Vanishing of Nil-terms and negative K-theory for additive categories
aa r X i v : . [ m a t h . K T ] J a n VANISHING OF NIL-TERMS AND NEGATIVE K -THEORY FORADDITIVE CATEGORIES BARTELS, A. AND L ¨UCK, W.
Abstract.
We extend the notion of regular coherence from rings to additivecategories and show that well-known consequences of regular coherence forrings also apply to additive categories. For instance the negative K -groups andall twisted Nil-groups vanish for an additive category if it is regular coherent.This will be applied to nested sequences of additive categories, motivated byour ongoing project to determine the algebraic K -theory of the Hecke algebraof a reductive p -adic group. Introduction
Background.
The Bass-Heller-Swan Theorem gives isomorphisms K n R [ t, t − ] ∼ = K n − ( R ) ⊕ K n ( R ) ⊕ f Nil n − ( R ) ⊕ f Nil n − ( R )for K-theory of rings. For regular rings all Nil groups f Nil n ( R ), n ∈ Z and neg-ative K-groups K n ( R ), n ∈ Z < vanish and this simplifies the Bass-Heller-Swanformula. Waldhausen [18, 19] proved far reaching extensions of the Bass-Heller-Swan formula for other group rings. He also introduced regular coherence for ringsand proved generalizations of the above vanishing results for regular coherent rings.Waldhausen’s motivation was that some group rings are regular coherent (but notregular) and this allowed him to bootstrap K-theory computations for group rings.The Bass-Heller-Swan Theorem is also an important ingredient in K-theory compu-tations via the Farrell-Jones conjecture. If R is regular, then so is R [ t, t − ], but wedo not know whether the same inheritance statement holds for regular coherence.This is one reason why we will not only concentrate on regular coherence here, butalso on regularity.The goal of this paper is to extend the notions of regularity and regular coherencefrom rings to additive categories and to extend the vanishing results in K -theory toadditive categories. The basic strategy will be to embed a given additive category A in the category of Z A -modules. The latter category is abelian. This mimics theadditive subcategory of finitely generated free R -modules of the abelian category ofall R -modules and allows the extension of arguments and definitions from rings toadditive categories. This is a standard construction and has been used for a longtime, for example to define Noetherian additive categories and global dimension foradditive categories.We also extract intrinsic characterizations on the level of additive categories. Forinstance, we call a sequence A f −→ A f −→ A in A exact at A , if f ◦ f = 0 and forevery object A and morphism g : A → A with f ◦ g = 0 there exists a morphism g : A → A with f ◦ g = g , see Definition 4.9. We show in Lemma 5.6 (iv) thatan idempotent complete additive category A is regular coherent if and only if for Date : January 2021.2010
Mathematics Subject Classification.
Key words and phrases. additive categories, K -theory, regularity properties. every morphism f : A → A we can find a sequence of finite length in A → A n f n −→ A n − f n − −−−→ · · · f −→ A f −→ A which is exact at A i for i = 1 , , . . . , n . It is l -uniformly regular coherent if thenumber n can be arranged to satisfy n ≤ l for every morphisms f . Our motivation.
In our experience it is often more convenient to work with ad-ditive categories in place of rings in connection with K-theory. Sometimes a minordrawback is that results for K-theory of rings have not been fully developed foradditive K-theory, although often they are really no more complicated. This papertakes care of the extension of regular coherence from rings to additive categoriesthat we expect to be helpful.More concretely, we rely on the present paper in our ongoing work aimed at thecomputation of the K-theory of Hecke algebras of reductive p -adic groups. Therewe apply regular coherence and the K-theory vanishing results to certain additivecategories that are defined in the spirit of controlled algebra. Namely, we consider adecreasing nested sequence of additive subcategories A ∗ = (cid:0) A ⊇ A ⊇ A ⊇ · · · (cid:1) ,see Definition 12.1, and associate to it the additive category S ( A ∗ ), called sequencecategory , and a certain quotient additive category L ( A ∗ ), called limit category , seeDefinition 12.2. An object in S ( A ∗ ) is a sequence A = ( A m ) m ≥ of objects in A such that for every l ∈ N almost all φ m lie in A l . A morphism φ : A → A ′ in S ( A ∗ ) consists of a sequence of morphisms φ m : A m → A ′ m in A such thatfor every l ∈ N almost all φ m lie in A l . . If the system A ∗ is constant, i.e., A m = A , then S ( A ∗ ) = Q m ∈ N A and L ( A ∗ ) is the quotient additive category Q m ∈ N A (cid:14)L m ∈ N A .Typically each A m will be regular, but this does not imply that S ( A ∗ ) or L ( A ∗ ) isregular as well, see Remark 10.4. The problem is that the property Noetherian doesnot pass to infinite products of additive categories, see Example 12.11. Thereforewe have to discard the condition Noetherian in our considerations. Main results.
As mentioned above we discuss various regularity properties whichare well-known for rings and extend them to additive categories. As long as we areconcerned with the notion regular or Noetherian, we follow the standard proof forrings which carry over to additive categories. This is done for the convenience ofthe reader.As described above, we need to discard the property Noetherian and stick to reg-ular coherence and the new notion of uniform regular coherence. These notions dopass to infinite products of additive categories, see Lemma 10.3, and more generallyunder a certain exactness condition about A ∗ to the additive categories S ( A ∗ ) and L ( S ∗ ), see Lemma 12.6. We remark that algebraic K -theory does commute withinfinite products for additive categories, see [3] and also [5, Theorem 1.2], but notwith infinite products of rings.We will show the vanishing of twisted Nil-terms and of the negative K -theoryfor regular coherent additive categories in Sections 6 and Section 11.The for us most valuable result is the technical Theorem 12.29. whose proof relieson the vanishing of twisted Nil-terms. It will be a key ingredient in our projectto extend the K -theoretic Farrell-Jones Conjecture for discrete groups to reductive p -adic groups, notably, when we want to reduce the family of subgroups, whichmap with a compact kernel to Z , to the family of compact open subgroups. Fordiscrete groups there is a well-known similar reduction relying also on regularityconditions. However, in the discrete case it typically suffices to use regularity for These categories come in our situation from controlled algebra; typically control conditionsget more restrictive with m → ∞ . ANISHING OF NIL-TERMS AND NEGATIVE K -THEORY 3 rings, while our approach to K -theory of reductive p -adic groups necessitates theuse of the weaker regularity conditions introduced in the present paper. Acknowledgements.
This paper is funded by the ERC Advanced Grant “KL2MG-interactions” (no. 662400) of the second author granted by the European ResearchCouncil, by the Deutsche Forschungsgemeinschaft (DFG, German Research Foun-dation) – Project-ID 427320536 - SFB 1442, as well as under Germany’s ExcellenceStrategy - GZ 2047/1, Projekt-ID 390685813, Hausdorff Center for Mathematics atBonn, and EXC 2044 - 390685587, Mathematics M¨unster: Dynamics - Geometry -Structure.The paper is organized as follows:
Contents
Introduction 1Background 1Our motivation 2Main results 2Acknowledgements 31. Z -categories, additive categories and idempotent completions 41.1. Z -categories 41.2. Additive categories 41.3. Idempotent completion 41.4. Twisted finite Laurent category 52. The algebraic K -theory of Z -categories 53. The Bass-Heller-Swan decomposition for additive categories 64. Z A -modules and the Yoneda embedding 74.1. Basics about Z A -modules 74.2. The Yoneda embedding 105. Regularity properties of additive categories 115.1. Definition of regularity properties in terms of the Yoneda embedding 115.2. The definitions of the regularity properties for rings and additivecategories are compatible 125.3. Intrinsic definitions of the regularity properties 156. Vanishing of Nil-terms 196.1. Nil-categories 196.2. Connective K -theory 196.3. Non-connective K -theory 227. Noetherian additive categories 238. Additive categories with finite global dimension 248.1. The characteristic sequence 248.2. Localization 268.3. Global dimension 299. Regular additive categories 3010. Directed union and infinite products of additive categories 3211. Vanishing of negative K -groups 3412. Nested sequences and the associated categories 3512.1. The sequence category 3612.2. Uniform regular coherence 3712.3. The algebraic K -theory of the sequence categories S ( A ∗ ), T ( A ∗ ) and L ( A ∗ ) 3912.4. Split embedding of lower K -theory in higher K -theory 4112.5. Additive categories with Z r -action 43 BARTELS, A. AND L¨UCK, W.
References 451. Z -categories, additive categories and idempotent completions Z -categories. A Z -category is a category A such that for every two objects A and A ′ in A the set of morphism mor A ( A, A ′ ) has the structure of a Z -modulefor which composition is a Z -bilinear map. Given a ring R , we denote by R the Z -category with precisely one object whose Z -module of endomorphisms is givenby R with its Z -module structure and composition is given by the multiplication in R .1.2. Additive categories. An additive category is a Z -category such that for anytwo objects A and A there is a model for their direct sum, i.,e., an object A together with morphisms i k : A k → A for k = 1 , B in A the Z -mapmor A ( A, B ) ∼ = −→ mor A ( A , B ) × mor A ( A , B ) , f ( f ◦ i , f ◦ i )is bijective.Given a ring R , the category R - MOD fgf of finitely generated free left R -modulescarries an obvious structure of an additive category.An equivalence F : A → A ′ of Z -categories or of additive categories respectivelyis a functor of Z -categories or of additive categories respectively such that for all ob-jects A , A in A the induced map F A ,A : mor A ( A , A ) ∼ = −→ mor A ( F ( A ) , F ( A ))sending f to F ( f ) is bijective, and for any object A ′ in A ′ there exists an object A in A such that F ( A ) and A ′ are isomorphic in A ′ . This is equivalent to the existenceof a functor F : A ′ → A of Z -categories or of additive categories respectively suchthat both composites F ◦ F ′ and F ′ ◦ F are naturally equivalent as such functorsto the identity functors.Given a Z -category, let A ⊕ be the associated additive category whose objectsare finite tuples of objects in A and whose morphisms are given by matrices ofmorphisms in A (of the right size) and the direct sum is given by concatenation oftuples and the block sum of matrices, see for instance [11, Section 1.3].Let R be a ring. Then we can consider the additive category R ⊕ . The obviousinclusion of additive categories(1.1) θ fgf : R ⊕ ≃ −→ R - MOD fgf is an equivalence of additive categories. Note that R ⊕ is small, in contrast to R - MOD fgf .1.3.
Idempotent completion.
Given an additive category A , its idempotent com-pletion Idem( A ) is defined to be the following additive category. Objects are mor-phisms p : A → A in A satisfying p ◦ p = p . A morphism f from p : A → A to p : A → A is a morphism f : A → A in A satisfying p ◦ f ◦ p = f . Thestructure of an additive category on A induces the structure of an additive categoryon Idem( A ) in the obvious way. The identity of an object ( A, p ) is given by themorphism p : ( A, p ) → ( A, p ). A functor of additive categories F : A → A ′ inducesa functor Idem( F ) : Idem( A ) → Idem( A ′ ) of additive categories by sending ( A, p )to ( F ( A ) , F ( p )).There is a obvious embedding η ( A ) : A →
Idem( A )sending an object A to id A : A → A and a morphism f : A → B to the mor-phism given by f again. An additive category A is called idempotent complete if ANISHING OF NIL-TERMS AND NEGATIVE K -THEORY 5 η ( A ) : A →
Idem( A ) is an equivalence of additive categories, or, equivalently, iffor every idempotent p : A → A in A there exists objects B and C and an isomor-phism f : A ∼ = −→ B ⊕ C in A such that f ◦ p ◦ f − : B ⊕ C → B ⊕ C is given by (cid:18) id B
00 0 (cid:19) . The idempotent completion Idem( A ) of an additive category A isidempotent complete.For a ring R , let R - MOD fgp be the additive category of finitely generated projec-tive R -modules. We get an equivalence of additive categories Idem( R - MOD fgf ) ≃ −→ R - MOD fgp by sending an object (
F, p ) to im( p ). It and the functor of (1.1) inducean equivalence of additive categories(1.2) θ fgp : Idem (cid:0) R ⊕ (cid:1) ≃ −→ R - MOD fgp
Notice that Idem (cid:0) R ⊕ (cid:1) is small, in contrast to R - MOD fgp .1.4.
Twisted finite Laurent category.
Let A be an additive category. LetΦ : A → A be an automorphism of additive categories.
Definition 1.3 (Twisted finite Laurent category A Φ [ t, t − ]) . Define the Φ -twistedfinite Laurent category A Φ [ t, t − ] as follows. It has the same objects as A . Giventwo objects A and B , a morphism f : A → B in A Φ [ t, t − ] is a formal sum f = P i ∈ Z f i · t i , where f i : Φ i ( A ) → B is a morphism in A from Φ i ( A ) to B and onlyfinitely many of the morphisms f i are non-trivial. If g = P j ∈ Z g j · t j is a morphismin A Φ [ t, t − ] from B to C , we define the composite g ◦ f : A → C by g ◦ f := X k ∈ Z (cid:18) X i,j ∈ Z ,i + j = k g j ◦ Φ j ( f i ) (cid:19) · t k . The direct sum and the structure of a Z -module on the set of morphism from A to B in A Φ [ t, t − ] are defined in the obvious way using the corresponding structuresof A .We sometimes also write A Φ [ Z ] instead of A Φ [ t, t − ]. Example 1.4.
Let R be a ring with an automorphism φ : R ∼ = −→ R of rings. Let R φ [ t, t − ] be the ring of φ -twisted finite Laurent series with coefficients in R . Weobtain from φ an automorphism Φ : R ∼ = −→ R of Z -categories. There is an obviousisomorphism of Z -categories(1.5) R Φ [ t, t − ] ∼ = −→ R φ [ t, t − ] . We obtain equivalences of additive categories( R ⊕ ) Φ [ t, t − ] ≃ −→ R φ [ t, t − ]- MOD fgf ;Idem (cid:0) ( R ⊕ ) Φ [ t, t − ] (cid:1) ≃ −→ R φ [ t, t − ]- MOD fgp . Definition 1.6 ( A Φ [ t ] and A Φ [ t − ]) . Let A Φ [ t ] and A Φ [ t − ] respectively be theadditive subcategory of A Φ [ t, t − ] whose set of objects is the set of objects in A andwhose morphism from A to B are given by finite formal Laurent series P i ∈ Z f i · t i with f i = 0 for i < i > The algebraic K -theory of Z -categories Let A be an additive category. One can interprete it as an exact category in thesense of Quillen or as a category with cofibrations and weak equivalence in the senseof Waldhausen and obtains the connective algebraic K -theory spectrum K ( A ) bythe constructions due to Quillen [14] or Waldhausen [20]. A construction of the BARTELS, A. AND L¨UCK, W. non-connective K -theory spectrum K ∞ ( A ) of an additive category can be foundfor instance in [10] or [13]. Definition 2.1 (Algebraic K -theory of Z -categories) . We will define the algebraic K -theory spectrum K ∞ ( A ) of the Z -category A to be the non-connective algebraic K -theory spectrum of the additive category A ⊕ . Define for n ∈ Z K n ( A ) := π n ( K ∞ ( A )) . The connective algebraic K -theory spectrum K ( A ) is defined to be the connectivealgebraic K -theory spectrum of the additive category A ⊕ .If A is an additive category and i ( A ) is the underlying Z -category, then there is acanonical equivalence of additive categories i ( A ) ⊕ → A . Hence there are canonicalweak homotopy equivalences K ( i ( A )) → K ( A ) and K ∞ ( i ( A )) → K ∞ ( A ).A functor F : A → A ′ of Z -categories induces a map of spectra(2.2) K ∞ ( F ) : K ∞ ( A ) → K ∞ ( A ′ ) . We call a full additive subcategory A of A ′ cofinal if for any object A ′ in A ′ thereis an object A in A together with morphisms i : A ′ → A and r : A ′ → A satisfying r ◦ i = id. Lemma 2.3.
Let I : A → A ′ be the inclusion of a full cofinal additive subcategory.(i) The induced map π n ( K ( I )) : π n ( K ( A )) → π n ( K ( A ′ )) is bijective for n ≥ ;(ii) The induced map K ∞ ( I ) : K ∞ ( A ) → K ∞ ( A ′ ) is a weak homotopy equivalence.Proof. (i) This is proved for A ′ = Idem( A ) in [17, Theorem A.9.1.]. Now the gen-eral case follows from the observation that Idem( A ) → Idem( A ′ ) is an equivalenceof additive categories.(ii) This follows from assertion (i) and [10, Corollary 3.7]. (cid:3) The Bass-Heller-Swan decomposition for additive categories
Denote by
Add - Cat the category of additive categories. Let us consider thegroup Z as a groupoid with one object and denote by Add - Cat Z the category offunctors Z → Add - Cat , with natural transformations as morphisms. Note that anobject of this category is a pair ( A , Φ) consisting of an additive category togetherwith an automorphism Φ : A ∼ = −→ A of additive categories. We recall from [11,Theorem 0.1 and Theorem 0.4] using the notation of this paper here and in thesequel: Theorem 3.1 (The Bass-Heller-Swan decomposition for non-connective K -theoryof additive categories) . Let
Φ :
A → A be an automorphism of additive categories.(i) There exists a weak homotopy equivalence of spectra, natural in ( A , Φ) , a ∞ ∨ b ∞ + ∨ b ∞− : T K ∞ (Φ − ) ∨ NK ∞ ( A Φ [ t ]) ∨ NK ∞ ( A Φ [ t − ]) ≃ −→ K ∞ ( A Φ [ t, t − ]) where T K ∞ (Φ − ) is the mapping torus of K ∞ (Φ − ) : K ∞ ( A ) → K ∞ ( A ) and NK ∞ ( A Φ [ t ± ]) is the homotopy fiber of the map K ∞ ( A Φ [ t − ]) → K ∞ ( A ) given by evaluation t = 0 ; ANISHING OF NIL-TERMS AND NEGATIVE K -THEORY 7 (ii) There exist a functor E ∞ : Add - Cat Z → Spectra and weak homotopyequivalences of spectra, natural in ( A , Φ) , Ω NK ∞ ( A Φ [ t ]) ≃ ←− E ∞ ( A , Φ); K ∞ ( A ) ∨ E ∞ ( A , Φ) ≃ −→ K ∞ Nil ( A , Φ) , where K ∞ Nil ( A , Φ) is the non-connective K -theory of a certain Nil-category Nil( A , Φ) . Theorem 3.2 (Fundamental sequence of K -groups) . Let A be an additive category.Then there exists for n ∈ Z a split exact sequence, natural in A (3.3) 0 → K n ( A ) ( k + ) ∗ ⊕− ( k − ) ∗ −−−−−−−−−→ K n ( A [ t ]) ⊕ K n ( A [ t − ]) ( l + ) ∗ ⊕ ( l − ) ∗ −−−−−−−−→ K n ( A [ t, t − ]) δ n −→ K n − ( A ) → , where ( k + ) ∗ , ( k − ) ∗ , ( l + ) ∗ , and ( l − ) ∗ are induced by the obvious inclusions k + , k − , l + , and l − and δ n is the composite of the inverse of the (untwisted) Bass-Heller-Swan isomorphism K n ( A ) ⊕ K n − ( A ) ⊕ NK n ( A [ t ]) ⊕ NK n ( A [ t − ]) ∼ = −→ K n ( A [ t, t − ]) , see Theorem 3.1, with the projection onto the summand K n − ( A ) .Proof. This follows directly from the untwisted version of Theorem 3.1. (cid:3)
There is also a version for the connective K -theory spectrum K . Denote by Add - Cat ic ⊂ Add - Cat the full subcategory of idempotent complete additive cate-gories.
Theorem 3.4 (The Bass-Heller-Swan decomposition for connective K -theory ofadditive categories) . Let A be an additive category which is idempotent complete.Let Φ :
A → A be an automorphism of additive categories.(i) Then there is a weak equivalence of spectra, natural in ( A , Φ) , a ∨ b + ∨ b − : T K (Φ − ) ∨ NK ( A Φ [ t ]) ∨ NK ( A Φ [ t − ]) ≃ −→ K ( A Φ [ t, t − ]) where T K (Φ − ) is the mapping torus of K (Φ − ) : K ( A ) → K ( A ) and NK ( A Φ [ t ± ]) is the homotopy fiber of the map K ( A Φ [ t − ]) → K ( A ) givenby evaluation t = 0 ;(ii) There exist a functor E : ( Add - Cat ic ) Z → Spectra and weak homotopyequivalences of spectra, natural in ( A , Φ) , Ω NK ( A Φ [ t ]) ≃ ←− E ( A , Φ); K ( A ) ∨ E ( A , Φ) ≃ −→ K (Nil( A , Φ)) , where K (Nil( A , Φ)) is the connective K -theory of a certain Nil-category Nil( A , Φ) . The purpose of the following sections is to find properties of A , which implyfor any automorphism Φ the vanishing of the Nil-terms above and are hopefullyinherited by the passage from A to A [ t, t − ].4. Z A -modules and the Yoneda embedding Basics about Z A -modules. Let A be a Z -category. We denote by Z A - MOD and
MOD - Z A respectively the abelian category of covariant or contravariant re-spectively functors of Z -categories A to Z - MOD . The abelian structure comes fromthe abelian structure in Z - MOD . For instance, a sequence F T −→ F T −→ F in MOD - Z A is declared to be exact if for each object A ∈ A the evaluation at A yields BARTELS, A. AND L¨UCK, W. an exact sequence of Z -modules F ( A ) T ( A ) −−−−→ F ( A ) T ( A ) −−−−→ F ( A ). The cokerneland kernel of a morphism T : F → F are defined by taking for each object A ∈ A the kernel or cokernel of the morphism T ( A ) : F ( A ) → F ( A ) in MOD - Z .In the sequel Z A -module means contravariant Z A -module unless specified ex-plicitly differently.Given an object A in A we obtain an object mor A (? , A ) in MOD - Z A by assigningto an object B the Z -module mor A ( B, A ) and to a morphism g : B → B the Z -homomorphism g ∗ : mor A ( B , A ) → mor A ( B , A ) given by precomposition with g . The elementary proof of the following lemma is left to the reader. Lemma 4.1 (Yoneda Lemma) . For each object A in A and each object M in MOD - Z A we obtain an isomorphism of Z -modules mor MOD - Z A (mor A (? , A ) , M (?)) ∼ = −→ M ( A ) , T T ( A )(id A ) . We call a Z A -module M free if it is isomorphic as Z A -module to L I mor A (? , A i )for some collection of objects { A i | i ∈ I } in A for some index set I . A Z A -module M is called projective if for any epimorphism p : N → N of Z A -modules and mor-phism f : M → N there is a morphism f : M → N with p ◦ f = f . A Z A -module M is finitely generated if there exists a collection of objects { A j | j ∈ J } in A for somefinite index set J and an epimorphism of Z A -modules L j ∈ J mor A (? , A j ) → M .Equivalently, M is finitely generated if there exists a finite collection of objects { A j | j ∈ J } in A together with elements x j ∈ M ( A j ) such that for any object A and any x ∈ M ( A ) there are morphisms ϕ : A → A j such that x = P j M ( ϕ j )( x j ).(The x j are the images of id A j under the above epimorphism.) Given a collection ofobjects { A i | i ∈ I } in A for some index set I , the free Z A -module L I mor A (? , A i )is finitely generated if and only if I is finite. A Z A -module M is finitely presented if there are finitely generated free Z A -modules F and F and an exact sequence F → F → M →
0. We say that a Z A -module has projective dimension ≤ d ,denoted by pdim Z A ( M ) ≤ d , for a natural number d if there exists an exact se-quence 0 → P d → P d − → · · · → P → P → M → Z A -module P i is projective. If we replace projective by free, we get an equivalent definition if d ≥
1. We call a Z A -module of type FL or of type
FP respectively if there exists anexact sequence of finite length 0 → F n → F n − → · · · → F → F → M → Z A -module F i is finitely generated free or finitely generated projectiverespectively. Remark 4.2.
Note the setting in this paper is different from the one appearingin [6], since here a Z A -module M satisfies M ( f + g ) = M ( f ) + M ( g ) for twomorphisms f, g : A → B which is not required in [6]. Nevertheless many of thearguments in [6] carry over to the setting of this paper because of the YonedaLemma 4.1, which replaces the corresponding Yoneda Lemma in [6, Subsection 9.16on page 167].However, the next result has no analogue in the setting of [6]. Lemma 4.3.
Let A be an additive category. For a Z A -module M and objects A , A , . . . , A n we obtain a natural isomorphism n M i =1 M (pr i ) : n M i =1 M ( A i ) ∼ = −→ M (cid:18) n M i =1 A i (cid:19) where pr j : L ni =1 A i → A j is the canonical projection for j = 1 , . . . , n . ANISHING OF NIL-TERMS AND NEGATIVE K -THEORY 9 Proof.
One easily checks using the fact that the functor M is compatible with the Z -module structures on the morphisms that the inverse is given M (cid:18) n M i =1 A i (cid:19) → n M i =1 M ( A i ) , x (cid:0) M ( k i )( x ) (cid:1) i , where k j : A j → L ni =1 A i is the inclusion of the j -the summand for j = 1 , . . . , n . (cid:3) Lemma 4.4.
Let A be a Z -category.(i) Every free Z A -module is projective;(ii) Let → M → M ′ → M ′′ → be an exact sequence of Z A -modules. If both M and M ′′ are free or projective respectively, then M ′ is free or projectiverespectively;(iii) Let → M → M ′ → M ′′ → be an exact sequence of Z A -modules. Iftwo of the Z A -modules M , M ′ and M ′′ are of type FL or FP respectively,then all three are of type FL or FP respectively;(iv) Let C ∗ be a projective Z A -chain complex i.e., a Z A -chain complex allwhose chain modules C n are projective. Then the following assertions areequivalent(a) Consider a natural number d . Let B d ( C ∗ ) be the image of c d +1 : C d +1 → C d and j : B d ( C ) → C d be the inclusion. There is a Z A -submodule C ⊥ d such that for the inclusion i : C ⊥ d → C d the map i ⊕ j : C ⊥ d ⊕ B d ( C ∗ ) → C d is an isomorphism. Moreover, the following chain mapfrom a d -dimensional projective Z A -chain complex to C ∗ is a Z A -chain homotopy equivalence · · · / / / / (cid:15) (cid:15) / / (cid:15) (cid:15) C ⊥ d c d ◦ i / / i (cid:15) (cid:15) C d − c d − / / id Cd − (cid:15) (cid:15) · · · c / / C C (cid:15) (cid:15) · · · c d +3 / / C d +2 c d +2 / / C d +1 c d +1 / / C d c d / / C d − c d − / / · · · c / / C (b) C ∗ is Z A -chain homotopy equivalent to a d -dimensional projective Z A -chain complex;(c) C ∗ is dominated by d -dimensional projective Z A -chain complex D ∗ ,i.e., there are Z A -chain maps i : C ∗ → D ∗ and r ∗ : D ∗ → C ∗ satisfy-ing r ∗ ◦ i ∗ ≃ id C ∗ ;(d) B d ( C ∗ ) is a direct summand in C d and H i ( C ∗ ) = 0 for i ≥ d + 1 ;(e) H d +1 Z A ( C ∗ ; M ) := H d +1 (hom Z A ( C ∗ , M )) vanishes for every Z A -module M and H i ( C ∗ ) = 0 for all i ≥ d + 1 ;(v) Let → M → M ′ → M ′′ → be an exact sequence of Z A -modules.If pdim Z A ( M ) , pdim Z A ( M ′′ ) ≤ d , then pdim Z A ( M ′ ) ≤ d ;If pdim Z A ( M ) , pdim Z A ( M ′ ) ≤ d , then pdim Z A ( M ′′ ) ≤ d + 1 ;If pdim Z A ( M ′ ) ≤ d , pdim Z A ( M ′′ ) ≤ d + 1 , then pdim Z A ( M ) ≤ d ;(vi) Suppose that A is an additive category. For two objects A and A in A together with a choice of a direct sum i k : A k → A ⊕ A for k = 0 , , theinduced Z -map i ∗ ⊕ i ∗ : mor A (? , A ) ⊕ mor A (? , A ) ∼ = −→ mor A (? , A ⊕ A ) is an isomorphism. In particular each finitely generated free Z A -moduleis isomorphic to Z A -module of the shape mor A (? , A ) for an appropriateobject A in A .Proof. (i) This follows from the Yoneda Lemma 4.1.(ii) This is obviously true. (iii) The proof is analogous to the one of [6, Lemma 11.6 on page 216].(iv) The proof is analogous to the one of [6, Proposition 11.10 on page 221].(v) This follows from (iv) for the projective dimension using the long exact (co)ho-mology sequence associated to a short exact sequence of (co)chain complexes, sinceevery Z A -module has a free resolution by the Yoneda Lemma 4.1.(vi) This is obvious and hence the proof of Lemma 4.4 is finished. (cid:3) If M and N are Z A -modules, then hom Z A ( M, N ) is the Z -module of Z A -homomorphisms M → N . Given a contravariant or covariant Z A -module and a Z -module T , then we obtain a covariant or contravariant Z A -module hom Z ( M, T )by sending an object A to hom Z ( M ( A ) , T ). Given a contravariant A -module M and covariant Z A -module N , their tensor product M ⊗ Z A N is the Z -module givenby L A ∈ ob( A ) M ( A ) ⊗ Z N ( A ) /T where T is the Z -submodule of L A ∈ ob( A ) M ( A ) ⊗ Z N ( A ) generated by elements of the form mf ⊗ n − m ⊗ f n for a morphism f : A → B in A , m ∈ M ( A ) and n ∈ N ( B ), where mf := M ( f )( m ) and f n = N ( f )( n ). It ischaracterized by the property that for any Z -module T , there are natural adjunctionisomorphisms hom Z ( M ⊗ Z A N, T ) ∼ = −→ hom Z A ( M, hom Z ( N, T ));(4.5) hom Z ( M ⊗ Z A N, T ) ∼ = −→ hom Z A ( N, hom Z ( M, T )) . (4.6)Let F : A → B be a functor of Z -categories. Then the restriction functor F ∗ : MOD - Z B →
MOD - Z A . is given by precomposition with F . The induction functor F ∗ : MOD - Z A →
MOD - Z B . sends a contravariant Z A -module M to M (?) ⊗ Z A mor B (?? , F (?)). We get for a Z B -module an identification F ∗ N = hom Z B (mor A (?? , F (?)) , N (??)) from the YonedaLemma 4.1. We conclude from (4.5)(4.7) hom Z B ( F ∗ M, N ) ∼ = −→ hom Z A ( M, F ∗ N )for a Z A -module M and a Z B -module N . The counit β ( N ) : F ∗ F ∗ ( N ) → N ofthe adjunction (4.7) is the adjoint of id F ∗ N and sends the equivalence class of x ⊗ f , with for x ∈ N ( F ( A )) and f ∈ mor B ( B, F ( A )) to xf = N ( f )( x ). Theunit α ( M ) : M → F ∗ F ∗ ( M ) is the adjoint of id F ∗ M and sends x ∈ M ( A ) to theequivalence class of x ⊗ id F ( A ) .The functor F ∗ is flat. The functor F ∗ is compatible with direct sums overarbitrary index sets, is right exact, see [21, Theorem 2.6.1. on page 51], and F ∗ mor Z A (? , C ) is Z B -isomorphic to mor Z B (? , F ( C )). In particular F ∗ respect theproperties finitely generated, free, and projective.4.2. The Yoneda embedding.
The
Yoneda embedding is the following covariantfunctor(4.8) ι : A →
MOD - Z A . It sends an object A to ι ( A ) = mor A (? , A ) and a morphism f : A → A to thetransformation ι ( f ) : mor A (? , A ) → mor A (? , A ) given by composition with f .Let MOD - Z A A be the full subcategory of MOD - Z A consisting of Z A -modulesmor A (? , A ) for any object A in A . Let MOD - Z A fgf be the full subcategory of MOD - Z A consisting of finitely generated free Z A -modules. Definition 4.9.
Let A be a Z -category. We call a sequence A f −→ A f −→ A in A exact at A , if f ◦ f = 0 and for every object A and morphism g : A → A with f ◦ g = 0 there exists a morphism g : A → A with f ◦ g = g . ANISHING OF NIL-TERMS AND NEGATIVE K -THEORY 11 Lemma 4.10. If A is a Z -category, the Yoneda embedding (4.8) induces an equiv-alence of Z -categories denoted by the same symbol ι : A →
MOD - Z A A . If A is an additive category, the Yoneda embedding (4.8) induces an equivalence ofadditive categories denoted by the same symbol ι : A →
MOD - Z A fgf . Both functors are faithfully flat.Proof.
This follows directly from the Yoneda Lemma 4.1 and Lemma 4.4 (vi). (cid:3)
The gain of Lemma 4.10 is that we have embedded A as a full subcategory ofthe abelian category MOD - Z A and we can now do certain standard homologicalconstructions in MOD - Z A which a priori make no sense in A .The elementary proof of the following lemma based on Lemma 4.10 is left to thereader. Lemma 4.11.
An additive category A is idempotent complete if and only if everyfinitely generated projective Z A -module is a finitely generated free Z A -module. Regularity properties of additive categories
Definition of regularity properties in terms of the Yoneda embed-ding.
Recall the following standard ring theoretic notions:
Definition 5.1 (Regularity properties of rings) . Let R be a ring and let l be anatural number.(i) We call R Noetherian , if any R -submodule of a finitely generated R -moduleis again finitely generated;(ii) We call R regular coherent , if every finitely presented R -module M is oftype FP;(iii) We call R l -uniformly regular coherent , if every finitely presented R -module M admits a l -dimensional finite projective resolution, i.e., there exist anexact sequence 0 → P l → P l − → · · · → P → M → P i is finitely generated projective;(iv) We call R von Neumann regular , if for any element r ∈ R there exists anelement s ∈ R with r = rsr ;(v) We call R regular , if it is Noetherian and regular coherent;(vi) We call R l -uniformly regular , if it is Noetherian and l -uniformly regularcoherent;(vii) We say that R has global dimension ≤ l if each R -module M has projectivedimension ≤ l .The notion von Neumann regular should not be confused with the notion regular.It stems from operator theory. A ring is von Neumann regular if and only if it is0-uniformly regular coherent. For more information about von Neumann regularrings, see for instance [7, Subsection 8.2.2 on pages 325-327].Let A be an additive category. Then we define analogously: Definition 5.2 (Regularity properties of additive categories) . Let A be a additivecategory and let l be a natural number.(i) We call A Noetherian if the category
MOD - Z A is Noetherian in the sensethat any Z A -submodule of a finitely generated Z A -module is again finitelygenerated, see [12, p.18] ; In [12, page 18] this is called left Noetherian; one obtains right Noetherian by working with Z A - MOD in place of
MOD - Z A (ii) We call A regular coherent , if every finitely presented Z A -module M is oftype FP;(iii) We call A l -uniformly regular coherent , if every finitely presented Z A -module M possesses an l -dimensional finite projective resolution, i.e., thereexist an exact sequence 0 → P l → P l − → · · · → P → M → P i is finitely generated projective;(iv) We call A regular , if it is Noetherian and regular coherent;(v) We call A l -uniformly regular , if is Noetherian and l -uniformly regularcoherent;(vi) We say that A has global dimension ≤ l , if each Z A -module M has pro-jective dimension ≤ l , see [12, page 42].5.2. The definitions of the regularity properties for rings and additivecategories are compatible.Lemma 5.3.
Let R be a ring. The functor F : R - MOD → MOD - Z R ⊕ sending M to hom R ( θ fgf ( − ) , M ) is an equivalence of additive categories, is faithfullyflat, and respects each of the properties finitely generated, free and projective, wherethe equivalence θ fgf has been defined in (1.1) Proof.
In the sequel we denote by [ n ] the n -fold direct sum in R ⊕ of the uniqueobject in R . Notice that θ ([ n ]) = R n . Define G : MOD - Z R ⊕ → R - MOD by sending M to M ( θ (1)). There is a natural equivalence G ◦ F → id R - MOD offunctors of additive categories, its value on the R -module M is given by evaluatingat 1 ∈ R = θ ([1]), G ◦ F ( M ) = hom R ( θ ([1]) , M ) ∼ = −→ M. Next we construct an equivalence S : F ◦ G → id R - MOD of functors of additivecategories. For a Z A -module N and objects A , . . . , A n we obtain from Lemma 4.3a natural isomorphism n M i =1 N (pr i ) : n M i =1 N ( A i ) ∼ = −→ N (cid:18) n M i =1 A i (cid:19) where pr j : L ni =1 A i → A j is the canonical projection for j = 1 , . . . , n .Recall that [ n ] is the n -fold direct sum of copies of [1], in other words, we havean identification [ n ] = L ni =1 [1]. It induces an isomorphism n M i =1 θ ([1]) ∼ = −→ θ ([ n ]) . Given an object [ n ] in R ⊕ and an R -module M , we define S ( M )([ n ]) by the followingcomposite of R -isomorphisms F ◦ G ( M )([ n ]) = hom R (cid:0) θ ([ n ]) , M ( θ (1)) (cid:1) ∼ = −→ hom R (cid:18) n M k =1 θ ([1]) , M ( θ (1)) (cid:19) ∼ = −→ n M k =1 hom R (cid:0) θ ([1]) , M ( θ (1)) (cid:1) = n M k =1 hom R (cid:0) R, M ( θ (1)) (cid:1) ∼ = −→ n M k =1 M ( θ (1)) ∼ = −→ M (cid:18) n M i =1 θ ([1]) (cid:19) = M ( θ ([ n ])) . ANISHING OF NIL-TERMS AND NEGATIVE K -THEORY 13 The functor F is faithfully exact, since for any object [ n ] in R ⊕ there is an R -isomorphism L ni =1 M ∼ = −→ F ( M )([ n ]), natural in M . Since F is compatible withdirect sums over arbitrary index sets and sends R to hom R ( θ ( − ) , R ) = mor R ⊕ (? , [1])it respects the properties finitely generated, free and projective. (cid:3) The following lemma implies in particular that the inclusion i : A →
Idem( A )induces equivalences MOD - Z A i ∗ / / MOD - Z Idem( A ) i ∗ o o . Lemma 5.4.
Let i : A → A ′ be a inclusion of an additive subcategory A of theadditive subcategory A ′ , which is full and cofinal, for instance A → A ′ = Idem( A ) .Then:(i) If M is a Z A -module, then the adjoint α ( M ) : M ∼ = −→ i ∗ i ∗ M of id i ∗ M under the adjunction (4.7) is an isomorphism of Z A -modules,natural in M ;(ii) The restriction functor i ∗ : MOD - Z A ′ → MOD - Z A is faithfully flat. Itsends a finitely generated Z A ′ -module to a finitely generated Z A -moduleand a projective Z A ′ -module to a projective Z A -module;(iii) The induction functor i ∗ : MOD - Z A →
MOD - Z A ′ is faithfully flat. Itsends a finitely generated Z A -module to a finitely generated Z A ′ -moduleand a projective Z A -module to a projective Z A ′ -module;(iv) If M ′ is a Z A ′ -module, then the adjoint β ( M ′ ) : i ∗ i ∗ M ′ ∼ = −→ M ′ of id i ∗ M ′ under the adjunction (4.7) is an isomorphism of Z A ′ -modules,natural in M ′ ;(v) A is Noetherian if and only A ′ is Noetherian;(vi) The category A is regular coherent or l -uniformly regular coherent respec-tively if and only if A ′ is regular coherent or l -uniformly regular coherent.(vii) The category A is of global dimension ≤ l if and only if A ′ is of globaldimension ≤ l .Proof. (i) This follows from the Yoneda-Lemma 4.1, namely, an inverse of α ( M )is given by i ∗ i ∗ M = M (?) ⊗ Z A mor Z A ′ ( i (? ′ ) , i (?)) = M (?) ⊗ Z A mor Z A (? ′ , ?) ∼ = −→ M (? ′ ) ,x ⊗ φ xφ = M ( φ )( x ) . (ii) Obviously i ∗ is flat.Consider a sequence of Z A ′ -modules M f −→ M f −→ M such that restrictionwith i yields the exact sequence of Z A -modules i ∗ M i ∗ f −−−→ i ∗ M i ∗ f −−−→ i ∗ M . Wehave to show for any object A ′ in A ′ that the sequence of R -modules M ( A ′ ) f ( A ′ ) −−−−→ M ( A ′ ) f ( A ′ ) −−−−→ M ( A ′ ) is exact. Since A is by assumption cofinal in A ′ , we canfind an object A in A and and morphisms j : A ′ → i ( A ) and r : i ( A ) → A ′ in A ′ satisfying r ◦ i = id A ′ . We obtain the following commutative diagram of R -modules M ( A ′ ) f ( A ′ ) / / M ( j ) (cid:15) (cid:15) M ( A ′ ) f ( A ′ ) / / M ( j ) (cid:15) (cid:15) M ( A ′ ) M ( j ) (cid:15) (cid:15) M ( i ( A )) f ( i ( A )) / / M ( r ) (cid:15) (cid:15) M ( i ( A )) f ( i ( A )) / / M ( r ) (cid:15) (cid:15) M ( i ( A )) M ( r ) (cid:15) (cid:15) M ( A ′ ) f ( A ′ ) / / M ( A ′ ) f ( A ′ ) / / M ( A ′ )such that the composite of the two vertical arrows appearing in each of the threecolumns is the identity. Since the middle horizontal sequence is exact, an easydiagram chase shows that the upper horizontal sequence is exact. This shows that i ∗ is faithfully flat.Consider an object A ′ in A ′ . Since A is by assumption cofinal in A , we canfind an object A in A and and morphism j : A ′ → i ( A ) and q : i ( A ) → A ′ in A ′ satisfying q ◦ j = id A ′ . Composition with q and j yield maps of Z A ′ -modules J : mor A ′ (? ′ , A ′ ) → mor A ′ (? ′ , i ( A )) and Q : mor A ′ (? ′ , i ( A )) → mor A ′ (? ′ , A ′ ) sat-isfying Q ◦ J = id mor A′ (? ′ ,A ′ ) . If we apply i ∗ , we obtain homomorphisms of Z A ′ -modules i ∗ J : i ∗ mor A ′ (? ′ , A ′ ) → i ∗ mor A ′ (? ′ , i ( A )) and i ∗ Q : i ∗ mor A ′ (? ′ , i ( A )) → i ∗ mor A ′ (? ′ , A ′ ) satisfying i ∗ Q ◦ i ∗ J = id i ∗ mor A′ (? ′ ,A ′ ) . Since i ∗ mor A ′ (? ′ , i ( A )) =mor A ′ ( i (? ′ ) , i ( A )) = mor A (? ′ , A ), the Z A -module i ∗ mor A ′ (? ′ , A ′ ) is a direct sum-mand in mor A (? ′ , A ) and hence a finitely generated projective Z A -module.Let M ′ be a finitely generated Z A ′ -module. Fix an epimorphism mor A (? ′ , A ′ ) → M ′ for some object A ′ in A ′ . We conclude that the Z A -module i ∗ M is a quotient ofmor A (? , A ) for some object A in A and hence finitely generated. Hence i ∗ respectsthe property finitely generated.Let P be a projective Z A ′ -module. Then we can find a collection of objects { A ′ k | k ∈ K } together with an epimorphism L k ∈ K mor A ′ (? , A ′ k ) → P by the YonedaLemma 4.1. Since P is projective, P is a direct summand in L k ∈ K mor A ′ (? , A ′ k ).This implies that i ∗ P is a direct summand in the direct sum L k ∈ K i ∗ mor A ′ (? , A ′ i )of projective Z A -modules and hence itself a projective Z A -module. Hence i ∗ re-spects the property projective.(iii) The faithful flatness follows from assertions (i) and (ii). Since i ∗ mor A (? , A ) =mor A ′ (? , i ( A )) holds for any object A in A , the functor i ∗ respects the propertiesfinitely generated and projective.(iv) We begin with the case M = mor A ′ (? ′ , i ( A )) = i ∗ mor A ′ (? , A ) for some ob-ject A in A . Then the claim follows from assertion (i) applied to the Z A -modulemor A ′ (? , A ) since in this case β ( M ) = i ∗ α ( M ). Consider an object A ′ in A ′ . Since A is by assumption cofinal in A , we can find an object A in A and and mor-phism j : A ′ → i ( A ) and q : i ( A ) → A ′ in A ′ satisfying q ◦ j = id A ′ . Compositionwith q and j yield maps of Z A ′ -modules J : mor A ′ (? ′ , A ′ ) → mor A ′ (? ′ , i ( A )) and Q : mor A ′ (? ′ , i ( A )) → mor A ′ (? ′ , A ′ ) satisfying Q ◦ J = id mor A′ (? ′ ,A ′ ) . Hence we get ANISHING OF NIL-TERMS AND NEGATIVE K -THEORY 15 a commutative diagram of Z A ′ -modules i ∗ i ∗ mor A ′ (? ′ , A ′ ) β (mor A′ (? ′ ,A ′ )) / / i ∗ i ∗ J (cid:15) (cid:15) mor A ′ (? ′ , A ′ ) J (cid:15) (cid:15) i ∗ i ∗ mor A ′ (? ′ , i ( A )) β (mor A′ (? ′ ,i ( A ))) / / i ∗ i ∗ Q (cid:15) (cid:15) mor A ′ (? ′ , i ( A )) Q (cid:15) (cid:15) i ∗ i ∗ mor A ′ (? ′ , A ′ ) β (mor A′ (? ′ ,A ′ )) / / mor A ′ (? ′ , A ′ )such that the composite of the two vertical maps in each of the two columns isthe identity and the middle arrow is an isomorphism. Hence the upper arrow is anisomorphism.For any Z A ′ -module M ′ we can find a collection of objects { A ′ k | k ∈ K } in A ′ together with an epimorphism f : F := L k ∈ K mor A ′ (? ′ , A ′ k ) → M ′ bythe Yoneda Lemma 4.1. Repeating this construction for ker( f ) instead of M ,we obtain another collection { A ′′ l | l ∈ L } of objects in A ′ together with a map f : F := L l ∈ L i ∗ mor A ′ (? , A ′ l ) → F whose image is ker( f ). We obtain fromassertions (ii) and (iii) a commutative diagram of Z A ′ -modules with exact rows i ∗ i ∗ F i ∗ i ∗ f / / β ( F ) (cid:15) (cid:15) i ∗ i ∗ F i ∗ i ∗ f / / β ( F ) (cid:15) (cid:15) i ∗ i ∗ M ′ / / β ( M ) (cid:15) (cid:15) F f / / F f / / M ′ / / β is compatible with direct sums over arbitrary index sets, the maps β ( F )and β ( F ) are isomorphisms. Hence β ( M ′ ) is an isomorphism.(v), (vi) and (vii) They follow now directly from assertions (i), (ii), (iii) and (iv). (cid:3) We conclude from Lemma 5.3 and Lemma 5.4 (v), (vi), and (vii).
Corollary 5.5.
Let R be a ring and let l be a natural number. Then the followingassertions are equivalent:(i) The ring R is Noetherian, regular coherent, l -uniformly regular coherent,regular, uniformly l -regular, or of global dimension ≤ l in the sense ofDefinition 5.1 respectively;(ii) The additive category R ⊕ is Noetherian, regular coherent, l -uniformly reg-ular coherent, regular, uniformly l -regular, or of global dimension ≤ d inthe sense of Definition 5.2 respectively;(iii) The additive category Idem( R ⊕ ) is Noetherian, regular coherent, l -uni-formly regular coherent, regular, uniformly l -regular, or of global dimension ≤ l in the sense of Definition 5.2 respectively. Intrinsic definitions of the regularity properties.
One can give an in-trinsic definition of the regularity properties above without referring to the Yonedaembedding. The situation is quite nice for regular coherent and l -uniformly regularcoherent for an idempotent complete additive category as as explained below. Lemma 5.6 (Intrinsic Reformulation of regular coherent) . Let A be an idempotentcomplete additive category.(i) Let l ≥ be a natural number. Then A is l -uniformly regular coherent ifand only if for every morphism f : A → A we can find a sequence oflength l in A → A l f l −→ A l − f l − −−−→ · · · f −→ A f −→ A
06 BARTELS, A. AND L¨UCK, W. which is exact at A i for i = 1 , , . . . , n ;(ii) A is -uniformly regular coherent if and only if for every morphism f : A → A we can find a factorization A f −→ B f −→ A of f such that f is sur-jective and f is injective;(iii) The following assertions are equivalent:(a) A is -uniformly regular coherent;(b) For every morphism f : A → A there exists a morphism f : A → A − such that A f −→ A f −→ A − → is exact;(c) For every morphism f : A → A there exists a morphism g : A → A satisfying f ◦ g ◦ f = f ;(iv) A is regular coherent if and only if for every morphism f : A → A wecan find a sequence of finite length in A → A n f n −→ A n − f n − −−−→ · · · f −→ A f −→ A which is exact at A i for i = 1 , , . . . , n .Proof. (i) it suffices to prove that the following statements are equivalent:(a) For any morphisms f : P → P of finitely generated projective Z A -modules we can find finitely generated projective Z A -modules P , P , . . . , P l and an exact sequence of Z A -modules0 → P l f l −→ P l − f l − −−−→ · · · f −→ P f −→ P ;(b) For any finitely presented Z A -module M there exists finitely generatedprojective Z A -modules P , P , . . . , P l and an exact sequence of Z A -modules0 → P l f l −→ P l − f l − −−−→ · · · f −→ P f −→ P f −→ M → . The implication implication (b) = ⇒ (a) is obvious since cok( f ) is a finitelypresented Z A -module. It remains to prove the implication (a) = ⇒ (b). Let f : P → P be a Z A -homomorphism of finitely generated projective Z A -modules.By assumption we can find an exact sequence of Z A -modules0 → Q l c l −→ Q l − c n − −−−→ · · · c −→ Q c −→ Q c −→ cok( f ) → . Let P ∗ be the 1-dimensional Z A -chain complex whose first differential is f . Let Q ∗ be the l -dimensional Z A -chain complex whose i th chain module is Q i for 0 ≤ i ≤ l and whose i th differential is c i : Q i → Q i − for 1 ≤ i ≤ l . One easily constructsa Z A -chain map u ∗ : P ∗ → Q ∗ such that H ( u ∗ ) is an isomorphism. Let cone( u ∗ )be the mapping cone. We conclude H i (cone( u ∗ )) = 0 for i = 2 from the longexact homology sequence associated to the exact sequence 0 → P ∗ i ∗ −→ cyl( u ∗ ) p ∗ −→ cone( u ∗ ) → q ∗ : cyl( u ∗ ) → Q ∗ is a Z A -chain homotopy equivalence with q ∗ ◦ i ∗ = u ∗ . Let D ∗ ⊆ cone( u ∗ ) be the Z A -subchain complex, whose i -th chain module is cone( u ∗ ) for i ≥
3, the kernelof the second differential of cone( u ∗ ) for i = 2 and { } for i = 0 ,
1. Then D i is finitely generated projective for i ≥ k ∗ : D ∗ → cone( u ∗ )induces isomorphisms on homology groups. Define the Z A -chain complex C ∗ bythe pullback C ∗ p ∗ / / k ∗ (cid:15) (cid:15) D ∗ k ∗ (cid:15) (cid:15) cyl( u ∗ ) p ∗ / / cone( u ∗ ) ANISHING OF NIL-TERMS AND NEGATIVE K -THEORY 17 This can be extended to a commutative diagram of Z A -chain complexes with exactrows 0 / / P ∗ i ∗ / / id (cid:15) (cid:15) C ∗ p ∗ / / k ∗ (cid:15) (cid:15) D ∗ k ∗ (cid:15) (cid:15) / / / / P ∗ i ∗ / / cyl( u ∗ ) p ∗ / / cone( u ∗ ) / / C ∗ is a l -dimensional Z A -chain complex whose Z A -chain modules are finitelygenerated projective. Since D i = 0 for i = 0 ,
1, we can identify P = C and P = C and the first differentials of P ∗ and C ∗ . Since k ∗ induces isomorphisms onhomology, the same is true for k ∗ . Hence C ∗ yields the desired extension of f toan exact sequence 0 → C l → C l − → · · · → C → P f −→ P This finishes the proof of assertion ((i)).(ii) Suppose that A is 1-uniformly regular coherent. Consider a morphism f : A → A . Let M be the finitely presented Z A -module given by the cokernel of the Z A -homomorphism ι ( f ) : ι ( A ) → ι ( A ). By assumption we can find an exact sequence0 → P → P → M → Z A -modules, where P and P are finitely gen-erated projective. We conclude from Lemma 4.4 (iv) that the image of ι ( f ) isfinitely generated projective. Hence we obtain a factorization if ι ( f ) as a composite ι ( f ) : ι ( A ) f ′ −→ im( ι ( f )) f ′ −→ ι ( A ) such that im( ι ( f )) is a finitely generated projec-tive Z A -module, f ′ is surjective, and f ′ is injective. We conclude from Lemma 4.10and Lemma 4.11 that im( f ) can be identified with ι ( B ) for some object B in A and there are morphisms f : A → B and f : B → A such that f ′ = ι ( f ) and f ′ = ι ( f ). Moreover, f is surjective, f is injective and f = f ◦ f .Suppose that for every morphism f : A → A we can find a factorization A f −→ B f −→ A of f such that f is surjective and f is injective. Consider any finitelypresented Z A -module M . We conclude from Lemma 4.10 that there is a morphism f : A → A in A and a morphism p : ι ( A ) → M of Z A -modules such that thesequence ι ( A ) ι ( f ) −−→ ι ( A ) p −→ M → f = f ◦ f such that f is surjective and f is injective. Let B be the domain of f . Weconclude from Lemma 4.10 that we obtain a short exact sequence 0 → ι ( B ) ι ( f ) −−−→ ι ( A ) p −→ M →
0. This is a 1-dimensional finite projection Z A -resolution of M .This finishes the proof of assertion (ii).(iii) We first show (iii)a = ⇒ (iii)c. Consider a morphism f : A → A . Let M be the finitely presented Z A -module given by the cokernel of ι ( f ) : ι ( A ) → ι ( A ).We obtain an exact sequence of Z A -modules ι ( A ) ι ( f ) −−→ ι ( A ) p −→ M →
0. Byassumption M is a finitely generated projective Z A -module. Let ι ( f ) : ι ( A ) q −→ im( ι ( f )) j −→ ι ( A ) be the obvious factorization of ι ( f ), Since M projective, im( f )is a direct summand in ι ( A ). We conclude from Lemma 4.10 and Lemma 4.11that we can identify im( ι ( f )) with ι ( B ) for an appropriate object B in A andcan find morphisms r : A → B and s : B → A in A such that ι ( r ) ◦ j = id ι ( B ) and q ◦ ι ( s ) = id ι ( B ) . Define g : A → A by g = s ◦ r . One easily checks that ι ( f ) ◦ ι ( g ) ◦ ι ( f ) = ι ( f ). Hence f ◦ g ◦ f = f .Next we show (iii)c = ⇒ (iii)b. Let f : A → A be a morphism in A . Choose amorphism h : A → A with f ◦ h ◦ f = f . Then f ◦ h : A → A is an idempotent. Since A is idempotent complete, we can find objects A − and A ⊥− and an isomor-phism u : A ∼ = −→ A − ⊕ A ⊥− in A such that u ◦ (id A − f ◦ h ) ◦ u − is (cid:18) id A −
00 0 (cid:19) .Define g : A → A − by the composite A u −→ A − ⊕ A ⊥− A − −−−−→ A − . One easilychecks that the sequence A f −→ A g −→ A − → ⇒ (iii)a. Consider a finitely presented Z A -module M .We conclude from Lemma 4.10 and that we an find a morphism f : A → A to-gether with an exact sequence of Z A -modules ι ( A ) ι ( f ) −−−→ ι ( A ) p −→ M →
0. Choosea morphism f : A → A − such that the sequence A f −→ A f −→ A − → A . Then we obtain an exact sequence of Z A -modules ι ( A ) ι ( f ) −−−→ ι ( A ) ι ( f ) −−−→ ι ( A − ) → M is Z A -isomorphic to ι ( A − ) andhence finitely generated projective. This finishes the proof of assertion (iii).(iv) This follows from assertion (i). This finishes the proof of Lemma 5.6. (cid:3) Next we deal with the property Noetherian. Consider two morphisms f : A → B and f ′ : A ′ → B . We write f ⊆ f ′ if there exists a morphism g : A → A ′ with f = f ′ ◦ g . Lemma 4.10 implies(5.7) f ⊆ f ′ ⇐⇒ im( f ∗ : mor A (? , A ) → mor A (? , B )) ⊆ im( f ′∗ : mor A (? , A ′ ) → mor A (? , B )) . Lemma 5.8 (Intrinsic Reformulation of Noetherian) . Let A be an additive category.Then the following assertions are equivalent:(i) A is Noetherian.(ii) Each object A has the following property: Consider any directed set I andcollections of morphisms { f i : A i → A | i ∈ I } with A as target such that f i ⊆ f j holds for i ≤ j . Then there exists i ∈ I with f i ⊆ f i for all i ∈ I .Proof. Suppose that (i) is true. Consider a directed set I and collections of mor-phisms { f i | i ∈ I } such that f i ⊆ f j holds for i ≤ j . Define M ⊆ mor A (?; A )by M = [ i ∈ I im(( f i ) ∗ ) . Since I is directed, we can find for two elements i and i another element j ∈ I with i , i ≤ j . Hence we have im(( f i ) ∗ ) , im(( f i ) ∗ ) ⊆ im(( f j ) ∗ ). This implies that M is Z A -submodule of mor A (? , A ). Since A is Noetherian, M is finitely generated.Hence there exists i ∈ I with im(( f i ) ∗ ) = M . Hence we get for every i ∈ I thatim(( f i ) ∗ ) ⊆ im(( f i ) ∗ ) and hence i ≤ i holds. Hence A satisfies (ii).Suppose that property (ii) holds. It remains to show for every finitely generated Z A -module N that every Z A -submodule M ⊆ N is finitely generated. Choose anobject A and a Z A -epimorphism u : mor A (? , A ) → N . Then u − ( M ) defined by u − ( M )(?) = u (?) − ( M (?)) is a Z A -submodule of mor A (? , A ) and u induces anepimorphism u − ( M ) → M . Therefore M is finitely generated if u − ( M ) is. Hencewe can assume without loss of generality N = mor A (? , A ).Let { M i | i ∈ I } the collection of finitely generated Z A -submodules of M directedby inclusion. For each i ∈ I we can choose an object A i ∈ A and a morphism f i : A i → A such that im(( f i ) ∗ ) = M i holds. Then { f i | i ∈ I } satisfies f i ⊆ f j for i ≤ j . By assumption there exists i ∈ I with f i ⊆ f i and hence im(( f i ) ∗ ) =im(( f i ) ∗ ) for all i ∈ I . This implies M = im(( f i ) ∗ ). Hence M is finitely generated. (cid:3) ANISHING OF NIL-TERMS AND NEGATIVE K -THEORY 19 Vanishing of Nil-terms
Nil-categories.
The next definition is taken from [11, Definition 7.1].
Definition 6.1 (Nilpotent morphisms and Nil-categories) . Let A be an additivecategory and Φ be an automorphism of A .(i) A morphism f : Φ( A ) → A of A is called Φ -nilpotent if for some n ≥ n -fold composite f ( n ) := f ◦ Φ( f ) ◦ · · · ◦ Φ n − ( f ) : Φ n ( A ) → A. is trivial;(ii) The category Nil( A , Φ) has as objects pairs (
A, φ ) where φ : Φ( A ) → A is a Φ-nilpotent morphism in A . A morphism from ( A, φ ) to (
B, µ ) is amorphism u : A → B in A such that the following diagram is commutative:Φ( A ) φ / / Φ( u ) (cid:15) (cid:15) A u (cid:15) (cid:15) Φ( B ) µ / / B The category Nil( A , Φ) inherits the structure of an exact category from A , asequence in Nil( A , Φ) is declared to be exact if the underlying sequence in A is splitexact.Let Φ : A ∼ = −→ A be an automorphism of an additive category A . It induces anautomorphism Φ − ∗ : MOD - Z A ∼ = −→ MOD - Z A of abelian categories by precompo-sition with Φ − : A ∼ = −→ A . It sends MOD - Z A fgf to itself, since Φ − ∗ mor A (? , A ) isisomorphic to mor A (? , Φ( A )). Thus we obtain an automorphism of additive cate-gories Φ − ∗ : MOD - Z A fgf ∼ = −→ MOD - Z A fgf Lemma 6.2.
There is an equivalence of exact categories ι : Nil( A ; Φ) ≃ −→ Nil(
MOD - Z A fgf ; Φ − ∗ ) Proof.
The desired functors ι sends an object ( A, f ) in Nil( A ; φ ) given by a mor-phism f : Φ( A ) → A to the object in Nil( MOD - Z A fgf ; Φ − ∗ ) given by the compositeΦ − ∗ mor A (? , A ) = mor A (Φ − (?) , A ) Φ −→ mor A (? , Φ( A )) mor A (? ,f ) −−−−−−→ mor A (? , A ) . A morphism u : ( A, f ) → ( A ′ , f ′ ) in Nil( A ; Φ), which given by a morphism u : A → A ′ in A satisfying f ′ ◦ Φ( u ) = u ◦ f , is sent to the morphism in Nil( MOD - Z A fgf ; Φ − ∗ )given by the morphism u ∗ : mor A (? , A ) → mor A (? , A ′ ). It defines indeed a mor-phism from ι ( A, f ) to ι ( A ′ , f ′ ) by the commutativity of the following diagrammor A (Φ − (?) , A ) Φ / / mor A (Φ − (?) ,u ) (cid:15) (cid:15) mor A (? , Φ( A )) mor A (? ,f ) / / mor A (? , Φ( u )) (cid:15) (cid:15) mor A (? , A ) mor A (? ,u ) (cid:15) (cid:15) mor A (Φ − (?) , A ′ ) Φ / / mor A (? , Φ( A ′ )) mor A (? ,f ′ ) / / mor A (? , A ′ )It is an equivalence of additive categories by Lemma 4.10. (cid:3) Connective K -theory.Lemma 6.3. Let A be an idempotent complete additive category. Suppose that A is regular coherent. Let Φ : A ∼ = −→ A be any automorphism of additive categories.Denote by J : A →
Nil( A , φ ) the inclusion sending an object A to the object ( A, .Then the induced map on connective K -theory K ( J ) : K ( A ) → K (Nil( A , Φ)) is a weak homotopy equivalence.Proof.
We abbreviate Ψ = Φ − ∗ We have the following commutative diagram A J / / ι (cid:15) (cid:15) Nil( A , Φ) ι (cid:15) (cid:15) MOD - Z A fgf J / / Nil(
MOD - Z A fgf , Ψ)where the vertical arrows are equivalences of exact categories given by Yonedaembeddings, see Lemma 4.10 and Lemma 6.2, and the lower horizontal arrow is theobvious analogue of the upper horizontal arrow. Hence it suffices to show that themap K ( J ) : K ( MOD - Z A fgf ) → K (Nil( MOD - Z A fgf , Ψ))is a weak homotopy equivalence.Denote by
MOD - Z A FL the full subcategory of MOD - Z A consisting of Z A -modules which are of type FL, i.e., possesses a finite dimensional resolution byfinitely generated free Z A -modules.Consider the following commutative diagram(6.4) K ( MOD - Z A fgf ) / / (cid:15) (cid:15) K (Nil( MOD - Z A fgf , Ψ)) (cid:15) (cid:15) K ( MOD - Z A FL ) / / K (Nil( MOD - Z A FL , Ψ))where all arrows are induced by the obvious inclusions of categories.The left vertical arrow in the diagram (6.4) is a weak homotopy equivalence bythe Resolution Theorem, see [16, Theorem 4.6 on page 41].Next we show that the lower horizontal arrow in the diagram (6.4) is a weakhomotopy equivalence. Consider an object (
M, f ) in Nil(
MOD - Z A FL , Ψ).Recall that nilpotent means that for some natural number n ≥ f ( n ) : Ψ n ( M ) Ψ n − ( f ) −−−−−→ Ψ n − ( M ) Ψ n − ( f ) −−−−−→ · · · Ψ( f ) −−−→ Ψ( M ) f −→ M is trivial. We get a filtration of (M,f) by subobjects( M, f ) ⊇ (im( f ) , f | Ψ(im( f )) ) ⊇ (im( f (2) ) , f | Ψ(im( f (2) )) ) ⊇ · · · ⊇ (im( f ( n − ) , f | Ψ(im( f ( n − )) ) ⊇ (im( f ( n ) ) , f | Ψ(im( f ( n ) )) ) = ( { } , id { } ) , where we consider Ψ(im( f ( i ) )) as a Z A -submodule of Ψ( M ) by the injective mapΨ(im( f ( i ) )) → Ψ( M ) which is obtained by applying Ψ to the inclusion im( f ( i ) ) → M . We get exact sequences of Z A -modules0 → im( f ( i ) ) → M → M/ im( f ( i ) ) → → im( f i +1 ) → im( f i ) → im( f i ) / im( f i +1 ) → . Since M is finitely presented and im( f i ) is finitely generated, M/ im( f ( i ) ) is finitelypresented. Since A is regular coherent and idempotent complete by assumption, M and M/ im( f ( i ) ) for all i are of type FL. We conclude by induction over i = 0 , , . . . from Lemma 4.4 (iii) that im( f ( i ) ) and im( f ( i ) ) / im( f ( i +1) ) belong to MOD - Z A FL again. The quotient of (im( f ( i ) ) , f | Ψ(im( f ( i ) )) ) by (im( f ( i +1) ) , f | Ψ(im( f ( i +1) )) ) is givenby (im( f ( i ) ) / im( f ( i +1) ) , MOD - Z A FL for all i . Now thelower horizontal arrow in diagram (6.4) is a weak homotopy equivalence by theDevissage Theorem, see [16, Theorem 4.8 on page 42]. ANISHING OF NIL-TERMS AND NEGATIVE K -THEORY 21 Next we show that the right vertical arrow in the diagram (6.4) induces splitinjections on homotopy groups. For this purpose we consider the following commu-tative diagram of exact categoriesNil(
MOD - Z A fgf , Ψ) I / / I + + ❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳ I ( ( PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP
HNil( Ch ( MOD - Z A fgf ) , Ψ)HNil( Ch res ( MOD - Z A fgf ) , Ψ) I O O H (cid:15) (cid:15) Nil(
MOD - Z A FL , Ψ)The category HNil( Ch ( MOD - Z A fgf ) , Ψ) is given by finite-dimensional chain com-plexes C ∗ over MOD - Z A fgf (with C i = 0 for i ≤ −
1) together with chain maps φ : C ∗ → C ∗ which are homotopy nilpotent, and HNil( Ch res ( MOD - Z A fgf ) , Ψ) isthe full subcategory of HNil( Ch ( MOD - Z A fgf ) , Ψ) consisting of those chain com-plexes for which H i ( C ∗ ) = 0 for i ≥
1. The maps I k for k = 1 , , , H is given by taking the zeroth homology group.The upper horizontal arrow induces a weak homotopy equivalence on connective K -theory by [11, page 173]. The functor H induces a weak homotopy equivalenceon connective K -theory by the Approximation Theorem of Waldhausen, see forinstance [11, Theorem 4.18]. Hence the map induced by I on connective K -theory,which is the right vertical arrow in the diagram (6.4), induces split injections onhomotopy groups.We conclude that all arrows appearing in the diagram (6.4) induce weak ho-motopy equivalences on connective algebraic K -theory. This finishes the proof ofLemma 6.3. (cid:3) Theorem 6.5 (The connective K -theory of additive categories) . Let A be an ad-ditive category which is idempotent complete and regular coherent. Consider anyautomorphism Φ : A ∼ = −→ A of additive categories.Then we get a map of connective spectra a : T K (Φ − ) → K ( A Φ [ t, t − ]) such that π n ( a ) is bijective for n ≥ .Proof. This follows from Theorem 3.4 since Lemma 6.3 implies π n ( E ( R, Φ)) = 0for n ≥ π n ( NK ( A Φ [ t ])) = π n ( NK ( A Φ [ t − ])) = 0 for all n ≥ (cid:3) We will need later the following consequence of Lemma 6.3, where we can dropthe assumption that A is idempotent complete. Lemma 6.6.
Let A be an additive category. Suppose that A is regular coherent.Let Φ : A ∼ = −→ A be any automorphism of additive categories. Denote by J : A →
Nil( A , φ ) the inclusion sending an object A to the object ( A, .Then the induced map π n ( K ( J )) : π n ( K ( A )) → π n ( K (Nil( A , Φ))) is bijective for n ≥ . Proof.
We have the obvious commutative diagram coming from the inclusion
A →
Idem( A ). π n ( K ( A )) / / (cid:15) (cid:15) π n ( K (Nil( A , Φ))) (cid:15) (cid:15) π n ( K (Idem( A ))) / / π n ( K (Nil(Idem( A )) , Idem(Φ)))The left vertical arrow is bijective for n ≥ n ≥ A ) is regular coherent byLemma 5.4 (vi). Hence we have to show that the right vertical arrow is bijective for n ≥
1. For this purpose it suffices to show because of Lemma 2.3 (i) that Nil( A , Φ)is a cofinal full subcategory of Nil(Idem( A ) , Idem(Φ)). This follows from the factthat A is a cofinal full subcategory of Idem( A ). (cid:3) Non-connective K -theory. In the sequel define A [ Z m ] inductively over m by A [ Z m ] := A [ Z m − ] id [ t, t − ], where A [ Z m − ] id [ t, t − ] is the (untwisted) finiteLaurent category associated to A [ Z m − ] and the automorphism given by the iden-tity, see Subsection 1.4. Lemma 6.7.
Let A be an additive category. Suppose that A [ Z m ] is regular coherentfor every m ≥ . Consider any automorphism Φ : A ∼ = −→ A of additive categories.Denote by J : A →
Nil( A , Φ) the inclusion sending an object A to the object ( A, .Then the induced map on non-connective K -theory K ∞ ( J ) : K ∞ ( A ) → K ∞ Nil (Nil( A , Φ)) is a weak homotopy equivalence.Proof.
Fix n ∈ Z . We have to show that π n ( K ∞ ( J )) is bijective. This follows fromLemma 6.6 for n ≥ n ∈ Z a commutative diagram π − n ( K ∞ ( A )) i (cid:15) (cid:15) / / π − n ( K ∞ Nil ( A , Φ)) j (cid:15) (cid:15) π n +1 ( K ∞ ( A [ Z ])) r (cid:15) (cid:15) / / π n +1 ( K ∞ Nil ( A [ Z ] , Φ[ Z ])) s (cid:15) (cid:15) π − n ( K ∞ ( A )) / / π − n ( K ∞ Nil ( A , Φ))where r ◦ i = id and j ◦ s = id and these maps are part of the corresponding(untwisted) Bass-Heller-Swan decompositions. Iterating this, one obtains for every m ≥ π − n ( K ∞ ( A )) i (cid:15) (cid:15) / / π − n ( K ∞ Nil ( A , Φ)) j (cid:15) (cid:15) π n + m ( K ∞ ( A [ Z m ])) r (cid:15) (cid:15) / / π n + m ( K ∞ Nil ( A [ Z m ] , Φ[ Z m ])) s (cid:15) (cid:15) π − n ( K ∞ ( A )) / / π − n ( K ∞ Nil ( A , Φ))where r ◦ i = id and j ◦ s = id holds. Now choose m such that n + m ≥ ANISHING OF NIL-TERMS AND NEGATIVE K -THEORY 23 version π n + m ( K ( A [ Z m ])) → π n + m ( K (Nil( A [ Z m ] , Φ[ Z m ]))) . Since this map is a bijection by Lemma 6.3 the upper horizontal arrow is a retractof an isomorphism and hence itself an isomorphism. (cid:3)
Theorem 6.8 (The non-connective K -theory of additive categories) . Let A bean additive category. Suppose that A [ Z m ] is regular coherent for every m ≥ .Consider any automorphism Φ : A ∼ = −→ A of additive categories.Then we get a weak homotopy equivalence of non-connective spectra a ∞ : T K ∞ (Φ − ) ≃ −→ K ∞ ( A Φ [ t, t − ]) . Proof.
This follows from Theorem 3.1 since Lemma 6.7 implies π n ( E ∞ ( R, Φ)) = 0and hence π n ( NK ∞ ( R Φ [ t ])) = π n ( NK ∞ ( A Φ [ t − ])) = 0 for all n ∈ Z . (cid:3) Noetherian additive categories
Theorem 7.1 (Hilbert Basis Theorem for additive categories) . Consider an additive category A together with an automorphism Φ : A ∼ = −→ A .(i) If the additive category A is Noetherian, then the additive categories A Φ [ t ] , A Φ [ t − ] , and A Φ [ t, t − ] are Noetherian;(ii) If the additive category A Φ [ t ] is Noetherian, then the additive category A Φ [ t, t − ] is Noetherian.Proof. (i) We only treat A Φ [ t ], the proof for A Φ [ t − ] is analogous. For A Φ [ t, t − ]the claim will follow then from (ii).We translate the usual proof of the Hilbert Basis Theorem for rings to additivecategories. Consider a finitely generated Z A Φ [ t ]-module N and a Z A Φ [ t ]-submodule M ⊆ N . We have to show that M is finitely generated. Lemma 4.4 (vi) implies thatthere is an epimorphisms φ : mor A Φ [ t ] (? , A ) → N for some object A . If φ − ( M )is finitely generated, then M is finitely generated since f induces an epimorphism f − ( M ) → M . Hence we can assume without loss of generality N = mor A Φ [ t ] (? , A ).Fix an object Z in A . Consider a non-trivial element f : Z → A in N ( Z ). Wecan write it as a finite sum P d ( f ) k =0 f k · t k , where f k : Φ k ( Z ) → A is a morphism in A and f d ( f ) = 0. We call the natural number d ( f ) the degree of f and R ( f ) = f d ( f ) : Φ d ( f ) ( Z ) → A the leading coefficient of f . We put d (0 : Z → A ) = −∞ and R (0 : Z → A ) = 0.We define now I d as the Z A -submodule of mor A (? , A ) that is generated byall R ( f ) with f ∈ M ( Z ) and d ( f ) = d for some object Z from A . We have I ⊆ I ⊆ I ⊆ · · · and define I to be the Z A -submodule S d ≥ I d . As A is byassumption Noetherian, I and all the I d are finitely generated. Therefore we finda finite collection of morphisms f i ∈ M ( Z i ) ⊆ mor A Φ [ t ] ( Z i , A ) such that the R ( f i )generate I . We abbreviate d i := d ( f i ). Since each f i lies in of the I d -s, we can finda natural number d such that I = I d = I d holds for d ≥ d . Hence we can alsoarrange for the f i to have the following property: for each d the R ( f i ) with d i ≤ d generate I d . We record that R ( f i ) ∈ I d i (Φ d i ( Z i )) ⊆ mor A (Φ d i ( Z i ) , A ).We will show that the f i generate M . Let f ∈ M ( Z ), f = 0. We abbreviate d := d ( f ). We have R ( f ) ∈ I d (Φ d ( Z )) ⊆ mor A (Φ d ( Z ) , A ). We can write R ( f ) = X i R ( f i ) ◦ ϕ i with ϕ i ∈ mor A (Φ d ( Z ) , Φ d i ( Z i ) and ϕ i = 0 whenever d ( f i ) > d ( f ). Set˜ ϕ i := Φ − d i ( ϕ i ) · t d − d i ∈ mor A φ [ t ] ( Z, Z i ) . Then R (cid:16) X i f i ◦ ˜ ϕ i (cid:17) = X i R ( f i ) ◦ Φ d i (Φ − d i ( ϕ i )) = X i R ( f i ) ◦ ϕ i . Thus d ( f − P i f i ◦ ˜ ϕ ) < d . Now we can repeat the argument for f ′ := f − P i f i ◦ ˜ ϕ .By induction on d ( f ) we now find that f belongs to the submodule of mor A φ [ t ](? ,A ) generated by the f i . Hence M is a finitely generated Z A Φ [ t ]-module.(ii) It suffices to show for a A Φ [ t, t − ]-submodule M of mor A Φ [ t,t − ] (? , A ) that M is finitely generated as A Φ [ t, t − ]-module. For Z ∈ A we have mor A Φ [ t ] ( Z, A ) ⊆ mor A Φ [ t,t − ] ( Z, A ). We define the Z A Φ [ t ]-module M ′ by M ′ ( Z ) := M ( Z ) ∩ mor A Φ [ t ] ( Z, A ) . Since A Φ [ t ] is Noetherian, we find a finite collection of morphisms f i ∈ M ′ ( Z i ) ⊆ M ( Z i ) ⊆ mor A Φ [ t,t − ] ( Z i , A ) that generate M ′ as an Z A Φ [ t ]-module. We claim thatthe f i also generate M as an Z A Φ [ t, t − ]-module. Let f ∈ M ( Z ) ⊆ mor A Φ [ t,t − ] ( Z, A ).For d ≥ Z · t d ∈ mor A Φ [ t,t − ] (Φ − d ( Z ) , Z ). For sufficiently large d we have f ◦ (id Z · t d ) ∈ mor A Φ [ t ] (Φ − d ( Z ) , A ) ∩ M (Φ − d ( Z )) = M ′ (Φ − d ( Z )). Thus f ◦ (id Z · t d )belongs to the Z A Φ [ t, t − ]-submodule of M generated by the f i . As (id Z · t d ) is anisomorphism in A Φ [ t, t − ], f also belongs to the Z A Φ [ t, t − ]-submodule of M gen-erated by the f i . (cid:3) Additive categories with finite global dimension
Let Φ :
A → A be an automorphism of the additive category A . Let Φ : A φ [ t ] →A Φ [ t ] be the automorphism of additive categories induced by Φ, which sends themorphisms P ∞ k =0 f k · t k : A → B to the morphism P ∞ k =0 Φ( f k ) · t k : Φ( A ) → Φ( B ).Denote by i : A → A Φ [ t ] the inclusion sending f : A → B to ( f · t ) : A → B .Obviously we have Φ ◦ i = i ◦ Φ.8.1.
The characteristic sequence.
Consider a Z A Φ [ t ]-module M . Let e : i ∗ i ∗ M → M be the Z A Φ [ t ]-morphism which is the adjoint of the Z A -homomorphism id : i ∗ M → i ∗ M under the adjunction (4.7). We get for every object A in A a morphismid Φ( A ) · t : A → Φ( A ) in A Φ [ t ]. It induces a Z A Φ [ t ]-morphism M (id φ ( A ) · t ) : M (Φ( A )) → M ( A ). Since for a morphisms u : A → B in A we have(id φ ( B ) · t ) ◦ i ( u ) = (id φ ( B ) · t ) ◦ ( u · t ) = Φ( u ) · t = (Φ( u ) · t ) ◦ (id Φ( A ) · t ) = i (Φ( u )) ◦ (id Φ( A ) · t ) , we obtain a morphism of Z A -modules(8.1) α ′ : Φ ∗ i ∗ M ∼ = −→ i ∗ M. By applying i ∗ we obtain a morphism of Z A Φ [ t ]-modules α : i ∗ Φ ∗ i ∗ M → i ∗ i ∗ M The morphism id Φ( A ) · t : A → Φ( A ) in A Φ [ t ] induces also a Z -map β ( A ) : i ∗ Φ ∗ i ∗ M ( A ) = i ∗ i ∗ M (Φ( A )) → i ∗ i ∗ M ( A ) . ANISHING OF NIL-TERMS AND NEGATIVE K -THEORY 25 Since for any morphism v = P ∞ k =0 f k · t k : A → B in A Φ [ t ] we have(id Φ( B ) · t ) ◦ v = (id Φ( B ) · t ) ◦ (cid:18) ∞ X k =0 f k · t k (cid:19) = ∞ X k =0 (id Φ( B ) · t ) ◦ ( f k · t k )= ∞ X k =0 Φ( f k ) · t k +1 = ∞ X k =0 (Φ( f k ) · t k ) ◦ (id Φ( A ) · t )= Φ (cid:18) ∞ X k =0 f k · t k (cid:19) ◦ (id Φ( A ) · t )= Φ( v ) ◦ (id Φ( A ) · t )we get a Z A Φ [ t ]-homomorphism denoted by β : i ∗ Φ ∗ i ∗ M → i ∗ i ∗ M. Define the so called characteristic sequence of Z A φ [ t ]-modules by(8.2) 0 → i ∗ Φ ∗ i ∗ M α − β −−−→ i ∗ i ∗ M e −→ M → . Given an object A ∈ A , ( α − β )( A ) is explicitly given by M (Φ(?)) ⊗ Z A mor A φ [ t ] ( A, ?) → M (?) ⊗ Z A mor A φ [ t ] ( A, ?) ,x ⊗ ( f k · t k : A → ?) M (id Φ(?) · t : ? → Φ(?))( x ) ⊗ ( f k · t k : A → ?) − x ⊗ (Φ( f k ) · t k +1 : A → Φ(?)) . and e ( A ) is explicitly given by M (?) ⊗ Z A mor A φ [ t ] ( A, ?) → M ( A ) , x ⊗ ( u : A → ?) M ( u )( x ) = xu. Lemma 8.3.
The characteristic sequence (8.2) is natural in M and exact.Proof. It is obviously natural in M . To prove exactness, it suffices to prove theexactness of the sequence of Z A -modules(8.4) 0 → i ∗ i ∗ Φ ∗ i ∗ M α − β −−−→ i ∗ i ∗ i ∗ M e −→ i ∗ M → . Let N be a Z A -module. We obtain a Z A -isomorphism(8.5) S ( N ) : ∞ M k =0 Φ k ( N ) ∼ = −→ i ∗ i ∗ N, which is defined for an object A in A by the Z -isomorphism S ( N )( A ) : ∞ M k =0 N (Φ k ( A )) ∼ = −→ i ∗ i ∗ N ( A ) = N (?) ⊗ Z A mor A Φ t ( i ( A ) , ?)sending ( x k ) k ≥ to P ∞ k =0 x ⊗ (cid:0) id Φ k ( A ) · t k : A → Φ k ( A ) (cid:1) . The inverse of S ( N )( A )sends y ⊗ (cid:18)P ∞ k =0 f k · t k : A → ? (cid:1) to P ∞ k =0 N ( f k )( y ). Applying this to N = i ∗ Φ ∗ M = Φ ∗ i ∗ M and N = i ∗ M , we get identifications i ∗ i ∗ i ∗ Φ ∗ M = ∞ M k =1 (Φ k ) ∗ i ∗ M ; i ∗ i ∗ i ∗ M = ∞ M k =0 (Φ k ) ∗ i ∗ M. Consider natural numbers m and n with m ≥ n . For an object A let the map s m,n ( A ) : (Φ m ) ∗ M ( A ) → (Φ n ) ∗ M ( A ) be the map obtained by applying M to themorphism id Φ m ( A ) · t m − n : Φ n ( A ) → Φ m ( A ) in A Φ [ t ]. This yields a map of Z A -modules s m,n : (Φ m ) ∗ i ∗ M → (Φ n ) ∗ i ∗ M. Under these identifications the Z A -sequence (8.4) becomes the sequence0 → ∞ M m =1 (Φ m ) ∗ i ∗ M − s , · · · id − s , · · · − s , · · · − s , · · · ... ... ... ... . . . −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→ ∞ M n =0 (Φ n ) ∗ i ∗ M (cid:16) id s , s , · · · (cid:17) −−−−−−−−−−−−−−−−→ i ∗ M → . Since s m,n ◦ s l,m = s l,n for l ≥ m ≥ n and s m,m = id hold, this sequence is splitexact, with a splitting given by ∞ M m =1 (Φ m ) ∗ M s , s , s , · · · s , s , · · · s , · · · · · · · · · ... ... ... ... ... . . . ←−−−−−−−−−−−−−−−−−−−−−−−−−− ∞ M n =0 (Φ n ) ∗ M id0000... ←−−−− M. (cid:3) Localization.Definition 8.6 (Local module) . We call a Z A Φ [ t ]-module M local if for any object A in A and any natural number k ∈ N the map M (id Φ k ( A ) · t k ) : M ( A ) → M (Φ k ( A ))induced by the morphism id Φ k ( A ) · t k : A → Φ k ( A ) in A Φ [ t ] is bijective.Let j : A Φ [ t ] → A Φ [ t, t − ] be the inclusion. Lemma 8.7. A Z A Φ [ t ] -module M is local if and only if there is a Z A Φ [ t, t − ] -module N such that M and j ∗ N are isomorphic as Z A Φ [ t ] -modules.Proof. Since the morphism id Φ k ( A ) · t k : A → Φ k ( A ) in A Φ [ t ] becomes invertiblewhen considered in A Φ [ t, t − ], a Z A Φ [ t ]-module M is local, if there is a Z A Φ [ t, t − ]-module N such that M and j ∗ N are isomorphic as Z A Φ [ t, t − ]-modules.Now consider a local Z A Φ [ t ]-module M . We have to explain how the Z A Φ [ t ]-structure extends to a Z A Φ [ t, t − ]-structure. Consider a morphism u : A → B ANISHING OF NIL-TERMS AND NEGATIVE K -THEORY 27 in A Φ [ t, t − ]. Then we can choose a natural number m such that the compos-ite A u −→ B id Φ m ( B ) · t m −−−−−−−−→ Φ m ( B ) is a morphism in A Φ [ t ]. Hence we have the Z -map M ((id Φ m ( B ) · t m ) ◦ u ) : M (Φ m ( B )) → M ( A ). Since M is local, the Z -map M (id Φ m ( B ) · t m ) : M (Φ m ( B )) → M ( B ) is a isomorphism. Now define M ( u ) : M ( B ) M (id Φ m ( B ) · t m ) − −−−−−−−−−−−−→ M (Φ m ( B )) M ((id Φ m ( B ) · t m ) ◦ u ) −−−−−−−−−−−−−→ M ( A ) . We leave it to the reader to check that the definition of M ( u ) is independent of thechoice of m and that we obtain the desired Z A Φ [ t, t − ]-structure on M extendingthe given Z A Φ [ t ]-structure. (cid:3) Let M be a Z A Φ [ t ]-module. We want to assign to it a Z A Φ [ t, t − ]-module S − M as follows. Consider an object A in A . Define the abelian group S − M ( A ) := { ( l, x ) | l ∈ Z , x ∈ M (Φ l ( A )) } / ∼ for the equivalence relation ∼ , where ( l , x ) and ( l , x ) are equivalent if and only ifthere is an integer l ∈ Z with l ≤ l , l such that the elements M (id Φ l ( A ) · t l − l )( x )and M (id Φ l ( A ) · t l − l )( x ) of M (Φ l ( A )) agree. Given a morphism u : A → B in A Φ [ t, t − ], we can choose a natural number m such that the composite A u −→ B id Φ m ( B ) · t m −−−−−−−−→ Φ m ( B ) is a morphism in A Φ [ t ]. Define S − ( M )( u ) : S − ( M )( B ) → S − M ( A ) by sending [ l, x ] to the class of (cid:0) l − m, M (cid:0) Φ l − m ((id Φ m ( B ) · t m ) ◦ u ) (cid:1) ( x ) (cid:1) .This is independent of the choice of the representative of [ l, x ], since we get for thedifferent representative ( l − , M (id Φ l ( B ) · t )( x )) S − ( M )([ l − , M (id Φ l ( B ) · t )( x )])= (cid:2) l − − m, M (cid:0) Φ l − − m ((id Φ m ( B ) · t m ) ◦ u ) (cid:1) ◦ M (id Φ l ( B ) · t )( x ) (cid:3) = (cid:2) l − − m, M (cid:0) (id Φ l ( B ) · t ) ◦ Φ l − − m ((id Φ m ( B ) · t m ) ◦ u ) (cid:1) ( x ) (cid:3) = (cid:2) l − − m, M (cid:0) (id Φ l ( B ) · t ) ◦ (id Φ l − B ) · t m ) ◦ Φ l − − m ( u ) (cid:1) ( x ) (cid:3) = (cid:2) l − − m, M (cid:0) (id Φ l ( B ) · t m ◦ id φ l − m ( B ) · t ) ◦ Φ l − − m ( u ) (cid:1) ( x ) (cid:3) = (cid:2) l − − m, M (cid:0) (id Φ l ( B ) · t m ) ◦ Φ l − m ( u ) (cid:1) ( x ) (cid:3) = S − ( M )([ l, x ]) . This is independent of the choice of m by the following calculation (cid:2) l − ( m + 1) , M (cid:0) Φ l − ( m +1) ((id Φ m +1 ( B ) · t m +1 ) ◦ u ) (cid:1) ( x ) (cid:3) = (cid:2) l − ( m + 1) , M (cid:0) Φ l − ( m +1) (id Φ m +1 ( B ) · t m +1 ) ◦ Φ l − ( m +1) ( u ) (cid:1) ( x ) (cid:3) = (cid:2) l − ( m + 1) , M (cid:0) (id Φ l ( B ) · t m +1 ) ◦ Φ l − ( m +1) ( u ) (cid:1) ( x ) (cid:3) = (cid:2) l − m − , M (cid:0) (id Φ l ( B ) · t m ) ◦ (id Φ l − m ( B ) · t ) ◦ Φ l − m − ( u ) (cid:1) ( x ) (cid:3) = (cid:2) l − m − , M (cid:0) (id Φ l ( B ) · t m ) ◦ Φ l − m ( u ) ◦ (id Φ l − m ( B ) · t ) (cid:1) ( x ) (cid:3) = (cid:2) l − m − , M (cid:0) Φ l − m (id Φ m ( B ) · t m ) ◦ Φ l − m ( u ) ◦ (id Φ l − m ( B ) · t ) (cid:1) ( x ) (cid:3) = (cid:2) l − m − , M (cid:0) Φ l − m ((id Φ m ( B ) · t m ) ◦ u ) ◦ (id Φ l − m ( B ) · t ) (cid:1) ( x ) (cid:3) = (cid:2) l − m − , M (id φ l − m ( B ) · t ) ◦ M (cid:0) Φ l − m ((id Φ m ( B ) · t m ) ◦ u ) (cid:1) ( x ) (cid:3) = (cid:2) l − m, M (cid:0) Φ l − m ((id Φ m ( B ) · t m ) ◦ u ) (cid:1) ( x ) (cid:3) . We leave it to the reader to check that S − M ( v ◦ u ) = S − M ( u ) ◦ S − M ( v ) holdsfor any two composable morphisms u : A → B and v : B → C in A Φ [ t, t − ] and S − M (id A ) = id S − M ( A ) holds for any object A in A . Notice that the A φ [ t ]-module j ∗ S − M is local by Lemma 8.7There is a natural map of A Φ [ t ]-modules I : M → j ∗ S − M which is given for an object A of A by the map I ( A ) : M ( A ) → S − M ( A ) sending x to (0 , x ). We claim that I is a localization in the sense that for any local A φ [ t ]-module N and any A φ [ t ]-homomorphism f : M → N there exists precisely one A φ [ t ]-homomorphism S − f : S − M → N .Firstly we explain that there is at most one such map S − f with these properties.Namely, consider an object A ∈ A and an element [ m, x ] ∈ S − ( M )( A ). If m ≥ S − f ( A )([ m, x ]) = S − ( A )([0 , M (id Φ m ( A ) · t m )( x )])(8.8) = S − ( A ) ◦ I ( A ) ◦ M (id Φ m ( A ) · t m )( x )= f ( A ) ◦ M (id Φ m ( A ) · t m )( x ) . Suppose m ≤
0. Since we have S − ( M )(id A · t − m )([ m, x ]) = [0 , x ], we compute for[ m, x ] ∈ S − M ( A ) S − ( N )(id A · t − m ) ◦ S − f ( A )([ m, x ])= S − f (Φ m ( A )) ◦ S − ( M )(id A · t − m )([ m, x ])= S − f (Φ m ( A ))([0 , x ])= S − f (Φ m ( A )) ◦ I ( A )( x )= f (Φ m ( A ))( x ) . Since the locality of N implies that S − ( N )(id φ m ( A ) · t m ) is an isomorphism, weconclude(8.9) S − f ( A )([ m, x ]) = S − ( N )(id A · t − m ) − ◦ f (Φ m ( A ))( x ) . Hence S − f ( A ) is determined by the equations (8.8) and (8.9). We leave it tothe reader to check that it makes sense to define the desired Z A [ t ]-homomorphism S − f ( A ) by the equations (8.8) and (8.9).The adjoint of I : M → j ∗ S − M under the adjunction (4.7) is denoted by(8.10) α : j ∗ M → S − M. The adjoint of id j ∗ M under the adjunction (4.7) is the Z A Φ [ t ]-homomorphism(8.11) λ : M → j ∗ j ∗ M which is explicitly given by M (??) → mor A Φ [ t,t − ] (?? , ?) ⊗ Z A Φ [ t ] M (?) sending u ∈ M (??) to id ?? ⊗ u . Given an Z A Φ [ t, t − ]-module N , the adjoint of id j ∗ N under theadjunction (4.7) is the Z A Φ [ t, t − ]-homomorphism(8.12) ρ : j ∗ j ∗ N → N which is explicitly given by N (?) ⊗ Z A Φ [ t ] mor A Φ [ t,t − ] (?? , ?) → N (??) sending x ⊗ u to N ( u )( x ) = xu . Lemma 8.13. (i) The Z A Φ [ t ] -homomorphism λ : M → j ∗ j ∗ M of (8.11) is alocalization;(ii) The Z A φ [ t, t − ] -homomorphism α : j ∗ M → S − M of (8.10) is an isomor-phism, which is natural in M ;(iii) Let N be a Z A Φ [ t, t − ] -module. Then Z A Φ [ t, t − ] -map ρ : j ∗ j ∗ N → N of (8.12) is an isomorphism. ANISHING OF NIL-TERMS AND NEGATIVE K -THEORY 29 Proof. (i) Let f : M → N be a Z A Φ [ t ]-map with a local Z A Φ [ t ]-module as target.Because of Lemma 8.7 there is a Z A Φ [ t, t − ]-module N ′ and a Z A Φ [ t ]-isomorphism u : N → j ∗ N ′ . Let the Z A Φ [ t, t − ]-map v : j ∗ M → N be the adjoint of u ◦ f under the adjunction (4.7). Because of the naturality of the adjunction (4.7) weget for the composite f : j ∗ j ∗ M j ∗ v −−−→ j ∗ N ′ u − −−→ N that f ◦ λ = f holds. Weconclude from the explicite description of λ and the fact that for any morphism u : A → B in A Φ [ t, t − ] there is a natural number such that the composite of u with id Φ m ( B ) · t m : Φ( M ) → Φ m ( B ) lies in A Φ [ t ] that f is uniquely determined by f ◦ λ = f .(ii) Obviously α is natural in M . The naturality of the adjunction (4.7) implies j ∗ α ◦ λ = I. Since both I : M → j ∗ S − M and λ : M → j ∗ j ∗ M a localizations, j ∗ α and hence α are bijective.(iii) It suffices to show that j ∗ ρ : j ∗ j ∗ j ∗ N → j ∗ N is bijective. Assertion (i) appliedto j ∗ N and the naturality of the adjunction (4.7) imply that j ∗ ρ : j ∗ N → j ∗ j ∗ j ∗ N isa localization. Since id j ∗ N : j ∗ N → j ∗ N is a localization, j ∗ ρ is an isomorphism. (cid:3) Lemma 8.14.
The functor j ∗ : MOD - Z A φ [ t ] → MOD - Z A φ [ t, t − ] is flat.Proof. Because of the adjunction (4.7) the functor j ∗ is right exact by a generalargument, see [21, Theorem 2.6.1. on page 51]. Hence it remains to show thatfor an injective Z A φ [ t ]-map i : M → N the Z A φ [ t, t − ]-map j ∗ i : j ∗ M → j ∗ N isinjective. In view of Lemma 8.13 (ii) it suffices to show that S − i : S − M → S − N is injective. Consider an object A in A and an element [ l, x ] in the ker-nel of S − ( i )( A ). Since S − i ([ l, x ]) = [ l, i (Φ l ( A ))( x )], there is a natural num-ber m ≤ l such that N (id Φ l ( A ) · t m − l )( i (Φ l ( A ))( x )) = 0. Since N (id Φ l ( A ) · t m − l ) ◦ i (Φ l ( A )) = i (Φ m − l ( A )) ◦ M (id Φ l ( A ) · t m − l ) and i (Φ m − l ( A )) is by assumption injec-tive, M (id Φ l ( A ) · t m − l )( x ) = 0. This implies [ l, x ] = 0. (cid:3) Global dimension.
Recall that an additive category A has global dimension ≤ d if the abelian category MOD - Z A has global dimension ≤ d , i.e., if every Z A -module has a projective d -dimensional resolution, see Definition 5.2 (vi). Theorem 8.15 (Global dimension and the passage from A to A Φ [ t ]) . Let A be anadditive category A and Φ :
A → A be an automorphism of additive categories.(i) Let M be a Z A Φ [ t ] -module. If pdim A ( i ∗ M ) ≤ d , then pdim A [ t ] ( M ) ≤ d +1 ;(ii) If A has global dimension ≤ d , then A Φ [ t ] has global dimension ≤ ( d + 1) . Theorem 8.15 is a version of the Hilbert syzygy theorem. Its the proof is notmuch different from the classical syzygy Theorem for rings. For a more generalversion see [12, Corollary 31.1 on page 119].
Proof of Theorem 8.15.
Obviously i ∗ : MOD - Z A Φ [ t ] → MOD - Z A is faithfully flatand is compatible with direct sums over arbitrary index sets. Next we showthat i ∗ sends projective Z A φ [ t ]-modules to projective Z A -modules. It suffices toshow that i ∗ mor A Φ [ t ] (? , A ) ∼ = i ∗ i ∗ mor A (? , A ) is free as a Z A -module for any ob-ject A . This follows from the Z A -isomorphism (8.5), since (Φ k ) ∗ mor A (? , A ) ∼ =mor A (? , Φ − k ( A )).The functor i ∗ : MOD - Z A →
MOD - Z A Φ [ t ] is compatible with direct sums overarbitrary index sets, is right exact and sends mor A (? , A ) to mor A Φ [ t ] (? , A ). Inparticular i ∗ respects the properties finitely generated, free, and projective. Nextwe want to show that i ∗ is faithfully flat. For this purpose it suffices to show that i ∗ ◦ i ∗ is faithfully flat. This is obvious since i ∗ ◦ i ∗ is the functor sending a morphism f : M → N to the morphism L k ∈ N (Φ k ) ∗ ( f ) : L k ∈ N (Φ k ) ∗ ( M ) → L k ∈ N (Φ k ) ∗ ( f )under the identification (8.5).Now consider a Z Φ [ t ]-module M with pdim A ( i ∗ M ) ≤ d . Since the Z A -modules i ∗ M and Φ ∗ i ∗ M are isomorphic, see (8.1), we get pdim A ( φ ∗ i ∗ M ) ≤ d . Since i ∗ isfaithfully flat and respects projective modules, we conclude pdim A Φ [ t ] ( i ∗ i ∗ M ) ≤ d and pdim A Φ [ t ] ( i ∗ Φ ∗ i ∗ M ) ≤ d . Now Lemma 4.4 (v) and Lemma 8.3 together implypdim A Φ [ t ] ( M ) ≤ ( d + 1).(ii) This follows directly from assertion (i). (cid:3) Theorem 8.16 (Global dimension and the passage from A φ [ t ] to A Φ [ t, t − ]) . Let A be an additive category A and Φ :
A → A be an automorphism of additive categories.(i) Let M be a Z A Φ [ t, t − ] -module. If we have pdim A [ t ] ( j ∗ M ) ≤ d , then weget pdim A [ t,t − ] ( M ) ≤ d ;(ii) If A Φ [ t ] has global dimension ≤ d , then A Φ [ t, t − ] has global dimension ≤ d .Proof. (i) Let M be a Z [ A ] Φ [ t, t − ]-module satisfying pdim A [ t ] ( j ∗ M ) ≤ d . Thefunctor j ∗ : MOD - Z A φ [ t ] → MOD - Z A φ [ t, t − ] is flat by Lemma 8.14. Since it re-spects the property projective, we get pdim A Φ [ t,t − ] ( j ∗ j ∗ M ) ≤ d . Lemma 8.13 (iii)implies pdim A [ t,t − ] ( M ) ≤ d .(ii) This follows from assertion (i). (cid:3) Regular additive categories
Regularity for additive categories A requires finite resolutions of finitely pre-sented modules, but not for arbitrary modules. In particular, regularity has noconsequence for global dimension and we cannot use Theorem 8.15 in the followingresult. Theorem 9.1 (Regularity and the passage from A to A Φ [ t ]) . Let A be an additivecategory A and Φ :
A → A be an automorphism of additive categories. Let l be anatural number.(i) Suppose that A is regular or l -uniformly regular respectively. Then A Φ [ t ] is regular or ( l + 2) -uniformly regular respectively;(ii) Suppose that A [ t ] is regular or l -uniformly regular respectively. Then A Φ [ t, t − ] is regular or l -uniformly regular respectively.Proof. (i) We know already that A φ [ t ] is Noetherian because of Theorem 7.1 (i).Let M be a finitely generated A φ [ t ]-module. We have to show that it has a finitelygenerated projective resolution which is finite-dimensional or ( l + 1)-dimensional.Since A φ [ t ] is Noetherian, there exists a finitely generated projective resolution of M which may be infinite-dimensional. We conclude from Theorem 4.4 (iv) thatit suffices to show the projective dimension of M is finite or bounded by ( l + 1)respectively. As M is finitely generated we find a finite collection of elements x j ∈ M ( Z j ) with objects Z j from A such that the x j generate M as an Z A Φ [ t ]-module.For d ≥ Z j · t d : Φ − d ( Z j ) → Z j in A Φ [ t ] and set x j [ d ] := M (id Z j · t d )( x j ) ∈ M (Φ − d ( Z j )). Let M n be the Z A -submodule of i ∗ M generatedby all x j [ d ] with d ≤ n . We obtain an increasing subsequence M ⊆ M ⊆ M ⊆ M ⊆ · · · of Z A -submodules of i ∗ M with i ∗ M = S n ≥ M n . Let T n : i ∗ M → Φ ∗ i ∗ M be the following Z A -morphism. For an object Z from A consider id Φ n ( Z ) · t n ∈ mor A Φ [ t ] ( Z, Φ n ( Z )) and define T Z : i ∗ M ( Z ) = M ( Z ) → Φ ∗ i ∗ M ( Z ) = M (Φ( Z ))to be M (id Φ n ( Z ) · t n ). Let pr n : (Φ n ) ∗ ( M n ) → (Φ n ) ∗ ( M n ) / (Φ n ) ∗ ( M n − ) be theprojection. The composition f n M T n −−→ (Φ n ) ∗ ( M n ) pr n −−→ (Φ n ) ∗ ( M n ) / (Φ n ) ∗ ( M n − ) ANISHING OF NIL-TERMS AND NEGATIVE K -THEORY 31 is surjective and we write K n for its kernel. We obtain an increasing sequence of A -submodules K ⊆ K ⊆ K ⊆ · · · of M . Since A is Noetherian and M isfinitely generated, there exists an integer n ≥ K n = K n holds for n ≥ n . Define g n : (Φ n ) ∗ M n / (Φ n ) ∗ M n − → (Φ n ) ∗ M n / (Φ n ) ∗ M n − for n ≥ n to be the map induced by Φ ∗ ( T n − n ) for n ≥ n . We obtain for every naturalnumber n with n ≥ n a commutative diagram of Z A -modules with exact rows0 / / K n / / ∼ = (cid:15) (cid:15) M f n / / id M (cid:15) (cid:15) (Φ n ) ∗ M n / (Φ n ) ∗ M n − / / g n (cid:15) (cid:15) / / K n / / M f n / / (Φ n ) ∗ M n / (Φ n ) ∗ M n − / / g n is an isomorphism of Z A -module n ≥ n . As Φ ∗ is an isomorphism wehavepdim Z A M n /M n − = pdim Z A (Φ n ) ∗ ( M n /M n − ) = pdim Z A (Φ n ) ∗ M n / (Φ n ) ∗ M n − ) . Thus for n ≥ n we have pdim Z A ( M n /M n − ) = pdim Z A ( M n /M n − ). We havethe short exact sequence 0 → M n − → M n → M n /M n − → Z A ( M n ) ≤ sup { pdim Z A ( M n − ) , pdim Z A ( M n /M n − ) } . This implies by induction over n ≥ n pdim Z A ( M n ) ≤ sup { pdim Z A ( M n − ) , pdim Z A ( M n /M n − ) } . Put D := sup (cid:8) sup { pdim Z A ( M k ) | k = 0 , , . . . , n − } , pdim Z A ( M n /M n − ) (cid:9) . Notice that
D < ∞ if A is regular and D ≤ l if A is uniformly l -regular. We getpdim Z A (cid:18)M n ∈ N M n (cid:19) ≤ sup { pdim Z A ( M n ) | n ≥ } ≤ D. We have the short exact sequence of A -modules0 → M n ∈ N M n → M n ∈ N M n → i ∗ M → , where the first map is given by ( x n ) n ≥ ( x , x − x .x , − x , . . . ) and the secondby ( x n ) n ≥ P n ≥ x n . We conclude from Lemma 4.4 (v)pdim Z A ( i ∗ M ) ≤ D + 1 . Now Theorem 8.15 (i) impliespdim Z A Φ [ t ] ( M ) ≤ D + 2 . This finishes the proof if assertion (i).(ii) We know already that A φ [ t, t − ] is Noetherian because of Theorem 7.1 (ii). Let M be a finitely generated A φ [ t, t − ]-module. We can find a finitely generated free Z A Φ [ t ]-module F and a free Z A Φ [ t ]-module F together with an exact sequenceof Z A Φ [ t, t − ]-modules j ∗ F f −→ j ∗ F e −→ M →
0. Here we write j for the inclusion A Φ [ t ] → A φ [ t, t − ]. By composing f with an appropriate automorphism of j ∗ F one can arrange that f = j ∗ g for some Z A Φ [ t ]-homomorphism g : F → F . Thecokernel of g is a finitely generated Z A Φ [ t ]-module N and there is an obvious exactsequence of Z A Φ [ t ]-modules F g −→ F → N →
0. Since the functor j ∗ is flatby Lemma 8.14 and respects the property projective, we obtain an Z A Φ [ t, t − ]-isomorphism j ∗ N ∼ = −→ M and have dim Z A Φ [ t,t − ] ( j ∗ N ) ≤ dim Z A Φ [ t ] ( N ). Hence weget dim Z A Φ [ t,t − ] ( M ) ≤ dim Z A Φ [ t ] ( N ). This finishes the proof of Theorem 9.1. (cid:3) Remark 9.2.
We do not know whether Theorem 9.1 remains true if we replaceregular by regular coherent. To our knowledge it is an open problem, whether fora regular coherent ring R the rings R [ t ] or R [ t, t − ] are regular coherent again.10. Directed union and infinite products of additive categories
A functor of additive categories F : A → B is called flat if for every exact se-quence A i −→ A p −→ A in A the sequence in F ( A ) F ( i ) −−−→ F ( A ) F ( p ) −−−→ F ( A ) in B is exact. It is called faithfully flat if a sequence A i −→ A p −→ A in A is exact if andonly if the sequence in F ( A ) F ( i ) −−−→ F ( A ) F ( p ) −−−→ F ( A ) in B is exact. Lemma 10.1.
Let i : A → A ′ and j : B → B ′ be inclusions of cofinal full additivesubcategories. Suppose that the following diagram of functors of additive categoriescommutes A F / / i (cid:15) (cid:15) B j (cid:15) (cid:15) A ′ F ′ / / B ′ Then(i) The inclusion i : A → A ′ is faithfully flat;(ii) F is flat or faithfully flat respectively if and only if F ′ is flat or faithfullyflat respectively.Proof. We first show that F ′ is exact or faithfully exact respectively provided that F is exact or faithfully exact respectively.Consider morphisms f ′ : A ′ → A ′ and g ′ : A ′ → A ′ in A . Choose objects A k in A and morphisms i k : A ′ k → A k and r k : A k → A ′ k in A ′ satisfying r k ◦ i k = id A k for k = 0 , ,
2. Define f : A → A and g : A → A by f = i ◦ f ′ ◦ r and g = i ◦ g ′ ◦ r .Then the following diagram of morphisms in A ′ commutes A ′ f ′ / / i (cid:15) (cid:15) A ′ g ′ / / i (cid:15) (cid:15) A ′ i ⊕ (cid:15) (cid:15) A f / / r (cid:15) (cid:15) A g ⊕ (id A − i ◦ r ) / / r (cid:15) (cid:15) A ⊕ A r ⊕ (cid:15) (cid:15) A ′ f ′ / / A ′ g ′ / / A ′ Next we check that the middle row is exact in A if and only if the upper row isexact in A ′ . Suppose that the middle row is exact in A . Consider a morphism v ′ : B ′ → A ′ in A ′ such that g ′ ◦ v ′ = 0. Choose an object B in A and maps j : B ′ → B and s : B → B ′ with s ◦ j = id B ′ . Then we have the morphism i ◦ v ′ ◦ s : B → A whose composite with g ⊕ (id A − i ◦ r ) : A → A ⊕ A iszero. Hence we can find a morphism u : B → A with f ◦ u = i ◦ v ′ ◦ s . Define u ′ : B ′ → A ′ by the composite r ◦ u ◦ j . One easily checks that f ′ ◦ u ′ = v ′ . Hencethe upper row is exact in A ′ .Suppose that the upper row is exact in A ′ . Consider a morphism v : B → A in A such that g ⊕ (id A − i ◦ r ) ◦ v = 0. Then g ◦ v = 0 and v = i ◦ r ◦ v . Weconclude g ′ ◦ ( r ◦ v ) = r ◦ i ◦ g ′ ◦ r ◦ v = r ◦ g ◦ i ◦ r ◦ v = r ◦ g ◦ v = r ◦ . ANISHING OF NIL-TERMS AND NEGATIVE K -THEORY 33 Since the upper row is exact, we can find u ′ : B → A ′ satisfying f ′ ◦ u ′ = r ◦ v .Define u : B → A by i ◦ u ′ . Then f ◦ u = f ◦ i ◦ u ′ = i ◦ f ′ ◦ u ′ = i ◦ r ◦ v = v. Hence the middle row is exact.If we apply F ′ and put i ′ k = F ′ ( i k ) and r ′ k = F ′ ( r k ), we get r ′ k ◦ i ′ k = id F ′ ( A ′ k ) and the commutative diagram F ′ ( A ′ ) F ′ ( f ′ ) / / i ′ (cid:15) (cid:15) F ′ ( A ′ ) F ′ ( g ′ ) / / i ′ (cid:15) (cid:15) F ′ ( A ′ ) i ′ ⊕ (cid:15) (cid:15) F ( A ) F ( f ) / / r ′ (cid:15) (cid:15) F ( A ) F ( g ) ⊕ (id F ( A − i ′ ◦ r ′ ) / / r ′ (cid:15) (cid:15) F ′ ( A ) ⊕ F ( A ) r ⊕ (cid:15) (cid:15) F ′ ( A ′ ) F ′ ( f ′ ) / / F ′ ( A ′ ) F ′ ( g ′ ) / / F ′ ( A ′ )and, by the same argument as above, the middle row is exact in B if and only ifthe upper row is exact in B ′ . We conclude that the functor F ′ is exact or faithfullyexact respectively, provided that F is exact or faithfully exact respectively.Since both id A and id B are faithfully flat, this special case shows that both i : A → A ′ and j : B → B ′ are faithfully flat.Suppose that F ′ is flat or faithfully flat respectively. Then j ◦ F = F ′ ◦ i is flat orfaithfully flat respectively. This implies that F is flat or faithfully flat respectively.This finishes the proof of Lemma 10.1. (cid:3) Lemma 10.2.
Let A = S i ∈ I A i be the directed union of additive subcategories A i for an arbitrary directed set I .(i) The idempotent completion Idem( A ) is the directed union of the idempotentcompletions Idem( A i ) ;(ii) Consider l ≥ .Suppose that A i is regular coherent or l -uniformly regular coherent re-spectively for every i ∈ I and for every i, j ∈ I with i ≤ j the inclusion A i → A j is flat. Then the inclusion Idem( A i ) → Idem( A j ) is flat forevery i, j ∈ I with i ≤ j and both A and Idem( A ) are regular coherent or l -uniformly regular coherent respectively;(iii) Suppose that A i is -uniformly regular coherent respectively for every i ∈ I .Then both A and Idem( A ) are -uniformly regular coherent respectively.Proof. (i) This is obvious.(ii) If the inclusion A i → A j is flat, then also the inclusion Idem( A i ) → Idem( A j )is flat by Lemma 10.1. In view of Lemma 5.4 (vi) and assertion (i), we can assumewithout loss of generality that each A i and A are idempotent complete. Hence wecan use the criterion for regular coherent given in Lemma 5.6 in the sequel. Wetreat only the case l ≥
2, the case l = 1 is proved analogouslyConsider a morphism f : A → A in A . Choose an index i such that f belongsto A i . Then we can find a sequence of morphisms0 → A n f n −→ A n − f n − −−−→ · · · f −→ A f −→ A which is in A i exact at A k for k = 1 , , . . . , n . It remains to show that this sequenceis exact at A at A k for k = 1 , , . . . , n . Fix k ∈ { , , . . . , n } . It remains to showfor any object A ∈ A and morphism g : A → A k with f k ◦ g = 0 that there exists amorphism g : A → A k +1 with f k +1 ◦ g = g . We can choose j ∈ I with i ≤ j suchthat g belongs to A j . Since A k +1 f k +1 −−−→ A k f k −→ A k − is exact in A i , we conclude from the assumptions that it is also exact in A j and hence we can construct thedesired lift g already in A j .(iii) In view of Lemma 5.4 (vi) and assertion (i), we can assume without loss ofgenerality that each A i and A are idempotent complete. Now the claim followsfrom the equivalence (iii)a ⇐⇒ (iii)c appearing in Lemma 5.6 (iii). (cid:3) Lemma 10.3.
Let l be a natural number. Let A = {A i | i ∈ I } be a collection of l -uniformly regular coherent additive categories A i for an arbitrary index set I .Then L i ∈ I A i and Q i ∈ I A i are l -uniformly regular coherent additive categories.Proof. Obviously Q i ∈ I A i inherits the structure of an additive category. Recallthat L i ∈ I A i is the full additive subcategory of Q i ∈ I A i consisting of those objects A i | i ∈ I } for which only finitely many of the objects A i are different from zero.Obviously Idem( M i ∈ I A i ) ∼ = M i ∈ I Idem( A i );Idem( Y i ∈ I A i ) ∼ = Y i ∈ I Idem( A i ) . Lemma 5.6 implies that L i ∈ I Idem( A i ) and Q i ∈ I Idem( A i ) are l -uniformly regularcoherent if each Idem( A i ) is l -uniformly regular coherent. Now the claim followsfrom Lemma 5.4 (vi). (cid:3) Lemma 10.3 will be generalized in Lemma 12.6.
Remark 10.4 (Advantage of the notion l -uniformly regular coherent) . The deci-sive advantage of the notion l -uniformly regular coherent is that it satisfies bothLemma 10.2 and Lemma 10.3. None of these lemmas hold for the properties Noe-therian, regular, or l -uniformly regular. Lemma 10.3 is not true if one replaces l -uniformly regular coherent by regular coherent unless I is finite.11. Vanishing of negative K -groups Theorem 11.1 (Vanishing of negative K -groups) . Let A be an additive category,such that A [ t , t , . . . , t m ] is regular coherent for every m ≥ .Then K n ( A ) = 0 holds for all n ≤ − .Proof. For an additive category B define G ′ ( Z B ) to be the abelian group withisomorphism classes [ M ] of finitely presented Z B -modules M as generators such thatfor each exact sequence of finitely presented Z B -modules 0 → M → M → M → M ] − [ M ] + [ M ] = 0. Define K ( Z B ) analogously but withfinitely presented replaced by finitely generated projective. A functor of additivecategories F : B → B ′ induces a homomorphism F ∗ : K ( Z B ) → K ( Z B ′ ) by sending[ M ] to [ F ∗ M ]. It induces a homomorphism F ∗ : G ′ ( Z B ) → G ′ ( Z B ′ ) by sending [ M ]to [ F ∗ M ], if F ∗ : MOD - Z A B → MOD - Z A B ′ is flat. There is the forgetful functor U : K ( Z B ) → G ′ ( Z B ′ ). If B is regular coherent, then U is a bijection by theResolution Theorem, see [16, Theorem 4.6 on page 41]. The Yoneda embeddinginduces an isomorphism K ( B ) ∼ = −→ K ( Z B ), natural in B .Suppose A [ t ] is regular coherent. We show that A [ t, t − ] is regular coherentand K − ( A ) = 0. The functor j ∗ : MOD - Z A [ t ] → MOD - Z A [ t, t − ] is flat byLemma 8.14. Let M ∗ be a finitely presented Z A [ t, t − ]-module. Then we can finda morphism f : A → A ′ in A [ t, t − ] together with an exact sequence of Z A [ t, t − ]-module mor A [ t,t − ] (? , A ) f ∗ −→ mor A [ t,t − ] (? , A ′ ) → M → . ANISHING OF NIL-TERMS AND NEGATIVE K -THEORY 35 Choose a natural number s and a morphism g : A → A ′ in A [ t ] such that (id A ′ · t s ) ◦ f = g holds in A [ t, t − ]. Since id A ′ · t s : A ′ ∼ = −→ A ′ is an isomorphism in A [ t, t − ], weobtain an exact sequence of Z A [ t, t − ]-modules j ∗ (mor A [ t ] (? , A )) j ∗ ( g ∗ ) −−−−→ j ∗ (mor A [ t ] (? , A ′ )) → M → . Let N be the finitely presented Z A [ t ]-module which is the cokernel of the Z A [ t ]-homomorphism g ∗ : mor A [ t ] (? , A ) → mor A [ t ] (? , A ′ ). Since j ∗ is flat and in particu-lar right exact, we obtain an isomorphism of finitely presented Z A [ t, t − ]-modules j ∗ N ∼ = −→ M . This implies that the homomorphism j ∗ : G ′ ( Z A [ t ]) → G ′ ( Z A [ t, t − ])is surjective and that A [ t, t − ] is regular coherent since A [ t ] is regular coherent byassumption.Hence we obtain a commutative diagram K ( A [ t ]) / / ∼ = (cid:15) (cid:15) K ( A [ t, t − ]) ∼ = (cid:15) (cid:15) K ( Z A [ t ]) / / ∼ = (cid:15) (cid:15) K ( Z A [ t, t − ]) ∼ = (cid:15) (cid:15) G ′ ( Z A [ t ]) / / G ′ ( Z A [ t, t − ])whose vertical arrows are bijections and whose lowermost horizontal arrow is sur-jective. Hence the uppermost horizontal arrow is surjective. We conclude fromTheorem 3.2 that K − ( A ) vanishes.Next we show by induction for n = 1 , . . . that K − m ( A ) vanishes for m =1 , , . . . , n . The induction beginning n = 1 has been taken care of above. Theinduction step from n ≥ n + 1 is done as follows. One shows using the claimabove by induction for i = 1 , , . . . , n that A [ Z i ][ t i +1 , . . . , t n +1 ] is regular coherent.In particular A [ Z n ][ t n +1 ] is regular coherent.We conclude from the n -times iterated Bass-Heller-Swan isomorphism, see The-orem 3.1 that K − n − ( A ) is a direct summand in K − ( A [ Z n ]). Hence it suffices toshow that K − ( A [ Z n ]) is trivial. This follows from the induction beginning appliedto A [ Z n ]. (cid:3) We conclude from Theorem 9.1 and Theorem 11.1
Corollary 11.2 (Vanishing of negative K -groups of regular additive categories) . Let A be an additive category which is regular.Then K n ( A ) = 0 holds for all n ≤ − . Remark 11.3.
As noted in Lemma 4.10 the Yoneda embedding identifies A withthe category of finitely generated free Z A -modules. If A is Noetherian, then thecategory of finitely generated Z A -modules is abelian. If it is in addition regularcoherent (i.e., if A is regular), then A is derived equivalent to this abelian category.Schlichting showed in [15, Theorem 6 on page 117] that K − of abelian categories istrivial. It follows that for regular A we can obtain Theorem 11.1 from Schlichting’sresult. Similarly, Corollary 11.2 can alternatively be deduced from [15, Theorem 7on page 118].12. Nested sequences and the associated categories
Definition 12.1 (Nested sequences of additive categories) . A nested sequence ofadditive categories A ∗ is a decreasing sequence of additive subcategories A ⊇ A ⊇ A ⊇ · · · . A morphism of nested sequences of additive categories F ∗ : A ∗ → A ′∗ is a sequenceof functors of additive categories F m : A m → A ′ m for m ∈ N such that F m restrictedto A m +1 is F m +1 .12.1. The sequence category.Definition 12.2 (The sequence category S ( A ∗ ) and the limit category L ( A ∗ )) . Define the additive category S ( A ∗ ), called sequence category , associated to thenested sequence of additive categories A ∗ as follows: • An object in S ( A ∗ ) is a sequence A = ( A m ) m ≥ of objects in A such thatthere exists a function (depending on A ) L : N → N with the property that A m belongs to A l for l, m ∈ N with m ≥ L ( l ); • A morphism φ : A → A ′ in S ( A ∗ ) consists of a sequence of morphisms φ m : A m → A ′ m in A such that there exists a function L : N → N with theproperty that φ m : A m → A ′ m belongs to A l for n ∈ N with m ≥ L ( l ); • Composition and the structure of an additive category on S ( A ∗ ) comesfrom the corresponding structures on A m for m ∈ N .Let T ( A ∗ ) be the full subcategory of S ( A ∗ ) consisting of objects A for whichthere exists a natural number M with A m = 0 for m ≥ M .The additive category L ( A ∗ ), called limit category , is defined to be the additivequotient category S ( A ∗ ) / T ( A ∗ ). Recall that for a subcategory U ⊆ B of an additivecategory the quotient category B / U has the same objects as B and that morphismsin B / U are equivalence classes of morphisms in B , where two morphisms φ, φ ′ areidentified if their difference φ − φ ′ can be factored through an object of U . For L ( A ∗ )this means that morphisms φ, φ ′ from S ( A ∗ ) are identified in L ( A ∗ ) iff φ m = φ ′ m for all but finitely many m .Obviously S ( A ∗ ) is an additive subcategory of Q m ∈ N A and is equal to it if thenested sequence is constant, i.e., A = A m for m ∈ N . Obviously T ( A ∗ ) can beidentified with L m ∈ N A m . Controlled categories appear for instance in proofs ofthe Farrell-Jones Conjecture. In for us important cases, this allows for translationsto nested sequences of additive categories, where the control condition becomessharper the larger the index m gets.For the notion of a Karoubi filtration and the associated weak homotopy fibrationsequence we refer for instance to [2] and [4, Definition 5.4]. One easily checks Lemma 12.3.
The inclusion T ( A ∗ ) ⊆ S ( A ∗ ) is Karoubi filtration and we have theweak homotopy fibration K ∞ ( T ( A ∗ )) → K ∞ ( S ( A ∗ )) → K ∞ ( L ( A ∗ )) . Definition 12.4.
We call a function I : N → N admissible if it has the followingproperties I ( m ) ≤ m for m ∈ N ; I ( m ) ≤ I ( m + 1) for m ∈ N ;lim m →∞ I ( m ) = ∞ . Let I be the set of admissible functions. It becomes a directed set by defining for I, J ∈ I I ≤ J ⇐⇒ I ( m ) ≥ J ( m ) for all m ∈ N . Note that I is indeed directed. For I, J ∈ I define K : N → N by K ( m ) =min { I ( m ) , J ( m ) } . Then K ∈ I and I, J ≤ K holds. Lemma 12.5. (i) Let φ be a morphism in S ( A ∗ ) . Then there exists an ad-missible function I ∈ I such that φ m ∈ A I ( m ) holds for all m ∈ N ; ANISHING OF NIL-TERMS AND NEGATIVE K -THEORY 37 (ii) Let φ be a sequence of morphisms φ m : A m → A ′ m in A . Suppose thatthere exists an admissible function I ∈ I such that φ m ∈ A I ( m ) holds forall m ∈ N . Then φ belongs to S ( A ∗ ) .Proof. (i) Choose a function L : N → N such that φ m belongs to A l if m ≥ L ( l ).Define a new function I ′ : N → N by I ′ ( m ) = max { i ∈ { , , . . . , m } | φ m ∈ A i } . It satisfies I ′ ( m ) ≤ m for m ∈ N ; l ≤ I ′ ( m ) for l, m ∈ N , m ≥ L ( l ) , m ≥ l ; φ m ∈ A I ′ ( m ) for m ∈ N . Define the function I : N → N by I ( m ) = min { I ′ ( j ) | j ∈ N , m ≤ j } . Then we get for all n ∈ N I ( m ) ≤ m for m ∈ N ; I ( m ) ≤ I ( m + 1) for m ∈ N ; l ≤ I ( m ) for l, m, ∈ N , m ≥ L ( l ) , m ≥ l ; φ m ∈ A I ( m ) for m ∈ N . The first three properties imply I ∈ I .(ii) Suppose that there exists I ∈ I satisfying φ m ∈ A I ( m ) for all m ∈ N . Definethe desired function L : N → N by L ( l ) = min { m ∈ N | l ≤ I ( m ) } . (cid:3) Uniform regular coherence.Lemma 12.6.
Consider the nested sequence A ∗ of additive categories A ⊇ A ⊇A ⊇ · · · . Suppose that for the natural number l ≥ each of the additive categories A m is l -uniformly regular coherent and that the inclusion A m → A m +1 is flat for m ∈ N .Then S ( A ∗ ) and L ( A ∗ ) are l -uniformly regular coherent.Proof. We first treat S ( A ∗ ). Let φ : A → A be a morphism in S ( A ∗ ). Becauseof Lemma 12.5 (i) we can choose I ∈ I with A m , A m , φ m ∈ A I ( m ) . By assumptionwe can find for each m ∈ N an exact sequence0 → A lm φ lm −−→ A l − m φ lm −−→ · · · φ m −−→ A m φ −→ A m in A I ( m ) . We conclude from Lemma 12.5 (ii) that the collection of these sequencesfor m = 0 , , . . . defines a sequence in S ( A ∗ )(12.7) 0 → A l φ l −→ A l − φ l − −−−→ · · · φ −→ A φ −→ A . Finally we show that the sequence (12.7) is exact as a sequence in S ( A ∗ ). We haveto solve for every j ∈ { , . . . , l } the following lifting problem in S ( A ∗ ).(12.8) A j +1 φ j +1 / / A j φ j / / A j − B ν a a ❉ ❉ ❉ ❉ ❉ µ O O = = ③③③③③③③③③ Because of Lemma 12.5 (i) we can choose J ∈ I such B m , µ m ∈ A J ( m ) holds.Choose K ∈ I with I, J ≤ K . Now consider the following lifting problem in A K ( m ) (12.9) A j +1 m φ j +1 m / / A jm φ jm / / A j − m B mν m b b ❊ ❊ ❊ ❊ µ m O O < < ②②②②②②②② As the inclusion A I ( m ) → A K ( m ) is flat by assumption, and the sequence A j +1 m φ j +1 m −−−→ A jm φ jm −−→ A j − m is by construction exact at A jm when considered in A I ( m ) , it is exactat A jm when considered in A K ( m ) . Hence (12.9) has a solution ν m : B m → A j +1 when considered in A K ( m ) . We conclude from Lemma 12.5 (ii) that the collectionof the morphisms ν m yields a morphism ν : B → A j +1 in S ( A ∗ ). Therefore ν is asolution to the lifting problem (12.8) in S ( A ∗ ). We conclude that (12.7) is an exactsequence in S ( A ∗ ). This finishes the proof of Lemma 12.6 for S ( A ∗ ).The proof for L ( A ∗ ) is the following modification of the one for S ( A ∗ ). Let φ : A → A be a morphism in L ( A ∗ ). Choose a representative φ : A → A in S ( A ∗ ). Now one proceeds as above and constructs the sequence (12.7) in S ( A ∗ ).However, instead of solving the lifting problem (12.8) in S ( A ∗ ) we have to solvethe lifting problem(12.10) A j +1 φ j +1 / / A j φ j / / A j − B ν a a ❉ ❉ ❉ ❉ ❉ µ O O = = ③③③③③③③③③ in L ( A ∗ ). Choose a representative µ for µ . There is a natural number M such that φ jm ◦ µ m = 0 holds for m ≥ M . We can change the representative µ by putting µ m = 0 for m < M and by leaving µ m unchanged for m ≥ M . Now we choosea solution ν m to the lifting problem (12.9) in A K ( m ) for m ≥ M . Put ν m = 0for m < M . Then we get a morphism ν : B → A j +1 in S ( A ∗ ) such that its class ν : B → A j +1 in L ( A ∗ ) is a solution to the lifting problem (12.10). This finishesthe proof of Lemma 12.6. (cid:3) Example 12.11 (The property Noetherian does not pass to the sequence category) . The analogue of Lemma 12.6 for the properties Noetherian, regular, or l -uniformlyregular instead of uniformly l -regular coherent does not hold as the following exam-ple shows. Suppose that none of the A m is the trivial additive category. Consideran object A of S ( A ∗ ) such that A m = { } for m ∈ N , and the Z S ( A ∗ )-module F = mor S ( A ∗ ) (? , A ) . Define a Z S ( A ∗ )-submodule V of F by V (?) = { φ ∈ F (?) | ∃ M ( φ ) ∈ N with φ m = 0 for m ≥ M ( φ ) } . Suppose that there exists an object B and an epimorphism f : mor S ( A ∗ ) (? , B ) → V .If we write f (id B ) = ψ ∈ V ( B ), then there must be a natural number M with ψ m = 0 for m ≥ M . This implies that for any φ ∈ V (?) we have φ m = 0 for m ≥ M . This is a contradiction, since M does not depend on φ . Hence V is notfinitely generated and S ( A ∗ ) is not Noetherian. This construction yields also acounterexample for L ( A ∗ ). ANISHING OF NIL-TERMS AND NEGATIVE K -THEORY 39 The algebraic K -theory of the sequence categories S ( A ∗ ) , T ( A ∗ ) and L ( A ∗ ) . Given I ∈ I , define a subcategory S ( A ∗ ) I of S ( A ∗ ) as follows. An object A in S ( A ∗ ) belongs to S ( A ∗ ) I if A m ∈ A I ( m ) holds for m ∈ N . A morphism φ : A → T in S ( A ∗ ) belongs to S ( A ∗ ) I if and only if φ m ∈ A I ( m ) holds for m ∈ N .Next we show S ( A ∗ ) I ⊆ S ( A ∗ ) J for I, J ∈ I , I ≤ J ;(12.12) S ( A ∗ ) = [ I ∈I S ( A ∗ ) I . (12.13)The first equation is obvious since A i ⊆ A j for i ≥ j . The second follows fromLemma 12.5.Define analogously the subcategory T ( A ∗ ) I of T ( A ∗ ). Then we get T ( A ∗ ) I ⊆ T ( A ∗ ) J for I, J ∈ I , I ≤ J ;(12.14) T ( A ∗ ) = [ I ∈I T ( A ∗ ) I . (12.15)One easily checks that the inclusion T ( A ∗ ) I ⊆ S ( A ∗ ) I is Karoubi filtration.Define L ( A ∗ ) I = S ( A ∗ ) I / T ( A ∗ ) I . (12.16) Lemma 12.17. (i) We get for
I, J ∈ I with I ≤ J functors L ( A ∗ ) I → L ( A ∗ ) J ; L ( A ∗ ) I → L ( A ∗ ) , and analogously for T and S ;(ii) The functors appearing in assertion (i) induce weak homotopy equiva-lences, natural in A ∗ hocolim I ∈I K ∞ ( T ( A ∗ ) I ) ≃ −→ K ∞ ( T ( A ∗ ));hocolim I ∈I K ∞ ( S ( A ∗ ) I ) ≃ −→ K ∞ ( S ( A ∗ ));hocolim I ∈I K ∞ ( L ( A ∗ ) I ) ≃ −→ K ∞ ( L ( A ∗ )) . Proof. (i) The desired functors come from (12.12), and (12.14).(ii) Note that homotopy colimits of weak homotopy fibrations are weak homotopyfibrations again in the category of spectra. Hence we obtain a commutative diagramof spectra whose rows are weak homotopy fibrationshocolim I ∈I K ∞ ( T ( A ∗ ) I ) / / (cid:15) (cid:15) hocolim I ∈I K ∞ ( S ( A ∗ ) I ) / / (cid:15) (cid:15) hocolim I ∈I K ∞ ( L ( A ∗ ) I ) (cid:15) (cid:15) K ∞ ( T ( A ∗ )) / / K ∞ ( S ( A ∗ )) / / K ∞ ( L ( A ∗ ))The first and the second vertical arrow are weak homotopy equivalences becauseof (12.13) (12.15), and [10, Corollary 7.2]. Hence the third vertical arrow is a weakhomotopy equivalence. (cid:3) Given I ∈ I , we define L m ∈ N A I ( m ) to be the full subcategory of Q m ∈ N A I ( m ) consisting of those objects A for which there exists a natural number M (dependingon A ) satisfying A m = 0 for m ≥ M . Let (cid:0)Q m ∈ N A m (cid:1)(cid:14)(cid:0)L m ∈ N A m (cid:1) be the quotientadditive category. Lemma 12.18.
Fix I ∈ I . There are weak homotopy equivalences, natural in A ∗ , K ∞ (cid:0) Y m ∈ N A I ( m ) (cid:1) ≃ −→ Y m ∈ N K ∞ (cid:0) A I ( m ) (cid:1) ; _ m ∈ N K ∞ ( A I ( m ) (cid:1) ≃ −→ K ∞ (cid:0) M m ∈ N A I ( m ) (cid:1) , and a zigzag of weak homotopy equivalences, natural in A ∗ , hocofib (cid:0) K ∞ (cid:0) M m ∈ N A I ( m ) (cid:1) → K ∞ (cid:0) Y m ∈ N A I ( m ) (cid:1)(cid:1) ≃ ←→ K ∞ Y m ∈ N A I ( m ) (cid:30) M m ∈ N A I ( m ) ! . Proof.
The first one is weak homotopy equivalence by [3], see also [5, Theorem 1.2]since the non-connective algebraic K -theory spectrum is indeed an Ω-spectrum.The second one is a weak homotopy equivalence since L m ∈ N A m is the union ofthe subcategories L nm =0 A i and hence we get a weak homotopy equivalencehocolim n →∞ K ∞ (cid:0) n M m =0 A I ( m ) (cid:1) ∼ = −→ K ∞ (cid:0) M m ∈ N A I ( m ) (cid:1) , and the natural map n _ m =0 K ( A I ( m ) ) ≃ −→ K ∞ (cid:0) n M m =0 A I ( m ) (cid:1) is a weak homotopy equivalence. The third map is a weak homotopy equivalencesince the inclusion L m ∈ N A I ( m ) ⊆ Q m ∈ N A I ( m ) is a Karoubi filtration. (cid:3) Lemma 12.19.
Given I ∈ I , there are weak homotopy equivalences, natural in A ∗ , K ∞ (cid:0) S ( A ∗ ) I (cid:1) ≃ −→ Y m ∈ N K ∞ (cid:0) A I ( m ) (cid:1) ; _ m ∈ N K ∞ ( A I ( m ) (cid:1) ≃ −→ K ∞ (cid:0) T ( A ∗ ) I (cid:1) ; and a zigzag of weak homotopy equivalences, natural in A ∗ , hocofib (cid:0) K ∞ (cid:0) M m ∈ N A I ( m ) (cid:1) → K ∞ (cid:0) Y m ∈ N A I ( m ) (cid:1)(cid:1) ≃ ←→ K ∞ ( L ( A ) I ) . Proof.
The are obvious identifications Y m ∈ N A I ( m ) = S ( A ∗ ) I ; M m ∈ N A I ( m ) = T ( A ∗ ) I ; Y m ∈ N A I ( m ) (cid:30) M m ∈ N A I ( m ) = L ( A ∗ ) I . Now the claim follows from Lema 12.18. (cid:3)
As a consequence of Lemma 12.17 (ii) and Lemma 12.19 we get
ANISHING OF NIL-TERMS AND NEGATIVE K -THEORY 41 Lemma 12.20 ( K -groups of S ( A ∗ ), T ( A ∗ ), and L ( A ∗ )) . There are zigzags of weakhomotopy equivalences of spectra, natural in A ∗ hocolim i ∈I Y m ∈ N K ∞ (cid:0) A I ( m ) (cid:1) ≃ ←→ K ∞ (cid:0) S ( A ∗ ) (cid:1) ;hocolim i ∈I _ m ∈ N K ∞ ( A I ( m ) (cid:1) ≃ ←→ K ∞ (cid:0) T ( A ∗ ) (cid:1) ;hocolim i ∈I hocofib (cid:0) K ∞ (cid:0) M m ∈ N A I ( m ) (cid:1) → K ∞ (cid:0) Y m ∈ N A I ( m ) (cid:1)(cid:1)! ≃ ←→ K ∞ ( L ( A )) . Split embedding of lower K -theory in higher K -theory. Fix a nat-ural number d . Let A ⊇ A ⊇ A ⊇ · · · be a nested sequence A ∗ of addi-tive categories. It induces another nested sequence A ∗ [ Z d ] of additive categories A [ Z d ] ⊇ A [ Z d ] ⊇ A [ Z d ] ⊇ · · · , where A [ Z d ] is given by the untwisted crossedproduct with Z d . Note that there are canonical inclusions ( S ( A ∗ ))[ Z d ] → S ( A ∗ [ Z d ])and ( L ( A ∗ ))[ Z d ] → L ( A ∗ [ Z d ]). One of our main technical difficulties comes fromthe fact that these inclusions are not equivalences of additive categories and do notinduce weak homotopy equivalences after applying the K -theory functor in general.Consider an additive category A . The (untwisted) Bass-Heller-Swan Theorem,see Theorem 3.1 (i), yields a weak homotopy equivalence, natural in A , bhs ′ : K ∞ ( A ) ∨ Σ K ∞ ( A ) ∨ NK ∞ ( A Φ [ t ]) ∨ NK ∞ ( A [ t − ]) ≃ −→ K ∞ ( A [ Z ]) . Fix a natural number d . If we iterate the construction above, we obtain a spectrum E ( A ) and a weak homotopy equivalence, natural in A , bhs : Σ d K ∞ ( A ) ∨ E ( A ) ≃ −→ K ∞ ( A [ Z d ]) . The inclusion and the projection onto the factor Σ d K ∞ ( A ) yields natural maps j : Σ d K ∞ ( A ) → Σ d K ∞ ( A ) ∨ E ( A ); p : Σ d K ∞ ( A ) ∨ E ( A ) → Σ d K ∞ ( A ) , such that p ◦ j is is the identity and j and p are natural in A .Given I ∈ I the maps above induce maps of spectra(12.21) BHS I ( A ∗ ) :hocofib _ m ∈ N (Σ d K ∞ ( A I ( m ) ) ∨ E ( A I ( m ) )) → Y m ∈ N (Σ d K ∞ ( A I ( m ) ) ∨ E ( A I ( m ) )) ! → hocofib _ m ∈ N K ∞ ( A I ( m ) [ Z d ]) → Y m ∈ N K ∞ ( A I ( m ) [ Z d ]) ! ;(12.22) J I ( A ∗ ) : hocofib _ m ∈ N Σ d K ∞ ( A I ( m ) ) → Y m ∈ N Σ d K ∞ ( A I ( m ) ) ! → hocofib _ m ∈ N (Σ d K ∞ ( A I ( m ) ) ∨ E ( A I ( m ) )) → Y m ∈ N (Σ d K ∞ ( A I ( m ) ) ∨ E ( A I ( m ) )) ! , and(12.23) P I ( A ∗ ) :hocofib _ m ∈ N (Σ d K ∞ ( A I ( m ) ) ∨ E ( A I ( m ) )) → Y m ∈ N (Σ d K ∞ ( A I ( m ) ) ∨ E ( A I ( m ) )) ! → hocofib _ m ∈ N Σ d K ∞ ( A I ( m ) ) → Y m ∈ N Σ d K ∞ ( A I ( m ) ) ! , such that BHS I ( A ∗ ) is a weak homotopy equivalence and P I ( A ∗ ) ◦ J I ( A ∗ ) = id.Put X ( A ∗ ) := hocolim I ∈I hocofib _ m ∈ N K ∞ ( A m ) → Y m ∈ N K ∞ ( A m ) !! ; X d ( A ∗ ) := hocolim I ∈I hocofib _ m ∈ N Σ d K ∞ ( A m ) → Y m ∈ N Σ d K ∞ ( A m ) !! ;and X ( A ∗ ) := hocolim I ∈I hocofib _ m ∈ N (Σ d K ∞ ( A m ) ∨ E ( A m )) → Y m ∈ N (Σ d K ∞ ( A m ) ∨ E ( A m )) !! . Define maps
BHS ( A ∗ ) = hocolim I ∈I BHS I ( A ∗ ) : X ( A ∗ ) ≃ −→ X ( A ∗ [ Z d ]);(12.24) J ( A ∗ ) = hocolim I ∈I J I ( A ∗ ) : X d ( A ∗ ) → X ( A ∗ );(12.25) P ( A ∗ ) = hocolim I ∈I P I ( A ∗ ) : X ( A ∗ ) → X d ( A ∗ ) . (12.26)such that BHS ( A ∗ ) is a weak homotopy equivalence and P ( A ∗ ) ◦ J ( A ∗ ) = id Σ d X ( A ∗ ) . Lemma 12.27.
Let A ∗ be a nested sequence of additive categories and d be anatural number. Then we obtain zigzags of weak homotopy equivalences, natural in A ∗ , A ( A ∗ ) : Σ d X ( A ∗ ) ≃ ←→ Σ d K ∞ ( L ( A ∗ )); A ( A ∗ ) : X ( A ∗ ) ≃ ←→ K ∞ ( L ( A ∗ [ Z d ])) , and maps, natural in A ∗ , J ( A ∗ ) : X d ( A ∗ ) ←→ X ( A ∗ ); P ( A ∗ ) : X ( A ∗ ) ←→ X d ( A ∗ ) , such that P ( A ∗ ) ◦ J ( A ∗ ) = id X d ( A ∗ ) holds and there is a weak homotopy equivalence F : Σ d X ( A ∗ ) ≃ −→ X d ( A ∗ ) . Proof.
We obtain from Lemma 12.20 a zigzag T ( A ∗ ) of weak homotopy equiva-lences of spectra from X ( A ∗ ) to K ∞ ( L ( A ∗ )), natural in A ∗ . Now define A ( A ∗ )by Σ d T ( A ∗ ) and A ( A ∗ ) by T ( A [ Z d ]) ◦ BHS , where
BHS has been introducedin (12.24). The maps J and P have been constructed in (12.25) and (12.26). SinceΣ commutes with homotopy cofibers, homotopy colimits and wedges, we obtain anatural homotopy equivalence F : Σ d X ( A ∗ ) ≃ −→ X d ( A ∗ ) . (cid:3) ANISHING OF NIL-TERMS AND NEGATIVE K -THEORY 43 The following commutative diagram(12.28) K ∞ ( L ( A ∗ [ Z d ])) O O A ( A ∗ ) ≃ (cid:15) (cid:15) X ( A ∗ ) P ( A ∗ ) ( ( ❘❘❘❘❘❘❘❘❘❘❘❘❘ X d ( A ∗ ) id / / J ( A ∗ ) ❧❧❧❧❧❧❧❧❧❧❧❧❧ O O F ≃ (cid:15) (cid:15) X d ( A ∗ ) O O F ≃ (cid:15) (cid:15) Σ d X ( A ∗ ) O O A ( A ∗ ) ≃ (cid:15) (cid:15) Σ d X ( A ∗ ) O O A ( A ∗ ) ≃ (cid:15) (cid:15) Σ d K ∞ ( L ( A ∗ )) Σ d K ∞ ( L ( A ∗ ))summarizes what we obtain from Lemma 12.27. Essentially it says that Σ d K ∞ ( L ( A ∗ ))is a retract of K ∞ ( L ( A ∗ [ Z d ])) up to zigzags of weak homotopy equivalences.12.5. Additive categories with Z r -action. Fix a natural number r and a nestedsequence of additive categories A ∗ . We have defined the additive category L ( A ∗ ) inDefinition 12.2. Suppose that each A m comes with a Z r -action Φ m : Z r → aut( A m )such that these are compatible with the inclusions A m → A m +1 for m ≥
0. Then weobtain a Z r -action Φ : Z r → aut( L ( A ∗ )) on L ( A ∗ ). We obtain a covariant functor,see for instance [1, Section 9], K ∞L ( A ∗ ) : Or ( Z r ) → Spectra . It determines a Z r -homology theory H Z r n ( − , K ∞L ( A ∗ ) ) with the property that forevery subgroup H ⊆ Z r and n ∈ Z we have the natural isomorphisms H Z r n ( Z r /H, K ∞L ( A ∗ ) ) ∼ = −→ K n ( L ( A ∗ ) ⋊ Φ | H H )as explained for instance in [1, Section 9]. Recall that the nested sequence ofadditive categories A ∗ yields for any natural number d another nested sequence ofadditive categories A ∗ [ Z d ] by A [ Z d ] ⊇ A [ Z d ] ⊇ A [ Z d ] ⊇ · · · , where A [ Z d ] isgiven by the untwisted crossed product with Z d . Moreover, L ( A ∗ [ Z d ]) inherits a Z r -action Φ[ Z d ] : Z r → aut( L ( A ∗ [ Z d ])).The main result of this section is Theorem 12.29.
Suppose:(i) For every natural number d there exists a natural number l ( d ) such thatfor any natural number m the additive category A m [ Z d ] is l ( d ) -uniformlyregular coherent;(ii) The inclusion A m +1 [ Z d ] → A m [ Z d ] is exact for any natural numbers d and m .Then the map induced by the projection E Z r → {•} H Z r n ( E Z r ; K ∞L ( A ∗ ) ) → H Z r n ( {•} ; K ∞L ( A ∗ ) ) = K n ( L ( A ∗ ) ⋊ φ Z r ) is bijective for all n ∈ Z .Proof. The Farrell-Jones Conjecture is known to be true for Z r and predicts thatthe map induced by the projection E VCYC ( Z r ) → {•} H Z r n ( E VCYC ( Z r ); K ∞L ( A ∗ ) ) → H Z r n ( {•} ; K ∞L ( A ∗ ) ) is an isomorphism for all n ∈ Z , where E VCYC ( Z r ) is the classifying space of thefamily VCYC of virtually cyclic subgroups of G , see for instance [8]. We also havethe map induced by the up to Z r -homotopy unique Z r -map E Z r → E VCYC ( Z r ).(12.30) H Z r n ( E Z r ; K ∞L ( A ∗ ) ) → H Z r n ( E VCYC ( Z r ); K ∞L ( A ∗ ) ) . Obviously it suffices to show that (12.30) is bijective for all n ∈ Z . By the Tran-sitivity Principle, see for instance [9, Theorem 65 on page 742], this boils down toshow that for any non-trivial virtually cyclic subgroup V of Z m the map inducedby the projection EV → {•} H Vn ( EV ; K ∞L ( A ∗ ) | V ) → H Vn ( {•} ; K ∞L ( A ∗ ) | V )is bijective for all n ∈ Z . Since any non-trivial virtually cyclic subgroup V of Z r is isomorphic Z , we have reduced the proof of Theorem 12.29 to the special case r = 1.We obtain from the diagram (12.28) coming from Lemma 12.27 for every naturalnumber d and every n ∈ Z a commutative diagram H Z n − d ( E Z ; K ∞L ( A ∗ ) ) (cid:15) (cid:15) / / H Z n − d ( {•} ; K ∞L ( A ∗ ) ) (cid:15) (cid:15) H Z n ( E Z ; K ∞L ( A ∗ [ Z d ]) ) (cid:15) (cid:15) / / H Z n ( {•} ; K ∞L ( A ∗ [ Z d ]) ) (cid:15) (cid:15) H Z n − d ( E Z ; K ∞L ( A ∗ ) ) / / H Z n − d ( {•} ; K ∞L ( A ∗ ) )where the composite of the two left vertical arrows and the composite of the tworight vertical arrows are isomorphisms. Obviously the upper horizontal arrow isbijective, if the middle horizontal arrows is bijective.Hence it remains to show that for every natural number d the middle arrow H Z n ( E Z ; K ∞L ( A ∗ [ Z d ]) ) → H Z n ( {•} ; K ∞L ( A ∗ [ Z d ]) )is bijective for n ∈ Z , n ≥ K Idem( L ( A ∗ [ Z d ])) be the connective version of K ∞ Idem( L ( A ∗ [ Z d ])) . Since we havedim( E Z ) ≤
1, we conclude from Lemma 2.3 that the vertical arrows appearing inthe commutative diagram H Z n ( E Z ; K Idem( L ( A ∗ [ Z d ])) ) ∼ = (cid:15) (cid:15) / / H Z n ( {•} ; K Idem( L ( A ∗ [ Z d ])) ) ∼ = (cid:15) (cid:15) H Z n ( E Z ; K ∞ Idem( L ( A ∗ [ Z d ])) ) / / H Z n ( {•} ; K ∞ Idem( L ( A ∗ [ Z d ])) ) H Z n ( E Z ; K ∞L ( A ∗ [ Z d ]) ) / / ∼ = O O H Z n ( {•} ; K ∞L ( A ∗ [ Z d ]) ) ∼ = O O are bijective for n ≥
2. Hence it suffices show that for every natural number d themap(12.31) H Z n ( E Z ; K Idem( L ( A ∗ [ Z d ])) ) → H Z n ( {•} ; K Idem( L ( A ∗ [ Z d ])) )is bijective for n ∈ Z , n ≥ A m [ Z d ] is uniformly l ( d )-regular coherent and the in-clusion A m [ Z d ] → A m +1 [ Z d ] is flat for every m ≥
0. We conclude from Lemma 12.6that L ( A ∗ [ Z d ]) is uniformly l ( d )-regular coherent. We conclude from Lemma 5.4 (vi) ANISHING OF NIL-TERMS AND NEGATIVE K -THEORY 45 that Idem( L ( A ∗ [ Z d ])) is uniformly l ( d )-regular coherent. Hence Idem( L ( A ∗ [ Z d ]))is idempotent complete and regular coherent. Now the bijectivity of (12.31) for n ∈ Z , n ≥ L ( A ∗ [ Z d ])) since forthe map a appearing there the homomorphism π n ( a ) can be identified with themap (12.31) for n ≥
2. This finishes the proof of Theorem 12.29. (cid:3)
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