Bounded G-theory with fibred control
aa r X i v : . [ m a t h . K T ] J un BOUNDED G -THEORY WITH FIBRED CONTROL GUNNAR CARLSSON AND BORIS GOLDFARB
Abstract.
We use filtered modules over a Noetherian ring and fibred boundedcontrol on homomorphisms to construct a new kind of controlled algebra withapplications in geometric topology. The resulting theory can be thought of asa “pushout” of bounded K -theory with fibred control and bounded G -theoryconstructed and used by the authors. Bounded G -theory was geared towardconstructing a G -theoretic version of assembly maps and proving the Novikovinjectivity conjecture for them. The G -theory with fibred control is needed inthe study of surjectivity of the assembly map. The relation between the K -and G -theories is the classical one: K -theory is meaningful, however G -theoryis easier to compute, and the relationship is expressed via a Cartan map. Thismap turns out to be an equivalence under very mild constraints in terms ofmetric geometry such as finite decomposition complexity. The fibred theory iscertainly more complicated than the absolute theory. This paper contains thenon-equivariant theory including fibred controlled excision theorems known tobe crucial for computations. Contents
1. Introduction 22. Elements of bounded G -theory 32.1. Basic definitions 32.2. Properties of filtered objects 62.3. Local finiteness property 92.4. Graded objects and their closure properties 103. Fibred bounded G -theory 133.1. Introduction of fibred control in G -theory 133.2. Properties of fibred objects 153.3. Fibrewise restriction 193.4. Fibrewise gradings 213.5. Localization fibration sequence 244. Fibrewise excision theorems 274.1. Waldhausen categories and K -theory 274.2. Controlled excision theorems 294.3. Fibred coarse coverings 324.4. Relative excision theorems 335. Conclusion 35References 35 Date : June 25, 2019. Introduction
The purpose of this paper is to use filtered modules over a Noetherian ring witha fibred bounded control on homomorphisms to construct a bounded G -theory withfibred control. This theory can be thought of as a “pushout” of the bounded K -theory with fibred control constructed by the authors in [8] and the controlled G -theory constructed in [6]. Here is a summary of this situation: K ( X, R ) / / (cid:15) (cid:15) K X ( Y ) (cid:15) (cid:15) G ( X, R ) / / G X ( Y ) Figure 1.
Bounded G -theory with fibred control as a “pushout”.Throughout this paper, metric spaces such as X and Y that appear in the squarewill be proper metric spaces in the sense that every closed bounded subspace is com-pact. The space or spectrum in the upper left corner represents the indispensablein modern geometric topology bounded K -theory of Pedersen and Weibel [14, 15],reviewed here in the beginning of section 2.1. This theory is defined for any ring ofcoefficients R . It is built out of free R -modules with generating sets parametrizedover the metric space X . This allows to impose geometric control conditions on thehomomorphisms f : F → G . The bounded control condition postulates that thereis number b ≥ F associated to somepoint x in X is spanned by basis elements in G that are referenced by points within b from x . We will review precise definitions shortly.The spectrum in the upper right corner K X ( Y ) is a generalization of this theoryto the situation when the modules are parametrized by the product of two metricspaces X and Y , and the control imposed on the homomorphisms is relaxed: it isessentially the bounded control across X but the bound is allowed to change in thecomplementary direction Y as one varies the X -coordinate. This theory becomesuseful when one considers “bundle phenomena”. For example, the space X can bethe universal cover of the tangent bundle of a manifold embedded in a Euclideanspace or even its discrete model such as the fundamental group with a word metric.The space Y can be the universal cover of the normal bundle to the embedding witha variety of useful metrics. This situation comes up the authors’ work in geometrictopology. The fibred K -theory is still defined for any coefficient ring R .To describe the bottom row in the square and for the rest of the paper, we restrictto Noetherian rings R .In place of parametrizations used to control homomorphisms between free mod-ules, one can use filtrations of arbitrary R -modules by subsets of the metric space X and impose control conditions in terms of the filtrations. This was done in [6] fora single space X . The result was the bounded G -theory spectrum G ( X, R ). The def-inition involved promoting the setting from the additive structure for free modulesin the definition of bounded K -theory to a specific non-split Quillen exact structureon a category of filtered R modules with morphisms satisfying control conditions -THEORY WITH FIBRED CONTROL 3 and the admissible morphisms satisfying further “bi-control” conditions. Regard-less of the significant change in techniques, literally every theorem about bounded K -theory has an exact (accidental pun) counterpart in G -theory.Now it is clear what the “pushout” G X ( Y ) is supposed to mean. We want to lookat the K -theory of a category built out of filtered modules over the product X × Y where the morphisms have the fibred control condition of the type described forfibred K -theory. This time we are interested in very specific excision results designedto deconstruct only the “fiber” direction. In our applications of this material we wantto perform what we call here relative excision in the normal bundle direction. This isgreatly facilitated by the hybrid conditions imposed on the objects themselves. Weinclude several remarks in the paper regarding the options and why our choices seemto be optimal. Long story short, we have resolved in this paper the problems thatmay be much harder to solve, if solvable at all, for the straightforward combinationof the theories in the corners of the diagram. We resolve them for a carefully craftedtheory that has all the desired properties and yet specializes to precisely G ( X, R )when localized near the subspace X × X × Y . Acknowledgement.
We would like to thank the referee for excellent commentsthat improved the narrative and the precision of the paper.2.
Elements of bounded G -theory Bounded G -theory defined in [6] is a variant of bounded K -theory of Pedersenand Weibel made applicable to more general, non-split exact structures. It wasdesigned by the authors for a different purpose than the one in this paper. The oldfocus was on the equivariant theory in addition to very basic excision that weresufficient for introducing an assembly map in G -theory for all finitely generatedgroups and proving the injectivity Novikov type theorem for a large class of groups.We will review and augment some material from [6] in the form best fit for thefibred theory.2.1. Basic definitions.
We start with a brief recollection of the bounded K -theorysetup. The coefficients R for this theory can be an arbitrary associative ring. The bounded category C ( X, R ) is the additive category of geometric R -modules whoseobjects are functions F : X → Free fg ( R ) which are locally finite assignments of freefinitely generated R -modules F x to points x of X . The local finiteness conditionrequires precisely that for any bounded subset S ⊂ X the restriction of F to X hasfinitely many nonzero values. Let d be the distance function in X . The morphismsin C ( M, R ) are the R -linear homomorphisms φ : M x ∈ X F x −→ M y ∈ X G y with the property that the components F x → G y are zero for d ( x, y ) > b for somefixed real number b = b ( φ ) ≥
0. The associated K -theory spectrum is denoted by K ( X, R ) and is called the bounded K-theory of X .2.1.1. Remark.
We would like to remind the reader that the original paper of Ped-ersen and Weibel [14] was already written in greater generality. If A is any additivecategory, it can be used as coefficients in this construction in place of finitely gener-ated free R -modules using the same formulas as above. The outcome is the boundedcategory C ( X, A ) which is again an additive category with the evident notion of GUNNAR CARLSSON AND BORIS GOLDFARB split exact sequences. It is now possible to iterate this construction: when thereare two metric spaces X and Y , Pedersen and Weibel built the additive category C ( X, C ( Y, R )). The objects of this category can be identified with the objects of C ( X × Y, R ) where the product is given a reasonable metric such as the max metric.The morphisms are, however, very different. They are R -homomorphisms whichare still controlled over X in the standard fashion but the non-zero components F x → G y are now allowed to vary in range with the X -coordinates of x and y . Incontrast, morphisms in C ( X × Y, R ) require one single number to work as a boundfor all of these components.Pedersen and Weibel were more interested in the product situation and, in fact,repaired the non-uniform boundedness properties of morphisms in C ( X, C ( Y, R ))by filtering the morphism sets. With that fix the category becomes isomorphic to C ( X × Y, R ). We, instead, embraced the flexibility of this construction in [8] withthe idea of exploiting the additional deformations in K ( X, C ( Y, R )), the
K-theorywith fibred control , that the construction allows. This is the spectrum that showsup in Figure 1 as K X ( Y ).2.1.2. Notation.
For a subset S ⊂ X and a real number r ≥ S [ r ] will stand forthe metric r -enlargement { x ∈ X | d ( x, S ) ≤ r } . In this notation, the metric ball ofradius r centered at x is { x } [ r ] or simply x [ r ].A variation of the basic construction of bounded K -theory is based on the fol-lowing observation. For every object F and a subset S there is a free R -module F ( S ) = L m ∈ S F m . In this context we say an element x ∈ F is supported on asubset S if x ∈ F ( S ). Now the restriction from arbitrary R -linear homomorphismsto the bounded ones can be described entirely in terms of these subobjects: φ iscontrolled as above precisely when there is a number b ≥ φF ( S ) ⊂ F ( S [ b ])for all choices of S .In the rest of this section and the rest of the paper, we will restrict to the caseof a Noetherian ring R .Let P ( X ) denote the power set of X partially ordered by inclusion and viewed asa category. If F is a left R -module, let I ( F ) denote the family of all R -submodulesof F partially ordered by inclusion.2.1.3. Definition. An X -filtered R -module is a module F together with a functor P ( X ) → I ( F ) from the power set of X to the family of R -submodules of F , bothordered by inclusion, such that the value on X is F . It will be most convenient tothink of F as the functor above and use notation F ( S ) for the value of the functoron S . We will call F reduced if F ( ∅ ) = 0.An R -homomorphism f : F → G of X -filtered modules is boundedly controlled if there is a fixed number b ≥ f ( F ( S )) is a submodule of G ( S [ b ]) for all subsets S of X .The objects of the category U ( X, R ) are the reduced X -filtered R -modules, andthe morphisms are the boundedly controlled homomorphisms.The category U ( X, R ) we constructed is clearly an additive category, but themore interesting structure for developing its K -theory is a certain Quillen exactstructure. For a good modern exposition of exact categories we refer to Keller [11];there is also a leisurely review of relevant basic theory in [6, section 2].Let us recall some standard terms. If a category has kernels and cokernels forall morphisms, it is called preabelian . If, in addition, the canonical map coim( f ) → -THEORY WITH FIBRED CONTROL 5 im( f ) for each morphism f is monic and epic but not necessarily invertible, we willsay the category is semi-abelian , cf. [16, pages 167-168] and [18]. It is abelian ifit is also balanced in the sense that the canonical map is an isomorphism. Recallalso that a category is called cocomplete if it contains colimits of arbitrary smalldiagrams, cf. Mac Lane [13, chapter V].2.1.4. Remark. If X is unbounded, U ( X, R ) is not a balanced category and there-fore not an abelian category. For an explicit description of a boundedly controlledmorphism in U ( Z , R ) which is an isomorphism of left R -modules but whose inverseis not boundedly controlled, we refer to [14, Example 1.5].It turns out that the kernels and cokernels in U ( X, R ) can be characterized usingan additional property a boundedly controlled morphism may or may not have.2.1.5.
Definition.
A morphism f : F → G in U ( X, R ) is called boundedly bicon-trolled if there exists a number b ≥ f ( F ( S )) ⊂ G ( S [ b ]) , there are inclusions f ( F ) ∩ G ( S ) ⊂ f F ( S [ b ])for all subsets S ⊂ X . In this case we will say that f has filtration degree b andwrite fil( f ) ≤ b .2.1.6. Definition.
We define the admissible monomorphisms in U ( X, R ) be theboundedly bicontrolled homomorphisms m : F → F such that the map F ( X ) → F ( X ) is a monomorphism. We define the admissible epimorphisms be the bound-edly bicontrolled homomorphisms e : F → F such that F ( X ) → F ( X ) is anepimorphism.Let the class E of exact sequences consist of the sequences F · : F ′ i −−→ F j −−→ F ′′ , where i is an admissible monomorphism, j is an admissible epimorphism, andim( i ) = ker( j ).There are numerous examples of semi-abelian categories that are classical orhave appeared recently in analysis and algebra that are listed in section 4 of [2] orin [16]. For us U ( X, R ) is of major interest.2.1.7.
Theorem. U ( X, R ) is a cocomplete semi-abelian category. The class of exactsequences E gives an exact structure on U ( X, R ) .Proof. This fact is contained in Proposition 2.6 and Theorem 2.13 of [6].We want to recall the explicit construction of kernels and cokernels U ( X, R ) forfuture reference. For a boundedly controlled morphism f : F → G , the kernel of f in Mod ( R ) has the X -filtration K where K ( S ) = ker( f ) ∩ F ( S ). This givesa kernel of f in U ( X, R ). Similarly, let I be the X -filtration of the image of f in Mod ( R ) by I ( S ) = im( f ) ∩ G ( S ). Let H ( X ) be the cokernel of f in Mod ( R ),which is the quotient G ( X ) /f F ( X ). Then there is a filtration H of H ( X ) defined by H ( S ) = im { G ( S ) /I ( S ) → H ( X ) } where the maps between quotients are inducedfrom the structure maps of G . The resulting epimorphism π : G ( X ) → H ( X ) givesa boundedly bicontrolled morphism of filtration 0. Lemma 2.5 of [6] verifies theuniversal properties of the cokernel H . (cid:3) GUNNAR CARLSSON AND BORIS GOLDFARB
Now suppose A is an arbitrary cocomplete semi-abelian category. All of thedefinitions and proofs presented so far can be interpreted verbatim to describe abounded category U ( X, A ) if instead of R -modules one uses objects from A . Weshould point out that the exposition in [6] is constructed in lesser generality with A assumed to be abelian. It is important for the fibred version to relax abelian tosemi-abelian. However, [6] can still serve as a good reference because throughoutthat paper only semi-abelian properties of A are used.In particular we have this conclusion.2.1.8. Theorem.
For any cocomplete semi-abelian category A , U ( X, A ) is a co-complete semi-abelian category. Of course, the category of R -modules Mod ( R ) is a cocomplete abelian categoryand so can serve as a basic example of cocomplete semi-abelian coefficients A . Inthis case U ( X, A ) is precisely U ( X, R ).2.2.
Properties of filtered objects.
In this section we will start imposing sev-eral conditions on objects in the bounded category U ( X, A ). These conditions areagnostic to the nature of A , and so we can use the simpler notation U ( X ) for thiscategory. These conditions and results about them will be used in the contexts ofboth module categories and abstract semi-abelian categories as coefficients.2.2.1. Definition.
Let F be an X -filtered R -module. • F is called lean or D - lean if there is a number D ≥ F ( S ) ⊂ X x ∈ S F ( x [ D ])for every subset S of X . • F is called split or D ′ - split if there is a number D ′ ≥ F ( S ) ⊂ F ( T [ D ′ ]) + F ( U [ D ′ ])whenever a subset S of X is written as a union T ∪ U . • F is called insular or d - insular if there is a number d ≥ F ( S ) ∩ F ( U ) ⊂ F ( S [ d ] ∩ U [ d ])for every pair of subsets S , U of X .2.2.2. Proposition.
The properties of being lean, split, and insular are preservedunder isomorphisms in U ( X ) . Also, a D -lean filtered module is D -split.Proof. If f : F → F is an isomorphism with fil( f ) ≤ b , and F is D -lean, D ′ -split,and d -insular, then F is ( D + b )-lean, ( D ′ + b )-split, and ( d + 2 b )-insular. For theother statement, we have F ( T ∪ U ) ⊂ X x ∈ T F ( x [ D ]) + X x ∈ U F ( x [ D ]) ⊂ F ( T [ D ]) + F ( U [ D ])since in general P x ∈ S F ( x [ D ]) ⊂ F ( S [ D ]). (cid:3) A collection of objects in an exact category is said to be closed under extensions if the middle term of an exact sequence belongs to the collection in case both ofthe extreme terms belong to the collection. -THEORY WITH FIBRED CONTROL 7
Lemma. (1)
Lean objects are closed under extensions. (2)
Insular objects are closed under extensions. (3)
Split objects are closed under extensions.Proof.
For an exact sequence E ′ f −→ E g −→ E ′′ in U ( X ), let b ≥ f and g . The first two statements follow from parts (1) and (2)of [6, Proposition 2.18]. It is shown there that if E ′ and E ′′ are D -lean then E is(4 b + D )-lean. Also, if both E ′ and E ′′ are d -insular then E is (4 b + 2 d )-insular.To prove (3), suppose both E ′ and E ′′ are D ′ -split. We have gE ( T ∪ U ) ⊂ E ′′ ( T [ b ] ∪ U [ b ]) , because in general ( T ∪ U )[ b ] ⊂ T [ b ] ∪ U [ b ]. So g ( T ∪ U ) ⊂ E ′′ ( T [ b + D ′ ]) + E ′′ ( U [ b + D ′ ]) ⊂ gE ( T [2 b + D ′ ]) + gE ( U [2 b + D ′ ]) . If z ∈ E ( T ∪ U ) then we can write g ( z ) = g ( z )+ g ( z ) where z ∈ E ( T [2 b + D ′ ]) and z ∈ E ( U [2 b + D ′ ]). Since z − z − z is an element of ker( g ) ∩ E ( T [2 b + D ′ ] ∪ U [2 b + D ′ ]),we have an element k ∈ E ′ ( T [3 b + D ′ ] ∪ U [3 b + D ′ ]) ⊂ E ′ ( T [3 b + 2 D ′ ]) + E ′ ( U [3 b + 2 D ′ ])such that z = f ( k ) + z + z ∈ E ( T [4 b + 2 D ′ ]) + E ( U [4 b + 2 D ′ ]) . So E is (4 b + 2 D ′ )-split. (cid:3) Lemma.
Let E ′ f −−→ E g −−→ E ′′ be an exact sequence in U ( X ) . (1) If the object E is lean then E ′′ is lean. (2) If E is split then E ′′ is split. (3) If E is insular then E ′ is insular. (4) If E is insular and E ′ is lean then E ′′ is insular. (5) If E is insular and E ′ is split then E ′′ is insular. (6) If E is split and E ′′ is insular then E ′ is split.Proof. Let b ≥ f and g . If E is D -lean, D ′ -split,or d -insular, it is easy to show that E ′′ is ( D + 2 b )-lean or ( D ′ + 2 b )-split and E ′ is( d + 2 b )-insular respectively, which verifies (1), (2), and (3).Statement (4) follows from the proof of part (3c) of [6, Proposition 2.18]. It isshown there that if E ′ is D -lean and E is d -insular then E ′′ is (4 b + D + d )-insular.The same proof actually shows statement (5). The only equation in that proof thatuses D -leanness of E ′ is only used to get a consequence that is in fact immediatefrom the assumption that E ′ is D -split.(6) Suppose E is D ′ -split and E ′′ is d -insular. Given z ∈ E ′ ( T ∪ U ), we have f ( z ) ∈ E ( T [ b ] ∪ U [ b ]). Now f ( z ) ∈ E ( T [ b + D ′ ]) + E ( U [ b + D ′ ]), so we can write GUNNAR CARLSSON AND BORIS GOLDFARB accordingly f ( z ) = y + y . Now f ( z ) ∈ ker( g ), because g ( y ) + g ( y ) = 0. Since E ′′ is d -insular, g ( y ) = − g ( y ) ∈ E ′′ ( T [2 b + D ′ + d ] ∩ U [2 b + D ′ + d ]) , so we are able to find y ∈ E ( T [3 b + D ′ + d ] ∩ U [3 b + D ′ + d ])such that g ( y ) = g ( y ) = − g ( y ), because generally ( S ∩ P )[ b ] ⊂ S [ b ] ∩ P [ b ]. Thus f ( z ) = y + y = ( y − y ) + ( y + y )and y − y ∈ E ( T [3 b + D ′ + d ]) , y + y ∈ E ( U [3 b + D ′ + d ]) . Let z = f − ( y − y ) and z = f − ( y + y ), and we have z = z + z such that z ∈ E ′ ( T [4 b + D ′ + d ]) , z ∈ E ′ ( U [4 b + D ′ + d ]) , so E ′ is (4 b + D ′ + d )-split. (cid:3) Corollary.
Let E ′ f −→ E g −→ E ′′ be an exact sequence in U ( X ) . If E is splitand insular then E ′′ is insular if and only if E ′ is split.Proof. This fact is the combination of parts (5) and (6) of the Lemma. (cid:3)
Remark.
The last Corollary is in contrast with the absence of the analogousgeneral fact if one substitutes the lean property for the split property. However,the analogue is true in the presence of certain geometric assumptions on the metricspace. For example, suppose X has finite asymptotic dimension. Then from themain theorem of [5], we have the following counterpart to part (6) of the Lemma:if E is lean and E ′′ is insular then E ′ is lean. This fact is not needed in thispaper. Here, the excision properties of the theory rely only on the properties of thecokernels. For the applications in [7], properties of the kernels become crucial indealing with coherence issues, and the geometric conditions need to be imposed.2.2.7. Definition.
We define L ( X ) as the full subcategory of U ( X ) on objects thatare lean and insular with the induced exact structure. Similarly, S ( X ) is the fullsubcategory of U ( X ) on objects that are split and insular.Exact structures in L ( X ) and S ( X ) can be induced from U ( X ). A full subcate-gory H of an exact category C is said to be closed under extensions or thick in C if (1) H contains the zero object, and(2) for any exact sequence C ′ → C → C ′′ in C , if C ′ and C ′′ are isomorphic toobjects from H then so is C .It is known (cf. [2, Lemma 10.20]) that a subcategory closed under extensions in C inherits the exact structure from C .2.2.8. Theorem. L ( X ) and S ( X ) are closed under extensions in U ( X ) . Therefore, L ( X ) and S ( X ) are exact subcategories of U ( X ) , so we have a sequence of exactinclusions L ( X ) −→ S ( X ) −→ U ( X ) . Proof.
The first fact follows from parts (1) and (2) of Lemma 2.2.3, the second from(2) and (3). (cid:3) -THEORY WITH FIBRED CONTROL 9
Local finiteness property.
Finally, there is an additional property that willconsider only in module categories.2.3.1.
Definition. An X -filtered R -module F is locally finitely generated if F ( S ) isa finitely generated R -module for every bounded subset S ⊂ X .The category BL ( X, R ) is the full subcategory of L ( X, R ) on the locally finitelygenerated objects. Similarly, the companion category BS ( X, R ) is the full subcate-gory of S ( X, R ) on the locally finitely generated objects.2.3.2.
Theorem.
The category BL ( X, R ) is closed under extensions in L ( X, R ) .Similarly, the category BS ( X, R ) is closed under extensions in S ( X, R ) .Proof. If f : F → G is an isomorphism with fil( f ) ≤ b and G is locally finitelygenerated, then F ( U ) are finitely generated submodules of G ( U [ b ]) for all bounded U , since R is a Noetherian ring. Suppose F ′ f −−→ F g −−→ F ′′ is an exact sequence and let b ≥ f and g . Assume that F ′ and F ′′ are locally finitely generated. For every bounded subset U ⊂ X the restriction g : F ( U ) → gF ( U ) is an epimorphism onto a submodule ofthe finitely generated R -module F ′′ ( U [ b ]). The kernel of g | F ( U ) is a submodule of F ′ ( U [ b ]), which is also finitely generated. So the extension F ( U ) is finitely generated. (cid:3) Corollary. BL ( X, R ) and BS ( X, R ) are exact categories. The additive cat-egory C ( X, R ) of geometric R -modules with the split exact structure is an exactsubcategory of BL ( X, R ) , so there is a sequence of exact inclusions C ( X, R ) −→ BL ( X, R ) −→ BS ( X, R ) −→ U ( X, R ) . Remark.
We want to briefly explain the roles played by the two condi-tions, lean and split, that distinguish the two categories BL ( X, R ) and BS ( X, R ). BL ( X, R ) was used exclusively in [6], where it was proven to have good excisionproperties. There is a separate important issue of homological coherence that stillrequires the lean condition for its resolution, cf. [5, 7]. The setting with the splitcondition in BS ( X, R ) is much more streamlined for the excision arguments buthas insufficient coherence properties. In the next section, we will pursue the goalof combining the two different conditions in the “base” and “fibre” in order toachieve required coherence in the base and “fibrewise” excision properties. The hy-brid lean/split condition will provide a considerable advantage because the fibredsetting is more complicated than the absolute case.Recall that a morphism e : F → F is an idempotent if e = e . Categories inwhich every idempotent is the projection onto a direct summand of F are called idempotent complete .2.3.5. Proposition. BL ( X, R ) and BS ( X, R ) are idempotent complete.Proof. First note that a regular preabelian category is idempotent complete. Theproof is exactly the same as for an abelian category: if e is an idempotent then itskernel is split by 1 − e . Since the restriction of an idempotent e to the image of e is the identity, every idempotent here is boundedly bicontrolled of filtration 0. Itfollows easily that the splitting of e in Mod ( R ) is in fact a splitting in BL ( X, R )or BS ( X, R ). (cid:3) Finally, we need to address (the lack of) inheritance in filtered modules. It isimmediate that a submodule of an insular filtered module is also insular with respectto the standard filtration induced on the submodule. However, a submodule of alean filtered module is not necessarily lean.2.3.6.
Definition. An X -filtered object F is called strict if there exists an orderpreserving function ℓ : P ( X ) → [0 , + ∞ ) such that for every S ⊂ X the submodule F ( S ) is ℓ S -lean and ℓ S -insular with respect to the standard X -filtration F ( S )( T ) = F ( S ) ∩ F ( T ).It is important to note that this property is not preserved under isomorphisms,so the subcategory of strict objects is not essentially full in BL ( X, R ).The bounded category B ( X, R ) was defined in [6] as the full subcategory of BL ( X, R ) on objects isomorphic to strict objects. Now this category is closed underexact extensions in U ( X, R ) according to [6, Theorem 2.22] and so is an exactcategory.A consequence of strictness, or more generally being isomorphic to a strict object,is the following feature. Given a filtered module F in B ( X, R ), a lean grading of F is a functor e F : P ( X ) → I ( F ) from the power set of X to the submodules of F such that(1) each e F ( S ) is an object of BL ( X, R ) when given the standard filtration,(2) there is a number K ≥ F ( S ) ⊂ e F ( S ) ⊂ F ( S [ K ])for all subsets S of X .Clearly, each e F ( S ) is an object of B ( X, R ). Also an actual strict object has a leangrading by e F ( S ) = F ( S ) with K = 0.We note for the interested reader that the theory in [6], including the excisiontheorems, could be alternately developed for modules with lean gradings in place of B ( X, R ). We do not require such theory in this paper. Instead, we develop a similarbut more relaxed notion of gradings in BS ( X, R ).2.4.
Graded objects and their closure properties.
Definition.
Given a filtered module F in BS ( X, R ), a grading of F is afunctor F : P ( X ) → I ( F ) such that(1) each F ( S ) is an object of BS ( X, R ) when given the standard filtration,(2) there is a number K ≥ F ( S ) ⊂ F ( S ) ⊂ F ( S [ K ])for all subsets S of X .We will say that a filtered module F is graded if it is possible to equip it with agrading, but there is no specific choice of grading that is specified.2.4.2. Proposition.
The graded objects are closed under isomorphisms.Proof. If f : F → F ′ is an isomorphism and F has a grading F , a grading for F ′ isgiven by F ′ ( C ) = f F ( C [ K + b ]), where b is a filtration bound for f . (cid:3) Definition.
We define G ( X, R ) as the full subcategory of BS ( X, R ) on thelocally finitely generated graded filtered modules. -THEORY WITH FIBRED CONTROL 11
Proposition. G ( X, R ) is closed under extensions in BS ( X, R ) . Therefore G ( X, R ) is an exact subcategory of BS ( X, R ) .Proof. Given an exact sequence F f −→ G g −→ H in BS ( X, R ), let b ≥ f and g as boundedly bicontrolled maps, and assume that F and H are graded modules in G ( X, R ) with the associated functors F and H .To define a grading for G , consider a subset S and suppose H ( S [ b ]) is D -split and d -insular. The induced epimorphism g : G ( S [2 b ]) ∩ g − H ( S [ b ]) → H ( S [ b ]) extendsto another epimorphism g ′ : f F ( S [3 b ]) + G ( S [2 b ]) ∩ g − H ( S [ b ]) −→ H ( S [ b ])with ker( g ′ ) = F ( S [3 b ]). Without loss of generality, suppose F ( S [3 b ]) is D -split and d -insular. We define G ( S ) = f F ( S [3 b ]) + G ( S [2 b ]) ∩ g − H ( S [ b ]) . From parts (2) and (3) of Lemma 2.2.3, the module G ( S ) with the standard filtrationis (4 b + 2 d )-split and (4 b + 2 d )-insular. Since G ( S ) ⊂ g − H ( S [ b ]), we have G ( S ) ⊂G ( S ). On the other hand, if the grading F has characteristic number K ≥ G ( S ) ⊂ G ( S [4 b + K ]). The last fact together with Theorem 2.3.2 shows that G ( S )is finitely generated. (cid:3) The relations between the categories in this section can be summarized as acommutative diagram of fully exact inclusions BL ( X, R ) / / BS ( X, R ) C ( X, R ) , , B ( X, R ) / / O O G ( X, R ) O O The advantage of working with the category G ( X, R ) is that one can readilylocalize to geometrically defined subobjects.2.4.5.
Lemma.
Suppose G is a graded X -filtered module with a grading G . Let F be a submodule which is split with respect to the standard filtration. Then F ( S ) = F ∩ G ( S ) is a grading of F . We will call the grading of a split submodule F obtained in Lemma 2.4.5 the standard grading of the submodule. Proof.
Of course, F ( S ) = F ∩ G ( S ) ⊂ F ∩ G ( S ) = F ( S ). On the other hand, thereis d ≥ G ( S ) ⊂ G ( S [ d ]), so F ( S ) ⊂ F ∩ G ( S [ d ]) = F ( S [ d ]).Consider the inclusion of modules i : F → G , and take the quotient q : G → H .Both F and G are split and insular, so H is split and insular by parts (2) and(4) of Lemma 2.2.4, with respect to the quotient filtration. We define H ( S ) as thepartial image q G ( S ) and give H ( S ) the standard filtration in H . Then H ( S ) is splitas the image of a split G ( S ) and insular since H is insular. Now the kernel of theepimorphism q | : G ( S ) → H ( S ), which is F ∩ G ( S ), is split by part (6) of Lemma2.2.4. Since F is insular, F ( S ) is also insular. This shows that F ( S ) gives a gradingfor F . (cid:3) This can be promoted to the following result.
Proposition.
Given a boundedly bicontrolled epimorphism g : G → H in BS ( X, R ) , suppose F is a submodule of G which is the kernel of g in Mod ( R ) . Itis given the standard filtration. If G is graded and F is split then both H and F aregraded.Proof. The grading for H is given by H ( S ) = g G ( S [ b ]), where b is a chosen bicontrolbound for g . Each H ( S ) is split and insular as in the proof of Lemma 2.4.5. Theinclusions H ( S ) ⊂ gG ( S [ b ]) ⊂ g G ( S [ b ]) = H ( S ) and g G ( S [ b ]) ⊂ gG ( S [ b + K ]) ⊂ H ( S [2 b + K ]) show that H is a grading. The same argument as in Lemma 2.4.5shows that F ( S ) = F ∩ G ( S [ b ]) gives a grading for F . (cid:3) Applying this fact, we are able to characterize admissible monomorphisms in G ( X, R ) as follows.2.4.7.
Proposition.
The inclusion of a subobject i : F → G in G ( X, R ) is anadmissible monomorphism if and only if F is split.Proof. The cokernel H of i in U ( X, R ) has the filtration described in the proof ofTheorem 2.1.7. From parts (2) and (5) of Lemma 2.2.4, H is split and insular. Infact, H is a cokernel of i in BS ( X, R ). From Proposition 2.4.6, H is graded, so itis also a cokernel of i in G ( X, R ). (cid:3) We will use the following convention. When d ≤
0, the notation S [ d ] will standfor the subset S \ ( X \ S )[ − d ].2.4.8. Corollary.
Given an object F in G ( X, R ) and a subset S of X , there isa number K ≥ and an admissible subobject i : F S → F in G ( X, R ) with theproperty that F S ⊂ F ( S [ K ]) . Moreover, the cokernel q : F → H has the propertythat H ( X ) = H (( X \ S )[2 D ′ ]) , where D ′ is a splitting constant for F .Proof. The object F has a grading F with a characteristic constant K . Property(1) in Definition 2.4.1 guarantees that F ( S ) is a split object for any subset S . Forthe first statement, choose F S = F ( S ) with the grading defined in Lemma 2.4.5and apply Proposition 2.4.7. The second statement is shown as follows. By part (2)of Lemma 2.2.4, since fil( q ) = 0, if F is D ′ -split then H is D ′ -split. Let T = S [ − D ′ ],then T [ D ′ ] ⊂ S , so H ( T [ D ′ ]) = qF ( T [ D ′ ]) ⊂ qF ( S ) ⊂ qF S = 0 . Using the decomposition X = T ∪ ( X \ T ) we can write H ( X ) = H ( T [ D ′ ]) + H (( X \ T )[ D ′ ]) = H (( X \ T )[ D ′ ]) = H (( X \ S )[2 D ′ ]) . (cid:3) The last three results can be summarized as follows.2.4.9.
Theorem.
Given a graded object F in G ( X, R ) and a subset S of X , weassume that F is D ′ -split and d -insular and is graded by F . The submodules F ( S ) have the following properties: (1) each F ( S ) is graded by F S ( T ) = F ( S ) ∩ F ( T ) ; (2) F ( S ) ⊂ F ( S ) ⊂ F ( S [ K ]) for some fixed number K ≥ ; (3) suppose q : F → H is the cokernel of the inclusion i : F ( S ) → F , then H issupported on ( X \ S )[2 D ′ ] ; (4) H ( S [ − D ′ − d ]) = 0 . -THEORY WITH FIBRED CONTROL 13 Proof.
Properties (1), (2), (3) are consequences of the last four results. (4) followsfrom the fact that a d -insular filtered module is 2 d -separated, in the sense thatfor any pair of subsets S and T such that S [2 d ] ∩ T = ∅ we have S [ d ] ∩ T [ d ] = ∅ so F ( S ) ∩ F ( T ) = 0. Now H ( S [ − D ′ − d ]) ∩ H (( X \ S )[2 D ′ ]) = 0, but H (( X \ S )[2 D ′ ]) = H ( X ), thus H ( S [ − D ′ − d ]) = 0. (cid:3) Remark.
Functoriality properties in controlled theories are well-understood.As expected, bounded G -theory is covariantly functorial in both variables in thesense that G ( X, R ) is a covariant functor from the category of proper metric spacesand uniformly expansive maps to exact categories when R is fixed and is similarlya covariant functor from Noetherian rings to exact categories when X is fixed.These details become important in the construction of the equivariant theory andin specific applications. We avoid questions of functoriality in this paper as weconcentrate on computational tools such as excision.3. Fibred bounded G -theory Introduction of fibred control in G -theory. Suppose X and Y are twoproper metric spaces and R is a Noetherian ring. The product X × Y is given theproduct metric d (( x, y ) , ( x ′ , y ′ )) = max { d ( x, x ′ ) , d ( y, y ′ ) } . There is certainly thesemi-abelian category U ( X × Y, R ), the exact category L ( X × Y, R ) and, further,the bounded category BL ( X × Y, R ).We wish to construct a larger fibred bounded category B X ( Y ). The result willinvolve a mix of features from BL ( X, A ) and BS ( Y, R ) and contain BL ( X × Y, R )as an exact subcategory.One definition can be made by simply imitating K -theory with fibred controlas described in Remark 2.1.1. It is obtained as an iterated construction from theend of section 2.1 on the level of unrestricted category as U ( X, U ( Y, R )). FromTheorem 2.1.8, U ( X, U ( Y, R )) is a complete semi-abelian category with completesemi-abelian coefficients A = U ( Y, R ). This concept has a transparent definition butis not well-suited for a fibred theory. In particular, while it does contain C ( X × Y, R ),it does not contain the fibred bounded category C ( X, C ( Y, R )) as a subcategory. Nowwe proceed to develop a different, more explicit set-up in terms of a category U X ( Y )that naturally contains U ( X × Y, R ) and C ( X, C ( Y, R )) as exact subcategories.3.1.1.
Definition.
Given an R -module F , an ( X, Y ) -filtration of F is a functor φ F : P ( X × Y ) → I ( F ) from the power set of the product metric space to thepartially ordered family of R -submodules of F ( X × Y ). Whenever F is given afiltration, and there is no ambiguity, we will denote the values φ F ( U ) by F ( U ). Weassume that F is reduced in the sense that the value on the empty subset is 0.The associated X -filtered R -module F X is given by F X ( S ) = F ( S × Y ) . Similarly, for each subset S ⊂ X , one has the Y -filtered R -module F S given by F S ( T ) = F ( S × T ) . In particular, F X ( T ) = F ( X × T ).We will use the following notation generalizing enlargements in a metric space.3.1.2. Notation.
Given a subset U of X × Y and a function k : X → [0 , + ∞ ), let U [ k ] = { ( x, y ) ∈ X × Y | there is ( x, y ′ ) ∈ U with d ( y, y ′ ) ≤ k ( x ) } . If in addition we are given a number K ≥ U [ K, k ] = { ( x, y ) ∈ X × Y | there is ( x ′ , y ) ∈ U [ k ] with d ( x, x ′ ) ≤ K } . So U [ k ] = U [0 , k ]. Notice that if U is a single point ( x, y ) then U [ K, k ] = x [ K ] × y [ k ( x )] = ( x, y )[ K, × ( x, y )[0 , k ( x )] . More generally, one can equivalently write U [ K, k ] = [ ( x,y ) ∈ U x [ K ] × y [ k ( x )] . If U is a product set S × T , it will be convenient to use the notation ( S, T )[ K, k ]in place of ( S × T )[ K, k ]. More generally, because the roles of the factors are verydifferent when working with (
X, Y )-filtrations, we will use the notation (
X, Y ) forthe product metric space so that the order of the factors is unambiguous. Similarly,we will use the notation (
S, T ) for the product subset S × T in ( X, Y ).3.1.3.
Definition.
We will refer to the pair (
K, k ) in the notation U [ K, k ] as the enlargement data .It is clear that when Y = pt, U [ K, k ] = U [ K ] for any function k under theidentification X × Y = X .3.1.4. Notation.
Let x be a chosen fixed point in X . Given a monotone function h : [0 , + ∞ ) → [0 , + ∞ ), there is a function h x : X → [0 , + ∞ ) defined by h x ( x ) = h ( d X ( x , x )) . Definition.
Given two (
X, Y )-filtered modules F and G , an R -homomorphism f : F ( X × Y ) → G ( X × Y ) is boundedly controlled if there are a number b ≥ θ : [0 , + ∞ ) → [0 , + ∞ ) such that( † ) f F ( U ) ⊂ G ( U [ b, θ x ])for all subsets U ⊂ X × Y and some choice of x ∈ X . It is clear that the conditionis independent of the choice of x .The unrestricted fibred bounded category U X ( Y ) has ( X, Y )-filtered modules asobjects and the boundedly controlled homomorphisms as morphisms.3.1.6.
Theorem. U X ( Y ) is a cocomplete semi-abelian category. First we require a very useful fact.A morphism f : F → G in U X ( Y ) is boundedly bicontrolled if there is filtrationdata b ≤ θ : [0 , + ∞ ) → [0 , + ∞ ) as in Definition 3.1.5, and in addition to ( † )one also has the containments f F ∩ G ( U ) ⊂ f F ( U [ b, θ x ]). In this case, we will usethe notation fil( f ) ≤ ( b, θ ).3.1.7. Lemma.
Let f : F → G , f : G → H be in U X ( Y ) and f = f f . (1) If f , f are boundedly bicontrolled morphisms and either f : F ( X × Y ) → G ( X × Y ) is an epi or f : G ( X × Y ) → H ( X × Y ) is a monic, then f isalso boundedly bicontrolled. (2) If f , f are boundedly bicontrolled and f is epic then f is also boundedlybicontrolled; if f is only boundedly controlled then f is also boundedlycontrolled. -THEORY WITH FIBRED CONTROL 15 (3) If f , f are boundedly bicontrolled and f is monic then f is also boundedlybicontrolled; if f is only boundedly controlled then f is also boundedlycontrolled.Proof. Suppose fil( f i ) ≤ ( b, θ ) and fil( f j ) ≤ ( b ′ , θ ′ ) for { i, j } ⊂ { , , } , then infact fil( f − i − j ) ≤ ( b + b ′ , θ + θ ′ ) in each of the three cases. For example, there arefactorizations f G ( U ) ⊂ f f F ( U [ b, θ x ]) = f F ( U [ b, θ x ]) ⊂ H ( U [ b + b ′ , θ x + θ ′ x ]) f G ( X ) ∩ H ( U ) ⊂ f F ( U [ b ′ , θ ′ x ]) = f f F ( U [ b ′ , θ ′ x ]) ⊂ f G ( U [ b + b ′ , θ x + θ ′ x ])which verify part 2 with i = 1, j = 3. (cid:3) Proof of Theorem . The additive properties are inherited from
Mod ( R ), sothe biproduct is given by the filtration-wise operation ( F ⊕ G )( U ) = F ( U ) ⊕ G ( U )in Mod ( R ). For any boundedly controlled morphism f : F → G , the kernel of f in Mod ( R ) has the standard ( X, Y )-filtration K where K ( S ) = ker( f ) ∩ F ( S ) whichgives the kernel of f in U X ( Y ). The canonical monic κ : K → F has filtration data(0 ,
0) and is therefore boundedly bicontrolled. It follows from part 3 of Lemma 3.1.7that K has the universal properties of the kernel in U X ( Y ).Similarly, let I be the standard ( X, Y )-filtration of the image of f in Mod ( R )by I ( U ) = im( f ) ∩ G ( U ) . Then there is a presheaf C over ( X, Y ) with C ( U ) = G ( U ) /I ( U ) for U ⊂ ( X, Y ). Of course C ( X × Y ) is the cokernel of f in Mod ( R ).Consider an ( X, Y )-filtered object C associated to C given by C ( U ) = im C ( U, X × Y ) . The canonical morphism π : G ( X × Y ) → C ( X × Y ) gives a boundedly bicon-trolled morphism π : G → C of filtration (0 ,
0) sinceim( πG ( U, X × Y )) = im C ( U, X × Y ) = C ( U ) . This in conjunction with part 2 of Lemma 3.1.7 also verifies the universal cokernelproperties of C and π in U X ( Y ). (cid:3) We mention one useful perspective on U X ( Y ).3.1.8. Proposition.
Suppose f : F → G in U X ( Y ) is boundedly controlled withcontrol data ( b, θ ) . Then (1) f is bounded by b when viewed as a morphism F X → G X in U ( X, R ) , and (2) for each bounded subset S ⊂ X , the restriction f | : F X ( S ) → G X ( S [ b ]) isbounded when viewed as a morphism F S → G S [ b ] of Y -filtered modules in U ( Y ) .Proof. If f : F → G is ( b, θ )-controlled then for any subset S ⊂ X we have f F X ( S ) ⊂ G (( S, Y )[ b, θ x ]) ⊂ G ( S [ b ] , Y ) = G X ( S [ b ]). So f : F X → G X is boundedby b . Now for a given subset S ⊂ X , let us define θ S = sup x ∈ S θ x ( x ). Then f F X ( S )( T ) = f F ( S, T ) ⊂ G ( S [ b ] , T [ θ S ]) = G X ( S [ b ])( T [ θ S ]) verifying that f | : F S → G S [ b ] is bounded by θ S . (cid:3) Properties of fibred objects.
Definition.
An (
X, Y )-filtered module F is called • lean or ( D, ∆)- lean if there is a number D ≥ , + ∞ ) → [0 , + ∞ ) so that F ( U ) ⊂ X ( x,y ) ∈ U F ( x [ D ] × y [∆ x ( x )]) = X ( x,y ) ∈ U F (( x, y )[ D, ∆ x ]) for any subset U of X × Y , • split or ( D ′ , ∆ ′ )- split if there is a number D ′ ≥ ′ : [0 , + ∞ ) → [0 , + ∞ ) so that F ( U ∪ U ) ⊂ F ( U [ D ′ , ∆ ′ x ]) + F ( U [ D ′ , ∆ ′ x ])for each pair of subsets U and U of X × Y , • lean/split or ( D, ∆ ′ )- lean/split if there is a number D ≥ ′ : [0 , + ∞ ) → [0 , + ∞ ) so that – the X -filtered module F X is D -lean, while – the ( X, Y )-filtered module F is ( D, ∆ ′ )-split, • insular or ( d, δ )- insular if there is a number d ≥ δ : [0 , + ∞ ) → [0 , + ∞ ) so that F ( U ) ∩ F ( U ) ⊂ F (cid:0) U [ d, δ x ] ∩ U [ d, δ x ] (cid:1) for each pair of subsets U and U of X × Y .3.2.2. Proposition.
Suppose F is an ( X, Y ) -filtered R -module. (1) If F is ( D, ∆) -lean then the corresponding X -filtered module F X defined byassigning F X ( S ) = F ( S × Y ) is D -lean. (2) Similarly, if F is ( d, δ ) -insular then F X is d -insular. (3) If F is ( D, ∆) -lean then it is ( D, ∆) -split and, further, ( D, ∆) -lean/split. (4) An ( X, Y ) -filtered module F which is lean/split and insular can be thoughtof as an object F X of L ( X, R ) .Proof. (1) Since ( x, y )[ D, ∆ x ] ⊂ x [ D ] × Y , we have F X ( S ) ⊂ X x ∈ S X y ∈ Y F (( x, y )[ D, ∆ x ]) ⊂ X x ∈ S F X ( x [ D ]) . (2) F X ( S ) ∩ F X ( T ) ⊂ F ( S [ d ] × Y ∩ T [ d ] × Y ) = F X ( S [ d ] ∩ T [ d ]).(3) The split property follows directly from definitions, and so the lean/splitproperty follows in view of part (1).(4) follows from (2). (cid:3) Definition.
There are two subcategories nested in U X ( Y ): • LS X ( Y ) is the full subcategory of U X ( Y ) on objects F that are lean/splitand insular, • B X ( Y ) is the full subcategory of LS X ( Y ) on objects F such that F ( U ) is afinitely generated submodule whenever U ⊂ X × Y is bounded. Equivalently,the subcategory B X ( Y ) is full on objects F such that all Y -filtered modules F S associated to bounded subsets S ⊂ X are locally finitely generated.Clearly, B X ( Y ) is a generalization of the bounded category B ( X, R ): if Y = ptthen B X ( Y ) is precisely B ( X, R ). On the other hand, if X = pt then B X ( Y ) is thefull subcategory of BS ( Y, R ) on locally finitely generated objects.We proceed to define appropriate exact structures in these categories.3.2.4.
Definition.
Let the admissible monomorphisms in U X ( Y ) be the boundedlybicontrolled homomorphisms m : F → F such that the module homomorphism F ( X × Y ) → F ( X × Y ) is a monomorphism. Let the admissible epimorphisms be the boundedly bicontrolled homomorphisms e : F → F such that F ( X × -THEORY WITH FIBRED CONTROL 17 Y ) → F ( X × Y ) is an epimorphism. The class E of exact sequences consists of thesequences F · : F ′ i −−→ F j −−→ F ′′ , where i is an admissible monomorphism, j is an admissible epimorphism, andim( i ) = ker( j ).3.2.5. Theorem. U X ( Y ) is a Quillen exact category.Proof. We will verify the axioms for exact structures due to Quillen with somesimplifications due to B. Keller [11, 12], cf. section 2 of [1].It follows from Lemma 3.1.7 that the collections of admissible monomorphismsand admissible epimorphisms are closed under composition and that any short exactsequence isomorphic to some sequence in E is also in E .Now suppose we are given an exact sequence F ′ i −→ F j −→ F ′′ in E and a morphism f : A → F ′′ in U X ( Y ). Let ( b j , θ j ) be some filtration data for j as a boundedlycontrolled epi and let ( b f , θ f ) be some contol data for f as a boundedly controlledmap. There is a base change diagram F ′ −−−−→ F × f A j ′ −−−−→ A = y y f ′ y f F ′ −−−−→ F j −−−−→ F ′′ where m : F × f A → F ⊕ A is the kernel of the epi j ◦ pr − f ◦ pr : F ⊕ A → F ′′ and f ′ = pr ◦ m , j ′ = pr ◦ m . The ( X, Y )-filtration on F × f A is the standardfiltration as a subobject of the product F × A . The induced map j ′ has the samekernel as j and is bounded by 0. In fact, f A ( U ) ⊂ E ′′ ( U [ b f , θ f,x ]) , so f A ( S ) ⊂ jE ( U [ b f + b j , θ f,x + θ j,x ]) , andim( j ′ ) ∩ A ( U ) ⊂ j ′ ( E × f A ) ( U [ b f + b j , θ f,x + θ j,x ]]) . This shows that j ′ is boundedly bicontrolled with filtration data ( b f + b j , θ f + θ j ).Therefore, the class of admissible epimorphisms is closed under base change byarbitrary morphisms in U X ( Y ). Cobase changes by admissible monomorphismsare similar. (cid:3) Proposition.
The admissible monomorphisms are precisely the morphismsisomorphic in U X ( Y ) to the filtration-wise monomorphisms and the admissible epi-morphisms are those morphisms isomorphic to the filtration-wise epimorphisms. Inother words, the exact structure E in U X ( Y ) consists of sequences isomorphic tothose E · : E ′ i −−→ E j −−→ E ′′ which possess filtration-wise restrictions E · ( U ) : E ′ ( U ) i −−→ E ( U ) j −−→ E ′′ ( U ) for all subsets U ⊂ ( X, Y ) , and each E · ( U ) is an exact sequence of R -modules.Proof. Each of the sequences E · in the statement is an exact sequence in U X ( Y )because the restriction i : E ′ ( U ) → E ( U ) is monic and j : E ( U ) → E ′′ ( U ) is epic,therefore i : E ′ → E and j : E → E ′′ are both bicontrolled of filtration (0 , Suppose F · is a sequence isomorphic to such E · , so there is a commutativediagram F ′ f −−−−→ F g −−−−→ F ′′∼ = y y ∼ = y ∼ = E ′ i −−−−→ E j −−−−→ E ′′ Then f and g are compositions of two isomorphisms (which are clearly boundedlybicontrolled) which are either preceded by a boundedly bicontrolled monic or fol-lowed by a boundedly bicontrolled epi. By part (1) of Lemma 3.1.7, both f and g are boundedly bicontrolled.Now suppose F · is an exact sequence in E . Let K = ker( g ) and C = coker( f ),then we obtain a commutative diagram F ′ f −−−−→ F g −−−−→ F ′′∼ = y y = x ∼ = K i −−−−→ F j −−−−→ C where the vertical maps are the canonical isomorphisms. By the construction ofkernels and cokernels in the proof of Theorem 3.1.6, there are exact sequences K ( U ) i −→ F ( U ) j −→ C ( U ) for all subsets U ⊂ ( X, Y ). (cid:3) Proposition.
In the exact category U X ( Y ) , (1) the lean/split objects are closed under extensions, (2) the insular objects are closed under extensions.Suppose E ′ f −→ E g −→ E ′′ is an exact sequence in U X ( Y ) . (3) If the object E is lean/split then E ′′ is lean/split. (4) If E is insular then E ′ is insular. (5) Suppose E is insular then E ′′ is insular if E ′ is lean/split.Proof. All parts are proved by adapting the proofs of Lemmas 2.2.3 and 2.2.4. Toillustrate, suppose that in the exact sequence, as given in the statement, ( b, θ ) iscommon filtration data for f and g and both E ′ and E ′′ are ( D, ∆ ′ )-lean/split. Forthe first statement of part (2), notice that E X is (4 b + D )-lean by part (1) of Lemma2.2.3, so we need to verify that split objects are closed under extensions. Considertwo subsets U and U of X × Y . Then gE ( U ) ⊂ E ′′ (( U ∪ U )[ b, θ x ])= E ′′ ( U [ b, θ x ] ∪ U [ b, θ x ]) ⊂ E ′′ ( U [ b + D, θ x + ∆ ′ x ]) + E ′′ ( U [ b + D, θ x + ∆ ′ x ]) . Therefore E ( U ) ⊂ E ( U [2 b + D, θ x + ∆ ′ x ]) + E ( U [2 b + D, θ x + ∆ ′ x ])+ f E ′ ( U [3 b + 2 D, θ x + 2∆ ′ x ]) + f E ′ ( U [3 b + 2 D, θ x + 2∆ ′ x ]) ⊂ E ( U [4 b + 2 D, θ x + 2∆ ′ x ]) + E ( U [4 b + 2 D, θ x + 2∆ ′ x ]) , showing that E is (4 b + 2 D, θ + 2∆ ′ )-lean/split. (cid:3) -THEORY WITH FIBRED CONTROL 19 Theorem. LS X ( Y ) is closed under extensions in U X ( Y ) . In turn, B X ( Y ) is closed under extensions in LS X ( Y ) . Therefore, B X ( Y ) is an exact category, andthe inclusion e : C X ( Y ) → B X ( Y ) is an exact embedding.Proof. The first statement follows from parts (2) and (3) of Proposition 3.2.7. Sup-pose f : F → G is an isomorphism with fil( f ) ≤ ( b, θ ) and G is locally finitely gen-erated, then F ( U ) is a finitely generated submodule of G ( U [ b, θ ]) for any boundedsubset U ⊂ X × Y since R is Noetherian.If F ′ f −→ F g −→ F ′′ is an exact sequence in LS X ( Y ), F ′ and F ′′ are locally finitelygenerated, and ( b, θ ) is common filtration data for f and g , then gF ( U ) is a finitelygenerated submodule of F ′′ ( U [ b, θ ]) for any bounded subset U . The kernel of therestriction of g to F ( U ) is a finitely generated submodule of F ′ ( U [ b, θ ]), so theextension F ( U ) is finitely generated. (cid:3) Remark. (1) There is certainly an exact embedding ι : B ( X × Y, R ) → B X ( Y ) which is given by the identity on objects. Because of the relaxation of thecontrol conditions on homomorphisms, the morphism sets in the image of ι arein general properly smaller than in B X ( Y ). However ι is also proper on objects.For example, the lean objects in BL ( X × Y, R ) are generated by the submodules f ( S × T ) where the diameters of S and T are uniformly bounded from above. Thisis different from the weaker condition in B X ( Y ).(2) While there is no functor between the categories U X ( Y ) and U ( X, U ( Y )),there is a “forgetful” function associating to objects of U X ( Y ) some objects of U ( X, U ( Y )). It is defined by φ ( F )( S ) = F ( S, Y ) with the Y -filtration given by F ( S, Y )( T ) = F ( S, T ).This relationship can be made much more fruitful if the nature of the objects in U ( X, U ( Y )) is shifted to be functors from the category of bounded subsets of X tosubobjects of U ( Y ) leading to a category that can be thought of as B ( X, B ( Y, R )).It turns out that on this level there is a well-defined exact functor B X ( Y ) → B ( X, B ( Y, R )). We don’t require this type of theory but it has been developed andapplied in [1, 3].3.3.
Fibrewise restriction.
We begin to prepare for the development of fibredlocalization exact sequences and fibred bounded excision theorems. The model forlocalization and fibration theorems in controlled G -theory [6, sections 3 and 4] canbe implemented here as well.There are two complementary ways to introduce support in B X ( Y ).(1) Let B Suppose F is a ( D, ∆) -lean/split object of B X ( Y ) . The follow-ing are equivalent statements. (1) F is an object of B X ( Y ) Proposition. B X ( Y ) 0, an order preserving function λ : B ( X ) → [0 , + ∞ ), and allbounded subsets S ⊂ X . Then f F x [ D ′ ] ⊂ G x [ D ′ + b ] ⊂ G x [ D ′ + b + k ] ( C [ λ ( x [ D ′ + b ])]) ⊂ G x [ D + k ] ( C [ λ ( x [ D ])]) , using the fact that λ is order preserving. Since G x [ D + k ] ( Y − C [ λ ( x [ D + k ]) + ∆( x [ D ]) + θ ( x [ D ′ ]) + 2 δ ( x [ D + k ])]) = 0 , we have F x [ D ′ ] ( Y − C [ λ ( x [ D + k ]) + ∆( x [ D ]) + 2 θ ( x [ D ′ ]) + 2 δ ( x [ D + k ])]) = 0 . Therefore F x [ D ′ ] ⊂ F x [ D ′ ] ( C [ λ ( x [ D ]) + ∆( x [ D ]) + ∆ ′ ( x [ D ]) + 2 θ ( x [ D ′ ]) + 2 δ ( x [ D + k ])] , so F , which is generated by F x [ D ′ ] , is also an object of B X ( Y ) The gradings from Definition 2.4.1 can be generalizedto gradings of objects from B X ( Y ).3.4.1. Definition. Given an object F of B X ( Y ), a grading of F is a covariantfunctor F : P ( X, Y ) → I ( F ) with the following properties:(1) if F ( C ) is given the standard filtration, it is an object of B X ( Y ),(2) there is an enlargement data ( K, k ) such that F ( C ) ⊂ F ( C ) ⊂ F ( C [ K, k x ]) , for all subsets C of ( X, Y ).3.4.2. Remark. If C = ( X, S ) then F ( C ) is an object of B X ( Y ) X, Y ).This makes the following partial gradings sufficient and easier to work with.3.4.3. Definition. Let M ≥ be the set of all monotone functions δ : [0 , + ∞ ) → [0 , + ∞ ). Let P X ( Y ) be the subcategory of P ( X, Y ) consisting of all subsets of theform ( X, C )[ D, δ x ] for some choices of a subset C ⊂ Y , a number D ≥ 0, and afunction δ ∈ M ≥ .Given an object F of B X ( Y ), a Y - grading of F is a functor F : P X ( Y ) → I ( F )with the following properties:(1) the submodule F (( X, C )[ D, δ x ]) with the standard filtration is an objectof B X ( Y ),(2) there is an enlargement data ( K, k ) such that F (( X, C )[ D, δ x ]) ⊂ F (( X, C )[ D, δ x ]) ⊂ F (( X, C )[ D + K, δ x + k x ]) , for all subsets in P X ( Y ).Since U [ D + K, δ x + k x ] = U [ D, δ x ][ K, k x ] for general subsets U , the third,largest submodule is independent of the choice of D , δ x .We say that an object F of B X ( Y ) is Y - graded if there exists a Y -grading of F ,but the grading itself is not specified, and define G X ( Y ) as the full subcategory of B X ( Y ) on Y -graded filtered modules.3.4.4. Proposition. The Y -graded objects in B X ( Y ) are closed under isomor-phisms. The subcategory G X ( Y ) is closed under extensions in B X ( Y ) . Therefore, G X ( Y ) is an exact subcategory of B X ( Y ) .Proof. Suppose f : F → F ′ is an isomorphism between a Y -graded module F and F ′ in B X ( Y ). If fil( f ) ≤ ( b, θ ) and F is a Y -grading for F then F ′ ( X, C )[ D, δ x ] = f F (( X, C )[ D + K + b, δ x + k x + θ x ]). The rest of the argument closely followsthe proof of Proposition 2.4.4. We want to spell out one detail for future reference.Let F f −−→ G g −−→ H be an exact sequence in B X ( Y ). Suppose fil( f ) ≤ ( b, θ ) and fil( g ) ≤ ( b, θ ) for thesame set of bicontrol bound data and suppose that F and H are graded modulesin G X ( Y ) with the associated functors F and H . The assignment G ( U ) = f F ( U [3 b, θ ]) + G ( S [2 b, θ ]) ∩ g − H ( S [ b, θ ])gives a grading of G as an object of G X ( Y ). (cid:3) As with the category G ( X, R ), the advantage of working with G X ( Y ) as opposedto B X ( Y ) is that we are able to localize to the grading subobjects associated tosubsets from the family P X ( Y ).3.4.5. Lemma. Let F be a submodule of a Y -filtered module G in G X ( Y ) whichis lean/split with respect to the standard filtration. Then F ( U ) = F ∩ G ( U ) is a Y -grading of F . We will call this induced Y -grading of F the standard Y -grading of the submod-ule. Proof. The proof is reduced to checking that F ( U ) is an object of B X ( Y ) for eachsubset U ∈ P X ( Y ). Suppose i : F → G is the inclusion and q : G → H is thequotient of i . Since F is insular by part (4) of Proposition 3.2.7, both F and G are -THEORY WITH FIBRED CONTROL 23 lean/split and insular. Thus H is lean/split and insular by parts (3) and (5) of 3.2.7.Let H ( U ) = q G ( U ) with the standard filtration in H . Then H ( U ) is lean/split bypart (3) and insular as a submodule of insular H . The kernel F ( U ) of the filtration(0 , 0) map q | : G ( U ) → H ( U ) is lean/split by part (5) of 3.2.7 and is insular as asubmodule of insular F . Locally finite generation of F ( U ) follows from the sameproperty of G ( U ). (cid:3) Proposition. Suppose g : G → H is a boundedly bicontrolled epimorphismin B X ( Y ) and suppose F is the kernel of g in Mod ( R ) . If G is Y -graded and F islean/split with respect to the standard Y -filtration then both H and F are Y -graded.Proof. The Y -grading for H is given by H ( U ) = g G ( U [ b, θ ]), where ( b, θ ) is a chosenset of filtration data for g . The argument for Lemma 3.4.5 applies directly to showthat H is indeed a grading of H , and the assignment F ( U ) = F ∩ G ( U [ b, θ ]) givesa Y -grading of F . (cid:3) This allows to characterize admissible monomorphisms in G X ( Y ).3.4.7. Proposition. The inclusion of a subobject i : F → G in G X ( Y ) is an ad-missible monomorphism if and only if F is lean/split.Proof. Let H be a cokernel of i in U X ( Y ). As verified in Lemma 3.4.5, H islean/split and insular and is, in fact, a cokernel of i in B X ( Y ). From Proposition3.4.6, H is graded, so it is also a cokernel of i in G ( X, R ). (cid:3) In the case K ≤ k is a non-positive function, we will use notation U [ K, k ]for the subset U \ (( X, Y ) \ U )[ − K, − k ] of ( X, Y ). The following is a direct analogueof Corollary 2.4.8 with exactly the same proof.3.4.8. Corollary. Given an object F in G X ( Y ) and a subset U from the fam-ily P X ( Y ) , there is a set of enlargement data ( K, k ) and an admissible subob-ject i : F U → F in G X ( Y ) with the property that F U ⊂ F ( U [ K, k ]) . If G is ( D, ∆ ′ ) -lean/split then the quotient q : F → H of the inclusion has the propertythat H ( X, Y ) = H ((( X, Y ) \ U )[2 D, ′ ]) . Now we can summarize the preceding results.3.4.9. Theorem. Given a graded object F in G X ( Y ) and a subset U from thefamily P X ( Y ) , we assume that F is ( D, ∆ ′ ) -split and ( d, δ ) -insular and is gradedby F . The submodule F ( U ) has the following properties: (1) F ( U ) is graded by F U ( T ) = F ( U ) ∩ F ( T ) , (2) F ( U ) ⊂ F ( U ) ⊂ F ( U [ K, k ]) for some fixed enlargement data ( K, k ) , (3) if q : F → H is the quotient of the inclusion i : F ( U ) → F and F is ( D, ∆ ′ ) -lean/split, then H is supported on ( X \ U )[2 D, ′ ] , (4) H ( U [ − D − d, − ′ − δ ]) = 0 .Proof. Property (1) follows from Lemma 3.4.5. Properties (2) and (3) follow fromCorollary 3.4.8. (4) follows from the fact that a d -insular filtered module is (2 d, δ )-separated, in the sense that for any pair of subsets U and V of ( X, Y ) such that U [2 d, δ ] ∩ V = ∅ we have U [ d, δ ] ∩ V [ d, δ ] = ∅ so F ( U ) ∩ F ( V ) = 0. Now H ( U [ − D − d, − ′ − δ ]) ∩ H ((( X, Y ) \ U )[2 D, ′ ]) = 0 , but H ((( X, Y ) \ U )[2 D, ′ ]) = H ( X, Y ), thus H ( U [ − D − d, − ′ − δ ]) = 0. (cid:3) Localization fibration sequence. We will use the localization theorem ofSchlichting [17] for Serre subcategories of exact categories. These techniques requirethe Serre subcategory to satisfy some additional assumptions that we verify next.3.5.1. Definition. A class of morphisms Σ in an additive category A admits acalculus of right fractions if(1) the identity of each object is in Σ,(2) Σ is closed under composition,(3) each diagram F f −→ G s ←−− G ′ with s ∈ Σ can be completed to a commuta-tive square F ′ f ′ −−−−→ G ′ y t y s F f −−−−→ G with t ∈ Σ, and(4) if f is a morphism in A and s ∈ Σ such that sf = 0 then there exists t ∈ Σsuch that f t = 0.In this case there is a construction of the localization A [Σ − ] which has the sameobjects as A . The morphism sets Hom( F, G ) in A [Σ − ] consist of equivalence classesof diagrams ( s, f ) : F s ←−− F ′ f −→ G with the equivalence relation generated by ( s , f ) ∼ ( s , f ) if there is a map h : F ′ → F ′ so that f = f h and s = s h . Let ( s | f ) denote the equivalence class of( s, f ). The composition of morphisms in A [Σ − ] is defined by ( s | f ) ◦ ( t | g ) = ( st ′ | gf ′ )where f ′ and t ′ fit in the commutative square F ′′ f ′ −−−−→ G ′ y t ′ y t F f −−−−→ G from axiom 3.3.5.2. Proposition. The localization A [Σ − ] is a category. The morphisms of theform (id | s ) where s ∈ Σ are isomorphisms in A [Σ − ] . The rule P Σ ( f ) = (id | f ) gives a functor P Σ : A → A [Σ − ] which is universal among the functors making themorphisms Σ invertible.Proof. The proofs of these facts can be found in Chapter I of [10]. The inverse of(id | s ) is ( s | id). (cid:3) From Proposition 3.3.3, given a subset C of Y , the category B X ( Y ) The restriction to Y -gradings in B X ( Y ) The category G is the exact subcategory of Y -graded objects in B X ( Y ). When the choice of the subset C ⊂ Y is understood, we will use notation C for the Serre subcategory G X ( Y ) Define the class of weak equivalences Σ( C ) in G to consist of allfinite compositions of admissible monomorphisms with cokernels in C and admissi-ble epimorphisms with kernels in C .We need the class Σ( C ) to admits calculus of right fractions. This follows from[17, Lemma 1.13] as soon as we prove the following fact.A Serre subcategory C of an exact category G is right filtering if each morphism f : F → F in G , where F is an object of C , factors through an admissibleepimorphism e : F → F , where F is in C .3.5.6. Lemma. The subcategory C = G X ( Y ) K, k ) for the grading F and any subset R we have f F ( R ) ⊂ f F ( R [ K, k x ]) ⊂ F ( R [ K + b, k x + θ x ]) . By part (3) of Theorem 3.4.9, F ( R [ K + b, k x + θ x ]) ∩ F (( X, C )[ r, ρ x ]) = 0 forany R such that R [ K + b + 2 D + 2 d, k x + θ x + 2∆ ′ x + 2 δ x ] ∩ ( X, C )[ r, ρ x ] = ∅ . If we choose R = ( X, Y ) \ ( X, C )[ K + b + 2 D + 2 d + r, k x + θ x + 2∆ ′ x + 2 δ x + ρ x ]and define E = F ( R ), then f E = 0. Let F be the cokernel of the inclusion E → F .Then F is lean/split and insular and has a grading given by F ( S ) = q F ( S [ b, θ x ]).Since F ( X, Y ) ⊂ F (( X, C )[ K + b + 2 D + 2 d + r, k x + θ x + 2∆ ′ x + 2 δ x + ρ x ]) , the quotient F is in C , and f factors as F → F → F in the right square in themap of exact sequences E −−−−→ F j ′ −−−−→ F i y y = y K k −−−−→ F f −−−−→ F as required. (cid:3) Definition. The category G / C is the localization G [Σ( C ) − ].It is clear that the quotient G / C is an additive category, and P Σ( C ) is an additivefunctor. In fact, we have the following.3.5.8. Theorem. The short sequences in G / C which are isomorphic to images ofexact sequences from G form a Quillen exact structure. Proof. This will be a consequence from [17, Proposition 1.16]. Since C is rightfiltering by Lemma 3.5.6, it remains to check that C right s-filtering in G in thefollowing sense. A subcategory C of an exact category G is right s-filtering if givenan admissible monomorphism f : F → F with F in C , there exist E in C and anadmissible epimorphism e : F → E such that the composition ef is an admissiblemonomorphism.Suppose that F and F have the same properties as in the proof of Lemma3.5.6, and fil( f ) ≤ ( b, θ ). Since F is in C , there are r ≥ ρ : [0 , + ∞ ) → [0 , + ∞ ) such that F ( X, Y ) ⊂ F (( X, C )[ r, ρ x ]). Then let F ′ = F ( T ) where T = ( X, Y ) \ ( X, C )[ K + b + 2 D + 2 d + r, k x + θ x + 2∆ ′ x + 2 δ x + ρ x ] . Define E as the cokernel of the inclusion F ′ → F and let e : F → E be the quotientmap. The composition ef is an admissible monomorphism with fil( ef ) = fil( f ) ≤ ( b, θ ). (cid:3) Notation. If C is a subset of Y as before, G X ( Y, C ) will stand for the exactcategory G / C and G X ( Y, C ) for its Quillen K -theory.The main tool in proving controlled excision theorems will be the following lo-calization sequence.3.5.10. Theorem (Theorem 2.1 of Schlichting [17]) . Let Z be an idempotent com-plete right s-filtering subcategory of an exact category E which is full and closedunder exact extensions. Then the sequence of exact categories Z → E → E / Z induces a homotopy fibration of Quillen K -theory spectra K ( Z ) −→ K ( E ) −→ K ( E / Z ) . Corollary. There is a homotopy fibration G X ( Y ) Theorem (Localization) . There is a homotopy fibration G X ( C ) −→ G X ( Y ) −→ G X ( Y, C ) . Theorem 3.5.12 is a consequence of Corollary 3.5.11 as soon as we show that G X ( C ) and G X ( Y ) Given a pair of proper metric spaces C ⊂ Y , there is a fullyfaithful embedding ǫ : G X ( C ) → G X ( Y ) . The Serre subcategory G X ( Y ) Suppose F is an object of G X ( C ). The embedding ǫ is given by ǫ ( F )( U ) = F (( X, C ) ∩ U ), ǫ ( F )( S ) = F (( X, C ) ∩ U ). It is clear that ǫ ( F ) is in G X ( Y ) Waldhausen categories and K -theory. Our main reference for Waldhausen K -theory terminology and notation is Thomason [19].A Waldhausen category D with weak equivalences w ( D ) is often denoted by w D as a reminder of the choice. A functor between Waldhausen categories is exactif it preserves the chosen zero objects, cofibrations, weak equivalences, and cobasechanges. Let D be a small Waldhausen category with respect to two categories ofweak equivalences v ( D ) ⊂ w ( D ) with a cylinder functor T both for v D and for w D satisfying the cylinder axiom for w D . Suppose also that w ( D ) satisfies the extensionand saturation axioms. Define v D w to be the full subcategory of v D whose objectsare F such that 0 → F ∈ w ( D ). Then v D w is a small Waldhausen category withcofibrations co( D w ) = co( D ) ∩ D w and weak equivalences v ( D w ) = v ( D ) ∩ D w .The cylinder functor T for v D induces a cylinder functor for v D w . If T satisfiesthe cylinder axiom then the induced functor does so too.4.1.1. Theorem (Approximation Theorem) . Let E : D → D be an exact functorbetween two small saturated Waldhausen categories. It induces a map of K -theoryspectra K ( E ) : K ( D ) −→ K ( D ) . Assume that D has a cylinder functor satisfying the cylinder axiom. If E satisfiestwo conditions: (1) a morphism f ∈ D is in w ( D ) if and only if E ( f ) ∈ D is in w ( D ) , (2) for any object D ∈ D and any morphism g : E ( D ) → D in D , there isan object D ′ ∈ D , a morphism f : D → D ′ in D , and a weak equivalence g ′ : E ( D ′ ) → D ∈ w ( D ) such that g = g ′ E ( f ) ,then K ( E ) is a homotopy equivalence.Proof. This is Theorem 1.6.7 of [20]. The presence of the cylinder functor with thecylinder axiom allows to make condition (2) weaker than that of Waldhausen, seepoint 1.9.1 in [19]. (cid:3) Definition. In any additive category, a sequence of morphisms E · : 0 −→ E d −−→ E d −−→ . . . d n − −−−−→ E n −→ (bounded) chain complex if the compositions d i +1 d i are the zero mapsfor all i = 1,. . . , n − 1. A chain map f : F · → E · is a collection of morphisms f i : F i → E i such that f i d i = d i f i . A chain map f is null-homotopic if there aremorphisms s i : F i +1 → E i such that f = ds + sd . Two chain maps f , g : F · → E · are chain homotopic if f − g is null-homotopic. Now f is a chain homotopy equivalence if there is a chain map h : E i → F i such that the compositions f h and hf are chainhomotopic to the respective identity maps.The Waldhausen structures on categories of bounded chain complexes are basedon homotopy equivalence as a weakening of the notion of isomorphism of chaincomplexes.A sequence of maps in an exact category is called acyclic if it is assembled outof short exact sequences in the sense that each map factors as the composition ofthe cokernel of the preceding map and the kernel of the succeeding map.It is known that the class of acyclic complexes in an exact category is closed underisomorphisms in the homotopy category if and only if the category is idempotentcomplete, which is also equivalent to the property that each contractible chaincomplex is acyclic, cf. [12, sec. 11].Given an exact category E , there is a standard choice for the Waldhausen struc-ture on the category E ′ of bounded chain complexes in E where the degree-wiseadmissible monomorphisms are the cofibrations and the chain maps whose map-ping cones are homotopy equivalent to acyclic complexes are the weak equivalences v ( E ′ ).The following fact is well-known, cf. point 1.1.2 in [19].4.1.3. Proposition. The category v E ′ is a Waldhausen category satisfying the ex-tension and saturation axioms and has cylinder functor satisfying the cylinder ax-iom. Example. There are two choices for the Waldhausen structure on the cate-gory of bounded chain complexes G ′ = G ( X, R ) ′ . One is the standard choice v G ′ as above. Given a subset C ⊂ Y , another choice for the weak equivalences w ( G ′ ) isthe chain maps whose mapping cones are homotopy equivalent to acyclic complexesin the quotient G / C .4.1.5. Corollary. The categories v G ′ and w G ′ are Waldhausen categories satisfy-ing the extension and saturation axioms and have cylinder functors satisfying thecylinder axiom.Proof. All axioms and constructions, including the cylinder functor, for w G ′ areinherited from v G ′ . (cid:3) The K -theory functor from the category of small Waldhausen categories D and exact functors to the category of connective spectra is defined in terms of S · -construction as in Waldhausen [20]. It extends to simplicial categories D withcofibrations and weak equivalences and inductively delivers the connective spectrum n 7→ | w S ( n ) · D | . We obtain the functor assigning to D the connective Ω-spectrum K ( D ) = Ω ∞ | w S ( ∞ ) · D | = colim −−−−→ n ≥ Ω n | w S ( n ) · D | representing the Waldhausen algebraic K -theory of D . For example, if D is the ad-ditive category of free finitely generated R -modules with the canonical Waldhausenstructure, then the stable homotopy groups of K ( D ) are the usual K -groups ofthe ring R . In fact, there is a general identification of the two theories. Recall thatfor any exact category E , the category E ′ of bounded chain complexes has theWaldhausen structure v E ′ as in Example 4.1.4. -THEORY WITH FIBRED CONTROL 29 Theorem. The Quillen K -theory of an exact category E is equivalent to theWaldhausen K -theory of v E ′ .Proof. The proof is based on repeated applications of the Additivity Theorem, cf.Thomason’s Theorem 1.11.7 from [19]. Thomason’s proof of his Theorem 1.11.7can be repeated verbatim here. It is in fact simpler in this case since his condition1.11.3.1 is not required. (cid:3) Controlled excision theorems. These are the major computational toolsin controlled K -theory. We develop excision results G X ( Y ) with respect to specificcoverings of the variable Y in this section.Suppose Y and Y are subsets of a proper metric space Y , and Y = Y ∪ Y .We use the notation G = G X ( Y ), G i = G X ( Y ) This is precisely the commutative diagram from Cardenas-Pedersen[4, section 8] transported from bounded K -theory to fibred G -theory. Cardenas andPedersen use Karoubi quotients and the Karoubi fibrations in order to generate theirdiagram. One of the crucial points in [4] is that the functor I between the Karoubiquotients is an isomorphism of categories. In fibred G -theory the situation is morecomplicated: I is not necessarily full and, therefore, not an isomorphism of cate-gories. We will use the Approximation Theorem to prove that K ( I ) is neverthelessan equivalence of spectra.4.2.2. Proposition. K ( w G ′ ) ≃ K ( G / C ) .Proof. This follows from Lemma 2.3 in [17] as part of the proof of Theorem 3.5.10where K ( w G ′ ) from Waldhausen’s Fibration Theorem is identified with the Quillen K -theory spectrum K ( G / C ). (cid:3) Lemma. If f · : F · → G · is a degreewise admissible monomorphism withcokernel in C then f · is a weak equivalence in w G ′ .Proof. The mapping cone Cf · is quasi-isomorphic to the cokernel of f · , by Lemma11.6 of [12], which is zero in G / C . (cid:3) The exact inclusion I induces the exact functor w G ′ → w G ′ .4.2.4. Lemma. The map K ( w G ′ ) → K ( w G ′ ) is a weak equivalence.Proof. Applying the Approximation Theorem, condition (1) is clear, so we need tocheck condition (2). Consider F · : 0 −→ F φ −−−→ F φ −−−→ . . . φ n − −−−−→ F n −→ in G and a chain map g : F · → G · for some complex G · : 0 −→ G ψ −−−→ G ψ −−−→ . . . ψ n − −−−−→ G n −→ G . Suppose all F i and G i are ( D, ∆ ′ )-lean/split and ( d, δ )-insular. Also assumethat there is a fixed number r ≥ ρ : [0 , + ∞ ) → [0 , + ∞ )such that F i ( X, Y ) ⊂ F i (( X, C )[ r, ρ x ]) holds for all 0 ≤ i ≤ n . If the pair ( b, θ )serves as bounded control data for all φ i , ψ i , and g i , we define the submodule F ′ i = G i (( X, Y )[ r + 3 ib, ρ x + 3 iθ x ])and define ξ i : F ′ i → F ′ i +1 to be the restrictions of ψ i to F ′ i . This gives a chainsubcomplex ( F ′ i , ξ i ) of ( G i , ψ i ) in G with the inclusion i : F ′ i → G i . Notice thatwe have the induced chain map g : F · → F ′ · in G so that g = iI ( g ).Once we establish that C · = coker( i ) is in G , K ( I ) is a weak equivalence byLemma 4.2.3.Since F ′ i ⊂ G i (( X, Y )[ r + 3 ib + K, ρ x + 3 iθ x + k x ]) , each C i is supported on( X, Y \ Y )[2 D + 2 d − r − ib − K, ′ x + 2 δ x − ρ x − iθ x − k x ] ⊂ ( X, Y )[2 D + 2 d, ′ x + 2 δ x ] , cf. Lemma 3.5.6. So the complex C · is indeed in G . (cid:3) The excision theorems are best stated in terms of non-connective deloopings ofthe K -theory spectra. Following Pedersen and Weibel we can use the same kind ofdiagram to first deloop K -theory and then reuse it to prove the excision theorem.Let R , R ≥ , and R ≤ denote the metric spaces of the reals, the nonnegative reals,and the nonpositive reals with the restriction of the usual metric on the real line R .Then there is the following instance of commutative diagram ( ♮ ) G X ( Y ) −−−−→ G X ( Y × R ≥ ) −−−−→ K ( G / G ) y y y K ( I ) G X ( Y × R ≥ ) −−−−→ G X ( Y × R ) −−−−→ K ( G / G )We already know that K ( I ) is an equivalence.4.2.5. Lemma. The spectra G X ( Y × R ≥ ) and G X ( Y × R ≤ ) are contractible.Proof. This follows from the fact that these controlled categories are flasque, thatis, the evident shift functor T in the positive (respectively negative) direction along R ≥ (respectively R ≤ ) interpreted in the obvious way is an exact endofunctor,and there is a natural equivalence 1 ⊕ ± T ∼ = ± T . Contractibility follows from theAdditivity Theorem, cf. Pedersen–Weibel [14]. (cid:3) In view of Lemma 4.2.4, we obtain a map G X ( Y ) → Ω G X ( Y × R ) which inducesisomorphisms of K -groups in positive dimensions. Weak equivalencesΩ k G X ( Y × R k ) −→ Ω k +1 G X ( Y × R k +1 )are obtained by iterating this construction for k ≥ -THEORY WITH FIBRED CONTROL 31 Definition. The nonconnective fibred bounded G -theory over the pair ( X, Y )is the spectrum G −∞ X ( Y ) def = hocolim −−−−→ k> Ω k G X ( Y × R k ) . Since BL ( X, R ) can be identified with B X (pt), this definition also gives a noncon-nective delooping of the G -theory of X : G −∞ ( X, R ) = hocolim −−−−→ k> Ω k G X ( R k ) . The subcategory G X ( Y × R k ) We define G −∞ X ( Y ) Let us write S k G for G X ( Y × R k ) whenever G is the fibred bounded cate-gory for a pair ( X, Y ). If C represents a family of coarsely equivalent subsets in acoarse covering U of Y , consider the fibration G X ( C ) −→ G X ( Y ) −→ K ( G / C )from Theorem 3.5.12. Notice that there is a map K ( G / C ) → Ω K ( S G /S C ) whichis an equivalence in positive dimensions by the Five Lemma. Defining G −∞ X ( Y, C ) = K −∞ ( G / C ) = hocolim −−−−→ k Ω k K ( S k G /S k C )gives an induced fibration G −∞ X ( C ) −→ G −∞ X ( Y ) −→ G −∞ X ( Y, C ) . The theorem follows from the commutative diagram G −∞ X ( Y ) To state the excision theorems properly in thecoarse geometric setting, we develop the language of fibred coarse coverings.Two subsets A , B of ( X, Y ) are called coarsely equivalent if there is a set ofenlargement data ( K, k ) such that A ⊂ B [ K, k x ] and B ⊂ A [ K, k x ]. We will usethe notation A k B for this equivalence relation.A family of subsets A is called coarsely saturated if it is maximal with respectto this equivalence relation. Given a subset A , we denote by S ( A ) the smallestboundedly saturated family containing A .A collection of subsets U = { U i } is a coarse covering of ( X, Y ) if ( X, Y ) = S S i for some S i ∈ S ( U i ). Similarly, U = {A i } is a coarse covering by coarsely saturatedfamilies if for some (and therefore any) choice of subsets A i ∈ A i , { A i } is a coarsecovering in the above sense.We will say that a pair of subsets A , B of ( X, Y ) are coarsely antithetic if for anytwo sets of enlargement data ( D , d ) and ( D , d ) there exist enlargement data( D, d ) such that A [ D , ( d ) x ] ∩ B [ D , ( d ) x ] ⊂ ( A ∩ B )[ D, d x ] . We will write A ♮ B to indicate that A and B are coarsely antithetic.Given two subsets A and B , we define S ( A, B ) = { A ′ ∩ B ′ | A ′ ∈ S ( A ) , B ′ ∈ S ( B ) , A ′ ♮ B ′ } . It is easy to see that S ( A, B ) is a coarsely saturated family.4.3.1. Proposition. S ( A, B ) is a coarsely saturated family.Proof. Suppose A , A ′ and A , A ′ are two coarsely antithetic pairs, and A ⊂ A [ D , ( d ) x ], A ′ ⊂ A ′ [ D ′ , ( d ′ ) x ] for some D , d , D ′ , and d ′ . Then A ∩ A ′ ⊂ A [ D , ( d ) x ] ∩ A ′ [ D ′ , ( d ′ ) x ] ⊂ ( A ∩ A ′ )[ D, d x ]for some ( D, d ). (cid:3) There is the straightforward generalization to the case of a finite number ofsubsets of ( X, Y ). Similarly, we write A ♮ . . . ♮ A k if for arbitrary sets of data( D i , d i ) there is a set of enlargement data ( D, d ) so that A [ D , ( d ) x ] ∩ . . . ∩ A k [ D k , ( d k ) x ] ⊂ ( A ∩ . . . ∩ A k )[ D, d x ]and define S ( A , . . . , A k ) = { A ′ ∩ . . . ∩ A ′ k | A ′ i ∈ S ( A i ) , A ♮ . . . ♮ A k } . -THEORY WITH FIBRED CONTROL 33 Identifying any coarsely saturated family A with S ( A ) for A ∈ A , one has the coarsesaturated family S ( A , . . . , A k ). We will refer to S ( A , . . . , A k ) as the coarse inter-section of A , . . . , A k . A coarse covering U is closed under coarse intersections if allcoarse intersections S ( A , . . . , A k ) are nonempty and are contained in U . If U is agiven coarse covering, the smallest coarse covering that is closed under coarse inter-sections and contains U will be called the closure of U under coarse intersections.All of the terms introduced above have absolute analogues obtained by simplyrestricting to the case Y = pt. So there are, in particular, finite coarse coverings ofa single metric space.4.3.2. Proposition. If U is a finite coarse antithetic covering of Y then ( X, U ) consisting of subsets ( X, U ) , U ∈ U , is a coarse antithetic covering of ( X, Y ) . If U is closed under coarse intersections, ( X, U ) is closed under coarse intersections.Proof. Suppose U = {A i } so that for A i ∈ A i , { A i } is a coarse covering of Y .Then { ( X, A i ) } is a covering of ( X, Y ). Suppose U is coarsely antithetic, so givennumbers d , d there is a number d so that A i [ d ] ∩ A j [ d ] ⊂ ( A i ∩ A j )[ d ]. If d , d are non-decreasing functions, these values give a non-decreasing function d . Nowgiven enlargement data ( D , d ) and ( D , d ), we have( X, A i )[ D , ( d ) x ] ∩ ( X, A j )[ D , ( d ) x ] ⊂ ( X, A i ∩ A j )[ D, h ] , where D can be any non-negative number, and h is the function h ( x ) = d ( d X ( x , x )+ D + D ). So ( X, U ) is a coarsely antithetic covering. A similar estimate gives thelast statement. (cid:3) Suppose U is a finite coarse covering of Y closed under coarse intersections. Wecan define the homotopy pushout G X ( Y ; U ) = hocolim −−−−→ U ∈U G −∞ X ( Y )
Apply Theorem 4.2.8 inductively to the sets in U . (cid:3) Relative excision theorems. Fibred G -theory has a useful relative version,and there are generalizations of the excision theorems to relative statements.4.4.1. Definition. Let Y ′ ∈ A for a coarse covering U of Y . Let G = G X ( Y ) < U and Y ′ = G X ( Y ) Theorem (Relative Fibrewise Excision, Version One) . If Y is the union oftwo subsets U and U , there is a homotopy pushout diagram of spectra G −∞ X ( Y, Y ′ )
Given a subset U of Y ′ , there is a weak equivalence G −∞ X ( Y, Y ′ ) ≃ G −∞ X ( Y − U, Y ′ − U ) . Proof. Consider the setup of Theorem 4.2.8 with Y = Y − U and Y = Y ′ , thenLemma 4.2.4 shows that the map G X ( Y ) < ( Y − U ) G X ( Y ) < ( Y − U ) ∩ G X ( Y ) Apply Theorem 4.4.2 inductively to the sets in U . (cid:3) -THEORY WITH FIBRED CONTROL 35 Conclusion It is a familiar fact that G -theoretic approximations to the usually more mean-ingful K -theoretic invariants are easier to compute. This paper confirms the patternin the controlled algebra setting. One approach to computing the K -theory leadsone to consider K -theory with fibred control. It is in this setting that the tools ofbounded K -theory become insufficient for computation. The paper [8, section 5.2]contains an explicit example of failure of the standard localization tools in bounded K -theory based on Karoubi filtrations. This paper, in contrast, uses a different tech-nology of exact and Waldhausen categories. The Fibrewise Excision Theorems fromsection 4 suffice to resolve the example from [8] and perform computations in moregeneral geometric settings. This material will appear in [9], while the relationshipbetween the K -theory of group rings for finitely generated groups and a G -theoreticanalogue based on controlled G -theory is studied in [5, 7]. For the purpose of stat-ing the results we restrict to regular coefficient rings R of finite global dimension.The conclusion is that the appropriate G -theory of the group ring is computableleveraging the results of this paper, while the Cartan comparison map from the K -theory is an equivalence for a remarkably large class of groups π including allgroups with finite K ( π, 1) and finite decomposition complexity. References [1] B. Bennett, Bounded algebra in symmetric monoidal categories , Ph.D. thesis, University atAlbany, SUNY, 2018.[2] T. Bühler, Exact categories , Expo. Math. (2010), 1–69.[3] P. Cahill, Bounded algebra over coarse spaces , Ph.D. thesis, University at Albany, SUNY,2017.[4] M. Cardenas and E.K. Pedersen, On the Karoubi filtration of a category , K -theory, (1997),165–191.[5] G. Carlsson and B. Goldfarb, On homological coherence of discrete groups , J. Algebra (2004), 502–514.[6] , Controlled algebraic G -theory, I , J. Homotopy Relat. Struct. (2011), 119–159.[7] , On modules over infinite group rings , Int. J. Algebra Comput. (2016), 1–16.[8] , K-theory of geometric modules with fibred control , preprint, 2018. arXiv:1404.5606 [9] , On relative lax limits , in preparation.[10] P. Gabriel and M. Zisman, Calculus of fractions and homotopy theory , Springer-Verlag (1967).[11] B. Keller, Chain complexes and stable categories , Manuscripta Math. (1990), 379–417.[12] , Derived categories and their uses , in Handbook of Algebra, Vol. 1 (M. Hazewinkel,ed.), 1996, Elsevier Science, 671–701.[13] S. Mac Lane, Categories for the working mathematician , Springer-Verlag (1971).[14] E.K. Pedersen and C. Weibel, A nonconnective delooping of algebraic K -theory , in Alge-braic and geometric topology (A. Ranicki, N. Levitt, and F. Quinn, eds.), Lecture Notes inMathematics , Springer-Verlag (1985), 166–181.[15] , K -theory homology of spaces , in Algebraic topology (G. Carlsson, R.L. Cohen,H.R. Miller, and D.C. Ravenel, eds.), Lecture Notes in Mathematics , Springer-Verlag(1989), 346–361.[16] W. Rump, Almost abelian categories , Cahiers Topologie Géom. Diffeérentielle Cateég. (2001), 163–225.[17] M. Schlichting, Delooping the K -theory of exact categories , Topology (2004), 1089–1103.[18] D. Sieg and S. Wegner, Maximal exact structures on additive categories , Math. Nachr. (2011), 2093–2100.[19] R.W. Thomason and Thomas Trobaugh, Higher algebraic K -theory of schemes and of derivedcategories , in The Grothendieck Festschrift , Vol. III , Progress in Mathematics , Birkhäuser(1990), 247–435. [20] F. Waldhausen, Algebraic K -theory of spaces , in Algebraic and geometric topology (A. Ranicki,N. Levitt, and F. Quinn, eds.), Lecture Notes in Mathematics , Springer-Verlag (1985),318–419. Department of Mathematics, Stanford University, Stanford, CA 94305 E-mail address : [email protected] Department of Mathematics and Statistics, SUNY, Albany, NY 12222 E-mail address ::