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(1) [MATH] is bounded by [MATH] when viewed as a morphism [MATH] in [MATH] , and (2) for each bounded subset [MATH] , the restriction |
[MATH] is bounded when viewed as a morphism [MATH] of [MATH] -filtered modules in [MATH] Proof. If [MATH] is [MATH] -controlled then for any subset [MATH] we have [MATH] . So [MATH] is bounded by [MATH] Now for a given subset [MATH] , let us define [MATH] Then [MATH] |
verifying that [MATH] is bounded by [MATH] 3.2. Properties of fibred objects 3.2.1 Definition An [MATH] -filtered module [MATH] is called |
lean or [MATH] lean if there is a number [MATH] and a monotone function [MATH] so that [EQUATION] for any subset [MATH] of [MATH] |
split or [MATH] split if there is a number [MATH] and a monotone function [MATH] so that [EQUATION] for each pair of subsets [MATH] and [MATH] of [MATH] |
lean/split or [MATH] lean/split if there is a number [MATH] and a monotone function [MATH] so that the [MATH] -filtered module [MATH] is [MATH] -lean, while |
the [MATH] -filtered module [MATH] is [MATH] -split, insular or [MATH] insular if there is a number [MATH] and a monotone function |
[MATH] so that [EQUATION] for each pair of subsets [MATH] and [MATH] of [MATH] 3.2.2 Proposition Suppose [MATH] is an [MATH] -filtered [MATH] -module. |
(1) If [MATH] is [MATH] -lean then the corresponding [MATH] -filtered module [MATH] defined by assigning [MATH] is [MATH] -lean. |
(2) Similarly, if [MATH] is [MATH] -insular then [MATH] is [MATH] -insular. (3) If [MATH] is [MATH] -lean then it is [MATH] -split and, further, [MATH] -lean/split. |
(4) An [MATH] -filtered module [MATH] which is lean/split and insular can be thought of as an object [MATH] of [MATH] Proof. (1) Since [MATH] , we have |
[EQUATION] (2) [MATH] (3) The split property follows directly from definitions, and so the lean/split property follows in view of part (1). |
(4) follows from (2). 3.2.3 Definition There are two subcategories nested in [MATH] [MATH] is the full subcategory of [MATH] on objects [MATH] that are lean/split and insular, |
[MATH] is the full subcategory of [MATH] on objects [MATH] such that [MATH] is a finitely generated submodule whenever [MATH] is bounded. Equivalently, the subcategory [MATH] is full on objects [MATH] such that all [MATH] -filtered modules [MATH] associated to bounded subsets [MATH] are locally finitely generated. |
Clearly, [MATH] is a generalization of the bounded category [MATH] : if [MATH] then [MATH] is precisely [MATH] . On the other hand, if [MATH] then [MATH] is the full subcategory of [MATH] on locally finitely generated objects. |
We proceed to define appropriate exact structures in these categories. 3.2.4 Definition Let the admissible monomorphisms in [MATH] be the boundedly bicontrolled homomorphisms [MATH] such that the module homomorphism [MATH] is a monomorphism. Let the admissible epimorphisms be the boundedly bicontrolled homomorphisms [M... |
exact sequences consists of the sequences [EQUATION] where [MATH] is an admissible monomorphism, [MATH] is an admissible epimorphism, and [MATH] |
3.2.5 Theorem [MATH] is a Quillen exact category. Proof. We will verify the axioms for exact structures due to Quillen with some simplifications due to B. Keller |
, cf. section 2 of It follows from Lemma 3.1.7 that the collections of admissible monomorphisms and admissible epimorphisms are closed under composition and that any short exact sequence isomorphic to some sequence in |
[MATH] is also in [MATH] Now suppose we are given an exact sequence [MATH] in [MATH] and a morphism [MATH] in [MATH] . Let [MATH] be some filtration data for [MATH] as a boundedly controlled epi and let [MATH] be some contol data for [MATH] as a boundedly controlled map. There is a base change diagram |
[EQUATION] where [MATH] is the kernel of the epi [MATH] and [MATH] [MATH] The [MATH] -filtration on [MATH] is the standard filtration as a subobject of the product [MATH] The induced map [MATH] has the same kernel as |
[MATH] and is bounded by [MATH] In fact, [MATH] so [MATH] and [EQUATION] This shows that [MATH] is boundedly bicontrolled with filtration data [MATH] Therefore, the class of admissible epimorphisms is closed under base change by arbitrary morphisms in [MATH] Cobase changes by admissible monomorphisms are similar. |
3.2.6 Proposition The admissible monomorphisms are precisely the morphisms isomorphic in [MATH] to the filtration-wise monomorphisms and the admissible epimorphisms are those morphisms isomorphic to the filtration-wise epimorphisms. In other words, the exact structure [MATH] in [MATH] consists of sequences isomorphic t... |
[EQUATION] which possess filtration-wise restrictions [EQUATION] for all subsets [MATH] , and each [MATH] is an exact sequence of [MATH] -modules. |
Proof. Each of the sequences [MATH] in the statement is an exact sequence in [MATH] because the restriction [MATH] is monic and [MATH] is epic, therefore [MATH] and [MATH] are both bicontrolled of filtration [MATH] |
Suppose [MATH] is a sequence isomorphic to such [MATH] , so there is a commutative diagram [EQUATION] Then [MATH] and [MATH] are compositions of two isomorphisms (which are clearly boundedly bicontrolled) which are either preceded by a boundedly bicontrolled monic or followed by a boundedly bicontrolled epi. By part (1... |
Now suppose [MATH] is an exact sequence in [MATH] . Let [MATH] and [MATH] , then we obtain a commutative diagram [EQUATION] where the vertical maps are the canonical isomorphisms. By the construction of kernels and cokernels in the proof of Theorem 3.1.6 , there are exact sequences |
[MATH] for all subsets [MATH] 3.2.7 Proposition In the exact category [MATH] (1) the lean/split objects are closed under extensions, |
(2) the insular objects are closed under extensions. Suppose [MATH] is an exact sequence in [MATH] (3) If the object [MATH] is lean/split then [MATH] is lean/split. |
(4) If [MATH] is insular then [MATH] is insular. (5) Suppose [MATH] is insular then [MATH] is insular if [MATH] is lean/split. Proof. |
All parts are proved by adapting the proofs of Lemmas 2.2.3 and 2.2.4 To illustrate, suppose that in the exact sequence, as given in the statement, [MATH] is common filtration data for [MATH] and [MATH] and both [MATH] and [MATH] are |
[MATH] -lean/split. For the first statement of part (2), notice that [MATH] is [MATH] -lean by part (1) of Lemma 2.2.3 , so we need to verify that split objects are closed under extensions. Consider two subsets [MATH] and [MATH] of [MATH] . Then |
[EQUATION] Therefore [EQUATION] showing that [MATH] is [MATH] -lean/split. 3.2.8 Theorem [MATH] is closed under extensions in [MATH] . In turn, [MATH] is closed under extensions in [MATH] . Therefore, [MATH] is an exact category, and the inclusion |
[MATH] is an exact embedding. Proof. The first statement follows from parts (2) and (3) of Proposition 3.2.7 Suppose [MATH] is an isomorphism with [MATH] and [MATH] is locally finitely generated, then [MATH] is a finitely generated submodule of [MATH] for any bounded subset [MATH] since [MATH] is Noetherian. |
If [MATH] is an exact sequence in [MATH] [MATH] and [MATH] are locally finitely generated, and [MATH] is common filtration data for [MATH] and [MATH] , then [MATH] is a finitely generated submodule of [MATH] for any bounded subset [MATH] The kernel of the restriction of [MATH] to [MATH] is a finitely generated submodul... |
3.2.9 Remark (1) There is certainly an exact embedding [MATH] which is given by the identity on objects. Because of the relaxation of the control conditions on homomorphisms, the morphism sets in the image of [MATH] are in general properly smaller than in [MATH] However [MATH] is also proper on objects. For example, th... |
where the diameters of [MATH] and [MATH] are uniformly bounded from above. This is different from the weaker condition in [MATH] |
(2) While there is no functor between the categories [MATH] and [MATH] , there is a “forgetful” function associating to objects of [MATH] some objects of [MATH] . It is defined by [MATH] with the [MATH] -filtration given by [MATH] |
This relationship can be made much more fruitful if the nature of the objects in [MATH] is shifted to be functors from the category of bounded subsets of [MATH] to subobjects of [MATH] leading to a category that can be thought of as [MATH] . It turns out that on this level there is a well-defined exact functor [MATH] .... |
3.3. Fibrewise restriction We begin to prepare for the development of fibred localization exact sequences and fibred bounded excision theorems. The model for localization and fibration theorems in controlled [MATH] -theory , sections 3 and 4] can be implemented here as well. |
There are two complementary ways to introduce support in [MATH] (1) Let [MATH] be the full subcategory of [MATH] on objects [MATH] supported near [MATH] when viewed as objects [MATH] in [MATH] In other words, [MATH] is an object of [MATH] if |
[MATH] for some number [MATH] (2) Let [MATH] be the full subcategory of [MATH] on objects [MATH] such that [EQUATION] for some number [MATH] and an order preserving function [MATH] |
The first version of support is a straightforward generalization of support for geometric modules that was exploited in In this paper we are more interested in the latter, fibrewise version (2) of support. |
3.3.1 Proposition Suppose [MATH] is a [MATH] -lean/split object of [MATH] The following are equivalent statements. (1) [MATH] is an object of [MATH] |
(2) There is a number [MATH] and an order preserving function [MATH] such that [EQUATION] for all bounded subsets [MATH] (3) There is a number [MATH] and a monotone function [MATH] |
such that [EQUATION] for all [MATH] Proof. [MATH] : If [MATH] satisfies (2) then [MATH] It suffices to define [MATH] such that [MATH] Since [MATH] and [MATH] is order preserving, one can take |
[MATH] In the opposite direction, given a bounded subset [MATH] [EQUATION] when [MATH] [MATH] : If [MATH] is in [MATH] then [MATH] so |
[EQUATION] If [MATH] is [MATH] -insular then [EQUATION] for [MATH] In the opposite direction, we have [EQUATION] for an object [MATH] of [MATH] satisfying (3). |
3.3.2 Definition Serre subcategory of an exact category is a full subcategory which is closed under exact extensions and closed under passage to admissible subobjects and admissible quotients. |
Note that this property is relative to the choice of exact structure. 3.3.3 Proposition [MATH] is a Serre subcategory of [MATH] Proof. |
First we show closure under exact extensions. Let [MATH] be an exact sequence in [MATH] , let [MATH] be common set of filtration data for [MATH] and [MATH] , and assume all objects be [MATH] -lean/split. We assume that [MATH] and [MATH] are objects of [MATH] , so there is a number [MATH] and a monotone function [MATH] ... |
[MATH] and [MATH] for some choice of a base point [MATH] in [MATH] Therefore [EQUATION] In particular, the image [MATH] with the standard filtration [MATH] is an object of [MATH] Now |
[EQUATION] Let [MATH] viewed as a subobject of [MATH] with the standard filtration. Since [MATH] for any submodule [MATH] with [MATH] , we have |
[EQUATION] so [MATH] is an object of [MATH] Suppose [MATH] is an admissible monomorphism in [MATH] , which is a boundedly bicontrolled monic with [MATH] [MATH] is [MATH] -lean/split, [MATH] is [MATH] -lean/split for some [MATH] , and |
[MATH] is [MATH] -insular. If [MATH] is an object of [MATH] , according to Proposition 3.3.1 [EQUATION] for some number [MATH] , an order preserving function [MATH] , and all bounded subsets [MATH] Then |
[EQUATION] using the fact that [MATH] is order preserving. Since [EQUATION] we have [EQUATION] Therefore [EQUATION] so [MATH] , which is [MATH] , is also an object of [MATH] |
On the other hand, let [MATH] be an admissible quotient with [MATH] and suppose [MATH] is an object of [MATH] so that there is a number [MATH] and a monotone function [MATH] such that |
[EQUATION] This implies that [EQUATION] so [MATH] is also in [MATH] 3.4. Fibrewise gradings The gradings from Definition 2.4.1 can be generalized to gradings of objects from [MATH] |
3.4.1 Definition Given an object [MATH] of [MATH] , a grading of [MATH] is a covariant functor [MATH] with the following properties: |
(1) if [MATH] is given the standard filtration, it is an object of [MATH] (2) there is an enlargement data [MATH] such that [EQUATION] |
for all subsets [MATH] of [MATH] 3.4.2 Remark If [MATH] then [MATH] is an object of [MATH] We are concerned with localizations to a specific type of subspaces of [MATH] This makes the following partial gradings sufficient and easier to work with. |
3.4.3 Definition Let [MATH] be the set of all monotone functions [MATH] Let [MATH] be the subcategory of [MATH] consisting of all subsets of the form [MATH] for some choices of a subset [MATH] , a number [MATH] , and a function [MATH] |
Given an object [MATH] of [MATH] , a [MATH] grading of [MATH] is a functor [MATH] with the following properties: (1) the submodule [MATH] with the standard filtration is an object of [MATH] |
(2) there is an enlargement data [MATH] such that [EQUATION] for all subsets in [MATH] Since [MATH] for general subsets [MATH] , the third, largest submodule is independent of the choice of [MATH] [MATH] |
We say that an object [MATH] of [MATH] is [MATH] graded if there exists a [MATH] -grading of [MATH] , but the grading itself is not specified, and define [MATH] as the full subcategory of [MATH] on [MATH] -graded filtered modules. |
3.4.4 Proposition The [MATH] -graded objects in [MATH] are closed under isomorphisms. The subcategory [MATH] is closed under extensions in [MATH] Therefore, [MATH] is an exact subcategory of [MATH] |
Proof. Suppose [MATH] is an isomorphism between a [MATH] -graded module [MATH] and [MATH] in [MATH] . If [MATH] and [MATH] is a [MATH] -grading for [MATH] then [MATH] . The rest of the argument closely follows the proof of Proposition 2.4.4 . We want to spell out one detail for future reference. Let |
[EQUATION] be an exact sequence in [MATH] . Suppose [MATH] and [MATH] for the same set of bicontrol bound data and suppose that [MATH] and [MATH] are graded modules in [MATH] with the associated functors [MATH] and [MATH] The assignment |
[EQUATION] gives a grading of [MATH] as an object of [MATH] As with the category [MATH] , the advantage of working with [MATH] as opposed to [MATH] is that we are able to localize to the grading subobjects associated to subsets from the family [MATH] |
3.4.5 Lemma Let [MATH] be a submodule of a [MATH] -filtered module [MATH] in [MATH] which is lean/split with respect to the standard filtration. Then [MATH] is a [MATH] -grading of [MATH] |
We will call this induced [MATH] -grading of [MATH] the standard [MATH] -grading of the submodule. Proof. The proof is reduced to checking that [MATH] is an object of [MATH] for each subset [MATH] Suppose [MATH] is the inclusion and [MATH] is the quotient of [MATH] Since [MATH] is insular by part (4) of Proposition 3.2... |
is lean/split by part (5) of 3.2.7 and is insular as a submodule of insular [MATH] Locally finite generation of [MATH] follows from the same property of [MATH] |
3.4.6 Proposition Suppose [MATH] is a boundedly bicontrolled epimorphism in [MATH] and suppose [MATH] is the kernel of [MATH] in [MATH] . If [MATH] is [MATH] -graded and [MATH] is lean/split with respect to the standard [MATH] -filtration then both [MATH] and [MATH] are [MATH] -graded. |
Proof. The [MATH] -grading for [MATH] is given by [MATH] , where [MATH] is a chosen set of filtration data for [MATH] The argument for Lemma 3.4.5 applies directly to show that [MATH] is indeed a grading of [MATH] , and the assignment [MATH] gives a [MATH] -grading of [MATH] |
This allows to characterize admissible monomorphisms in [MATH] 3.4.7 Proposition The inclusion of a subobject [MATH] in [MATH] is an admissible monomorphism if and only if [MATH] is lean/split. |
Proof. Let [MATH] be a cokernel of [MATH] in [MATH] . As verified in Lemma 3.4.5 [MATH] is lean/split and insular and is, in fact, a cokernel of [MATH] in [MATH] From Proposition 3.4.6 [MATH] is graded, so it is also a cokernel of [MATH] in [MATH] |
In the case [MATH] and [MATH] is a non-positive function, we will use notation [MATH] for the subset [MATH] of [MATH] . The following is a direct analogue of Corollary 2.4.8 with exactly the same proof. |
3.4.8 Corollary Given an object [MATH] in [MATH] and a subset [MATH] from the family [MATH] , there is a set of enlargement data [MATH] and an admissible subobject [MATH] in [MATH] with the property that [MATH] If [MATH] is [MATH] -lean/split then the quotient [MATH] of the inclusion has the property that [MATH] |
Now we can summarize the preceding results. 3.4.9 Theorem Given a graded object [MATH] in [MATH] and a subset [MATH] from the family [MATH] we assume that [MATH] is [MATH] -split and [MATH] -insular and is graded by [MATH] The submodule [MATH] has the following properties: |
(1) [MATH] is graded by [MATH] (2) [MATH] for some fixed enlargement data [MATH] (3) if [MATH] is the quotient of the inclusion [MATH] and [MATH] is [MATH] -lean/split, then [MATH] is supported on [MATH] |
(4) [MATH] Proof. Property (1) follows from Lemma 3.4.5 Properties (2) and (3) follow from Corollary 3.4.8 (4) follows from the fact that a [MATH] -insular filtered module is [MATH] -separated, in the sense that for any pair of subsets [MATH] and [MATH] of [MATH] such that [MATH] we have |
[MATH] so [MATH] Now [EQUATION] but [MATH] , thus [MATH] 3.5. Localization fibration sequence We will use the localization theorem of Schlichting |
for Serre subcategories of exact categories. These techniques require the Serre subcategory to satisfy some additional assumptions that we verify next. |
3.5.1 Definition A class of morphisms [MATH] in an additive category [MATH] admits a calculus of right fractions if (1) the identity of each object is in [MATH] |
(2) [MATH] is closed under composition, (3) each diagram [MATH] with [MATH] can be completed to a commutative square [EQUATION] with [MATH] , and |
(4) if [MATH] is a morphism in [MATH] and [MATH] such that [MATH] then there exists [MATH] such that [MATH] In this case there is a construction of the localization |
[MATH] which has the same objects as [MATH] . The morphism sets [MATH] in [MATH] consist of equivalence classes of diagrams [EQUATION] |
with the equivalence relation [MATH] if there is a map [MATH] so that [MATH] and [MATH] . Let [MATH] denote the equivalence class of [MATH] . The composition of morphisms in [MATH] is defined by |
[MATH] where [MATH] and [MATH] fit in the commutative square [EQUATION] from axiom 3. 3.5.2 Proposition The localization [MATH] is a category. The morphisms of the form [MATH] where |
[MATH] are isomorphisms in [MATH] . The rule [MATH] gives a functor [MATH] which is universal among the functors making the morphisms [MATH] |
invertible. Proof. The proofs of these facts can be found in Chapter I of . The inverse of [MATH] is [MATH] From Proposition 3.3.3 , given a subset [MATH] of [MATH] , the category [MATH] is a Serre subcategory of [MATH] |
3.5.3 Proposition The restriction to [MATH] -gradings in [MATH] gives a full exact subcategory [MATH] which is a Serre subcategory of [MATH] |
Proof. One only needs to observe that if the modules at the ends of the exact sequence in the proof of Proposition 3.4.4 have [MATH] -gradings in [MATH] then the displayed grading of [MATH] shows that [MATH] is an object of [MATH] |
The following shorthand notation is convenient when the choice of [MATH] is clear. 3.5.4 Notation The category [MATH] is the exact subcategory of [MATH] -graded objects in [MATH] When the choice of the subset [MATH] is understood, we will use notation [MATH] for the Serre subcategory |
[MATH] of [MATH] 3.5.5 Definition Define the class of weak equivalences [MATH] in [MATH] to consist of all finite compositions of admissible monomorphisms with cokernels in [MATH] and admissible epimorphisms with kernels in [MATH] |
We need the class [MATH] to admits calculus of right fractions. This follows from 17 , Lemma 1.13] as soon as we prove the following fact. |
A Serre subcategory [MATH] of an exact category [MATH] is right filtering if each morphism [MATH] in [MATH] , where [MATH] is an object of [MATH] , factors through an admissible epimorphism [MATH] , where [MATH] is in [MATH] |
3.5.6 Lemma The subcategory [MATH] of [MATH] is right filtering. Proof. For a morphism [MATH] in [MATH] with [MATH] in [MATH] , we assume that both [MATH] and [MATH] are |
[MATH] -lean/split and [MATH] -insular. Suppose [MATH] is bounded by [MATH] and let [MATH] and [MATH] be a monotone function such that |
[EQUATION] Now for any characteristic set of data [MATH] for the grading [MATH] and any subset [MATH] we have [EQUATION] By part (3) of Theorem 3.4.9 |
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