text stringlengths 128 2.05k |
|---|
2.2. Properties of filtered objects In this section we will start imposing several conditions on objects in the bounded category [MATH] . These conditions are agnostic to the nature of [MATH] , and so we can use the simpler notation [MATH] for this category. These conditions and results about them will be used in the c... |
2.2.1 Definition Let [MATH] be an [MATH] -filtered [MATH] -module. [MATH] is called lean or [MATH] lean if there is a number [MATH] such that |
[EQUATION] for every subset [MATH] of [MATH] [MATH] is called split or [MATH] split if there is a number [MATH] such that we have |
[EQUATION] whenever a subset [MATH] of [MATH] is written as a union [MATH] [MATH] is called insular or [MATH] insular if there is a number [MATH] such that |
[EQUATION] for every pair of subsets [MATH] [MATH] of [MATH] 2.2.2 Proposition The properties of being lean, split, and insular are preserved under isomorphisms in [MATH] . Also, a [MATH] -lean filtered module is [MATH] -split. |
Proof. If [MATH] is an isomorphism with [MATH] , and [MATH] is [MATH] -lean, [MATH] -split, and [MATH] -insular, then [MATH] is [MATH] -lean, [MATH] -split, and [MATH] -insular. For the other statement, we have |
[EQUATION] since in general [MATH] 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 of the extreme terms belong to the collection. |
2.2.3 Lemma [MATH] Lean objects are closed under extensions. [MATH] Insular objects are closed under extensions. [MATH] Split objects are closed under extensions. |
Proof. For an exact sequence [MATH] in [MATH] , let [MATH] be a common filtration degree for [MATH] and [MATH] The first two statements follow from parts (1) and (2) of , Proposition 2.18] . It is shown there that if [MATH] and [MATH] are [MATH] -lean then [MATH] is [MATH] -lean. Also, if both [MATH] and [MATH] are [MA... |
To prove (3), suppose both [MATH] and [MATH] are [MATH] -split. We have [EQUATION] because in general [MATH] So [EQUATION] If [MATH] then we can write [MATH] where [MATH] and [MATH] Since [MATH] is an element of [MATH] , we have an element |
[EQUATION] such that [EQUATION] So [MATH] is [MATH] -split. 2.2.4 Lemma Let [EQUATION] be an exact sequence in [MATH] (1) If the object [MATH] is lean then [MATH] is lean. |
(2) If [MATH] is split then [MATH] is split. (3) If [MATH] is insular then [MATH] is insular. (4) If [MATH] is insular and [MATH] is lean then [MATH] is insular. |
(5) If [MATH] is insular and [MATH] is split then [MATH] is insular. (6) If [MATH] is split and [MATH] is insular then [MATH] is split. |
Proof. Let [MATH] be a common filtration degree for [MATH] and [MATH] If [MATH] is [MATH] -lean, [MATH] -split, or [MATH] -insular, it is easy to show that [MATH] is [MATH] -lean or [MATH] -split and [MATH] is [MATH] -insular respectively, which verifies (1), (2), and (3). |
Statement (4) follows from the proof of part (3c) of , Proposition 2.18] . It is shown there that if [MATH] is [MATH] -lean and [MATH] is [MATH] -insular then [MATH] is [MATH] -insular. The same proof actually shows statement (5). The only equation in that proof that uses [MATH] -leanness of [MATH] is only used to get ... |
(6) Suppose [MATH] is [MATH] -split and [MATH] is [MATH] -insular. Given [MATH] , we have [MATH] Now [MATH] , so we can write accordingly [MATH] Now [MATH] , because [MATH] Since [MATH] is [MATH] -insular, |
[EQUATION] so we are able to find [EQUATION] such that [MATH] , because generally [MATH] Thus [EQUATION] and [EQUATION] Let [MATH] and [MATH] , and we have [MATH] such that |
[EQUATION] so [MATH] is [MATH] -split. 2.2.5 Corollary Let [MATH] be an exact sequence in [MATH] If [MATH] is split and insular then [MATH] is insular if and only if [MATH] is split. |
Proof. This fact is the combination of parts (5) and (6) of the Lemma. 2.2.6 Remark The last Corollary is in contrast with the absence of the analogous general 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 metric s... |
, we have the following counterpart to part (6) of the Lemma: if [MATH] is lean and [MATH] is insular then [MATH] is lean. This fact is not needed in this paper. Here, the excision properties of the theory rely only on the properties of the cokernels. For the applications in |
, properties of the kernels become crucial in dealing with coherence issues, and the geometric conditions need to be imposed. 2.2.7 Definition |
We define [MATH] as the full subcategory of [MATH] on objects that are lean and insular with the induced exact structure. Similarly, [MATH] is the full subcategory of [MATH] on objects that are split and insular. |
Exact structures in [MATH] and [MATH] can be induced from [MATH] A full subcategory [MATH] of an exact category [MATH] is said to be closed under extensions or thick in [MATH] if |
(1) [MATH] contains the zero object, and (2) for any exact sequence [MATH] in [MATH] if [MATH] and [MATH] are isomorphic to objects from [MATH] then so is |
[MATH] It is known (cf. , Lemma 10.20] ) that a subcategory closed under extensions in [MATH] inherits the exact structure from [MATH] |
2.2.8 Theorem [MATH] and [MATH] are closed under extensions in [MATH] Therefore, [MATH] and [MATH] are exact subcategories of [MATH] so we have a sequence of exact inclusions |
[EQUATION] Proof. The first fact follows from parts (1) and (2) of Lemma 2.2.3 , the second from (2) and (3). 2.3. Local finiteness property |
Finally, there is an additional property that will consider only in module categories. 2.3.1 Definition An [MATH] -filtered [MATH] -module [MATH] is locally finitely generated if [MATH] is a finitely generated [MATH] -module for every bounded subset [MATH] |
The category [MATH] is the full subcategory of [MATH] on the locally finitely generated objects. Similarly, the companion category [MATH] is the full subcategory of [MATH] on the locally finitely generated objects. |
2.3.2 Theorem The category [MATH] is closed under extensions in [MATH] Similarly, the category [MATH] is closed under extensions in [MATH] |
Proof. If [MATH] is an isomorphism with [MATH] and [MATH] is locally finitely generated, then [MATH] are finitely generated submodules of [MATH] for all bounded [MATH] , since [MATH] is a Noetherian ring. Suppose |
[EQUATION] is an exact sequence and let [MATH] be a common filtration degree for both [MATH] and [MATH] Assume that [MATH] and [MATH] are locally finitely generated. For every bounded subset [MATH] the restriction [MATH] is an epimorphism onto a submodule of the finitely generated [MATH] -module |
[MATH] . The kernel of [MATH] is a submodule of [MATH] , which is also finitely generated. So the extension [MATH] is finitely generated. |
2.3.3 Corollary [MATH] and [MATH] are exact categories. The additive category [MATH] of geometric [MATH] -modules with the split exact structure is an exact subcategory of [MATH] , so there is a sequence of exact inclusions |
[EQUATION] 2.3.4 Remark We want to briefly explain the roles played by the two conditions, lean and split, that distinguish the two categories [MATH] and [MATH] [MATH] was used exclusively in |
, where it was proven to have good excision properties. There is a separate important issue of homological coherence that still requires the lean condition for its resolution, cf. |
The setting with the split condition in [MATH] is much more streamlined for the excision arguments but has insufficient coherence properties. In the next section, we will pursue the goal of combining the two different conditions in the “base” and “fibre” in order to achieve required coherence in the base and “fibrewise... |
Recall that a morphism [MATH] is an idempotent if [MATH] . Categories in which every idempotent is the projection onto a direct summand of [MATH] are called idempotent complete |
2.3.5 Proposition [MATH] and [MATH] are idempotent complete. Proof. First note that a regular preabelian category is idempotent complete. The proof is exactly the same as for an abelian category: if [MATH] is an idempotent then its kernel is split by [MATH] Since the restriction of an idempotent [MATH] to the image of ... |
[MATH] is in fact a splitting in [MATH] or [MATH] Finally, we need to address (the lack of) inheritance in filtered modules. It is immediate that a submodule of an insular filtered module is also insular with respect to the standard filtration induced on the submodule. However, a submodule of a lean filtered module is ... |
2.3.6 Definition An [MATH] -filtered object [MATH] is called strict if there exists an order preserving function [MATH] such that for every [MATH] the submodule [MATH] is |
[MATH] -lean and [MATH] -insular with respect to the standard [MATH] -filtration [MATH] It is important to note that this property is not preserved under isomorphisms, so the subcategory of strict objects is not essentially full in [MATH] |
The bounded category [MATH] was defined in as the full subcategory of [MATH] on objects isomorphic to strict objects. Now this category is closed under exact extensions in [MATH] according to , Theorem 2.22] and so is an exact category. |
A consequence of strictness, or more generally being isomorphic to a strict object, is the following feature. Given a filtered module [MATH] in [MATH] , a lean grading of [MATH] is a functor |
[MATH] from the power set of [MATH] to the submodules of [MATH] such that (1) each [MATH] is an object of [MATH] when given the standard filtration, |
(2) there is a number [MATH] such that [EQUATION] for all subsets [MATH] of [MATH] Clearly, each [MATH] is an object of [MATH] Also an actual strict object has a lean grading by [MATH] with [MATH] |
We note for the interested reader that the theory in , including the excision theorems, could be alternately developed for modules with lean gradings in place of [MATH] We do not require such theory in this paper. Instead, we develop a similar but more relaxed notion of gradings in [MATH] |
2.4. Graded objects and their closure properties 2.4.1 Definition Given a filtered module [MATH] in [MATH] , a grading of [MATH] is a functor |
[MATH] such that (1) each [MATH] is an object of [MATH] when given the standard filtration, (2) there is a number [MATH] so that |
[EQUATION] for all subsets [MATH] of [MATH] We will say that a filtered module [MATH] is graded if it is possible to equip it with a grading, but there is no specific choice of grading that is specified. |
2.4.2 Proposition The graded objects are closed under isomorphisms. Proof. If [MATH] is an isomorphism and [MATH] has a grading [MATH] , a grading for [MATH] is given by |
[MATH] where [MATH] is a filtration bound for [MATH] 2.4.3 Definition We define [MATH] as the full subcategory of [MATH] on the locally finitely generated graded filtered modules. |
2.4.4 Proposition [MATH] is closed under extensions in [MATH] Therefore [MATH] is an exact subcategory of [MATH] Proof. Given an exact sequence [MATH] in [MATH] let [MATH] be a common filtration degree for both [MATH] and [MATH] as boundedly bicontrolled maps, and assume that |
[MATH] and [MATH] are graded modules in [MATH] with the associated functors [MATH] and [MATH] To define a grading for [MATH] consider a subset [MATH] and suppose [MATH] |
is [MATH] -split and [MATH] -insular. The induced epimorphism [MATH] extends to another epimorphism [EQUATION] with [MATH] Without loss of generality, suppose [MATH] is [MATH] -split and [MATH] -insular. We define |
[EQUATION] From parts (2) and (3) of Lemma 2.2.3 , the module [MATH] with the standard filtration is [MATH] -split and [MATH] -insular. Since [MATH] , we have [MATH] On the other hand, if the grading [MATH] has characteristic number [MATH] then |
[MATH] The last fact together with Theorem 2.3.2 shows that [MATH] is finitely generated. The relations between the categories in this section can be summarized as a commutative diagram of fully exact inclusions |
[EQUATION] The advantage of working with the category [MATH] is that one can readily localize to geometrically defined subobjects. |
2.4.5 Lemma Suppose [MATH] is a graded [MATH] -filtered module with a grading [MATH] . Let [MATH] be a submodule which is split with respect to the standard filtration. Then [MATH] is a grading of [MATH] |
We will call the grading of a split submodule [MATH] obtained in Lemma 2.4.5 the standard grading of the submodule. Proof. Of course, [MATH] On the other hand, there is [MATH] such that [MATH] , so [MATH] |
Consider the inclusion of modules [MATH] , and take the quotient [MATH] Both [MATH] and [MATH] are split and insular, so [MATH] is split and insular by parts (2) and (4) of Lemma 2.2.4 , with respect to the quotient filtration. We define [MATH] as the partial image [MATH] and give [MATH] the standard filtration in [MAT... |
This can be promoted to the following result. 2.4.6 Proposition Given a boundedly bicontrolled epimorphism [MATH] in [MATH] , suppose [MATH] is a submodule of [MATH] which is the kernel of [MATH] in [MATH] . It is given the standard filtration. If [MATH] is graded and [MATH] is split then both [MATH] and [MATH] are gra... |
Proof. The grading for [MATH] is given by [MATH] , where [MATH] is a chosen bicontrol bound for [MATH] Each [MATH] is split and insular as in the proof of Lemma 2.4.5 The inclusions [MATH] and |
[MATH] show that [MATH] is a grading. The same argument as in Lemma 2.4.5 shows that [MATH] gives a grading for [MATH] Applying this fact, we are able to characterize admissible monomorphisms in [MATH] as follows. |
2.4.7 Proposition The inclusion of a subobject [MATH] in [MATH] is an admissible monomorphism if and only if [MATH] is split. Proof. |
The cokernel [MATH] of [MATH] in [MATH] has the filtration described in the proof of Theorem 2.1.7 . From parts (2) and (5) of Lemma 2.2.4 [MATH] is split and insular. In fact, [MATH] is a cokernel of [MATH] in [MATH] From Proposition 2.4.6 [MATH] is graded, so it is also a cokernel of [MATH] in [MATH] |
We will use the following convention. When [MATH] , the notation [MATH] will stand for the subset [MATH] 2.4.8 Corollary Given an object [MATH] in [MATH] and a subset [MATH] of [MATH] , there is a number [MATH] and an admissible subobject [MATH] in [MATH] with the property that [MATH] Moreover, the cokernel [MATH] has ... |
Proof. The object [MATH] has a grading [MATH] with a characteristic constant [MATH] Property (1) in Definition 2.4.1 guarantees that [MATH] is a split object for any subset [MATH] For the first statement, choose [MATH] with the grading defined in Lemma 2.4.5 and apply Proposition 2.4.7 The second statement is shown as ... |
[EQUATION] Using the decomposition [MATH] we can write [EQUATION] The last three results can be summarized as follows. 2.4.9 Theorem |
Given a graded object [MATH] in [MATH] and a subset [MATH] of [MATH] we assume that [MATH] is [MATH] -split and [MATH] -insular and is graded by [MATH] The submodules [MATH] have the following properties: |
(1) each [MATH] is graded by [MATH] (2) [MATH] for some fixed number [MATH] (3) suppose [MATH] is the cokernel of the inclusion [MATH] , then [MATH] is supported on [MATH] |
(4) [MATH] Proof. Properties (1), (2), (3) are consequences of the last four results. (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] such that [MATH] we have |
[MATH] so [MATH] Now [MATH] , but [MATH] , thus [MATH] 2.4.10 Remark Functoriality properties in controlled theories are well-understood. As expected, bounded [MATH] -theory is covariantly functorial in both variables in the sense that [MATH] is a covariant functor from the category of proper metric spaces and uniforml... |
3. Fibred bounded -theory 3.1. Introduction of fibred control in -theory Suppose [MATH] and [MATH] are two proper metric spaces and [MATH] is a Noetherian ring. The product [MATH] is given the product metric |
[MATH] There is certainly the semi-abelian category [MATH] , the exact category [MATH] and, further, the bounded category [MATH] |
We wish to construct a larger fibred bounded category [MATH] The result will involve a mix of features from [MATH] and [MATH] and contain [MATH] as an exact subcategory. |
One definition can be made by simply imitating [MATH] -theory with fibred control as described in Remark 2.1.1 . It is obtained as an iterated construction from the end of section 2.1 on the level of unrestricted category as [MATH] From Theorem 2.1.8 [MATH] is a complete semi-abelian category with complete semi-abelian... |
3.1.1 Definition Given an [MATH] -module [MATH] , an [MATH] -filtration of [MATH] is a functor [MATH] from the power set of the product metric space to the partially ordered family of [MATH] -submodules of [MATH] Whenever [MATH] is given a filtration, and there is no ambiguity, we will denote the values [MATH] by [MATH... |
in the sense that the value on the empty subset is [MATH] The associated [MATH] -filtered [MATH] -module [MATH] is given by [EQUATION] |
Similarly, for each subset [MATH] , one has the [MATH] -filtered [MATH] -module [MATH] given by [EQUATION] In particular, [MATH] |
We will use the following notation generalizing enlargements in a metric space. 3.1.2 Notation Given a subset [MATH] of [MATH] and a function [MATH] , let |
[EQUATION] If in addition we are given a number [MATH] then [EQUATION] So [MATH] . Notice that if [MATH] is a single point [MATH] then |
[EQUATION] More generally, one can equivalently write [EQUATION] If [MATH] is a product set [MATH] , it will be convenient to use the notation [MATH] in place of [MATH] More generally, because the roles of the factors are very different when working with [MATH] -filtrations, we will use the notation [MATH] for the prod... |
3.1.3 Definition We will refer to the pair [MATH] in the notation [MATH] as the enlargement data It is clear that when [MATH] [MATH] for any function [MATH] under the identification [MATH] |
3.1.4 Notation Let [MATH] be a chosen fixed point in [MATH] Given a monotone function [MATH] , there is a function [MATH] defined by |
[EQUATION] 3.1.5 Definition Given two [MATH] -filtered modules [MATH] and [MATH] , an [MATH] -homomorphism [MATH] is boundedly controlled if there are a number [MATH] and a monotone function [MATH] such that |
[EQUATION] for all subsets [MATH] and some choice of [MATH] It is clear that the condition is independent of the choice of [MATH] |
The unrestricted fibred bounded category [MATH] has [MATH] -filtered modules as objects and the boundedly controlled homomorphisms as morphisms. |
3.1.6 Theorem [MATH] is a cocomplete semi-abelian category. First we require a very useful fact. A morphism [MATH] in [MATH] is boundedly bicontrolled if there is filtration data [MATH] and [MATH] as in Definition 3.1.5 , and in addition to ( [MATH] ) one also has the containments |
[MATH] In this case, we will use the notation [MATH] 3.1.7 Lemma Let [MATH] [MATH] be in [MATH] and [MATH] (1) If [MATH] [MATH] are boundedly bicontrolled morphisms and either [MATH] is an epi or [MATH] is a monic, then [MATH] is also boundedly bicontrolled. |
(2) If [MATH] [MATH] are boundedly bicontrolled and [MATH] is epic then [MATH] is also boundedly bicontrolled; if [MATH] is only boundedly controlled then [MATH] is also boundedly controlled. |
(3) If [MATH] [MATH] are boundedly bicontrolled and [MATH] is monic then [MATH] is also boundedly bicontrolled; if [MATH] is only boundedly controlled then [MATH] is also boundedly controlled. |
Proof. Suppose [MATH] and [MATH] for [MATH] , then in fact [MATH] in each of the three cases. For example, there are factorizations |
[EQUATION] which verify part 2 with [MATH] [MATH] Proof of Theorem 3.1.6 The additive properties are inherited from [MATH] , so the biproduct is given by the filtration-wise operation |
[MATH] in [MATH] . For any boundedly controlled morphism [MATH] , the kernel of [MATH] in [MATH] has the standard [MATH] -filtration [MATH] where |
[MATH] which gives the kernel of [MATH] in [MATH] . The canonical monic [MATH] has filtration data [MATH] and is therefore boundedly bicontrolled. It follows from part 3 of Lemma |
3.1.7 that [MATH] has the universal properties of the kernel in [MATH] Similarly, let [MATH] be the standard [MATH] -filtration of the image of |
[MATH] in [MATH] by [MATH] Then there is a presheaf [MATH] over [MATH] with [MATH] for [MATH] Of course [MATH] is the cokernel of [MATH] in [MATH] . Consider an [MATH] -filtered object [MATH] associated to [MATH] given by |
[MATH] The canonical morphism [MATH] gives a boundedly bicontrolled morphism [MATH] of filtration [MATH] since [EQUATION] This in conjunction with part 2 of Lemma 3.1.7 also verifies the universal cokernel properties of [MATH] and [MATH] in [MATH] |
We mention one useful perspective on [MATH] 3.1.8 Proposition Suppose [MATH] in [MATH] is boundedly controlled with control data [MATH] Then |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.