and by Y the complementof β ( H ) in G . The group H acts on both sides on X and Y and X and Y are free H -bisets. Moreover we have: S = R [ X ] S = R [ Y ] S = S = 0and then: S = R [ X ] ⊕ R [ Y ]
63n the HNN extension case we have two monomorphisms α : H −→ G and β : H −→ G . Denote by X the complement of α ( H ) in G and by Y the complementof β ( H ) in G . Denote also by U (resp. V) the set G where H acts on the left by β and on the right by α (resp. H acts on the right by β and on the left by α ). Thesesets X , Y , U and V are free H -bisets. In this case, the bimodule S is characterizedby the conditions: S = R [ X ] S = R [ Y ] S = R [ U ] S = R [ V ]and then: S = R [ X ] ⊕ R [ Y ] ⊕ R [ U ] ⊕ R [ V ] Consider the HNN extension case. For every x ∈ G we set:Γ( x ) = β − ( xα ( H ) x − ) Γ ′ ( x ) = α − ( xβ ( H ) x − )For each x ∈ G , Γ( x ) and Γ ′ ( x ) are subgroups of H and Γ( x ) is regular coherent.Moreover, we have a group homomorphism λ x : Γ( x ) −→ H such that: ∀ γ ∈ Γ( x ) , β ( γ ) = xα ( λ x ( γ )) x − It is easy to see that λ x is an isomorphism from Γ( x ) to Γ ′ ( x − ). Thus groups Γ ′ ( x )are also regular coherent.Let: U = ` i U i be the decomposition of U by orbits. Then, for every i , there exists an element x ∈ U such that: U i = β ( H ) xα ( H )Let H (resp. x H ) be the ( H, Γ( x ))-biset (resp. the (Γ( x ) , H )-biset) H , where H actsin the standard way on the left (resp. on the right) and Γ( x ) acts by the inclusion onthe right (resp. by the morphism λ x on the left). Then the map: ( u, v ) β ( u ) xα ( v )from H × H to β ( H ) xα ( H ) induces an isomorphism of H -bisets: H × Γ( x ) x H ≃ β ( H ) xα ( H )and we have: R [ U i ] ≃ R [ H × Γ( x ) x H ]= ⇒ R [ U i ] ≃ R [ H ] ⊗ R [Γ( x )] R [ x H ] Moreover the ring R [Γ( x )] is regular coherent.Hence, because of theorem 6, we have a homotopy equivalence of spectra: N il ( C × C, R [ X ] ⊕ R [ Y ] ⊕ R [ V ] ) ∼ −→ N il ( C × C, R [ X ] ⊕ R [ Y ] ⊕ R [ U ] ⊕ R [ V ] )64e proceed the same with the biset V and we get a homotopy equivalence ofspectra: N il ( C × C, R [ X ] ⊕ R [ Y ] ) ∼ −→ N il ( C × C, R [ X ] ⊕ R [ Y ] ⊕ R [ V ] )Hence, in both amalgamated case and HNN extension case, we have a homotopyequivalence: N il ( C × C, R [ X ] ⊕ R [ Y ] )) ∼ −→ N il ( C × C, S )and, because of theorem 4, we have a homotopy equivalence:
N il ( C × C, S ) ≃ N il ( C, R [ X × H Y ])Denote by Z j the orbits of the biset X × X Y . Then, for each j , there is an element( x, y ) ∈ X × Y such that: Z j = α ( H ) xyβ ( H )For each x ∈ X and each y ∈ Y we have the groups:Γ ( x ) = α − ( xα ( H ) x − ) Γ ( y ) = β − ( yβ ( H ) y − ) H ( x, y ) = Γ ( x ) ∩ Γ ( y )We have group homomorphisms λ x : Γ ( x ) −→ H and µ y : Γ ( y ) −→ H defined by: ∀ γ ∈ Γ ( x ) , α ( λ x ( γ )) = xα ( γ ) x − ∀ γ ∈ Γ ( y ) , β ( γ ) = yβ ( µ y ( γ )) y − Denote by H x the ( H, H ( x, y ))-biset H where H acts in the standard way onthe left and H ( x, y ) acts via λ x on the right. Denote also by y H the ( H ( x, y ) , H )-biset where H acts in the standard way on the right and H ( x, y ) acts via µ y in theleft. Then the map ( u, v ) α ( u ) xyβ ( v ) from H × H to α ( H ) xyβ ( H ) induces anisomorphism: H x × H ( x,y ) y H ∼ −→ α ( H ) xyβ ( H )where H ( x, y ) is regular coherent. Hence, because of theorem 6, the spectrum N il ( R [ H ] , R [ X × H Y ]) is contractible and so is N il ( R [ H ] × R [ H ] , S ). For every group G in Cl and every ring R which is regular noetherianon the right, the Whitehead spectrum W h R ( G ) is contractible. Proof:
Denote by Cl the class of groups G such that W h R ( G ) is contractible forevery ring R which is regular noetherian on the right. Because of theorem 4.2, Cl contains the class Cl.Since the functor W h commutes with filtered colimits, the class Cl is stableunder filtered colimits. Therefore it’s enough to prove that Cl is stable under takingamalgamated free products and HNN extensions of regular coherent diagrams. Butthat follows directly from theorems 4.3 and 4.4.65 .7 Example: Consider the group H with two generators x and t and the followingrelations: ∀ n ∈ Z , xt n xt − n = t n xt − n x For every integer p = 0, the correspondence x x and t t p induces a monomor-phism f p : H −→ H . Consider a monomorphism of groups α : H −→ G and denoteby Γ the amalgamated free product of the diagram: H f p −−−−→ H α y G For every ring R which is regular noetherian on the right, the mor-phism G −→ Γ induces a homotopy equivalence of spectra: W h R ( G ) ∼ −→ W h R (Γ) .Moreover, if G belongs to Cl , then the group Γ is also in Cl . Proof:
Denote by H ′ the normal closure of x in H . This group is commutativeand freely generated by the elements: x n = t n xt − n for n ∈ Z . The correspondence. x n x n +1 is an automorphism τ : H ′ ∼ −→ H ′ and H is the semidirect product of H ′ and Z , or equivalently, the HNN extension of H ′ with morphisms Id, τ : H ′ −→ H ′ .On the other hand, R [ H ′ ] is regular coherent on the right (but not noetherian) and H belongs to the class Cl. Hence the Whitehead spectrum W h R ( H ) is contractible.Denote by H p the image of f p : H −→ H and by X its complement in H . Forevery z ∈ H , denote by Γ( z ) the subgroup f − p ( zf p ( H ) z − ) of H . We have thefollowing formula: Γ( f p ( a ) zf p ( b )) = a Γ( z ) a − for every a, b, z in H and the conjugacy class of Γ( z ) depends only on the class of z in the set Y = H p \ H/H p . Let z be an element in X . A direct computation showsthe following: • if z is congruent in Y to an element in H ′ then Γ( z ) is the group H ′ • if z is congruent in Y to a power of t then Γ( z ) is conjugate to the subgroup of H generated by t • in the other cases Γ( z ) is the trivial group.Therefore Γ( z ) is always a free abelian group. Hence, for every y in G \ α ( H ),the group Γ( z, y ) = Γ( z ) ∩ α − ( yα ( H ) y − ) is also a free abelian group and the ring R [Γ( z, y )] is regular coherent on the right. Then theorem 4.3 applies and the resultfollows.The class Cl seems to be strictly bigger than the class Cl. For example the66malgamated free product Γ of the diagram: H f p −−−−→ H f q y H with p, q >
1, belongs to the class Cl . But in this case, Waldhausen’s theoremscannot be used to prove that Γ belongs to the class Cl because of the following result: The ring Z [ H ] is not regular coherent. Proof:
Let f : Z [ H ] ⊕ Z [ H ] −→ Z [ H ] be the following morphism:( U, V ) f ( U, V ) = (1 − t + tx ) U − (1 − t + t xt − ) V and K be its kernel. We’ll prove that K is not finitely generated and that will implythat Z [ H ] is not coherent and therefore not regular coherent.Denote by A the ring Z [ H ′ ]. This ring is the ring of Laurent polynomials in the x i ’s. Then A is an integral domain and every element u ∈ Z [ H ] can be written in aunique way on a finite sum: u = X i ∈ Z t i u i with each u i in A . So u may be considered as a Laurent polynomial in t and hasa valuation ν ( u ) and a degree ∂ ◦ u (at least if u is not zero). If u = 0, we set: ν ( u ) = + ∞ and ∂ ◦ u = −∞ .Define the elements y i and z i in A by: ∀ i ∈ Z , y i = 1 − x i = 1 − t i xt − i z i = y i − y i − = x i − − x i and, for every integer n ≥
0, we have the following elements in Z [ H ]: U n = z − n − t n +1 z y y − . . . y − n V n = z − n + X
Denote by E the Z [ H ]-submodule of Z [ H ] ⊕ Z [ H ] generated by the set { W , W , W , . . . } . Since each W n is in K , we have an inclusion E ⊂ K and we haveto prove that this inclusion is an equality.67or each integer n ≥
0, denote by K n the set of the elements ( U, V ) in K satisfyingthe following: ν ( U ) ≥ ∂ ◦ U ≤ n ν ( V ) ≥ ∂ ◦ V ≤ n We denote also by I n the ideal ( z , z − , z − , . . . , z − n ) ⊂ A and by J n the right idealof Z [ H ] generated by I n .The quotient B n = A/I n is the quotient of A by the relations x = x − = x − = . . . = x − n and B n is a Laurent polynomial ring where z , z − n and all y i are not zero.Suppose we have proven that K n − is contained in E . Let U = P ≤ i ≤ n t i u i and V = P ≤ i ≤ n t i v i be two elements in Z [ H ], with u i and v i in A . For W = ( U, V ) we havethe following equivalences: W ∈ K n ⇐⇒ (1 − ty ) U = (1 − ty ) V ⇐⇒ X i t i ( u i − v i ) = ty X i t i u i − ty X i t i v i ⇐⇒ X i t i ( u i − v i ) = X i t i +1 y − i u i − X i t i +1 y − i v i ⇐⇒ ∀ i, u i − v i = y − i u i − − y − i v i − And these conditions are equivalent to the following: v = u v = u + z u v = u + z u + z y u . . .v n = u n + X ≤ i 0. On the other hand, for every W ∈ K ,there is some integer p such that W t p belongs to some K n . Hence K is contained in E and the lemma is proven. The module K is not finitely generated. Proof: For each integer n > 0, denote by E n the submodule of E generated by { W , W , W , . . . , W n − } .Let F n : Z [ H ] n −→ Z [ H ] ⊕ Z [ H ] be the morphism:( c , c , c , . . . , c n − ) X ≤ i 69s zero in K and induces a well defined element X ( p, q ) ∈ R n (for every n > q ) and,for every p with 0 ≤ p < n − 1, we have: π n ( X ( p, n − z − p .Let n > J ′ n is not contained in J n . Then there is anelement X ∈ R n such that π n ( X ) doesn’t belong to J n . Set: d = ∂ ◦ π n ( X ) anddenote by Z the set of X ∈ R n such that: ∂ ◦ π n ( X ) = d and π n ( X ) − π n ( X ) ∈ J n .For each X = ( c , c , . . . , c n − ) ∈ Z we can associate three integers α, β, γ definedthis way: • α = ν ( c n − ) • β is the lowest ν ( c k ), for k = 0 , , . . . , n − • γ is the highest integer k such that ν ( c k ) = β .The triple χ ( X ) = ( α, β, γ ) will be called the complexity of X . This complexitybelongs to the set C of triple ( α, β, γ ) ∈ Z satisfying the following conditions: β ≤ α ≤ d and 0 ≤ γ < n The lexicographical order of ( d − α, α − β, γ ) induces a well order relation on C and we have:( α, β, γ ) < ( α ′ , β ′ , γ ′ ) ⇐⇒ α > α ′ or α = α ′ and β > β ′ or ( α, β ) = ( α ′ , β ′ ) and γ < γ ′ Since C is well ordered, there is an element in Z with a minimal complexity. Let X = ( c , c , . . . , c n − ) be such an element.For each integer k ∈ { , , . . . , n − } , we have a decomposition: c k = X β ≤ i c ki t i with c ki ∈ A .The condition X ∈ R n implies the following: X ≤ k ≤ γ z − k c kβ = 0and that implies the congruence z − γ c γβ ≡ B γ = A/I γ . But z − γ is not a zerodivisor in B γ and c γβ belongs to I γ . So we have a decomposition in A : c γβ = X ≤ j<γ z − j a j and we get a new element in R n : X ′ = X − X ≤ j<γ X ( j, γ ) a j t β It is easy to see that π n ( X ′ ) ≡ π n ( X ) ≡ π n ( C ) mod J n and that X ′ belongs to Z . Moreover we have the following: χ ( X ′ ) < χ ( X ). But that’s impossible because X was chosen with a minimal complexity.70ence we get a contradiction and the module J ′ n is contained in J n . As a conse-quence, by killing all the z i ’s, we get epimorphisms: Z [ H ] /J ′ n −→ Z [ H ] /J n −→ Z [ x ± , t ± ]and Z [ H ] /J ′ n is not trivial. Hence the sequence E ⊂ E ⊂ E ⊂ . . . is strictlyincreasing and E =Ker( f ) is not finitely generated. Therefore the category of finitelypresented right Z [ H ]-modules is not abelian and Z [ H ] is not coherent. The resultfollows. References: [B] H. Bass – Algebraic K-theory , Benjamin (1968).[C] P. M. Cohn – Free ideal rings , J. Algebra (1964) 47–69.[HW] A. Hatcher and J. Wagoner – Pseudo isotopies of compact manifolds , Ast´erisque , (1973).[K] B. Keller – Chain complexes and stable categories , Manuscripta Mathematica. (1990), 379-417. doi:10.1007/BF02568439.[KV] M. Karoubi and O. Villamayor – K-th´eorie alg´ebrique et K-th´eorie topologique ,Math. Scand. (1971), 265–307.[Q] D. Quillen – Higher algebraic K-theory I , Proc. Conf. alg. K-theory, LectureNotes in Math. (1973), 85–147.[TT] R. W. Thomason and T. Trobaugh – Higher algebraic K-Theory of schemes andof derived categories , The Grothendieck Festschrift III, Progress in Math., ,Birkh¨auser Boston, Boston, MA (1990) 247-435. MR1106918[Wa1] F. Waldhausen – Algebraic K-theory of generalized free products, part 1 & 2 ,Annals of Math. (1978), 135–256.[Wa2] F. Waldhausen – Algebraic K-theory of spaces , Algebraic and geometric topology(New Brunswick, N.J., 1983), 318-419, Lecture Notes in Math. Springer,Berlin, (1985).[We] C. A. Weibel – The K-book: An introduction in algebraic K-theory , GraduateStudies in Mathematics,145