K -theory of locally compact modules over orders
Oliver Braunling, Ruben Henrard, Adam-Christiaan van Roosmalen
aa r X i v : . [ m a t h . K T ] J un K -THEORY OF LOCALLY COMPACT MODULES OVER ORDERS OLIVER BRAUNLING, RUBEN HENRARD, AND ADAM-CHRISTIAAN VAN ROOSMALEN
Abstract.
We present a quick approach to computing the K -theory of the category of locally compactmodules over any order in a semisimple Q -algebra. We obtain the K -theory by first quotienting outthe compact modules and subsequently the vector modules. Our proof exploits the fact that the pair(vector modules plus compact modules, discrete modules) becomes a torsion theory after we quotient outthe finite modules. Treating these quotients as exact categories is possible due to a recent localizationformalism. Contents
1. Introduction 12. Localizations of exact categories 22.1. One-sided exact categories 22.2. Strictly percolating subcategories 32.3. Quotients by strictly percolating subcategories 33. Structure theory of locally compact modules over an order 44. K -theory of locally compact modules over an order 54.1. The localization Q C : LCA A → LCA A / LCA A , C Q R : LCA A / LCA A , C → F F ≃
Mod A / mod A Introduction
Suppose A is a finite-dimensional semisimple Q -algebra and A ⊂ A is any Z -order. We write mod( − )for the category of finitely generated right modules, A R for A ⊗ Q R , and LCA A for the exact category oflocally compact topological A -modules [8]. We give a new proof for the following theorem. Theorem 1.1.
For every localizing invariant K : Cat Ex ∞ → D (where D is any stable ∞ -category), thereis a canonical fiber sequence: K (mod( A )) → K (mod( A R )) → K ( LCA A ) , where the first map is induced by the natural embedding − ⊗ Z R : A → A R = A R . For localizing invariants and Cat Ex ∞ we refer to the framework and notation of [2]. The principalexample is non-connective K -theory taking values in spectra (and we will indicatively always denote theinvariant by K throughout the paper). When convenient, and for example in the above statement, wewrite K ( C ) even if C is an exact (or one-sided exact) category, for K ( D b ∞ ( C )), where D b ∞ ( C ) is the stable ∞ -category of bounded complexes attached to C .The first theorem of the above kind is due to Clausen [5, Theorem 3.4], who proved it in the special case A = Q and A = Z (with an additional, but ultimately inconsequential, restriction to second-countabletopologies) with an eye to applications in class field theory. To this end, he set up a cone constructionon the level of stable ∞ -categories. The above version stems from [3, Theorem 11.4]. It was basedon Schlichting’s Localization Theorem from [13]. In our new approach, we use the recent LocalizationTheorem of [6, 7], which uses the additional flexibility of one-sided exact categories. These devices Date : June 22, 2020.1991
Mathematics Subject Classification.
Key words and phrases.
Locally compact modules, K -theory, exact category.The first author was supported by DFG GK1821 “Cohomological Methods in Geometry”.The third author was supported by FWO (12.M33.16N). can be thought of as convenient tools to avoid having to handle the underlying stable ∞ -categories (ortriangulated categories) manually.The proof of the main theorem is given in §
4. We start by considering the quotient of
LCA A bythe subcategory of compact modules LCA A , C . While this subcategory does not satisfy the s -filteringconditions of Schlichting’s Localization Theorem, it does satisfy the conditions of the recent LocalizationTheorem of [6, 7]. The latter theory endows the quotient E := LCA A / LCA A , C with the structure of aone-sided exact category (in the sense of [1, 12]), which can then canonically be embedded in its exacthull E ex . It is from this exact hull that we take a further quotient, this time by the subcategory V of vector modules of LCA A . Finally, we show that the resulting category F := E ex / V is equivalent toMod A / mod A ; it is from this equivalence that we obtain the sequence in Theorem 1.1.The equivalence F ≃
Mod A / mod A is based on the universal properties of the aforementioned quo-tients and the exact hull, as well as on the following observation (see Theorem 3.8): after quotienting thefinite modules out, the Structure Theorem of LCA A (Theorem 3.4) implies that the pair ( LCA A , C R , LCA A , D )becomes a torsion pair. This means that the sequences “ M compact ⊕ M vector M ։ M discrete ” given bythe Structure Theorem become essentially unique in the quotient.Using these new tools, our proof of Theorem 1.1 is considerably shorter and technically less involved.We never seriously leave the world of 1-categories, do not use the ∞ -categorical cone construction of [5],nor do we need the tedious verifications of Schlichting’s left/right s -filtering conditions done in [3].2. Localizations of exact categories
This section is preliminary in nature. We summarize the results of [6, 7] about localizations of exactcategories. One salient feature of the localizations we consider is that the resulting category need not beexact, but will be one-sided exact (in the sense of [1, 12]).2.1.
One-sided exact categories.Definition 2.1. A conflation category is an additive category C together with a chosen class of kernel-cokernel pairs (closed under isomorphisms), called conflations . The kernel part of a conflation is calledan inflation and the cokernel part of a conflation is called a deflation . We depict inflations by anddeflations by ։ .An additive functor F : C → D between conflation categories is called conflation-exact if conflationsare mapped to conflations.
Definition 2.2.
A conflation category E is called an inflation-exact or left exact category if E satisfiesthe following axioms: L0 For each X ∈ E , the map 0 → X is an inflation. L1 The composition of two inflations is again an inflation. L2 The pushout of any morphism along an inflation exists, moreover, inflations are stable underpushouts.Dualizing the above axioms yields the notion of a deflation-exact category. A
Quillen exact category issimply a conflation category which is both inflation-exact and deflation-exact by [9, Appendix A].Let E be one-sided exact category. Analogous to exact categories, one can define the bounded derivedcategory D b ( E ) as the Verdier localization K b ( E ) / h Ac ( E ) i thick of the bounded homotopy category by thethick closure of the triangulated subcategory of acyclic complexes (see [1, Corollary 7.3]). The canonicalembedding i : E → D b ( E ), mapping objects to stalk complexes in degree zero, is a fully faithful embeddingmapping conflations to triangles. We write D b ∞ ( E ) for the corresponding stable ∞ -category; in particular,the homotopy category of D b ∞ ( E ) recovers D b ( E ).The derived category allows a construction of the exact hull E ex of E (see [6]): the exact hull is givenby the extension closure of E in the (bounded) derived category D b ( E ). A sequence X → Y → Z in E ex is a conflation if and only if it fits in a triangle X → Y → Z → Σ X. The following proposition is shownin [6].
Proposition 2.3.
Let j : E → E ex be the embedding of an inflation-exact category in its exact hull.(1) The embedding j : E → E ex is -universal among conflation-exact functors to exact categories.(2) The embedding j lifts to an equivalence D b ∞ ( E ) ≃ → D b ∞ ( E ex ) of stable ∞ -categories. -THEORY OF LOCALLY COMPACT MODULES 3 Strictly percolating subcategories.Definition 2.4.
Let E be an exact category. A full subcategory A ⊆ E is called a strictly inflation-percolating subcategory if the following properties are satisfied: A1 The category A is a Serre subcategory of E , i.e. for any conflation X Y ։ Z in E , we have Y ∈ A ⇔ X, Z ∈ A . A2 Every morphism f : A → X with A ∈ A is strict , i.e. factors as A ։ im( f ) X , and im( f ) ∈ A . Remark 2.5. If A ⊆ E is a strictly inflation-percolating subcategory, then A is a fully exact abeliansubcategory of E .The following observation will be of use later. Proposition 2.6.
Let E be an inflation-exact category and let V ⊆ E be a full additive subcategorysatisfying axioms A1 and A2 . If every object of V is injective in E , then V is a strictly inflation-percolating subcategory of the exact hull E ex .Proof. As each V ∈ V ⊆ E is injective, Hom E ( − , V ) is exact and hence Ext E ( − , V ) = 0 . As E ex is theextension-closure of E in E ex , it follows that V is injective in E ex as well.Note that E ex = S n ≥ E n where E = E and E n for n ≥ E n − . As V ⊆ E ex consists of injective objects, it follows that V ⊆ E ex is an extension-closedsubcategory.We now show axiom A2 . Let f : V → Y be a map in E ex with V ∈ V . By definition, there is an n such that Y ∈ E n . We proceed by induction on n ≥
0. If n = 0, the result follows as V ⊆ E satisfiesaxiom A2 . If n ≥
1, then there is a conflation X ι Y ρ ։ Z in E ex with X, Z ∈ E n − . By the inductionhypothesis, the composition ρ ◦ f factors as V p ։ V ′′ h Z with V ′′ ∈ V . As V ⊆ E ex is an abeliansubcategory, ( V ′ :=) ker( p ) ∈ V . Note that there is an induced map g : V ′ → X such that f i = ιg . Againthe induction hypothesis yields that g factors as V ′ g ′ ։ U g ′′ X with U ∈ V . Taking the pushout of g ′ along i in E ex yields the following commutative diagram (where the rows are conflations): V ′ / / i / / g ′ (cid:15) (cid:15) (cid:15) (cid:15) V p / / / / f ′ (cid:15) (cid:15) (cid:15) (cid:15) V ′′ U / / i ′ / / (cid:15) (cid:15) g ′′ (cid:15) (cid:15) W p ′ / / / / f ′′ (cid:15) (cid:15) V ′′ (cid:15) (cid:15) h (cid:15) (cid:15) X / / ι / / Y ρ / / / / Z Here the upper-left square is bicartesian, f ′′ f ′ = f and W ∈ V as V ⊆ E ex is extension-closed. It followsfrom [4, Corollary 3.2] that f ′′ is an inflation in E ex . This shows axiom A2 .To show axiom A1 , it remains to show that given a conflation X ι V ρ ։ Z in E ex with V ∈ V , X, Z belong to V as well. By axiom A2 , ρ is admissible with image in V . It follows that Z ∈ V . As X is thekernel of a morphism in V and V is an abelian subcategory of E , we know that X belongs to V as well.This concludes the proof. (cid:3) Quotients by strictly percolating subcategories.
The next definition is based on [13, Defini-tion 1.12].
Definition 2.7.
Let E be an inflation-exact category and let A ⊆ E be a strictly inflation-percolatingsubcategory. A morphism f : X → Y in E is called a weak A -isomorphism (or simply a weak isomorphism )if f is strict and ker( f ) , coker( f ) ∈ A . The set of weak A -isomorphisms is denoted by S A .The following theorem summarizes the main results of [6, 7]. Theorem 2.8.
Let A be a strictly inflation-percolating subcategory of an exact category E .(1) The set S A of weak A -isomorphisms is a left multiplicative system.(2) The natural localization functor Q : E → E [ S − A ] endows the localization E [ S − A ] with an inflation-exact structure such that Q preserves and reflects conflations.(3) The localization functor Q is also a quotient in the category of conflation categories, i.e. it satisfiesthe following universal property: if F : E → F is a conflation-exact functor between conflationcategories such that F ( A ) = 0 , then F factors uniquely through Q via a conflation-exact functor F ′ : E / A = E [ S − A ] → F . OLIVER BRAUNLING, RUBEN HENRARD, AND ADAM-CHRISTIAAN VAN ROOSMALEN
Moreover, the localization sequence A ֒ → E Q → E / A induces a Verdier localization sequence on thebounded derived categories D b A ( E ) → D b ( E ) → D b ( E / A ) where D b A ( E ) is the thick subcategory of D b ( E ) generated by A under the canonical embedding E ֒ → D b ( E ) .If A has enough E -injectives, then the natural embedding D b ( A ) ֒ → D b A ( E ) is a triangle equivalence,and there is an exact sequence in Cat Ex ∞ : D b ∞ ( A ) → D b ∞ ( E ) → D b ∞ ( E / A ) . Structure theory of locally compact modules over an order
Let
LCA be the exact category of locally compact abelian groups, cf. [8]. Let A denote a finite-dimensional semisimple Q -algebra and A ⊂ A is a Z -order, i.e. a subring of A which is a finitely generated Z -module such that Q · A = A . In this section, we have a closer look at the category LCA A of locallycompact right A -modules. Definition 3.1.
We define the category
LCA A of locally compact right modules over A as follows:(1) An object M ∈ LCA A is a right A -module such that the additive group ( M, +) ∈ LCA is a locallycompact group and such that right multiplication by any α ∈ M is a continuous endomorphismof ( M, +).(2) A morphism f : M → N is a continuous right A -module map.The following theorem is standard, see [3, Proposition 3.4 and Lemma 3.6] (based on the earlier [8]). Theorem 3.2.
The category
LCA A is a quasi-abelian category; the inflations are given by closed injectionsand deflations are given by open surjections. Definition 3.3.
We consider the following subcategories of
LCA A :(1) LCA A , C denotes the full subcategory of compact A -modules.(2) LCA A , D denotes the full subcategory of discrete A -modules.(3) LCA A , R denotes the full subcategory of vector A -modules, i.e. those A -modules whose underlyinglocally compact abelian group is isomorphic to R n for some finite n ≥ LCA A , C R denotes the full subcategory of A -modules which are a direct sum of a compact and avector A -module.(5) LCA A , f denotes the full subcategory of finite A -modules.The Structure Theorem for locally compact abelian groups extends to A -modules in the following sense(see [3, Lemma 6.5]). Theorem 3.4 (Structure Theorem for locally compact modules) . For each M ∈ LCA A , there exists a(non-canonical) conflation C M ⊕ V M i M M p M ։ D M with C M ∈ LCA A , C , D M ∈ LCA A , D and V M ∈ LCA A , R . It is well known that vector groups are both injective and projective in
LCA (see for example [11,Corollary 3 to Theorem 3.3]). This result extends to A -modules (see [3, Theorem 5.13]). Theorem 3.5.
The vector A -modules are both injective and projective in LCA A . The next lemma will be useful later.
Lemma 3.6.
Let X ∈ LCA A , C R , thus X ∼ = C ⊕ V with C ∈ LCA A , C and V ∈ LCA A , R . (1) If X Y ։ Z is a conflation with Z ∈ LCA A , C , then Y ∈ LCA A , C R . (2) If f : X ։ Y is a deflation, then Y ∈ LCA A , C R .(3) If g : Y X is an inflation such that Y ∈ LCA A , D , then Y is a finitely generated A -module.Proof. (1) As V is injective, the conflation X Y ։ Z is a direct sum of conflations V V ։ C C ′ ։ Z . Since LCA A , C is closed under extensions, we find that C ′ ∈ LCA A , C , as required.(2) Applying the Structure Theorem to Y yields a conflation C ′ ⊕ V ′ Y ։ D , with D ∈ LCA A , D . From the previous statement, we see that it suffices to show that D ∈ LCA A , C (thus, D isfinite). Write h for the composition X ։ Y ։ D and consider the conflation ker h X ։ D. As X ∼ = C ⊕ V and Hom( V, D ) = 0, we see that this conflation is the direct sum of conflations V V ։ K C ։ D. As C is compact, we find that D is compact as well. -THEORY OF LOCALLY COMPACT MODULES 5 (3) It suffices to show that Y is finitely generated as an abelian group. As Y is a closed subgroupof X , the Pontryagin dual of [10, Chapter 2, Corollary 2 to Theorem 7] implies that Y is of theform R m ⊕ Z l ⊕ D with D a finite (discrete) group. As Y is discrete by assumption, we see that Y is a finitely generated group. (cid:3) We now interpret these results in the quotient category
LCA A := LCA A / LCA A , f . We write LCA A , D for the full subcategory of LCA A consisting of those objects which are discrete groups; the subcategory LCA A , C R of LCA A is defined similarly. Proposition 3.7.
The category
LCA A , f is a strictly two-sided percolating subcategory of LCA A . Moreover,the following hold:(1) the quotient
LCA A / LCA A , f is quasi-abelian,(2) the localization Q f : LCA A → LCA A / LCA A , f commutes with finite limits and colimits,(3) the subcategories LCA A , D and LCA A , C R of LCA A are closed under isomorphisms,(4) LCA A , D = LCA A , D / LCA A , f and LCA A , C R = LCA A , C R / LCA A , f , (5) any morphism X → Y in LCA A , D is strict.Proof. It is easy to verify that
LCA A , f is a strictly inflation- and deflation-percolating in LCA A . It followsfrom [7] that
LCA A / LCA A , f is quasi-abelian. As the set S LCA A , f is a left and a right multiplicative set, thelocalization Q f commutes with finite limits and colimits. For (3), recall from Theorem 2.8 that S LCA A , f is saturated. So, we can reduce to showing that for any weak isomorphism s : X ∼ → Y , we have that X ∈ LCA A , D (or in LCA A , C R ) if and only if Y ∈ LCA A , D (or in LCA A , C R ). As every weak isomorphismis a composition of inflations and deflations in (with cokernel and kernel in LCA A , f ), we can furthermoreassume that s is of this form. These cases are then easily handled separately.The last two statements follow easily from (3). (cid:3) Theorem 3.8 (Structure Theorem for
LCA A ) . The pair ( LCA A , C R , LCA A , D ) is a torsion pair in LCA A ,meaning that Hom(
LCA A , C R , LCA A , D ) = 0 and every M ∈ LCA A fits in a conflation C M i M M p M ։ D M with C M ∈ LCA A , C R and D M ∈ LCA A , D . Such a conflation is then unique up to unique isomorphism.Proof.
Directly from Theorem 3.4. (cid:3)
Corollary 3.9.
Let X f Y g ։ Z be a conflation in LCA A .(1) If Y ∈ LCA A , C R , then Z ∈ LCA A , C R .(2) If Y ∈ LCA A , D , then X, Z ∈ LCA A , D . (3) If Y ∈ LCA A , C R and X ∈ LCA A , D , then X is finitely generated.Proof. We only prove the first statement. The other statements can be proven in an analogous way. Let X ∼ ← X ′ f ′ → Y be a roof in LCA A representing f ; in LCA A , we have coker f ∼ = coker f ′ . It follows fromLemma 3.6 that coker f ′ ∈ LCA A , C R and hence coker f ∈ LCA A , C R by Proposition 3.7.(3). (cid:3) K -theory of locally compact modules over an order Throughout this section, A denotes a finite-dimensional semisimple Q -algebra and A ⊂ A is a Z -order.The aim of this section is to show Theorem 1.1 from the introduction. We proceed in four steps.4.1. The localization Q C : LCA A → LCA A / LCA A , C . The following proposition (see [6]) reduces thestudy of localizing invariants of
LCA A (such as non-connective K -theory) to that of the quotient category LCA A / LCA A , C , which we shall call E . Proposition 4.1.
The subcategory
LCA A , C ⊆ LCA A is a strictly inflation-percolating subcategory. Thequotient Q C : LCA A → E (= LCA A / LCA A , C ) induces an exact sequence of stable ∞ -categories D b ∞ ( LCA A , C ) → D b ∞ ( LCA A ) → D b ∞ ( E ) . As every object in D b ∞ ( LCA A , C ) can be trivialized using an Eilenberg swindle with infinite products, forany localizing invariant K , there is an equivalence K ( LCA A ) ≃ K ( E ) . OLIVER BRAUNLING, RUBEN HENRARD, AND ADAM-CHRISTIAAN VAN ROOSMALEN
The functor Q R : LCA A / LCA A , C → F . We now write V for the full additive subcategory of E generated by the vector A -modules. Our first goal is to show that V is a strictly inflation-percolatingsubcategory of E ex , the exact hull of E , so that we can consider the quotient F := E ex / V . We start withthe following lemma.
Lemma 4.2. (1) For any vector A -module V , the localization functor Q C induces a natural equivalence Q C : Hom LCA A ( − , V ) → Hom E ( Q C ( − ) , Q C ( V )) . In particular, it follows that V is injective in E .(2) The category V is equivalent to the category mod( A R ) where A R := A ⊗ Z R . In particular, V is afully exact abelian subcategory of E .Proof. (1) Let f ∈ Hom
LCA A ( X, V ) such that Q C ( f ) = 0. There exists a weak isomorphism t : V ∼ → Y in S LCA A , C (thus, with ker f ∈ LCA A , C ) such that t ◦ f = 0 in LCA A . As the only compactsubmodule of V is trivial, t is a monomorphism. It follows that f = 0 in LCA A . This showsHom LCA A ( X, V ) → Hom E ( X, V ) is an injection.To show that it is a surjection, let g ∈ Hom E ( X, V ) be represented by a roof X f → Y s ← V with s ∈ S LCA A , C . Note that s is an inflation (as s is strict and V only has the trivial compactsubmodule). As V is injective in LCA A , the inflation V s Y is a coretraction. Let t : Y ։ V bethe corresponding retraction, i.e. ts = 1 V . It follows that Q C ( t ◦ f ) = g . This shows the desiredbijection.As V is injective in LCA A , Hom LCA A ( − , V ) is an exact functor. Hence Hom E ( − , V ) is exactas well and thus V is injective in E (this characterization of injective objects remains valid forinflation-exact categories, see [6, Proposition 3.22]).(2) This follows immediately from the above equivalence. (cid:3) Proposition 4.3.
The category V is a strictly inflation-percolating subcategory of the exact hull E ex of E .Proof. By Proposition 2.6 and Lemma 4.2.(1), it suffices to show that
V ⊆ E satisfies axioms A1 and A2 .We first show axiom A2 . Let f ∈ Hom E ( V, X ) be represented by a roof V g → Y s ← X with s ∈ S LCA A , C .As the image of a connected space is connected, the Structure Theorem yields the following commutativediagram in LCA A : D Y V g / / : : h $ $ Y p Y O O O O X s o o C Y ⊕ V Y O O i Y O O Here V Y is a vector A -module. As the projection C Y ⊕ V Y ։ V Y is an isomorphism in E and the compo-sition V → V Y is strict, we know that V → C Y ⊕ V Y is strict in E . As E is an inflation-exact category,axiom L1 yields that the composition of inflations is an inflation. It follows that the composition i Y ◦ h is strict in E . This shows axiom A2 .We now show axiom A1 , i.e. that V is a Serre subcategory of E . Let X ι Y ρ ։ Z be a conflationin E . Assume that Y ∈ V . By axiom A2 , ρ is strict with image in V , thus Z ∈ V . As X ∼ = ker( ρ ) is thekernel of a morphism in V and V ⊆ E is an abelian subcategory by Lemma 4.2.(2), X ∈ V .Conversely, assume that X, Z ∈ V . By Lemma 4.2.(1), X is injective in E . It follows that the conflation( ι, ρ ) splits and thus Y ∼ = X ⊕ Z belongs to V . (cid:3) Corollary 4.4.
The quotient functor Q R : E ex → F (:= E ex / V ) induces a fibre sequence K ( V ) → K ( E ex ) → K ( F ) , where K is any localizing invariant.Proof. By Lemma 4.2, we know that V contains enough E ex -injective objects. The statement then followsfrom Theorem 2.8. (cid:3) -THEORY OF LOCALLY COMPACT MODULES 7 Q A : Mod A Q A ,f / / (cid:15) (cid:15) Mod A / fmod A Q ′ A / / R (cid:15) (cid:15) Mod A / mod A Φ (cid:15) (cid:15) Q C R : LCA A Q f / / LCA A / LCA A , f Q ′ / / D (cid:15) (cid:15) F Ψ (cid:15) (cid:15) Mod A / fmod A Q ′ A / / Mod A / mod A Figure 1.
Overview of the functors from Construction 4.8.
Proposition 4.5.
The functor Q C R : LCA A → E → E ex → F is 2-universal with respect to the conflation-exact functors F : LCA A → C with C exact and F ( LCA A , C R ) = 0 , thus the functor − ◦ Q C R : Fun( F , C ) → Fun(
LCA A , C ) is a fully faithful functor whose essential image consists of those F : LCA A → C for which F ( LCA A , C R ) = 0 .Proof. From combining each of the universal properties of
LCA A → E → E ex → F . (cid:3) Remark 4.6.
Note that in E , we have A ∼ = R ⊗ Z A ∈ V . In particular, Q C R ( A ) = 0 . Hence, Q C R sendsevery finitely generated discrete A -module to zero.4.3. The equivalence
F ≃
Mod A / mod A . In order to complete the proof of Theorem 1.1, we showthat
F ≃
Mod A / mod A (see Proposition 4.9). For this, consider the localization functors Q A : Mod A → Mod A / mod A and Q A ,f : Mod A → Mod A / fmod A , where fmod A is the full subcategory of Mod A consisting of finite A -modules. It follows from Proposition 3.7.(4) that Q f ( LCA A , D ) ≃ Mod A / fmod A . Moreover, the universal property of Q A ,f shows that there is a unique functor Q ′ A : Mod A / fmod A → Mod A / mod A such that Q A = Q ′ A ◦ Q A ,f .The torsion-free part functor D : LCA A → LCA A , D : M D M from Theorem 3.8 need not beconflation-exact. This can be seen by setting A = Z and starting from the conflation Z R ։ R / Z . However, we need not change much to obtain a conflation-exact functor.
Proposition 4.7.
The functor Q ′ A ◦ D : LCA A / LCA A , f → Mod A / mod A is conflation-exact.Proof. Let X Y ։ Z be a conflation in LCA A / LCA A , f . The Structure Theorem of
LCA A gives thefollowing commutative diagram C X / / / / (cid:15) (cid:15) (cid:15) (cid:15) X / / / / (cid:15) (cid:15) (cid:15) (cid:15) D Xg (cid:15) (cid:15) C Y / / / / Y / / / / D Y where the left vertical arrow is an inflation by the dual of [4, Proposition 7.6] and the rightmost verticalarrow is strict by Proposition 3.7.(5). Applying the Short Snake Lemma ([4, Corollary 8.13]), we findexact sequences ker g i C Y /C X → Z ։ coker g and ker g D X g → D Y ։ coker g. It follows from Corollary 3.9 that C Y /C X ∈ LCA A , C R and hence coker i ∈ LCA A , C R . Likewise, we findthat coker g ∈ LCA A , D . This shows that the conflation coker i Z ։ coker g is the torsion / torsion-freeconflation of Z from Theorem 3.8, hence coker g ∼ = D Z . Moreover, it follows from Corollary 3.9 that ker g is finitely generated and discrete. Hence, we find aconflation Q ′ A ( D X ) Q ′ A ( D Y ) ։ Q ′ A ( D Z ) in Mod A / mod A , as required. (cid:3) Construction 4.8.
We now construct the diagram given in Figure 1. We start with the rows. Thefunctor Q ′ A is the unique functor such that Q A = Q ′ A ◦ Q A ,f , and Q ′ is the unique functor such that Q C R = Q ′ ◦ Q f ; these are induced by the universal properties of Q A ,f and Q f , respectively.For the columns, the functor Mod A → LCA A is the functor mapping an A -module to the correspondingdiscrete A -module. The functor R is the unique functor making the top-left square commute (it existsby the universal property of Q A ,f ). The (essential) image of R corresponds to the torsion-free part of thetorsion pair in Theorem 3.8, hence R has a left adjoint D : LCA A / LCA A , f → Mod A / fmod A , given bymapping any object X ∈ LCA A / LCA A , f to its torsion-free part D X . By construction, D ◦ R ∼ = 1 . In the last column, the functor Φ : Mod A / mod A → F is the unique functor making the top rectanglecommute; it exists by the universal property of Q A : Mod A → Mod A / mod A (see Remark 4.6). Notethat Φ is also the unique functor such that Φ ◦ Q ′ A = Q ′ ◦ R. OLIVER BRAUNLING, RUBEN HENRARD, AND ADAM-CHRISTIAAN VAN ROOSMALEN
The functor Ψ :
F →
Mod A / mod A is a functor such that Ψ ◦ Q C R ∼ = Q ′ A ◦ D ◦ Q f ; it exists by theuniversal property of Q C R (see Proposition 4.5, note that Q ′ A ◦ D ◦ Q f is conflation-exact by Proposition4.7). Proposition 4.9.
The functors Ψ and Φ are quasi-inverses.Proof. For each M ∈ LCA , the map p M : M R ( D M ) corresponds to the unit of the adjunction D ⊣ R .As Q ′ ( p M ) is an isomorphism, we find that Φ ◦ Ψ ◦ Q C R ∼ = Q ′ ◦ R ◦ D ◦ Q f is isomorphic to Q C R = Q ′ ◦ Q f . It follows from the universal property of Q C R that 1 F → Φ ◦ Ψ is a natural equivalence.For the other direction, we start from Q ′ A ∼ = Q ′ A ◦ D ◦ R = Ψ ◦ Φ ◦ Q ′ A , so that the universal propertyof Q ′ A yields that Ψ ◦ Φ ∼ = 1, as required. (cid:3) Proof of Theorem 1.1.
We are now in a position to prove the main theorem.
Proof of Theorem 1.1.
Consider the essentially commutative diagrammod( A ) / / (cid:15) (cid:15) Mod( A ) Q A / / (cid:15) (cid:15) Mod( A ) / mod( A ) ≃ Φ (cid:15) (cid:15) V / / E ex / / F of functors, lifting to an essentially commutative diagram of the bounded derived ∞ -categories (wherethe rows are exact sequences). It was shown in Lemma 4.2.(2) that V ≃ mod A R ; the leftmost downwardsarrow is given by M R ⊗ Z M . This induces a bicartesian square of stable ∞ -categories D b ∞ (mod A ) / / (cid:15) (cid:15) D b ∞ (Mod A ) (cid:15) (cid:15) D b ∞ (mod A R ) / / D b ∞ ( E ex ) . Using the Eilenberg swindle with direct sums, shows that every object in D b ∞ (Mod A ) gets trivial-ized under a localizing invariant K : Cat Ex ∞ → A . Hence, for each such K , there is a fiber sequence K ( D b ∞ (mod A )) → K ( D b ∞ (mod A R )) → K ( D b ∞ ( E ex )) . Combining Theorem 2.3 and Proposition 4.1, wefind that K ( D b ∞ ( LCA A )) ≃ K ( D b ∞ ( E )) ≃ K ( D b ∞ ( E ex )) . Using our convention to suppress D b ∞ in the no-tation whenever convenient, this yields the required fiber sequence as formulated in the introduction. (cid:3) References [1] Silvana Bazzoni and Septimiu Crivei,
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E-mail address : [email protected] Ruben Henrard, Universiteit Hasselt, Campus Diepenbeek, Departement WNI, 3590 Diepenbeek, Belgium
E-mail address : [email protected] Adam-Christiaan van Roosmalen, Universiteit Hasselt, Campus Diepenbeek, Departement WNI, 3590 Diepen-beek, Belgium
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