The "fundamental theorem" for the algebraic K -theory of strongly Z -graded rings
aa r X i v : . [ m a t h . K T ] M a r THE “FUNDAMENTAL THEOREM” FOR THEALGEBRAIC K -THEORY OF STRONGLY Z -GRADED RINGS THOMAS HÜTTEMANN
Abstract.
The “fundamental theorem” for algebraic K -theory expresses the K -groups of a Laurent polynomial ring L [ t, t − ] as a direct sum of two copiesof the K -groups of L (with a degree shift in one copy), and certain “nil” groupsof L . It is shown here that a modified version of this result generalises tostrongly Z -graded rings; rather than the algebraic K -groups of L , the splittinginvolves groups related to the shift actions on the category of L -modules com-ing from the graded structure. (These actions are trivial in the classical case).The nil groups are identified with the reduced K -theory of homotopy nilpotenttwisted endomorphisms, and analogues of Mayer - Vietoris and localisationsequences are established.
Contents
Introduction 2 · The fundamental theorem for the algebraic K -theory of rings 2 · Relation with other work 3 · Structure of the paper 3 · Acknowledgements 31. Notation and main results 3 · Notation and conventions 3 · Strongly Z -graded rings 4 · The groups R NK ± q ( R ) · Shifts and shift differences 5 · The first negative K -group 5 · The fundamental theorem 5 · The Mayer-Vietoris sequence 6 · Homotopy nilpotent twisted endomorphisms and the localisationsequence 6 · Noetherian regular rings 72. Induced modules and chain complexes 7 · The Grothendieck group of a ring 7 · Induced and stably induced modules 7 · Stabilisation of chain complexes 8 · Induced and stably induced chain complexes. The chain complexlifting problem 8 · The chain complex lifting problem for acyclic complexes 9 · The chain complex lifting problem for well-behaved ringhomomorphisms 103. Finite domination 114. The projective line associated with a strongly Z -graded ring 125. The algebraic K -theory of the projective line 13 Date : April 1, 2020.2010
Mathematics Subject Classification.
Primary 19D50; Secondary 19D35 16E20 18G35.
6. The nil terms 14 · The category of twisted endomorphisms 14 · The characteristic sequence of a twisted endomorphism 15 · Chain complexes of twisted endomorphisms 16 · Homotopy nilpotent twisted endomorphisms 16 · The nil category 187. The fundamental square 21 · · The fibres of the fundamental square 21 · Auxiliary categories 24 · The corners of the fundamental square 268. Establishing the Mayer-Vietoris sequence 279. Proof of the fundamental theorem 28 · The K -theory of the projective line revisited 28 · The modified Mayer-Vietoris sequence 29 · The map ηβα · The kernel of ηβα · The cokernel of ηβα K -theory of homotopy nilpotent twisted endomorphisms 33References 35 Introduction
The fundamental theorem for the algebraic K -theory of rings. The “fun-damental theorem”, also know as the
Bass - Heller - Swan formula, expresses thealgebraic K -groups of a Laurent polynomial ring R [ t, t − ] as a direct sum K q (cid:0) R [ t, t − ] (cid:1) ∼ = K q R ⊕ K q − ( R ) ⊕ R NK + q ( R ) ⊕ R NK − q ( R ) . (0.1)It will be shown that this result largely depends on the structure of R [ t, t − ] as a Z -graded ring, and that a similar splitting can be established in much greatergenerality.By way of analogy we think of any Z -graded ring R = L k ∈ Z R k as a substitutefor a Laurent polynomial ring, and consider the subrings R ≤ = L k ≤ R and R ≥ = L k ≥ R k as substitutes for the polynomial rings R [ t − ] and R [ t ] . If thering R is strongly graded in the sense that R k R ℓ = R k + ℓ for all integers k and ℓ , theanalogy is appropriate, and many results known for ( Laurent ) polynomial ringscan be proved to hold for the more general setting, e.g. , the characterisation of finitedomination via
Novikov homology [HS17], the splitting for the algebraic K -theoryof the projective line [HM18], and the connection between finite domination andnon-commutative localisation [Hüt18].This is not idle play: the class of strongly Z -graded rings is much bigger than theclass of Laurent polynomial rings. One specific example of a strongly Z -gradedrings is K [ A, B, C, D ] / ( AB + CD = 1) where K is a field, deg( A ) = deg( C ) = 1 and deg( B ) = deg( D ) = − . (The only units in this ring are the non-zero elementsof K in degree .) A natural infinite family of examples is formed by the Leavitt path algebras associated with row-finite directed graphs without sink satisfying acertain condition Y [NÖ20, Theorem 1.3]; this includes all Leavitt path algebrasassociated with finite directed graphs without sink.
HE FUNDAMENTAL THEOREM FOR STRONGLY Z -GRADED RINGS 3 Back to the fundamental theorem, the graded structure of R = L k ∈ Z R k induces“shift functors” s k : M M ⊗ R R k on categories of R -modules; in case of a strongly graded ring, these functors preserve projectivity and satisfy the relation s k ◦ s ℓ ∼ = s k + ℓ . The key point is to measure how non-trivial the resulting Z -action onalgebraic K -groups is, which is done by considering the kernel A q and cokernel B q of the shift difference map sd ∗ = id − s − : K q R ✲ K q R . It turns out that the groups A q − and B q play the role of K q − R and K q R in theclassical formulation (0.1) of the fundamental theorem. More precisely, K q R willbe shown to be an extension of A q − by a direct sum of B q with two appropriatelydefined nil terms (Theorem 1.7); the nil terms are identified as the reduced algebraic K -theory of categories of homotopy nilpotent endomorphisms (Theorem 11.1). Relation with other work.
The “classical” fundamental theorem for the higheralgebraic K -theory of rings has been proved by Quillen and
Grayson [Gra76].It has been extended from the K -theory of rings to the K -theory of schemes by Thomason and
Trobaugh [TT90, Theorem 6.6], and to the algebraic K -theoryof spaces [HKV +
01] by
Klein , Vogell , Waldhausen , Williams and the au-thor. A version for skew
Laurent polynomial rings has been discussed by
Yao [Yao95]. More recently, the result has been established by
Lück and
Steimle forskew
Laurent extensions of additive categories [LS16], and by
Fontes and
Ogle [FO18] in the context of S -algebras. Most of the recent accounts follow the patternlaid out in [HKV + Laurent polynomial rings tothe essential information contained in the graded structure of the ring extension R ⊂ R . Structure of the paper.
The paper is structured in a way that avoids forwardreferences in proofs. §1 provides an overview of notation and main results. §2discusses induced chain complexes. §3 introduces finite domination, an importantfiniteness condition for chain complexes. In §§4–5 the “projective line” and its K -theory are reviewed. §6 is devoted to an analysis of the “nil terms” in thefundamental theorem. §7 contains the proof that the “fundamental square” of theprojective line (roughly speaking relating the K -groups of R ≤ , R and R ≥ withthose of the projective line) is homotopy cartesian, which leads to a proof of the Mayer - Vietoris sequence in §8 and a proof of the fundamental theorem in §9.In §10 we establish a “localisation sequence” for algebraic K -theory, and finishthe paper by identifying the nil groups as the reduced K -theory of categories ofhomotopy nilpotent twisted endomorphisms in §11. Acknowledgements.
The basic ideas for this paper were developed during a re-search visit of the author to Beijing Institute of Technology. Their hospitality andfinancial support is greatly appreciated.1.
Notation and main results
Notation and conventions.
The word “ring” will always refer to an associativeunital ring, homomorphisms of rings respect the unit, and “modules” are understoodto be unital and right, unless otherwise specified. Let R = L k ∈ Z R k be a Z -gradedunital ring, so that R k R ℓ ⊆ R k + ℓ for all k, ℓ ∈ Z . (Here R k R ℓ is the set of finitesums of products xy with x ∈ R k and y ∈ R ℓ .) The component R is a subring THOMAS HÜTTEMANN of R with the same unit element [Dad80, Proposition 1.4]. Two further subrings ofnote are R ≤ = M k ≤ R k and R ≥ = M k ≥ R k . There are ring inclusions R ≤ ✛ i − R i + ✲ R ≥ and R ≤ j − ✲ R ✛ j + R ≥ , and ring homomorphisms given by projection R ≤ p − ✲ R ✛ p + R ≥ ; they satisfy the relations j − ◦ i − = j + ◦ i + , p − ◦ i − = id R and p + ◦ i + = id R . These various maps are used to define induction functors for modules, i −∗ : P P ⊗ R R ≤ and its relatives i + ∗ , j ∓∗ and p ∓∗ ; the resulting maps on algebraic K -groups aredenoted by the same symbols as the functors. Strongly Z -graded rings. The Z -graded ring R is called strongly graded if R k R ℓ = R k + ℓ for all k, ℓ ∈ Z , or equivalently, if R R − = R = R − R . This ensuresthat the ring multiplication yields R -bimodule isomorphisms R k ⊗ R R − k ∼ = R and, more generally, R k ⊗ R R ℓ ∼ = R k + ℓ ; consequently, each R k is an invertible R -bimodule, and hence a finitely generated projective (left and right) R -module[HS17, Proposition 1.6]. Similarly, one verifies [HS17, Lemma 1.9] that R ≤ q = M k ≤ q R k and R ≥− p = M k ≥− p R k are finitely generated projective (left and right) modules over R ≤ and R ≥ , re-spectively, for all p, q ∈ Z . The groups R NK ± q ( R ) . Let R be a Z -graded ring. Definition 1.1.
We define the nil groups of R relative to R as R NK − q ( R ) = coker (cid:0) i −∗ : K q ( R ) ✲ K q ( R ≤ ) (cid:1) and R NK + q ( R ) = coker (cid:0) i + ∗ : K q ( R ) ✲ K q ( R ≥ ) (cid:1) . Thus we have a split short exact sequence ✲ K q ( R ) i + ∗ ✲✛ p + ∗ K q ( R ≥ ) ✲ R NK + q ( R ) ✲ resulting in isomorphisms K q ( R ≥ ) ∼ = K q ( R ) ⊕ R NK + q ( R ) and R NK + q ∼ = ker (cid:0) p + ∗ : K q ( R ≥ ) ✲ K q ( R ) (cid:1) . (1.2)Of course these remarks hold for R NK − q ( R ) and K q ( R ≤ ) mutatis mutandis . HE FUNDAMENTAL THEOREM FOR STRONGLY Z -GRADED RINGS 5 Shifts and shift differences.
The Z -graded over-ring R of the ring R definesendofunctors of the category of R -modules, the shift functors , by s j : P P ⊗ R R j ( j ∈ Z ) . If R is strongly graded then R j is an invertible R -bimodule whence s j defines anauto-equivalence of the category of finitely generated projective R -modules, withresulting isomorphisms on algebraic K -groups s j : K q R ✲ K q R . Definition 1.3.
Suppose that R is a strongly Z -graded ring. The q th shift differ-ence map of R relative to R is defined as sd ∗ = id − s − : K q R ✲ K q R . (1.4)The kernel of sd ∗ : K q R ✲ K q R is called the q th shift kernel of R relativeto R , and is denoted by the symbol R ker K q R . The cokernel of this map is calledthe q th shift cokernel of R relative to R , and is denoted by the symbol R coker K q R .It might be worth pointing out that we could just as well use the shift functor s in place of s − ; in view of the R -bimodule isomorphism R − ⊗ R R ∼ = R ,this would lead to the same description of the shift kernel, and to an isomorphicdescription of the shift cokernel. In any case, the exact sequence ✲ R ker K q R ✲ K q R ∗ ✲ K q R ✲ R coker K q R ✲ determines the q th shift kernel and q th shift cokernel up to canonical isomorphism. Remark 1.5.
If there is an R -bimodule isomorphism R − ∼ = R then sd ∗ is thezero map whence R ker K q R = R coker K q R = K q R . This happens, for example, incase R is a Laurent polynomial ring R = R [ t, t − ] with a central indeterminate t . The first negative K -group. Let R = L k ∈ Z R k be a Z -graded ring. Definition 1.6.
We define the first negative K -group of R relative to R as R K − ( R ) = coker( j −∗ + j + ∗ ) , where the ring inclusions j − and j + induce the map j −∗ + j + ∗ : K ( R ≤ ) ⊕ K ( R ≥ ) ✲ K ( R ) . In the case of a
Laurent polynomial ring R ⊆ R [ t, t − ] this recovers the Bass K -group K − ( R ) = R [ t,t − ] K − ( R ) . The fundamental theorem.
The fundamental theorem expresses K q (cid:0) R [ t, t − ] (cid:1) as a direct sum of groups K q ( R ) , K q − ( R ) , R NK + q ( R ) and R NK − q ( R ) . In themore general context of R ⊆ R with R strongly Z -graded, the fundamental resultreads as follows: Theorem 1.7 (The fundamental theorem for the algebraic K -theory of strongly Z -graded rings) . Let R be a strongly Z -graded ring. There are short exact sequencesof abel ian groups ✲ R NK − q ( R ) ⊕ R coker K q ( R ) ⊕ R NK + q ( R ) ✲ K q ( R ) ✲ R ker K q − ( R ) ✲ (for q > ) (1.7a) THOMAS HÜTTEMANN and ✲ R NK − ( R ) ⊕ R coker K ( R ) ⊕ R NK +0 ( R ) ✲ K ( R ) ✲ R K − R ✲ . (1.7b) The Mayer-Vietoris sequence.
Let R be a strongly Z -graded ring. By analogywith algebraic geometry, one can consider a “projective line” which is obtained by“gluing spec R ≤ and spec R ≥ along their intersection spec R ”; more precisely, onecan define the analogue of the category of quasi-coherent sheaves on the projectiveline as the category of certain diagrams of modules. The projective line P in thissense was introduced and its K -theory computed by Montgomery and the author[HM18]; the relevant parts of the theory will be surveyed in §4 below.
Theorem 1.8 ( Mayer - Vietoris sequence) . Let R be a strongly Z -graded ring.There is a long exact sequence of algebraic K -groups . . . . . . γ ✲ K q +1 ( R ) δ ✲ K q ( P ) β ✲ K q ( R ≤ ) ⊕ K q ( R ≥ ) γ ✲ K q ( R ) δ ✲ K q − ( P ) β ✲ K q − ( R ≤ ) ⊕ K q − ( R ≥ ) γ ✲ K q − ( R ) ... δ ✲ K ( P ) β ✲ K ( R ≤ ) ⊕ K ( R ≥ ) γ ✲ K ( R ) ✲ R K − ( R ) ✲ . Homotopy nilpotent twisted endomorphisms and the localisation se-quence.
Let R be a strongly Z -graded ring. We define R Nil + ( R ) to be thecategory of pairs ( Z, ζ ) , with Z an R -finitely dominated bounded chain complexesof projective R -modules, and ζ : Z ⊗ R R ✲ Z ⊗ R R a homotopy nilpotenttwisted endomorphism, cf. §6. Let R Nil + q ( R ) denote the q th algebraic K -groupof R Nil + ( R ) (with respect to the quasi-isomorphisms as weak equivalences andthe levelwise split monomorphisms as cofibrations). Theorem 1.9.
Let R be a strongly Z -graded ring. The forgetful functor ( Z, ζ ) Z ,defined on R Nil + ( R ) , induces a homomorphism R Nil + q ( R ) ✲ K q R with ker-nel R NK − q +1 ( R ) . The groups R Nil + q ( R ) fit into a long exact sequences of algebraic K -groups . . . ✲ K q +1 R ✲ R Nil + q ( R ) φ ✲ K q R ≥ ✲ K q R ✲ . . . ✲ R Nil +0 ( R ) φ ✲ K R ≥ ✲ K R with φ induced by the functor sending ( Z, ζ ) to the R ≥ -module complex Z with R ≥ acting through ζ . There is a symmetric version involving a category R Nil − ( R ) , using R − inplace of R , and its K -groups R Nil − q ( R ) . The complete version of the result isformulated and proved in Theorems 10.1 and 11.1 below. HE FUNDAMENTAL THEOREM FOR STRONGLY Z -GRADED RINGS 7 Noetherian regular rings.
Suppose that R is a Z -graded ring (possibly notstrongly graded). Suppose further that R is right noether ian and regular. Forthis situation, van den Bergh [VdB86] established an exact sequence . . . ✲ K q +1 R ✲ K gr q R σ ✲ K gr q R ✲ K q R ✲ . . . , where K gr q R denotes the “graded K -theory of R ”, that is, the K -theory of thecategory of finitely generated projective Z -graded R -modules; the map σ is thedifference of the maps induced by the identity and the shift map M M (1) . Incase R is strongly graded, the categories of finitely generated projective R -modulesand of finitely generated projective graded R -modules are equivalent via the functor P P ⊗ R R ( Dade [Dad80, Theorem 2.8]). Thus we have the long exact sequence . . . ✲ K q +1 R ✲ K q R ∗ ✲ K q R ✲ K q R ✲ . . . which we can split at the shift difference map to obtain: Theorem 1.10.
Suppose that R is a strongly Z -graded right noether ian regularring. There are short exact sequences (for q > ) ✲ R coker K q ( R ) ✲ K q ( R ) ✲ R ker K q − ( R ) ✲ . (cid:3) Comparison with Theorem 1.7 suggests that R NK ± q ( R ) = 0 in this situation.We will not pursue the issue here.2. Induced modules and chain complexes
The Grothendieck group of a ring.
Let L be a rings. The group K ( L ) is,by definition, the Grothendieck group of the category P ( L ) of finitely generatedprojective L -modules; we denote the element corresponding to the module P by thesymbol [ P ] . The cokernel of the map K ( Z ) ✲ K ( L ) induced by the inductionfunctor M M ⊗ Z L is the reduced Grothendieck group K ( Z ↓ L ) of L , moreusually denoted by the symbol ˜ K ( L ) . We write the element corresponding to themodule P by α (cid:0) [ P ] (cid:1) . The following equivalences are well known: [ P ] = [ Q ] in K ( L ) ⇔ ∃ k ≥ P ⊕ L k ∼ = Q ⊕ L k (2.1a)and α (cid:0) [ P ] (cid:1) = α (cid:0) [ Q ] (cid:1) in K ( Z ↓ L ) ⇔ ∃ k, ℓ ≥ P ⊕ L k ∼ = Q ⊕ L ℓ (2.1b)In particular, α (cid:0) [ P ] (cid:1) = 0 if and only if P is stably free.We will repeatedly make use of the fact that K ( L ) can be described in otherways, for example using the machinery of Waldhausen K -theory applied to thecategory Ch ♭ P ( L ) of bounded complexes of finitely generated projective L -modules(with quasi-isomorphisms as weak equivalences, and levelwise split monomorphismsas cofibrations). This description is such that a chain complex C in Ch ♭ P ( L ) givesrise to the element [ C ] = χ ( C ) = P k ( − k [ C k ] of K ( L ) . If C is contractible then [ C ] = 0 . Induced and stably induced modules.
Let f : L ✲ S be a ring homomor-phism, with induced map f ∗ : K ( L ) ✲ K ( S ) , [ P ] [ P ⊗ L S ] . We will use the notation f ∗ also for the induction functor - ⊗ L S so that f ∗ (cid:0) [ P ] (cid:1) = (cid:2) f ∗ ( P ) (cid:3) . THOMAS HÜTTEMANN
Definition 2.2.
Let Q be a finitely generated projective S -module.(1) We say that Q is induced from L if there exists a finitely generated projective L -module P such that f ∗ ( P ) ∼ = Q .(2) The module Q is called stably induced from L if there exist a number q ≥ and a finitely generated projective L -module P such that f ∗ ( P ) ∼ = Q ⊕ S q .Algebraic K -theory provides an obstruction to modules being stably induced,the obstruction group being K ( L ↓ S ) = coker( f ∗ ) . It comes equipped with acanonical map α : K ( S ) ✲ K ( L ↓ S ) . (2.3) Proposition 2.4.
Let Q be a finitely generated projective S -module. The followingare equivalent: (1) The module Q is stably induced from L . (2) The element α (cid:0) [ Q ] (cid:1) of K ( L ↓ S ) is trivial. For example, if Q is a stably free S -module then Q is certainly stably inducedfrom L so that α (cid:0) [ Q ] (cid:1) = 0 in K ( L ↓ S ) Proof of Proposition 2.4. If Q ⊕ S s = P ⊗ L S then [ Q ] = f ∗ (cid:0) [ P ] (cid:1) − f ∗ (cid:0) [ L s ] (cid:1) lies inthe image of f ∗ whence α (cid:0) [ Q ] (cid:1) = 0 ∈ K ( L ↓ S ) .Conversely, suppose that α (cid:0) [ Q ] (cid:1) = 0 . Then there exists a ∈ K ( L ) such that [ Q ] = f ∗ ( a ) in K ( S ) . We can find a finitely generated projective L -module M anda number r ≥ such that a = [ M ] − [ L r ] in K ( L ) . By applying f ∗ and re-arrangingwe find the equality [ Q ⊕ S r ] = [ Q ] + [ S r ] = f ∗ ( a ) + f ∗ (cid:0) [ L r ] (cid:1) = f ∗ (cid:0) [ M ] (cid:1) ∈ K ( S ) . This in turn implies that there exists k ≥ with Q ⊕ S r ⊕ S k ∼ = f ∗ ( M ) ⊕ S k = f ∗ ( M ⊕ L k ) . This shows Q to be stably induced. (cid:3) Stabilisation of chain complexes.
As a matter of notation, we write D ( k, M ) for the chain complex concentrated in degrees k and k − with non-trivial entries M and differential the identity map of M . Definition 2.5.
Let D be a chain complex of S -modules. We say that the chaincomplex D ′ is a stabilisation of D if D ′ = D ⊕ L k D ( k, F k ) for finitely many finitelygenerated free S -modules F k .If D ′ is a stabilisation of D then there are mutually homotopy inverse chainhomotopy equivalences s : D ✲ D ′ (the inclusion) and r : D ′ ✲ D (the pro-jection) such that rs = id D , and such that coker( s ) = ker( r ) = L k D ( k, F k ) isa contractible bounded complex with finitely generated free chain, boundary andcycle modules. Induced and stably induced chain complexes. The chain complex liftingproblem.
Let f : L ✲ S be a ring homomorphism. Definition 2.6.
Let D be a bounded complex of finitely generated projective S -modules.(1) We say that D is induced from L if there exists a bounded complex offinitely generated projective L -module C such that f ∗ ( C ) ∼ = D . HE FUNDAMENTAL THEOREM FOR STRONGLY Z -GRADED RINGS 9 (2) The complex D is called stably induced from L if there exist a boundedcomplex C of finitely generated projective L -modules such that f ∗ ( C ) isisomorphic to a stabilisation of D .Again, let D be a bounded complex of finitely generated projective S -modules;suppose that all chain modules D k are induced from L . The chain complex liftingproblem is to decide whether D is homotopy equivalent to a complex induced from L .We will address two variations of the theme below and show that • if D is acyclic, then there exists an acyclic bounded complex C of finitelygenerated projective L -modules such that f ∗ ( C ) is isomorphic to a stabili-sation of D ; • if f satisfies a certain strong flatness condition, then D is stably inducedfrom L . The chain complex lifting problem for acyclic complexes.Theorem 2.7.
Every acyclic complex of stably induced modules is stably inducedfrom an acyclic complex. — More precisely, let D be an acyclic bounded chaincomplex of finitely generated projective S -modules concentrated in chain levels to n such that each chain module D k is stably induced from L . Then there exista stabilisation D ′ of D and an acyclic bounded complex C ′ of finitely generatedprojective L -modules, both concentrated in chain levels to n , such that f ∗ ( C ′ ) isisomorphic to D ′ .Proof. As D is acyclic there are finitely generated projective S -modules E k , for ≤ k ≤ n , such that D ∼ = n M k =1 D ( k, E k ) . (2.8)By construction we have E = D and consequently α (cid:0) [ E ] (cid:1) = 0 , since D is stablyinduced from L by hypothesis. By iteration, assuming that α (cid:0) [ E k − ] (cid:1) = 0 is known,we infer from D k − = E k − ⊕ E k that α (cid:0) [ E k ] (cid:1) = 0 as well.Thus for ≤ k ≤ n we can choose numbers j k ≥ and finitely generatedprojective L -modules P k such that E k ⊕ S j k ∼ = f ∗ ( P k ) . We define chain complexes D ′ = n M k =1 D ( k, E k ⊕ S j k ) and C ′ = n M k =1 D ( k, P k ) ; thus C ′ and D ′ are acyclic complexes concentrated in chain levels to n . Thecomputation D ′ = n M k =1 D ( k, E k ⊕ S j k ) = n M k =1 (cid:16) D ( k, E k ) ⊕ D ( k, S j k ) (cid:17) ∼ = (2.8) D ⊕ n M k =1 D ( k, S j k ) confirms D ′ as a stabilisation of D . Finally, by construction we have isomorphismsof chain complexes f ∗ ( C ′ ) = n M k =1 D (cid:0) k, f ∗ ( P k ) (cid:1) ∼ = n M k =1 D ( k, E k ⊕ S j k ) = D ′ as required. (cid:3) The chain complex lifting problem for well-behaved ring homomorphisms.
As before, let f : L ✲ S be a ring homomorphism. We consider S as an L - L -bimodule, with L acting via f . To solve the chain complex lifting problem, we willassume that the following condition is satisfied:The L - L -bimodule S is a filtered colimit of L - L -sub-bimodules S j whichare finitely generated projective right L -modules such that S j ⊗ L S = S . (2.9)In particular, S is a flat right L -module in this case. Example 2.10.
The ring inclusion f : L = R ≤ ⊆ ✲ R = S with R a strongly Z --graded ring satisfies condition (2.9) since R = S k ≥ R ≤ k with R ≤ k a finitelygenerated projective (left and right) R ≤ -module [HM18, Lemma I.2.2] such that R ≥ k ⊗ R ≤ R = R [HM18, Lemma I.2.4]. Proposition 2.11.
Let D be a chain complex consisting of finitely generated pro-jective S -modules concentrated in chain levels to n . Suppose that each chainmodule D k is induced from L so that D k ∼ = f ∗ ( C ′ k ) for a finitely generated projec-tive L -module C ′ k . Suppose further that the ring homomorphism f satisfies con-dition (2.9) . Then there exists a chain complex C of finitely generated projective L -modules, concentrated in chain levels to n , such that f ∗ ( C ) ∼ = D .Proof. We denote the differentials of D by d k : D k ✲ D k − . Set C k = { } for k < and for k > n , and choose C n = C ′ n .Since C n is finitely generated, the composite map C n ✲ C n ⊗ L S ∼ = D n d n ✲ D n − = f ∗ ( C ′ n − ) = (2.9) colim j C ′ n − ⊗ L S j factors through some stage j n of the colimit system, resulting in a map ∂ n : C n ✲ C ′ n − ⊗ L S j n =: C n − with target a finitely generated projective L -module, by our hypotheses on f , suchthat f ∗ ( ∂ n ) ∼ = d n .Since C n − is finitely generated, the composite map C n − ✲ C n − ⊗ L S ∼ = D n − d n − ✲ D n − = f ∗ ( C ′ n − ) = (2.9) colim j C ′ n − ⊗ L S j factors through some stage j of the colimit system, resulting in a map ˜ ∂ n − : C n − ✲ C ′ n − ⊗ L S j . By replacing j with a larger index j n − (where “larger” refers to the filtered prop-erties of the indexing category), we may assume that the composite with ∂ n is thezero map. In other words, we have a map ˜ ∂ n − : C n − ✲ C ′ n − ⊗ L S j n − =: C n − with target a finitely generated projective L -module, by our hypotheses on f , suchthat f ∗ ( ∂ n ) ∼ = d n and ∂ n − ◦ ∂ n = 0 .The process is repeated iteratively, until we have constructed C and ∂ . (cid:3) Proposition 2.12.
Let D be a chain complex consisting of finitely generated pro-jective S -modules concentrated in chain levels ≤ k ≤ n , such that each chain HE FUNDAMENTAL THEOREM FOR STRONGLY Z -GRADED RINGS 11 module D k is stably induced from L . Suppose that the ring homomorphism f sat-isfied condition (2.9) . Then there exist a stabilisation D ′ of D and a chain com-plex C ′ of finitely generated projective L -modules, both concentrated in chain levels ≤ k ≤ n + 1 , such that f ∗ ( C ′ ) ∼ = D ′ .Proof. For ≤ k ≤ n choose a number s k ≥ such that D k ⊕ S s k is inducedfrom L , and choose a finitely generated projective L -module C k with f ∗ ( C k ) ∼ = D k ⊕ S s k . Set D ′ = D ⊕ L k D ( k + 1 , S s k ) . By construction, D ′ is concentratedin chain levels to n + 1 , and each chain module D ′ is induced from L (as it isthe direct sum of a module induced from L with a finitely generated free module).Hence Proposition 2.11 yields a bounded complex C ′ of finitely generated projective L -modules, concentrated in chain levels to n + 1 , such that f ∗ ( C ′ ) ∼ = D ′ . (cid:3) Finite domination
Let C be a (possibly unbounded) chain complex of K -modules, for some uni-tal ring K . We say that C is K -finitely dominated , or of type (FP) over K , if K is homotopy equivalent to a bounded complex of finitely generated projective K -modules.Finite domination satisfies the “2-of-3 property” with respect to short exact se-quences of chain complexes. We include a proof for completeness. Lemma 3.1.
Let C , D and E be bounded below complexes of projective K -modules,and suppose that there is a short exact sequence ✲ C f ✲ D g ✲ E ✲ . (3.2) If any two of the complexes are K -finitely dominated then so is the third.Proof. The given sequence gives rise to a short exact sequence of bounded complexesof projective K -modules ✲ D ✲ cyl( g ) h ✲ cone( g ) ✲ , together with homotopy equivalences E ✲ cyl( g ) and q : C [1] ✲ cone( g ) . Byiteration, there is a short exact sequence ✲ cyl( g ) ✲ cyl( h ) ✲ cone( h ) ✲ , together with homotopy equivalences C [1] ✲ cyl( h ) and D [1] ✲ cone( h ) .As finite domination is invariant under suspension and homotopy equivalences, itsuffices to prove that if C and D are K -finitely dominated so is E .Since we are dealing with bounded below complexes of projective modules, thecanonical map cone( f ) ✲ E is a homotopy equivalence. As C and D are K -finitely dominated there exist bounded complexes C ′ and D ′ of finitely gen-erated projective K -modules and chain homotopy equivalences α : C ✲ C ′ and β : D ✲ D ′ . Choose a homotopy inverse α ′ of α , and let h : id C ≃ α ′ α be ahomotopy. The chain map (cid:18) αβf h β (cid:19) : cone( f ) ✲ cone( βf α ′ ) is a quasi-isomorphism (since α and β are) and hence a homotopy equivalence;its target is a bounded complex of finitely generated projective K -modules. Thus E ≃ cone( f ) ≃ cone( βf α ′ ) shows that E is K -finitely dominated. (cid:3) The projective line associated with a strongly Z -graded ring The projective line associated with a strongly Z -graded ring has been introducedby Montgomery and the author [HM18]. We recall definitions and K -theoreticalresults which will take a central place when establishing the fundamental theorem. From now on we will assume throughout that R = L k ∈ Z R k is a strongly Z -graded ring unless other hypotheses are specified.Definition 4.1 (Sheaves and vector bundles on the projective line [HM18, Defini-tions II.1.1 and II.2.1]) . A quasi-coherent sheaf on P , or just sheaf for short, is adiagram Y = (cid:16) Y − υ − ✲ Y ✛ υ + Y + (cid:17) (4.2)where Y − , Y and Y + are modules over R ≤ , R and R ≥ , respectively, with an R ≤ -linear homomorphisms υ − and an R ≥ -linear homomorphism υ + , such thatthe diagram of the adjoint R -linear maps Y − ⊗ R ≤ R υ − ♯ ∼ = ✲ Y ✛ υ + ♯ ∼ = Y + ⊗ R ≥ R (4.3)consists of isomorphisms. This latter condition will be referred to as the sheaf con-dition . A morphism f = ( f − , f , f + ) : Y ✲ Z between sheaves is a commutativediagram of the form Y Y − υ − ✲ Y ✛ υ + Y + Z ❄ Z − f − ❄ ζ − ✲ Z f ❄✛ ζ + Z + f + ❄ with f − , f and f + homomorphisms of modules over R ≤ , R and R ≥ , respectively.We call the sheaf Y a vector bundle if its constituent modules are finitely gener-ated projective modules over their respective ground rings. The category of vectorbundles (and all morphisms of sheaves between them) is denoted by Vect( P ) .Specific examples of vector bundles are the twisting sheaves O ( k, ℓ ) = (cid:16) R ≤ k ⊆ ✲ R ✛ ⊇ R ≥− ℓ (cid:17) where k and ℓ are integers. (Note that the diagrams O ( k, ℓ ) may not be sheavesif the Z -graded ring R fails to be strongly graded.) — Taking tensor product withtwisting sheaves defines an interesting operation on the category of sheaves on P : Definition 4.4 (Twisting [HM18, Definition II.2.4]) . Let Y be a sheaf, and let k, ℓ ∈ Z . We define the ( k, ℓ ) th twist of Y , denoted Y ( k, ℓ ) , to be the sheaf Y ( k, ℓ ) = (cid:16) Y − ⊗ R ≤ R ≤ k ✲ Y ⊗ R R ✛ Y + ⊗ R ≥ R ≥− ℓ (cid:17) , with structure maps induced by those of Y and the inclusion maps. Definition 4.5.
Let k, ℓ ∈ Z . The ( k, ℓ ) th canonical sheaf functor Ψ k,ℓ is definedby Ψ k,ℓ : P ( R ) ✲ Vect( P ) , P P ⊗ O ( k, ℓ ) HE FUNDAMENTAL THEOREM FOR STRONGLY Z -GRADED RINGS 13 where P ( R ) is the category of finitely generated projective R -modules, and thesymbol P ⊗ O ( k, ℓ ) denotes the sheaf P ⊗ R R ≤ k ✲ P ⊗ R R ✛ P ⊗ R R ≥− ℓ . Definition 4.6.
The sheaf cohomology modules of the sheaf Y of (4.2) are definedby the exact sequence ✲ H Y ✲ Y − ⊕ Y + υ − − υ + ✲ Y ✲ H Y ✲ , that is, H Y = ker( υ − − υ + ) and H Y = coker( υ − − υ + ) . We will also use thenotation Γ Y for H Y and speak of global sections of Y .The R -modules H Y and H Y depend functorially on Y . Considering Y as adiagram of R -modules we have isomorphisms H q Y = lim ← q Y for q = 0 , .One can explicitly compute the sheaf cohomology of twisting sheaves by directinspection [HM18, Proposition II.3.4]. Similarly, one can show: Proposition 4.7.
For P ∈ P ( R ) and k, ℓ ∈ Z there is an isomorphism ΓΨ k,ℓ P ∼ = k M j = − ℓ s j P where s j P = P ⊗ R R j . The isomorphism is natural in P so that there are isomor-phisms of functors Γ ◦ Ψ k,ℓ ∼ = 0 if k + ℓ < , Γ ◦ Ψ , ∼ = id , Γ ◦ Ψ k, − k ∼ = s k . Moreover, if k + ℓ ≥ − then H ◦ Ψ k,ℓ Y = 0 . (cid:3) The algebraic K -theory of the projective line We let Ch ♭ Vect( P ) denote the category of bounded chain complexes of vec-tor bundles; similarly, we denote by Ch ♭ Vect( P ) the category of bounded chaincomplexes in the category Vect( P ) . A map f of vector bundles will be calledan h -equivalence , or a quasi-isomorphism , if f ? is a quasi-isomorphism of chaincomplexes of modules for each decoration ? ∈ {− , , + } . As weak equivalencesare defined homologically, they satisfy the saturation and extension axioms. Allcategories mentioned have a cylinder functor given by the usual mapping cylinderconstruction which satisfies the cylinder axiom. Definition 5.1 ([HM18, Definition III.1.1]) . The K -theory space of the projectiveline is defined to be K ( P ) = Ω | h S • Ch ♭ Vect( P ) | , where “ h ” stands for the category of h -equivalences.It is technically more convenient to use a certain subcategory of Vect( P ) , onwhich the global sections functor Γ is exact: Definition 5.2.
The category
Vect( P ) is the full subcategory of Vect( P ) con-sisting of those objects Y satisfying H Y ( k, ℓ ) = 0 for all k, ℓ ∈ Z with k + ℓ ≥ . Lemma 5.3 ([HM18, Corollary III.1.4]) . The inclusion
Vect( P ) ⊆ Vect( P ) in-duces a homotopy equivalence h S • Ch ♭ Vect( P ) ≃ ✲ h S • Ch ♭ Vect( P ) , and hence a homotopy equivalence Ω | h S • Ch ♭ Vect( P ) | ≃ ✲ K ( P ) . (cid:3) For k + ℓ ≥ − the functor Ψ k,ℓ : Ch ♭ P ( R ) ✲ Vect( P ) is an exact functorbetween Waldhausen categories (with quasi-isomorphisms and h -equivalences asweak equivalences, and cofibrations the monomorphisms with cokernel an object ofthe category under consideration). Theorem 5.4 ([HM18, Theorem III.5.1]) . Suppose that R = L k ∈ Z R k is a strongly Z -graded ring. There is a homotopy equivalence of K -theory spaces K ( R ) × K ( R ) ✲ K ( P ) induced by the functor Ψ − , + Ψ , : Ch ♭ P ( R ) × Ch ♭ P ( R ) ✲ Ch ♭ Vect( P ) ( C, D ) Ψ − , ( C ) ⊕ Ψ , ( D ) . (cid:3) The nil terms
The category of twisted endomorphisms.
Let M be an R -module. A (posi-tive) twisted endomorphism of M is an R -linear map α : M ⊗ R R ✲ M ⊗ R R . The collection of such pairs ( M, α ) , for various modules M and their twisted en-domorphisms, forms a category Tw + End( R ) ; a morphism f : ( M, α ) ✲ ( N, β ) is an R -linear map f : M ✲ N with β ◦ ( f ⊗ id R ) = ( f ⊗ id R ) ◦ α . — Thecategory Tw − End( R ) of (negative) twisted endomorphisms of the form M ⊗ R R − ✲ M ⊗ R R is defined analogously.In case of a strongly Z -graded ring R we have the equality R n +1 = R n R , or(what is the same) the isomorphism R n +1 ∼ = R n ⊗ R R . Given an object ( M, α ) of Tw + End( R ) , we can thus recursively define the n th iteration of α to be themap α ( n ) : M ⊗ R R n ✲ M ⊗ R R ( n ≥ determined by α (0) = id and α (1) = α , with α ( n +1) being the composition M ⊗ R R n +1 ∼ = M ⊗ R R n ⊗ R R α ( n ) ⊗ id R ✲ M ⊗ R R ⊗ R R ∼ = M ⊗ R R α ✲ M ⊗ R R . We say that the twisted endomorphism α of M is nilpotent if α ( n ) = 0 for n ≫ .Every R ≥ -module M determines an object ( M, µ ) of Tw + End( R ) , with M considered as an R -module by restriction of scalars, and µ defined by m ⊗ r mr ⊗ for m ∈ M and r ∈ R . There results a functor Φ = Φ + : Mod - R ≥ ✲ Tw + End( R ) ; HE FUNDAMENTAL THEOREM FOR STRONGLY Z -GRADED RINGS 15 an analogous construction provides us with a functor Φ − : Mod - R ≤ ✲ Tw − End( R ) . Lemma 6.1.
The functors
Φ = Φ + and Φ − are isomorphisms of categories.Proof. This is just the theory of modules over tensor rings, since (thanks to thestrong grading) R ≥ is the tensor ring of R over R , and R ≤ is the tensor ringof R − over R . (cid:3) As a matter of notation, for ( M, α ) ∈ Tw + End( R ) we will denote the R ≥ -module Φ − ( M, α ) simply by the symbol M ; if we wish to stress the twisted endomorphism,we shall write M α . The module structure of M α and the maps α ( n ) are related bythe square diagram (6.2) below; the vertical maps, induced by inclusions, are iso- M ⊗ R R n α ( n ) ✲ M ⊗ R R M α ⊗ R ≥ R ≥ n ∼ = ❄ ✲ M α ⊗ R ≥ R ≥ ∼ = ❄ (6.2)morphisms of R -modules ([HM18, Proposition I.2.12]), and the bottom horizontalmap is the obvious one, mapping x ⊗ r ∈ M α ⊗ R ≥ R ≥ n to x ⊗ r ∈ M α ⊗ R ≥ R ≥ .The diagram commutes. The characteristic sequence of a twisted endomorphism.
For a strongly Z -graded ring R we have the equality R R − = R , by definition of strongly graded.In particular, there exist finitely many elements x (1) j ∈ R and y ( − j ∈ R − so that P j x (1) j y ( − j = 1 . For any R ≥ -module M there results a well-defined map of R ≥ -modules χ M = χ : M ⊗ R R ≥ ✲ M ⊗ R R ≥ , m ⊗ r m ⊗ r − X j mx (1) j ⊗ y ( − j r (6.3)which does not depend on the specific choice of elements x (1) j and y ( − j [Hüt18, §6]. Proposition 6.4 (Characteristic sequence of a twised endomorphism) . Let ( M, α ) be an object of Tw + End( R ) . There is a short exact sequence of R ≥ -modules,natural in ( M, α ) , ✲ M α ⊗ R R ≥ χ M ✲ M α ⊗ R R ≥ π ✲ M α ✲ , (6.5) with π ( m ⊗ r ) = mr . The sequence is split exact as a sequence of R -modules. This generalises the usual characteristic sequence of a module equipped with anendomorphism [Bas68, Proposition XII.1.1].
Proof.
The sequence (6.5) is nothing but the “canonical resolution” of [Hüt18,Lemma 6.2] for the R ≥ -module M α associated with ( M, α ) . (cid:3) Chain complexes of twisted endomorphisms.
A bounded chain complex inthe category Tw + End( R ) , that is, an object of the category Ch ♭ Tw + End( R ) ,consists of a pair ( C, α ) where C is a bounded chain complex of R -modules, and α : C ⊗ R R ✲ C ⊗ R R is a map of R -module complexes. Definition 6.6.
The mapping half-torus of ( C, α ) ∈ Ch ♭ Tw + End( R ) is the com-plex of R ≥ -modules H ( C, α ) = cone (cid:16) χ C : C ⊗ R R ≥ ✲ C ⊗ R R ≥ (cid:17) , with χ C = χ defined in (6.3).The mapping half-torus provides an exact additive functor defined on the cate-gory Tw + End( R ) to the category of chain complexes of R ≥ -modules. It comesequipped with a natural R ≥ -linear map H ( C, α ) ✲ C α induced from the shortexact sequence (6.5). Proposition 6.7 ([Hüt18, Corollaries 6.5 and 6.8]) . Suppose that ( C, α ) is abounded chain complex in Tw + End( R ) . (1) The map H ( C, α ) ✲ C α is a quasi-isomorphism. (2) If C is a complex of projective R -modules, the map H ( C, α ) ✲ C α is achain homotopy equivalence of R -module complexes. (3) If C is an R -finitely dominated complex of projective R ≥ -modules, then H ( C, α ) is R ≥ -finitely dominated. (cid:3) Homotopy nilpotent twisted endomorphisms.
Let ( C, α ) be a chain complexin the category Tw + End( R ) . We say that α is homotopy nilpotent if the chainmap α ( n ) : C ⊗ R R n ✲ C ⊗ R R (6.8)is null homotopic for some (equivalently, all) sufficiently large n ≥ . Using dia-gram (6.2), this amounts to saying that the map of R -module complexes C α ⊗ R ≥ R ≥ n ✲ C α ⊗ R ≥ R ≥ = C α (induced by the inclusion map R ≥ n ⊆ R ≥ ) is null homotopic for n ≫ . Since R ≥ n is an invertible R ≥ -bimodule with inverse R ≥− n , this in turn is equivalent to C α = C α ⊗ R ≥ R ≥ ✲ C α ⊗ R ≥ R ≥− n (6.9)being null homotopic for n ≫ . Under suitable finiteness assumptions, this isequivalent to the stronger condition that the map (6.9) is null homotopic for n ≫ as a map of R ≥ -module complexes: Lemma 6.10.
Let C be an R ≥ -finitely dominated complex of R ≥ -modules. Thefollowing statements are equivalent: (1) The induced complex C ⊗ R ≥ R is acyclic. (2) For n ≫ the obvious map ν ,n : C ⊗ R ≥ R ≥ ✲ C ⊗ R ≥ R ≥− n , consid-ered as a map of R ≥ -module chain complexes, is null homotopic. (3) For n ≫ the obvious map ν ,n : C ⊗ R ≥ R ≥ ✲ C ⊗ R ≥ R ≥− n , consid-ered as a map of R -module chain complexes, is null homotopic.If these equivalent conditions hold, the complex C is R -finitely dominated. HE FUNDAMENTAL THEOREM FOR STRONGLY Z -GRADED RINGS 17 Proof.
All three statements and the conclusion on finite domination are invariantunder homotopy equivalence of complexes of R ≥ -modules. As C is R ≥ -finitelydominated we may, and will, assume from the outset that C is a bounded complexof finitely generated projective R ≥ -modules.We use the notation ν : C ⊗ R ≥ R ≥ ✲ C ⊗ R ≥ R and ν k,n : C ⊗ R ≥ R ≥− k ✲ C ⊗ R ≥ R ≥− k − n for the obvious R ≥ -linear maps sending x ⊗ r to x ⊗ r .(1) ⇒ (2): The map ν is certainly null homotopic as its target is contractible byhypothesis, and a choice of null homotopy yields a map H : cyl(id C ) ✲ C ⊗ R ≥ R = [ n ≥ C ⊗ R ≥ R ≥− n restricting to ν and , respectively, on the “ends” of the cylinder. Since the sourceis a bounded complex of finitely generated projective modules, the map H factorsthrough a finite stage of the increasing union, resulting in a number N ≥ and amap H ′ : cyl(id C ) ✲ C ⊗ R ≥ R ≥− N . (6.11)By construction, H ′ is a null homotopy of ν ,N , and hence ν ,n ≃ for all n ≥ N .(2) ⇒ (3): This is a tautology.(3) ⇒ (1): Let n be sufficiently large so that ν ,n ≃ as a map of R -modulecomplexes. As ν k,n is isomorphic to ν ,n ⊗ R R − k , as a map of R -modules ([HM18,Proposition I.2.12]), we conclude that ν k,n ≃ for all k ≥ . In particular, the maps ν k,n induce the trivial map on homology. Since C ⊗ R ≥ R = colim n ≥ C ⊗ R ≥ R ≥− n ,and since homology commutes with filtered colimits, we conclude by cofinality that H ∗ ( C ⊗ R ≥ R ) = colim n ≥ H ∗ ( C ⊗ R ≥ R ≥− n ) = colim k ≥ H ∗ ( C ⊗ R ≥ R ≥− kn ) , the last colimit taken with respect to the maps H ∗ ( ν kn,n ) which are trivial. Itfollows that the colimit is trivial, whence C ⊗ R ≥ R is acyclic.Suppose now that the equivalent conditions of the Lemma are satisfied. Goingback to (6.11) and the notation used there, we know that the chain map ν ,N : C ∼ = C ⊗ R ≥ R ≥ ✲ C ⊗ R ≥ R ≥− N is null homotopic. Hence its mapping cone is homotopy equivalent to the mappingcone M = C ⊕ Σ (cid:0) C ⊗ R ≥ R ≥− N (cid:1) of the zero map between the same complexes,which contains C as a direct summand. On the other hand, the map ν ,N is injective(since C is assumed to consist of projective R ≥ -modules), hence its mapping coneis quasi-isomorphic to the cokernel K of ν ,N . The ℓ th chain module K ℓ of K is isomorphic, as an R ≥ -module, to C ℓ ⊗ R ≥ ( R ≥− N /R ≥ ) . Since C ℓ is a direct summand of R n ℓ ≥ , for suitable n ℓ ≥ , the module K ℓ is a direct summand of R n ℓ ≥ ⊗ R ≥ ( R ≥− N /R ≥ ) ∼ = (cid:16) R ≥ ⊗ R ≥ ( R ≥− N /R ≥ ) (cid:17) n ℓ ∼ = (cid:0) R ≥− N /R ≥ (cid:1) n ℓ ∼ = (cid:16) − M q = − N R q (cid:17) n ℓ , the last isomorphism being R -linear. Since R is strongly graded each R q is afinitely generated projective R -module so that K ℓ is a finitely generated projective R -module as well. Thus the (bounded) complex K is R -finitely dominated. Since C consists of projective R -modules so does M , and as M is quasi-isomorphic to K by the above arguments there is in fact an R -linear homotopy equivalence M ≃ K .It follows that M is R -finitely dominated, hence so is its direct summand C . —The finiteness result also follows from Theorem 8.1 of [Hüt18]; in the notation usedthere the ring R ∗ (( t − )) contains R so that C ⊗ R ≥ R ∗ (( t − )) ∼ = C ⊗ R ≥ R ⊗ R R ∗ (( t − )) is acyclic as required. (cid:3) From Lemma 6.10, and from the fact that the maps (6.8) and (6.9) are isomorphicas R -linear maps, we conclude: Corollary 6.12.
Let ( C, α ) be an object of Ch ♭ Tw + End( R ) . Suppose that theassociated R ≥ -module complex C α is R ≥ -finitely dominated. The twisted endo-morphism α is homotopy nilpotent if and only if C α ⊗ R ≥ R is acyclic. (cid:3) For later use we record the following useful fact:
Lemma 6.13.
Let ( Z, ζ ) be an object of Ch ♭ Tw + End( R ) . Suppose that the asso-ciated R ≥ -module complex Z ζ is an R ≥ -finitely dominated complex of projective R ≥ -modules. Suppose further that Z ζ ⊗ R ≥ R is acyclic. Then Z is an R -finitelydominated bounded complex of projective R -modules.Proof. Since R is strongly Z -graded the complex Z consists of projective R -modules.By hypothesis, Z ζ is homotopy equivalent to a bounded complex C of finitely gen-erated projective R ≥ -modules. Since Z ζ ⊗ R ≥ R is acyclic so is C ⊗ R ≥ R . Thisforces C , hence Z ζ , to be R -finitely dominated in view of Lemma 6.10. (cid:3) The nil category.
We are now in a position to define the nil category, the cate-gory of homotopy nilpotent endomorphisms of chain complexes satisfying a suitablefiniteness constraint. In fact there are two variants, corresponding to the two sub-rings R ≥ and R ≤ of R . Definition 6.14.
The positive nil category R Nil + ( R ) of R relative to R is thefull subcategory of Ch ♭ (Tw + End( R )) consisting of those chain complexes ( C, α ) such that(1) the underlying R -module complex C consists of projective R -modules,and is R -finitely dominated;(2) the chain map α : C ⊗ R R ✲ C ⊗ R R is homotopy nilpotent in thesense that α ( n ) ≃ for n ≫ .The negative nil category R Nil − ( R ) of R relative to R is defined analogously,using the category Tw − End( R ) of negative twisted endomorphisms in place of Tw + End( R ) . HE FUNDAMENTAL THEOREM FOR STRONGLY Z -GRADED RINGS 19 Remark 6.15.
Let ( C, α ) be an object of R Nil + ( R ) . By Proposition 6.7, the R ≥ -linear map H ( C, α ) ✲ C α is a quasi-isomorphism with R ≥ -finitely dom-inated source. The same map is an R -linear homotopy equivalence. Thus thesecond condition in the definition of R Nil + ( R ) is equivalent to the condition H ( C, α ) ⊗ R ≥ R ≃ , by Corollary 6.12.A morphism in the category R Nil + ( R ) is called a cofibration if it is injectiveand its cokernel consists of projective R -modules. We say the morphism is a weakequivalence , or a q -equivalence , if it is a quasi-isomorphism of R -module complexes. Lemma 6.16.
These definitions equip R Nil + ( R ) with the structure of a categorywith cofibrations and weak equivalences.Proof. This is mostly straightforward; the gluing lemma holds, for example, sinceit holds in the category of bounded complexes of projective R -modules. Whatneeds explicit verification is that the requisite pushouts exist within the cate-gory R Nil + ( R ) . Using the functor Φ and its inverse ( A, α ) A α we identify R Nil + ( R ) with a full subcategory of the category K of bounded chain com-plexes of R ≥ -modules. Let ( A, α ) , ( B, β ) and ( C, γ ) be objects in R Nil + ( R ) ,let f : ( A, α ) ✲ ( B, β ) be a cofibration, and let g : ( A, α ) ✲ ( C, γ ) be anarbitrary morphism in R Nil + ( R ) . We can then form the pushout diagram A α f ✲ B β C γ g ❄ f ′ ✲ P π ❄ in the category K , and claim that ( P, π ) is on object of Ch ♭ (Tw + End( R )) .As f is a cofibration in R Nil + ( R ) it is an injective map, and the same isthus true for its pushout f ′ . Let K κ denote the cokernel of f , associated with ( K, κ ) ∈ Ch ♭ (Tw + End( R )) . From general properties of pushout squares we knowthat coker( f ′ ) ∼ = K κ . Hence we obtain a commutative ladder diagram of shortexact sequences of chain complexes: ✲ A α ✲ B β ✲ K κ ✲ ✲ C γ ❄ ✲ P π ❄ ✲ K κ ∼ = ❄ ✲ Again, since f is a cofibration its cokernel K κ consists of projective R -modules.Thus both sequences are levelwise split short exact sequences of R -module com-plexes, and P consists of projective R -modules. By Lemma 3.1, applied to thetop row, the complex K κ is R -finitely dominated, which implies, by applyingLemma 3.1 to the bottom row, that so is P π . Application of the exact half-torus function yields another commutative ladderdiagram of short exact sequences, ✲ H ( A, α ) ✲ H ( B, β ) ✲ H ( K, κ ) ✲ ✲ H ( C, γ ) ❄ ✲ H ( P, π ) ❄ ✲ H ( K, κ ) ∼ = ❄ ✲ consisting of bounded complexes of finitely generated projective R ≥ -modules; inparticular, both short exact sequences are levelwise split. By Remark 6.15 thechain complexes H ( A, α ) , H ( B, β ) and H ( C, γ ) are R ≥ -finitely dominated. Hence H ( K, κ ) and H ( P, π ) are R ≥ -finitely dominated as well, by two applications ofLemma 3.1.The bottom row yields a short exact sequence ✲ H ( C, γ ) ⊗ R ≥ R ✲ H ( P, π ) ⊗ R ≥ R ✲ H ( K, κ ) ⊗ R ≥ R ✲ . The first and third entry are acyclic, by Corollary 6.12, hence so is the middleentry. Applying the Corollary again leads us to conclude that ( P, π ) is an objectof R Nil + ( R ) . (cid:3) With this lemma in place, we can now introduce the notation R Nil + q ( R ) = K q (cid:0) R Nil + ( R ) (cid:1) = π q Ω | q S • R Nil + ( R ) | (6.17a)and R Nil − q ( R ) = K q (cid:0) R Nil − ( R ) (cid:1) = π q Ω | q S • R Nil − ( R ) | . (6.17b)For L a unital ring write FD ( L ) for the category of L -finitely dominated boundedcomplexes of projective L -modules. It is well-known, and can be verified with-out difficulty using Waldhausen ’s approximation theorem, that the inclusion Ch ♭ P ( L ) ⊆ ✲ FD ( L ) induces a homotopy equivalence on K -theory spaces K ( L ) = Ω | q S • Ch ♭ P ( L ) | ≃ ✲ Ω | q S • FD ( L ) | , (6.18)where “ q ” stands for quasi-isomorphisms as usual, and the cofibrations are theinjective maps with levelwise projective cokernel. Hence the forgetful functors o ∓ : R Nil ∓ ( R ) ✲ FD ( R ) , ( Z, ζ ) Z yield group homomorphisms o −∗ : R Nil − q ( R ) ✲ K q ( R ) and o + ∗ : R Nil + q ( R ) ✲ K q ( R ) , (6.19)and we will establish in Theorem 11.1 that there are isomorphisms R NK − q +1 ( R ) ∼ = ker( o + ) and R NK + q +1 ( R ) ∼ = ker( o − ) , identifying the groups R NK ∓ q +1 ( R ) with the q th reduced algebraic K -group of thecategory R Nil ± ( R ) . HE FUNDAMENTAL THEOREM FOR STRONGLY Z -GRADED RINGS 21 The fundamental square
The fundamental square of the projective line.
Let f = ( f − , f , f + ) bea morphism Y ✲ Z in the category Ch ♭ Vect( P ) of bounded chain complexes ofvector bundles on the projective line associated with a strongly Z -graded ring R .We say that f is an h ? -equivalence , for the decoration ? ∈ {− , , + } , if the com-ponent f ? is a quasi-isomorphism of chain complexes. Together with the previousnotion of cofibrations, viz. , levelwise split injections in each component, this equips Ch ♭ Vect( P ) with three new structures of a category with cofibrations and weakequivalences. Note that an h − -equivalence is automatically an h -equivalence aswell since in the commutative square Y − ⊗ R ≤ R υ − ♯ ✲ Y Z − ⊗ R ≤ Rf − ⊗ R ❄ ζ − ♯ ✲ Z f ❄ the horizontal maps are isomorphisms (sheaf condition), and the left vertical mapis a quasi-isomorphism because R is a flat left R ≤ -module thanks to the stronggrading. — Similarly, every h + -equivalence is an h -equivalence.Thus the identity functor yields a commutative square of K -theory spaces h S • Ch ♭ Vect( P ) ✲ h + S • Ch ♭ Vect( P ) h − S • Ch ♭ Vect( P ) ❄ ✲ h S • Ch ♭ Vect( P ) ❄ (7.1)which we call the fundamental square of the projective line. Associated with it, bytaking vertical homotopy fibres, is the map α : h S • Ch ♭ Vect( P ) h − ✲ h + S • Ch ♭ Vect( P ) h (7.2)where the notation is as in Waldhausen ’s fibration theorem [Wal85, Theorem 1.6.4].Explicitly, Ch ♭ Vect( P ) h − is the full subcategory of Ch ♭ Vect( P ) consisting of thoseobjects Y for which Y ✲ is an h − -equivalence, that is, which satisfy Y − ≃ ∗ and Y ≃ ∗ , and Ch ♭ Vect( P ) h is the full subcategory of Ch ♭ Vect( P ) consistingof those objects Y for which Y ✲ is an h -equivalence, that is, which satisfy Y ≃ ∗ . The fibres of the fundamental square.
The (homotopy) fibres of the maps inthe fundamental square can be identified explicitly: they are homotopy equivalentto K -theory spaces of categories of homotopy nilpotent twisted endomorphisms.We will use this identification presently to conclude that the fundamental square ishomotopy cartesian. Theorem 7.3 (Fibres of the fundamental square) . The functor F : Ch ♭ Vect( P ) h ✲ R Nil + ( R ) , Y = (cid:0) Y − ✲ Y ✛ Y + (cid:1) Φ( Y + ) , defined on the category of bounded chain complexes of vector bundles Y with Y ≃ ∗ ,induces a homotopy equivalence h + S • Ch ♭ Vect( P ) h ∼ ✲ q S • R Nil + ( R ) , (7.3a) and, by restriction, a homotopy equivalence h S • Ch ♭ Vect( P ) h − ∼ ✲ q S • R Nil + ( R ) (7.3b) where the letter “ q ” stands for weak equivalences in the category R Nil + ( R ) . By symmetry, there are analogous homotopy equivalences h − S • Ch ♭ Vect( P ) h ∼ ✲ q S • R Nil − ( R ) , (7.4a)and, by restriction, h S • Ch ♭ Vect( P ) h + ∼ ✲ q S • R Nil − ( R ) . (7.4b) Proof.
The functor F is well defined. Indeed, the component Y + of Y is a boundedcomplex of finitely generated projective R ≥ -modules, hence Y + consists of pro-jective R -modules. The hypothesis Y + ⊗ R ≥ R ∼ = Y ≃ ∗ implies that Y + is R -finitely dominated by Lemma 6.13. Moreover, the associated twisted endomor-phism Y + ⊗ R R ✲ Y + ⊗ R R is homotopy nilpotent by Corollary 6.12.We will employ Waldhausen ’s approximation theorem [Wal85, Theorem 1.6.7],combined with the standard observation that the map required by property (App 2)does not have to be a cofibration (since it can be replaced by one in the presenceof cylinder functors). — We can in fact treat cases (7.3a) and (7.3b) at the sametime as the proofs are almost identical.To start with, a morphism a = ( a − , a , a + ) in Ch ♭ Vect( P ) h is an h + -equiva-lence, by definition, if and only if F ( a ) = a + is a quasi-isomorphism. A morphism b = ( b − , b , b + ) in Ch ♭ Vect( P ) h − is an h -equivalence if and only if F ( b ) = b + isa quasi-isomorphism since b − and b are maps between acyclic complexes, henceare quasi-isomorphisms in any case. In other words, the functor F satisfies prop-erty (App 1) in both cases under consideration.Let ( Z, ζ ) ∈ R Nil + ( R ) and Y + ∈ Ch ♭ Vect( P ) h , and let f : Φ( Y + ) ✲ ( Z, ζ ) be a morphism in R Nil + ( R ) . We can equivalently consider f as an R ≥ -linearmap of chain complexes Y + ✲ Z ζ . By Remark 6.15, the chain complex Z ζ isquasi-isomorphic to the R ≥ -finitely dominated complex H ( Z, ζ ) so that there existsa bounded complex E of finitely generated projective R ≥ -modules and a quasi-isomorphism e : E ✲ Z ζ . We can lift f up to homotopy to an R ≥ -linear map g : Y + ✲ E so that eg is homotopic to f . A choice of homotopy f ≃ eg deter-mines a map h ′ : cyl( g ) ✲ Z ζ such that the composite Y + i ✲ cyl( g ) h ′ ✲ Z ζ coincides with f , and such that the composite E ≃ i ✲ cyl( g ) h ′ ✲ Z ζ coincideswith e . In particular, h ′ is a quasi-isomorphism of R ≥ -module complexes. (Here i and i are the front and back inclusion into the mapping cylinder.)We can now form a complex X + by attaching to cyl( g ) a direct sum of con-tractible complexes of the type D ( ℓ, M ) , with M a finitely generated projective R ≥ -module, so that all chain modules of X + except possibly one are finitely HE FUNDAMENTAL THEOREM FOR STRONGLY Z -GRADED RINGS 23 generated free modules. We let a + denote the composition of i with the in-clusion cyl( g ) ✲ X + , and let h + denote the composition of the projection X + ✲ cyl( g ) with h ′ . We have constructed a commutative diagram Y + a + ✲ X + Z ζ ≃ h + ❄ f ✲ where X + is a bounded complex of finitely generated projective R ≥ -modules, and h + is a quasi-isomorphism.We will now extend X + to a vector bundle X = ( X − , X , X + ) and a + to amap of vector bundles a = ( a − , a , a + ) : Y ✲ X . This may involve enlarging X + further, by taking the direct sum with contractible complexes of the form D ( k, R k ≥ ) and composing a + and h + with the obvious inclusion and projection maps. We shallkeep the notation X + and h + even for the modified data.Let X = X + ⊗ R ≥ R . Recall that Z ζ is quasi-isomorphic to X + , via the map h + ,and to the half-torus H ( Z, ζ ) , by Remark 6.15. Both X + and H ( Z, ζ ) are boundedcomplexes of finitely generated projective R ≥ -modules, so these complexes arehomotopy equivalent. It follows that X ≃ H ( Z, ζ ) ⊗ R ≥ R is contractible, byRemark 6.15 again; this implies that [ X ] = P k ( − k [ X k ] = 0 ∈ K ( R ) . As allchain modules of X are free with the possible exception of a single module, weconclude that X consists of stably free modules, and hence consists of moduleswhich are stably induced from R ≤ . We now appeal to Theorem 2.7: we canmodify X + by taking direct sum with finitely many contractible complexes of theform D ( k, R j ≥ ) , thereby modifying X in an analogous manner, so that X isisomorphic to X − ⊗ R ≤ R for a bounded acyclic complex X − of finitely generatedprojective R ≤ -modules.We have thus constructed a vector bundle X = ( X − ✲ X ✛ X + ) together with a quasi-isomorphism h + : X + ✲ Z ζ and a map a + : Y + ✲ X + such that h + ◦ a + = f . The map a + induces a compatible map a = ξ + ♯ ◦ ( a + ⊗ id) ◦ (cid:0) ξ + ♯ (cid:1) − , see the diagram in Fig. 1. For each chain level ℓ the composite map Y − υ − ✲ Y ✛ ξ + ♯ ∼ = Y + ⊗ R ≥ RX − ξ − ✲ X a ❄✛ ξ + ♯ = X + ⊗ R ≥ Ra + ⊗ id ❄ Figure 1.
Diagram used in proof of Theorem 7.3 Y − ℓ υ − ✲ Y ℓ a ✲ X ℓ ∼ = X − ℓ ⊗ R ≤ R = [ q ≥ X − ℓ ⊗ R ≤ R ≤ q factors through some term X − ℓ ⊗ R ≤ R ≤ q so that, by choosing q ≫ , the map a ◦ υ − factorises as Y − a − ✲ X − ⊗ R ≤ R ≤ q ✲ X , the second map in this composition given by x ⊗ r ξ − ( x ) · r . That is, there exists q ≥ and a − : Y − ✲ X − ⊗ R ≤ R ≤ q such that a = ( a − , a , a + ) : Y ✲ X ( q, is a map of vector bundles, and such that h + ◦ F ( a ) = f . Since X − and X are con-tractible by construction, X ( q, is an object of Ch ♭ Vect( P ) h − ⊆ Ch ♭ Vect( P ) h .This proves that F satisfies property (App 2) in addition to (App 1), and theapproximation theorem applies. (cid:3) Corollary 7.5.
The map α of (7.2) is a homotopy equivalence. Thus, the funda-mental square (7.1) is homotopy cartesian.Proof. The previous Theorem asserts that in the chain of maps h S • Ch ♭ Vect( P ) h − α ✲ h + S • Ch ♭ Vect( P ) h F ✲ q S • R Nil + ( R ) both F and F ◦ α are homotopy equivalences. It follows that α is a homotopyequivalence as well. Since the fundamental square consists of connected spaces itis homotopy cartesian. (cid:3) Auxiliary categories.
We let C + denote the category of bounded chain complexesof “vector bundles on spec( R ≥ ) ”, that is, diagrams of the form Y ✛ υ + Y + with Y and Y + being bounded complexes of finitely generated projective over R and R ≥ , respectively, subject to the condition that the associated adjointmap Y ✛ Y + ⊗ R ≥ R be an isomorphism. A morphism g = ( g , g + ) from Y ✛ υ + Y + to Z ✛ ζ + Z + consists of an R -linear map g : Y ✲ Z and an R ≥ -linear map g + : Y + ✲ Z + such that ζ + ◦ g + = g ◦ υ + .By D + we denote the full subcategory of C + consisting of objects Y ✛ υ + Y + such that all chain modules Y k are stably induced from R ≤ , that is, such that [ Y k ] ∈ im (cid:0) K ( R ≤ ) ✲ K ( R ) (cid:1) , or equivalently, such that α (cid:0) [ Y k ] (cid:1) = 0 in K ( R ≤ ↓ R ) for all k . Lemma 7.6.
Every object of D + stably extends to an object of Ch ♭ Vect( P ) . Thatis, given an object Y ✛ υ + Y + of D + there exist an object Z − ζ − ✲ Z ✛ ζ + Z + HE FUNDAMENTAL THEOREM FOR STRONGLY Z -GRADED RINGS 25 of Ch ♭ Vect( P ) and finitely many numbers j k ≥ together with a commutativediagram Y ⊕ M k D ( k, R j k ) ✛ υ + ⊕ inc Y + ⊕ M k D ( k, R j k ≥ ) Z = ❄✛ ζ + Z + , = ❄ where “ inc ” denotes the obvious inclusion map based on the ring inclusion R ⊇ R ≥ .Proof. We apply Proposition 2.12 to the ring inclusion f : R ≤ ✲ R and thechain complex D = Y . We obtain a stabilisation D ′ = Y ⊕ M k D ( k, R j k ) of D = Y and a bounded complex of finitely generated projective R ≤ -modules Z − = C ′ such that there is an isomorphism i : Z − ⊗ R ≤ R ✲ D ′ . Write Z inplace of D ′ , let ζ − : Z − ✲ Z be the R ≤ -linear map adjoint to i , and define Z + = Y + ⊕ M k D ( k, R j k ≥ ) ; with ζ + = υ + ⊕ inc the data satisfies all the requirements of the Lemma. (cid:3) A morphism g = ( g , g + ) in C + is called a cofibration if both g and g + are lev-elwise injections such that coker( g ) is an object of C + . We call g an h + -equivalence if g + is a quasi-isomorphism (and hence so is g ∼ = g + ⊗ R ≥ R ). Lemma 7.7.
With these definitions, both C + and D + are categories with cofibra-tions and weak equivalences.Proof. This is clear for C + since this category is equivalent to the category ofbounded chain complexes of finitely generated projective R ≥ -modules, via thefunctors (cid:0) Y ✛ Y + (cid:1) Y + and M + (cid:0) M + ⊗ R ≥ R ✛ M + (cid:1) . It remains to observe that the cokernel of a cofibration g in D + is automatically anobject of D + since, given g k : Y k ✲ Z k , we have the equality α (cid:0) [coker g k ] (cid:1) = α (cid:0) [ Z k ] (cid:1) − α (cid:0) [ Y k ] (cid:1) = 0 in the group K ( R ≤ ↓ R ) . (cid:3) We make the analogous symmetric definitions for C − and D − . The category C is defined to be the category of bounded chain complexes of finitely generatedprojective R -modules, with D the full subcategory of those objects Y such thatall chain modules Y k are stably induced from both R ≤ and R ≥ , that is, such that [ Y k ] ∈ im (cid:0) K R ≤ ✲ K R (cid:1) ∩ im (cid:0) K R ≥ ✲ K R (cid:1) for all k . Lemmas 7.6 and 7.7 are valid mutatis mutandis for these categories.
Lemma 7.8.
The forgetful functor (cid:0) Y ✛ Y + (cid:1) Y + induces isomorphisms on algebraic K -groups K q D + ∼ = ✲ K q R ≥ ( q > and an injection K D + ⊆ ✲ K R ≥ . — These statements hold mutatis mutandisfor the analogous maps K q D − ✲ K q R ≤ and K q D ✲ K q R .Proof. As remarked before, the functor
Y 7→ Y + establishes an equivalence of C + with the category of bounded chain complexes of finitely generated projective R ≥ -modules, and D + corresponds to the subcategory of complexes Y + such that Y + ⊗ R ≥ R consists of modules which are stably induced from R ≤ . Let D ′ and C ′ denote the full subcategories of complexes concentrated in chain level . Then themap of K -groups in question can be computed using Quillen ’s Q -construction,applied to the inclusion of exact categories D ′ ⊆ C ′ . The result is now an immediateconsequence of Grayson cofinality [Gra79, Theorem 1.1] since D ′ is closed underextension in C ′ ; indeed, an object Y ∈ C ′ is in D ′ if and only if α (cid:0) [ Y ] (cid:1) = 0 ∈ K ( R ≤ ↓ R ) , and K -theory is additive on short exact sequences. To see thatthe former category is cofinal in the latter it suffices to observe that every finitelygenerated projective module can be complemented to a finitely generated free one,which is automatically stably induced. (cid:3) The corners of the fundamental square.
We can now identify the corners ofthe fundamental square as the algebraic K -theory spaces of the categories D − , D and D + . Lemma 7.9.
The forgetful functor Φ + : Ch ♭ Vect( P ) ✲ D + , given by Y = (cid:0) Y − ✲ Y ✛ Y + (cid:1) Φ + ( Y ) = (cid:0) Y ✛ Y + (cid:1) , induces a homotopy equivalence on S • -constructions with respect to h + -equivalences: h + S • Ch ♭ Vect( P ) ≃ ✲ h + S • D + The statement holds mutatis mutandis for D and D − as well.Proof. By definition of h + -equivalences the forgetful functor respects and detectsweak equivalences. Let X = (cid:0) X − ξ − ✲ X ✛ ξ + X + (cid:1) be an object of Ch ♭ Vect( P ) , and let ←−Y = (cid:0) Y ✛ υ + Y + (cid:1) be an object of D + . Given a morphism g = ( g , g + ) : Φ + ( X ) ✲ ←−Y we construct the object Z ∈ Ch ♭ Vect( P ) associated with ←−Y as described inLemma 7.6; denote the composition of g with the inclusion ←−Y ✲ Φ + ( Z ) by h = ( h , h + ) . The composite map X − ξ − ✲ X h ✲ Z HE FUNDAMENTAL THEOREM FOR STRONGLY Z -GRADED RINGS 27 factors as X − h − ✲ Z − ⊗ R ≤ R ≤ q ⊆ ✲ Z − ⊗ R ≤ R ∼ = Z for sufficiently large q ≥ ; this is because X − is a bounded complex of finitelygenerated R ≤ -modules, and because Z ∼ = Z − ⊗ R ≤ R = Z − ⊗ R ≤ [ k ≥ R ≤ k = [ k ≥ Z − ⊗ R ≤ R k . This results in a map h = ( h − , h , h + ) : X ✲ Z ( q, in Ch ♭ Vect( P ) (we identify the canonically isomorphic complexes Φ + ( Z ) = Z + and Φ + (cid:0) Z ( q, (cid:1) = Z + ⊗ R ≥ R here). The projection map p : Φ + (cid:0) Z ( q, (cid:1) ✲ ←−Y is an h + -equivalence, and satisfies the condition p ◦ Φ + ( h ) = g . By Waldhausen ’sapproximation theorem [Wal85, Theorem 1.6.7], Φ + induces a homotopy equiva-lence on S • -constructions. (cid:3) Establishing the Mayer-Vietoris sequence
We will now establish the
Mayer - Vietoris sequence of Theorem 1.8. As beforelet R be a strongly Z -graded ring. The induction functors j −∗ = ( - ⊗ R ≤ R ) and j + ∗ = ( - ⊗ R ≥ R ) give rise to maps γ = j −∗ − j + ∗ : K q ( R ≤ ) ⊕ K q ( R ≥ ) ✲ K q ( R ) , for q ≥ .In view of Lemma 7.9, the homotopy cartesian fundamental square of the pro-jective line (7.1) yields a Mayer - Vietoris sequence of algebraic K -groups . . . γ ✲ K q +1 D δ ✲ K q P β =( β − ,β + ) ✲ K q D − ⊕ K q D + γ = γ − − γ + ✲ K q D δ ✲ . . . , for q ≥ , ending with a surjective homomorphism K D − ⊕ K D + ✲ K D . ByLemma 7.8 this sequence coincides, for q > , with the one of Theorem 1.8, so weonly need to look at its tail end. Using Lemma 7.8 again we construct the followingcommutative diagram: K ( R ) ✲ K ( P ) ✲ K ( R ≤ ) ⊕ K ( R ≥ ) ¯ γ ✲ K ( R ) ✲ R K − ( R ) ✲ K D = ✻ ✲ K ( P )= ✻ β ✲ K D − ⊕ K D + ⊆ ✻ γ ✲ K D ⊆ ✻ ✲ We know that the bottom row is exact, and argue that this remains true for thetop row. — Replacing the target of β by a larger group does not change ker( β ) , sothe top row is exact at K ( P ) . It is exact at K ( R ) and R K − ( R ) by definitionof the latter group. Thus it remains to verify exactness at the third entry.So let ( x, y ) ∈ ker ¯ γ be given. We can find finitely generated projective modules P and Q over R ≤ and R ≥ , respectively, and numbers p, q ≥ , such that x = [ P ] − [ R p ≤ ] ∈ K ( R ≤ ) and y = [ Q ] − [ R q ≥ ] ∈ K ( R ≥ ) . The condition ¯ γ ( x, y ) = 0 translates into the equality [ P ⊗ R ≤ R ] − [ R p ] = [ Q ⊗ R ≥ R ] − [ R q ] in K ( R ) . Consequently, there exists s ≥ such that (cid:0) P ⊗ R ≤ R (cid:1) ⊕ R q ⊕ R s ∼ = (cid:0) Q ⊗ R ≥ R (cid:1) ⊕ R p ⊕ R s . As the right-hand module is induced from R ≥ this shows that P ⊗ R ≤ R is stablyinduced from R ≥ so that the diagram P ✲ P ⊗ R ≤ R defines an object of D − . Since R p ≤ ✲ R p is an object of D − as well we concludethat x ∈ K D − . By a symmetric argument we can show y ∈ K D + . As γ ( x, y ) = 0 (the fourth vertical map is injective), and as the bottom row is exact, we know that ( x, y ) = β ( z ) for some z ∈ K ( P ) , which shows exactness of the top row. Thiscompletes the proof of Theorem 1.8. (cid:3) Proof of the fundamental theorem
The K -theory of the projective line revisited. Recall from Theorem 5.4 thatthere are isomorphisms of K -groups ˜ α = (cid:0) Ψ − , Ψ , (cid:1) : K q R ⊕ K q R ✲ K q P , (cid:0) [ P ] , [ Q ] (cid:1) (cid:2) Ψ − , ( P ) (cid:3) + (cid:2) Ψ , ( Q ) (cid:3) , where Ψ k,ℓ stands for the canonical sheaf functors (Definition 4.5). By pre-com-posing ˜ α with the invertible map (cid:0) − (cid:1) , we obtain modified isomorphisms α = (cid:0) − Ψ − , + Ψ , Ψ , (cid:1) : K q R ⊕ K q R ✲ K q P , (cid:0) [ P ] , [ Q ] (cid:1)
7→ − (cid:2) Ψ − , ( P ) (cid:3) + (cid:2) Ψ , ( P ) (cid:3) + (cid:2) Ψ , ( Q ) (cid:3) . (9.1)(This map cannot be confused with the map α from (2.3).) As we have a cylinderfunctor at our disposal, the additivity theorem implies that we can model the minussign by taking suitable “homotopy cofibres” of maps of functors. Concretely, theisomorphisms α are induced by the functor A : Ch ♭ P ( R ) × Ch ♭ P ( R ) ✲ Vect( P ) , ( C, D ) cone (cid:0) Ψ − , ( C ) ✲ Ψ , ( C ) (cid:1) ⊕ Ψ , ( D ) (9.2)and the ensuing homotopy equivalence of K -theory spaces cone(Ψ − , ✲ Ψ , ) + Ψ , : K ( R ) × K ( R ) ✲ K ( P ) , where “ + ” refers to the H -space structure given by direct sum. Alternatively, wecan use the functor A ′ : Ch ♭ P ( R ) × Ch ♭ P ( R ) ✲ Vect( P ) , ( C, D ) ΣΨ − , ( C ) ⊕ Ψ , ( C ) ⊕ Ψ , ( D ) ; the maps induced by A and A ′ are homotopic, by the additivity theorem.It will be convenient to have an explicit homotopy inverse for α . We record thefollowing fact: Lemma 9.3.
The functor
Ξ : Vect( P ) ✲ Ch ♭ P ( R ) × Ch ♭ P ( R ) , Y 7→ (cid:18) ΣΓ Y (1 , ⊕ Γ Y ⊕ Γ Y (1 , − Y (1 , − ⊕ Γ Y (1 , (cid:19) induces a homotopy inverse on the level of K -theory spaces; here Γ is the “globalsections” functor from Definition 4.6. HE FUNDAMENTAL THEOREM FOR STRONGLY Z -GRADED RINGS 29 Proof.
The functor A is known to induce a homotopy equivalence. Hence in viewof the above remarks on A ′ it is enough to show that the map induced by Ξ ◦ A ′ ishomotopic to the identity. Recalling the natural isomorphisms ΓΨ k,ℓ P ∼ = k M j = − ℓ s j P ⊗ R R j and (cid:0) Ψ k,ℓ P (cid:1) ( a, b ) ∼ = Ψ k + a,ℓ + b P , from Proposition 4.7, where s j P = P ⊗ R R j is the j th shift of P , one calculatesreadily that the first component of Ξ ◦ A ′ ( C, D ) is Σ Ψ , ( C ) ⊕ ΣΓΨ , ( C ) ⊕ ΣΓΨ , ( D ) ⊕ ΣΓΨ − , ( C ) ⊕ ΓΨ , ( C ) ⊕ Γ Y , ( D ) ⊕ ΣΓΨ , − ( C ) ⊕ ΓΨ , − ( C ) ⊕ ΓΨ , − ( D ) ∼ = Σ C ⊕ Σ C ⊕ Σ s C ⊕ Σ D ⊕ Σ s D ⊕ C ⊕ D ⊕ s C ⊕ s D , while the second component is Σ Ψ , − ( C ) ⊕ ΣΓΨ , − ( C ) ⊕ ΣΓΨ , − ( D ) ⊕ ΣΓΨ , ( C ) ⊕ ΓΨ , ( C ) ⊕ Γ Y , ( D ) ∼ = Σ s C ⊕ Σ s D ⊕ Σ C ⊕ C ⊕ s C ⊕ D ⊕ s D .
Recalling that suspension represents the H -space structure inverse on K -theoryspaces, this shows that the composition Ξ ◦ A ′ induces a map that is homotopic tothe identity as required. (cid:3) The modified Mayer-Vietoris sequence.
On a much more elementary level,we have automorphisms η = (cid:18) id − i −∗ p + ∗ (cid:19) : K q ( R ≤ ) ⊕ K q ( R ≥ ) ✲ K q ( R ≤ ) ⊕ K q ( R ≥ ) ( q ≥ with inverse given by η − = (cid:18) id i −∗ p + ∗ (cid:19) : K q ( R ≤ ) ⊕ K q ( R ≥ ) ✲ K q ( R ≤ ) ⊕ K q ( R ≥ ) . Both maps are induced by functors, with the minus sign modelled by suspension(making implicit use of the additivity theorem and the presence of a cylinder functoragain). Explicitly, η is induced by the functor Ch ♭ P ( R ≤ ) × Ch ♭ P ( R ≥ ) ✲ Ch ♭ P ( R ≤ ) × Ch ♭ P ( R ≥ ) , ( C, D ) (cid:0) C ⊕ Σ i −∗ p + ∗ D, D (cid:1) . We use the map η and its inverse to construct a modified Mayer - Vietoris se-quence . . . γη − ✲ K q +1 R α − δ ✲ K q R ⊕ K q R ηβα ✲ K q R ≤ ⊕ K q R ≥ γη − ✲ K q R α − δ ✲ . . . , (9.4) ending with the exact sequence K R ⊕ K R ηβα ✲ K R ≤ ⊕ K R ≥ γη − ✲ K R ✲ R K − R ✲ . Note that any exact sequence of the form A f ✲ B g ✲ C gives rise to a shortexact sequence ✲ A/ ker( f ) ✲ B ✲ im( g ) ✲ in this way, sequence (9.4) can be split into short exact sequences ✲ (cid:0) K q R ≤ ⊕ K q R ≥ ) (cid:14) ker( γη − ) γη − ✲ K q R α − δ ✲ im( α − δ ) ✲ q > and ✲ (cid:0) K R ≤ ⊕ K R ≥ ) (cid:14) ker( γη − ) γη − ✲ K R ✲ R K − R ✲ . From exactness of the modified
Mayer - Vietoris sequence (9.4) again we havethe equalities ker( γη − ) = im( ηβα ) and im( α − δ ) = ker( ηβα ) , so we obtain shortexact sequences ✲ coker( ηβα ) ✲ K q R ✲ ker( ηβα ) ✲ q > (9.5)and ✲ coker( ηβα ) ✲ K R ✲ R K − R ✲ . (9.6) The map ηβα . On the level of categories the effect of the map βα is to send ( P, Q ) ∈ Ch ♭ P ( R ) × Ch ♭ P ( R ) to ( C, D ) ∈ Ch ♭ P ( R ≤ ) × Ch ♭ P ( R ≥ ) where C = cone (cid:0) P ⊗ R R ≤− ⊆ ✲ P ⊗ R R ≤ (cid:1) ⊕ (cid:0) Q ⊗ R R ≤ (cid:1) = i −∗ cone( s − P ✲ P ) ⊕ i −∗ Q and D = cone (cid:0) P ⊗ R R ≥ ✲ P ⊗ R R ≥ (cid:1) ⊕ (cid:0) Q ⊗ R R ≥ (cid:1) ≃ Q ⊗ R R ≥ = i + ∗ Q ; here s − denotes the shift functor s − : Q Q ⊗ R R − which maps the category of finitely generated projective R -modules to itself. —To determine the effect of the map η on ( C, D ) , recall first that weakly equivalentfunctors induce homotopic maps on K -theory spaces. So we can, for example,replace D by i + ∗ Q . The first component of ηβα ( P, Q ) then is, up to homotopy, i −∗ cone( s − P ✲ P ) ⊕ i −∗ Q ⊕ Σ i −∗ p + ∗ i + ∗ Q .
Since p + ∗ i + ∗ Q = Q , and since suspension represents a homotopy inverse for directsum [Wal85, Proposition 1.6.2], we can thus model the effect of ηβα by the functor ( P, Q ) (cid:0) i −∗ cone( s − P ✲ P ) , i + ∗ Q (cid:1) . On the level of K -groups, this means that the effect of ηβα is described by thediagonal matrix ηβα = (cid:18) i −∗ (id − s − ) 00 i + ∗ (cid:19) : K q R ⊕ K q R ✲ K q R ≤ ⊕ K q R ≥ . (9.7) HE FUNDAMENTAL THEOREM FOR STRONGLY Z -GRADED RINGS 31 The kernel of ηβα . In view of (9.7), ker( ηβα ) = ker (cid:0) i −∗ ( s − − id) (cid:1) ⊕ ker( i −∗ ) .Now the maps i ∓∗ are split injective (with left inverse p ∓∗ ), thus ker( ηβα ) ∼ = ker(sd ∗ ) ⊕ { } = R ker K q R , (9.8)where sd ∗ denotes the shift difference map sd ∗ = id − s − : K q R ✲ K q R from (1.4) with kernel R ker K q R (Definition 1.3). The cokernel of ηβα . From the specific representation of ηβα in (9.7) we readoff that coker( ηβα ) = coker( i −∗ ◦ sd ∗ ) ⊕ coker( i + ∗ ) , the second summand being nothing but R NK + q R by definition. To identify thefirst summand we compose i −∗ ◦ sd ∗ with the splitting isomorphism (1.2): K q ( R ) sd ∗ ✲ K q ( R ) i −∗ ✲ K q ( R ≤ ) S =( c,p −∗ ) ∼ = ✲ R NK − q ⊕ K q ( R ) (Here c : K q ( R ≤ ) ✲ R NK − q is the canonical projection onto the cokernel of i −∗ .)Thus coker( i −∗ ◦ sd ∗ ) ∼ = coker( S ◦ i −∗ ◦ sd ∗ ) . Since c ◦ i −∗ = 0 , and since p −∗ ◦ i −∗ is the identity, we conclude that this group is isomorphic to the cokernel of thecomposition K q R ∗ ✲ K q R ⊆ ✲ R NK − q R ⊕ K q R , which is R NK − q ⊕ R coker K q R − . In total, this results in an isomorphism coker( ηβα ) ∼ = R NK − q R ⊕ R coker K q R ⊕ R NK + q R . (9.9)Putting the identifications (9.8) and (9.9) of the kernel and cokernel of ηβα intothe sequences (9.5) and (9.6) gives precisely the advertised Fundamental Theo-rem 1.7. (cid:3) The localisation sequence
In this section we will establish the long exact “localisation” sequence of Theo-rem 1.9. There are in fact two “mirror-symmetric” versions, which we state in fullfor completeness.
Theorem 10.1 (“Localisation sequence”) . Let R be a strongly Z -graded ring. Thereare long exact sequences of algebraic K -groups . . . ✲ K q +1 R ✲ R Nil + q ( R ) φ ✲ K q R ≥ ✲ K q R ✲ . . . ✲ R Nil +0 ( R ) φ ✲ K R ≥ ✲ K R (10.1a) and . . . ✲ K q +1 R ✲ R Nil − q ( R ) φ ✲ K q R ≤ ✲ K q R ✲ . . . ✲ R Nil − ( R ) φ ✲ K R ≤ ✲ K R (10.1b) with φ induced by the forgetful functor ( Z, ζ ) Z on the category R Nil ± ( R ) . Thegroups R Nil ± q ( R ) are as defined in (6.17a) and (6.17b) . Proof.
We will show that the sequence (10.1a) is exact; the argument for se-quence (10.1b) is similar.We apply
Waldhausen ’s fibration theorem to the right-hand vertical map inthe fundamental square (7.1). In view of Lemma 7.9 this results in exact sequencesof K -groups . . . ✲ K q +1 D + ✲ K q +1 D ✲ π q Ω | h + Vect( P ) h | k ′ ✲ K q D + ✲ K q D for q ≥ . In view of Lemma 7.8 and Theorem 7.3, this proves exactness of thesequence (10.1a) in the q ≥ range down to the term K R ≥ . Lemma 7.8 also statesthat in the commutative diagram in Fig. 2 the two vertical maps labelled f and gK R ≥ ✲ K R ✲ π Ω | h + Vect( P ) h | k ✲ K R ≥ j + ∗ ✲ K RK D + = ✻ ✲ K D = ✻ ✲ π Ω | h + Vect( P ) h | = ✻ k ′ ✲ K D + ⊆ f ✻ j ′ ✲ K D ⊆ g ✻ Figure 2.
Diagram used to establish the tail end of the localisa-tion sequenceare injective. As the bottom row is exact, the top row is automatically exact as wellexcept possibly at K R ≥ . Let x ∈ K R ≥ be an element with j + ∗ ( x ) = 0 . We canwrite x = [ P ] − [ Q ] , for finitely generated projective R ≥ -modules P and Q . Thecondition j + ∗ [ P ] − j + ∗ [ Q ] = j + ∗ ( x ) = 0 ∈ K R , that is, [ P ⊗ R ≥ R ] = [ Q ⊗ R ≥ R ] ∈ K R , means that there exists a number ℓ ≥ together with an isomorphism j + ∗ ( P ) ⊕ R ℓ = (cid:0) P ⊗ R ≥ R (cid:1) ⊕ R ℓ ∼ = (cid:0) Q ⊗ R ≥ R (cid:1) ⊕ R ℓ = j + ∗ ( Q ) ⊕ R ℓ . Let P ′ be a complement of P so that P ′ ⊕ P is a finitely generated free R ≥ -module.Then j + ∗ ( P ′ ⊕ P ) ⊕ R ℓ is a finitely generated free R -module, and j + ∗ ( P ′ ⊕ P ) ⊕ R ℓ = j + ∗ ( P ′ ) ⊕ j + ∗ ( P ) ⊕ R ℓ ∼ = j + ∗ ( P ′ ) ⊕ j + ∗ ( Q ) ⊕ R ℓ = j + ∗ ( P ′ ⊕ Q ) ⊕ R ℓ so that j + ∗ ( P ′ ⊕ Q ) ⊕ R ℓ is a finitely generated free R module as well. Consequently,both j + ∗ ( P ′ ⊕ P ) and j + ∗ ( P ′ ⊕ Q ) stably extend to R ≤ ; thus the two diagrams p = (cid:16) j + ∗ ( P ′ ⊕ P ) ✛ P ′ ⊕ P (cid:17) and q = (cid:16) j + ∗ ( P ′ ⊕ Q ) ✛ P ′ ⊕ Q (cid:17) are objects of D + . Let us consider the element z = [ p ] − [ q ] ∈ K D + . We calculate f ( z ) = f (cid:0) [ p ] − [ q ] (cid:1) = [ P ′ ⊕ P ] − [ P ′ ⊕ Q ] = [ P ] − [ Q ] = x (see Lemma 7.8 for the effect of f ). As x ∈ ker( j + ∗ ) , and as g is an injection, thisimplies z ∈ ker( j ′ ) , and by exactness of the lower horizontal sequence we infer that z = k ′ ( y ) for some y ∈ π Ω | h + Vect( P ) h | . Then k ( y ) = f k ′ ( y ) = f ( z ) = x so that x ∈ im k . This proves the top row to be exact at K R ≥ , and establishes togetherwith Theorem 7.3 the tail end of the long exact sequence.It remains to identify the map φ in the sequence (10.1a). To this end, let FD ( R ≥ ) denote the category of bounded complexes of R ≥ -modules which are HE FUNDAMENTAL THEOREM FOR STRONGLY Z -GRADED RINGS 33 quasi-isomorphic to a bounded complex of finitely generated projective R ≥ -mod-ules, and consist of projective R -modules. The inclusion functor Ch ♭ P ( R ≥ ) ✲ FD ( R ≥ ) is exact and yields isomorphisms on algebraic K -groups (with respect to weakequivalences the quasi-isomorphisms, and cofibrations the monomorphisms withlevelwise R -projective cokernel) as can be checked with the help of Waldhausen ’sapproximation theorem. This inclusion is the right hand vertical map in the squarediagram of Fig. 3. The top horizontal arrow maps the complex of vector bundles Z Ch ♭ Vect( P ) h ✲ Ch ♭ P ( R ≥ ) R Nil + ( R ) ❄ ✲ FD ( R ≥ ) ⊆ ❄ Figure 3.
Diagram used in proof of localisation sequenceto its component Z + . The left-hand vertical map is the one discussed in Theo-rem 7.3, identifying the group π q Ω | h + Vect( P ) h | with R Nil + q ( R ) ; it is given bythe assignment Z = (cid:16) Z − ✲ Z ✛ ζ + Z + (cid:17) ( Z + , ζ ) with ζ the composition Z + ⊗ R R ∼ = Z + ⊗ R ≥ R ≥ ✲ Z + ⊗ R ≥ R ≥ ∼ = Z + ⊗ R R .Note that Z + consists of projective R -modules since R is strongly Z -graded, that Z + is R -finitely dominated by Lemma 6.13, and that ζ is homotopy nilpotentby Corollary 6.12. Using the functor ( Z, ζ ) Z ζ in the lower horizontal posi-tion renders the square diagram commutative; the complex Z ζ is indeed an objectof FD ( R ≥ ) by Remark 6.15. The induced map on algebraic K -groups is the map φ occurring in the sequence (10.1a). — The proof is complete. (cid:3) The K -theory of homotopy nilpotent twisted endomorphisms The nil groups R NK ∓ R can be interpreted as obstruction groups for finitelygenerated projective modules over R ≤ or R ≥ , respectively, to be stably inducedfrom R (Proposition 2.4). For q > the groups R NK ± q R are isomorphic to the(reduced) algebraic K -groups of the category of nilpotent twisted endomorphisms,as will be shown in this section. This will also complete the proof of Theorem 1.9. Theorem 11.1 (Nil groups and K -theory of nilpotent endomorphisms) . For q > there are natural isomorphisms of algebraic K -groups R NK + q ( R ) ∼ = ✲ ker (cid:16) o −∗ : R Nil − q − R ✲ K q − R (cid:17) (11.1a) and R NK − q ( R ) ∼ = ✲ ker (cid:16) o + ∗ : R Nil + q − R ✲ K q − R (cid:17) , (11.1b) with the groups R Nil ∓ q − R from (6.17a) and (6.17b) , and maps o ∓∗ as introducedin (6.19) induced by the forgetful functors o ∓ : ( Z, ζ ) Z . Proof.
Upon taking the homotopy fibre of the top horizontal map in the funda-mental square (7.1) we obtain a long exact sequence of K -groups containing thesnippet K q P ✲ K q D + ∇ ✲ K q − Vect( P ) h + ι ✲ K q − P ✲ K q − D + , with ι induced by the inclusion functor Ch ♭ Vect( P ) h + ✲ Ch ♭ Vect( P ) ; thesymbol ∇ stands for a connecting homomorphism in the long exact sequence of ho-motopy groups associated with the fibration. — Using the isomorphism α : K q R ⊕ K q R ✲ K q P from (9.1), and with K k ( R ≥ ) in place of K k D + (Lemma 7.8),we obtain the exact sequence K q R ⊕ K q R ǫ ✲ K q R ≤ ∇ ✲ K q − Vect( P ) h + α − ι ✲ K q − R ⊕ K q − R ǫ ✲ K q − R ≤ . The map ǫ is the difference of maps induced by the functors ( P, Q ) (cid:0) P ⊗ R R ≥ (cid:1) ⊕ (cid:0) Q ⊗ R R ≥ (cid:1) and ( P, Q ) P ⊗ R R ≥ , thus ǫ is the usual split injection induced from i + : Q Q ⊗ R R ≥ on the secondsummand, and is the zero map on the first. Hence the image of the map α − ι , whichequals the kernel of ǫ , is K q − R ⊕ { } , and there results another exact sequence ✲ { } ⊕ K q R i + ∗ ✲ K q R ≥ ∇ ✲ K q − Vect( P ) h + pr α − ι ✲ K q − R ✲ . By Theorem 7.3 we have an isomorphism ω : K q − Vect( P ) h + ∼ = ✲ K q − R Nil − ( R ) . Claim:
The composition pr α − ιω − is inducedby the forgetful functor o − : ( Y, υ ) Y . (11.2)Assuming the claim for the moment, we obtain an exact sequence K q R i + ∗ ✲ K q R ≥ ω ∇ ✲ K q − R Nil − q − R o − ✲ K q − R , giving the isomorphism R NK + q = coker i + ∗ = K q R ≥ / ker( ω ∇ ) ∼ = im( ω ∇ ) = ker( o − ) . This establishes (11.1a). The isomorphism of (11.1b) is verified using a symmetricargument, employing the isomorphism K q R ⊕ K q R ✲ K q P , (cid:0) [ P ] , [ Q ] (cid:1) (cid:2) Ψ , ( P ) (cid:3) − (cid:2) Ψ , − ( P ) (cid:3) + (cid:2) Ψ , ( Q ) (cid:3) , (11.3)in place of α .It remains to verify Claim (11.2). By Lemma 9.2 the isomorphism α − is inducedby the functor Ξ : Vect( P ) ✲ Ch ♭ P ( R ) × Ch ♭ P ( R ) , Y 7→ (cid:18) ΣΓ Y (1 , ⊕ Γ Y ⊕ Γ Y (1 , − Y (1 , − ⊕ Γ Y (1 , . (cid:19) Now consider the diagram in Fig. 4. It is not a commutative diagram of functors,but on the level of K -theory spaces it is homotopy commutative. Since all verticalarrows marked as inclusion maps induce identity maps on K -groups, and since ω induces an isomorphism on K -groups, this establishes Claim (11.2). — Thevertical arrows a and b result in homotopy equivalences on K -theory spaces with HE FUNDAMENTAL THEOREM FOR STRONGLY Z -GRADED RINGS 35 Vect( P ) h + ι ✲ Vect( P ) pr Ξ ✲ Ch ♭ P ( R )Vect( P ) h + a ⊆ ❄ ι ✲ Vect( P ) b ⊆ ❄ R Nil − ( R ) ω ❄ o − ✲ FD ( R ) i ⊆ ❄ Figure 4.
Diagram used to establish Claim (11.2)respect to h -equivalences; this is analogous to Lemma 5.3), and the argumentsof Lemma III.1.3 and Corollary III.1.4 of [HM18] carry over verbatim. For thearrow i see (6.18). As to the alleged homotopy commutativity, recall that for Z ∈
Vect( P ) we have lim ← Z ( k, ℓ ) = H Z ( k, ℓ ) = 0 when k + ℓ ≥ , by definitionof the category Vect( P ) ; consequently, the map Γ Z ( k, ℓ ) ✲ holim Z ( k, ℓ ) is aquasi-isomorphism when k + ℓ ≥ , where “ holim ” stands for the homotopy limit orhomotopy pullback construction, which is the dual of the double mapping cylinder.Furthermore, for Z ∈
Vect( P ) h + we have Z ( k, ℓ ) + = Z + ⊗ R ≥ R ≥− ℓ ≃ ∗ and Z ( k, ℓ ) = Z ⊗ R R ≃ ∗ , which for k + ℓ ≥ implies Γ Z ( k, ℓ ) ≃ holim (cid:0) Z − ⊗ R ≤ R ≤ k ✲ Z ⊗ R R ✛ Z + ⊗ R ≥ R ≥− ℓ (cid:1) ≃ holim (cid:0) Z − ⊗ R ≤ R ≤ k ✲ ∗ ✛ ∗ (cid:1) ≃ Z − ⊗ R ≤ R ≤ k . Thus the composition i pr Ξ ι , that is, the functor that sends Y ∈
Vect( P ) h + to ΣΓ Y (1 , ⊕ Γ Y ⊕ Γ Y (1 , − , is weakly equivalent to the functor that sends Y to (cid:0) Σ Y − ⊗ R ≤ R ≤ (cid:1) ⊕ Y − ⊕ (cid:0) Y − ⊗ R ≤ R ≤ (cid:1) ; by the additivity theorem, the induced map on K -theory spaces is homotopic tothe map induced by Y 7→ Y − since suspension models an inverse for the H -spacestructure. But the formula Y 7→ Y − describes the effect of o − ◦ ω as well. Thisfinishes the proof. (cid:3) References [Bas68] Hyman Bass.
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Thomas Hüttemann, Queen’s University Belfast, School of Mathematics andPhysics, Mathematical Sciences Research Centre, Belfast BT7 1NN, UK
E-mail address : [email protected] URL : https://t-huettemann.github.io/ URL ::