CCYCLIC HOMOLOGY FOR BORNOLOGICAL COARSE SPACES
LUIGI CAPUTI
Abstract.
We define Hochschild and cyclic homologies for bornological coarse spaces: for afixed field k and group G , these are lax symmetric monoidal functors X HH Gk and X HC Gk fromthe category of equivariant bornological coarse spaces G BornCoarse to the cocomplete stable ∞ -category of chain complexes Ch ∞ . We relate these equivariant coarse homology theoriesto coarse algebraic K -theory X K Gk and to coarse ordinary homology X H G by constructing atrace-like natural transformation X K Gk → X H G that factors through coarse Hochschild (orcyclic) homology. We further compare the forget-control map for coarse Hochschild homologywith the associated generalized assembly map. Contents
Introduction 11. Equivariant coarse homotopy theory 42. Keller’s cyclic homology for dg-categories 103. Equivariant coarse Hochschild and cyclic homology 164. From coarse algebraic K -theory to coarse ordinary homology 21References 25 Introduction
Coarse geometry is the study of metric spaces from a large-scale point of view [Roe93,Roe96, Roe03]. A new axiomatic and homotopic approach to coarse geometry and coarsehomotopy theory has been recently developed by Bunke and Engel [BE16]. In this set-up,the main objects are called bornological coarse spaces [BE16, Def. 2.5] (for example, everymetric space is a bornological coarse space in a canonical way). In the equivariant setting, if G denotes a group, G -bornological coarse spaces are bornological coarse spaces with a G -actionby automorphisms [BEKW17, Def. 2.1]. Among various invariants of G -bornological coarsespaces we are interested in equivariant coarse homology theories , i.e., functors E : G BornCoarse → C from the category of G -bornological coarse spaces G BornCoarse to a cocomplete stable ∞ -category C , satisfying some additional axioms: coarse invariance, flasqueness, coarseexcision and u-continuity [BEKW17, Def. 3.10]. Examples of coarse homology theoriesarise as coarsifications of locally finite homology theories. Among others, there are coarseversions of ordinary homology and of topological K -theory [BE16], of equivariant algebraic K -homology and of topological Hochschild homology [BEKW17, BC17], and of Waldhausen’s A -theory [BKW18]. Date : July 8, 2019. a r X i v : . [ m a t h . K T ] J u l LUIGI CAPUTI
Hochschild and cyclic homologies have classically been defined as homology invariants ofalgebras, and have been then extended to invariants of dg-algebras, schemes, additive categoriesand exact categories [Kel99, McC94]. The aim of the paper is to give a concrete construction ofcoarse homology theories defining Hochschild and cyclic homology for bornological coarse spaces.These coarse homology theories can be defined abstractly by using a universal equivariant coarsehomology theory [BC17]. However, we choose to give a more concrete definition that might bemore suitable for computations (see, e.g., the construction of the natural transformation tocoarse ordinary homology, Theorem 4.8). We now explain the main results of the paper.Let k be field and let G be a group. We denote by C HH ∗ and C HC ∗ the chain complexescomputing Hochschild homology and cyclic homology (of k -algebras) respectively. The G -bornological coarse space G can , min denotes a canonical bornological coarse space associatedto the group G (see Example 1.2). Let Ch ∞ be the ∞ -category of chain complexes. Thefollowing is a combination of Theorem 3.11, Proposition 3.14 and Proposition 3.15: Theorem A.
There are lax symmetric monoidal functors X HH Gk : G BornCoarse → Ch ∞ and X HC Gk : G BornCoarse → Ch ∞ satisfying the following properties:(i) X HH Gk and X HC Gk are G -equivariant coarse homology theories;(ii) there are equivalences of chain complexes X HH Gk ( ∗ ) (cid:39) C HH ∗ ( k ) and X HC Gk ( ∗ ) (cid:39) C HC ∗ ( k ) between the evaluations of X HH Gk and X HC Gk at the one point bornological coarse space {∗} , endowed with the trivial G -action, and the chain complexes computing Hochschildand cyclic homology of k ;(iii) there are equivalences X HH Gk ( G can , min ) (cid:39) C HH ∗ ( k [ G ]; k ) and X HC Gk ( G can , min ) (cid:39) C HC ∗ ( k [ G ]; k ) of chain complexes between the evaluations at the G -bornological coarse space G can , min and the chain complexes computing Hochschild and cyclic homology of the k -algebra k [ G ] . The construction of the functors X HH Gk and X HC Gk uses a cyclic homology theory for dg-categories that satisfies certain additive and localizing properties in the sense of Tabuada [Tab07].This is Keller’s cone construction
Mix : dgcat → Mix , for dg-categories [Kel99], which is a functor from the category dgcat of small dg-categoriesto Kassel’s category Mix of mixed complexes [Kas87]. Hochschild and cyclic homologies fordg-categories are then defined in terms of mixed complexes, consistently with the classicaldefinitions for k -algebras [Kas87]. We consider the functor (with values to the category ofsmall k -linear categories Cat k ) V Gk : G BornCoarse → Cat k , that associates to every G -bornological coarse space X a suitable k -linear category V Gk ( X ) of G -equivariant X -controlled (finite dimensional) k -vector spaces [BEKW17, Def. 8.3]; a k -linearcategory is a dg-category in a standard way. We prove that the resulting functor X Mix Gk : G BornCoarse Cat k dgcat k Mix Mix ∞ V Gk ι Mix loc
YCLIC HOMOLOGY FOR BORNOLOGICAL COARSE SPACES 3 (see Definition 3.1) with values in the cocomplete stable ∞ -category of mixed complexes Mix ∞ is a coarse homology theory (Theorem 3.2). Coarse Hochschild X HH Gk and coarse cyclichomology X HC Gk are then defined by post-composition of the Hochschild and cyclic homologyfunctors for mixed complexes with the functor X Mix Gk (see Definition 3.10).Let Sp be the ∞ -category of spectra. Besides giving an explicit construction, our mainmotivation is a better understanding of the spectra-valued equivariant coarse algebraic K -homology X K Gk : G BornCoarse → Sp [BEKW17, Def. 8.8]. Algebraic K -theory comesequipped with trace maps ( e.g., the Dennis trace map from algebraic K -theory of rings toHochschild homology, or the refined version, the cyclotomic trace, from the algebraic K -theoryspectrum to the topological cyclic homology spectrum) and these trace maps have been offundamental importance in the understanding of algebraic K -theory [BHM93,DGM13]. Inspiredby the classical case, we define trace maps from equivariant coarse algebraic K -homology toequivariant coarse Hochschild and cyclic homology and from equivariant coarse Hochschild andcyclic homology to equivariant coarse ordinary homology X H G : G BornCoarse → Sp (seeProposition 4.10, Theorem 4.8 and Proposition 4.9): Theorem B. (1) The classical Dennis trace map induces a natural transformation ofequivariant coarse homology theories: K X Gk → X HH Gk ; (2) There is a natural transformation Φ X HH Gk : X HH Gk → X H G of G -equivariant coarse homology theories, which induces an equivalence of spectra whenevaluated at the one point space {∗} . By composition of these two natural transformations, we get a natural transformation K X Gk → X HH Gk −→ X H G from equivariant coarse algebraic K -homology to equivariant coarse ordinary homology. Thislast coarse homology theory is defined in terms of equivariant locally finite controlled maps X n +1 → k (see Definition 1.6) and it is easier to compute then coarse algebraic K -homology.We believe that the study of this transformation can be useful for the understanding anddetection of coarse K -theory classes. Conventions.
We freely employ the language of ∞ -categories. More precisely, we model ∞ -categories as quasi-categories [Cis, Lur09, Lur14]. When not otherwise specified, G willdenote a group, k a field, ⊗ the tensor product over k . Without further comments, we alwaysconsider an additive category as a dg-category in the canonical way (Example 2.3). Structure of the paper.
In Section 1 we review the basic definitions in coarse homotopytheory: bornological coarse spaces, coarse homology theories and the category of controlledobjects. In Section 2, we introduce the (cocomplete stable ∞ -category) of mixed complexes andKeller’s definition of cyclic homology. In Section 3 we define the functors X Mix Gk , X HH Gk and X HC Gk and we prove that they are equivariant coarse homology theories. In the last sectionSection 4, we construct the natural transformations from coarse algebraic K -homology to coarseHochschild homology and from coarse Hochschild homology to coarse ordinary homology. LUIGI CAPUTI Equivariant coarse homotopy theory
The main purpose of this section is to recollect notations and definitions describing thecategory G BornCoarse of G -equivariant bornological coarse spaces and of G -equivariant coarsehomology theories. We will freely use the terminology of [BE16, Sec. 2] and [BEKW17, Sec. 2& Sec. 3].1.1. Equivariant bornological coarse spaces. A bornology on a set X is a subset B ⊆ P ( X ) of the power set of X that is closed under taking subsets and finite unions, and such that X = ∪ B ∈B B .A coarse structure on a set X is a subset C ⊆ P ( X × X ) which contains the diagonal ∆ X := { ( x, x ) ∈ X × X | x ∈ X } and is closed under taking subsets, finite unions, inverses,and compositions. The elements of C are called entourages . If U is an entourage of a coarsespace X and B is any subset of X , the U -thickening of B is the subset of X :(1.1) U [ B ] := { x ∈ X | ∃ b ∈ B, ( x, b ) ∈ U } ⊆ X A bornology B and a coarse structure C on a set X are compatible if every controlledthickening of every bounded set B ∈ B is bounded. Definition 1.1. [BE16, Definition 2.5] A bornological coarse space is a triple ( X, C , B ) givenby a set X , a bornology B and a coarse structure C on X , such that B and C are compatible.Morphisms of bornological coarse spaces are maps for which pre-images of bounded sets arebounded and images of entourages are entourages. The category of bornological coarse spacesis denoted by BornCoarse .Let G be a group acting by automorphisms on a bornological coarse space X . The G -actionon X induces a G -action on the set of entourages of X . Let C G be the partially orderedsubset of C consisting of the set-wise G -fixed entourages. A G -bornological coarse space[BEKW17, Definition 2.1] is a bornological coarse space ( X, C , B ) equipped with a G -actionby automorphisms such that the set of invariant entourages C G is cofinal in C . We denoteby G BornCoarse the category of G -bornological coarse spaces and G -equivariant, propercontrolled maps. Example 1.2. (i) Let ( X, d ) be a metric space. The family B d := (cid:104){ B ( x, r ) | x ∈ X, r ≥ }(cid:105) is the bornology generated by the d -bounded balls B ( x, r ) . The family of subsets U r := { ( x, y ) | d ( x, y ) ≤ r ) } , for every r ≥ , generates a coarse structure that dependson the metric d denoted by C d . Then, the triple ( X, C d , B d ) is a bornological coarse space.(ii) Let G be a group, B min be the minimal bornology on its underlying set and let C can := (cid:104){ G ( B × B ) | B ∈ B min }(cid:105) be the coarse structure on G generated by the G -orbits. Thespace G can , min := ( G, C can , B min ) is a G -bornological coarse space.(iii) Let X be a G -bornological coarse space and let Z be a G -invariant subset of X . We definethe induced coarse structure and bornology on Z by restriction: C Z := { ( Z × Z ) ∩ U | U ∈ C} and B Z := { Z ∩ B | B ∈ B} . Then, Z X := ( Z, C Z , B Z ) is a G -bornological coarsespace and the inclusion Z (cid:44) → X is a morphism of G -bornological coarse spaces.(iv) Let X be a G -bornological coarse space and let U be a G -invariant entourage of X . Wedefine the induced coarse structure C U as the coarse structure on X generated by U . Thisis compatible with the bornology B , and X U := ( X, C U , B ) is a G -bornological coarsespace. YCLIC HOMOLOGY FOR BORNOLOGICAL COARSE SPACES 5
Equivariant coarse homology theories.
Equivariant coarse homology theories arefunctors from the category of G -bornological coarse spaces to a stable cocomplete ∞ -category C (e.g. the ∞ -category of chain complexes Ch or the ∞ -category of spectra Sp ), satisfyingadditional axioms: coarse invariance, flasqueness, coarse excision and u-continuity, that wenow recall.Let f , f : ( X, C , B ) → ( X (cid:48) , C (cid:48) , B (cid:48) ) be morphisms between bornological coarse spaces. Wesay that f and f are close to each other if the image of the diagonal ( f , f )(∆ X ) is anentourage of X (cid:48) . A morphism f : ( X, C , B ) → ( X (cid:48) , C (cid:48) , B (cid:48) ) is an equivalence of bornologicalcoarse spaces if there exists an inverse g : ( X (cid:48) , C (cid:48) , B (cid:48) ) → ( X, C , B ) such that the compositions g ◦ f and f ◦ g are close to the identity maps. In this case, the spaces X and X (cid:48) are called coarsely equivalent . Two morphisms between G -bornological coarse spaces are close to eachother if they are close as morphisms between the underlying bornological coarse spaces. Definition 1.3. [BEKW17, Def. 3.8] A G -bornological coarse space X is called flasque if itadmits a morphism f : X → X such that:(i) f is close to the identity map;(ii) for every entourage U , the subset (cid:83) k ∈ N ( f k × f k )( U ) is an entourage of X ;(iii) for every bounded set B in X there exists k such that f k ( X ) ∩ GB = ∅ . Definition 1.4. [BEKW17, Def. 3.5 & 3.7] Let ( X, C , B ) be a G -bornological coarse space.(1) A big family Y = ( Y i ) i ∈ I on X is a filtered family of subsets of X satisfying thefollowing: ∀ i ∈ I, ∀ U ∈ C , ∃ j ∈ I such that U [ Y i ] ⊆ Y j An equivariant big family is a big family consisting of G -invariant subsets.(2) A pair ( Z, Y ) consisting of a subset Z of X and of a big family Y on X is called a complementary pair if there exists an index i ∈ I for which Z ∪ Y i = X . It is an equivariant complementary pair if Z is a G -invariant subset and Y is an equivariantbig family.Let Z be a subset of X . If Y is a big family on X , then the intersection Z ∩ Y := ( Z ∩ Y i ) i ∈ I is a big family on Z .Let Y = ( Y i ) i ∈ I be a filtered family of G -invariant subsets of X . If E : G BornCoarse → C is a functor with values in a cocomplete ∞ -category C , we define the value of E at the family Y as the filtered colimit E ( Y ) := colim i ∈ I E ( Y i ) . There is an induced map from E ( Y ) to E ( X ) . Definition 1.5. [BEKW17, Definition 3.10] Let G be a group and let G BornCoarse bethe category of G -bornological coarse spaces. Let C be a cocomplete stable ∞ -category. A G - equivariant C -valued coarse homology theory is a functor E : G BornCoarse −→ C with the following properties:i. Coarse invariance: E sends equivalences X → X (cid:48) of G -bornological coarse spaces toequivalences E ( X ) → E ( X (cid:48) ) of C ;ii. Flasqueness: if X is a flasque G -bornological coarse space, then E ( X ) (cid:39) ; LUIGI CAPUTI iii.
Coarse excision: E ( ∅ ) (cid:39) , and for every equivariant complementary pair ( Z, Y ) on X ,the diagram E ( Z ∩ Y ) E ( Z ) E ( Y ) E ( X ) is a push-out square;iv. u-continuity: for every G -bornological coarse space ( X, C , B ) , the canonical morphisms X U → X induce an equivalence E ( X ) (cid:39) colim U ∈C G E ( X U ) .Examples of (equivariant) coarse homology theories are coarse ordinary homology (1.4) andcoarse topological K-theory [BE16], coarse algebraic K-theory (Definition 1.20) and coarsetopological Hochschild homology [BEKW17, BC17], coarse Hochschild and cyclic homology(Theorem 3.11).1.3. Coarse ordinary homology.
We recall the construction of coarse ordinary homology.More details are given in [BE16, Sec. 6.3] and [BEKW17, Sec. 7.1].Let X be a G -bornological coarse space, n ∈ N a natural number, B a bounded set of X , and x = ( x , . . . , x n ) a point of X n +1 . We say that x meets B if there exists an index i ∈ { , . . . , n } such that x i belongs to B . If U is an entourage of X , we say that x is U -controlled if, for each i and j in { , . . . , n } , the pair ( x i , x j ) belongs to U .An n -chain c on X is a function c : X n +1 → Z ; its support supp( c ) is defined as the set ofpoints for which the function c is non-zero:(1.2) supp( c ) = { x ∈ X n +1 | c ( x ) (cid:54) = 0 } . We say that an n -chain c is U -controlled if every point x of supp( c ) is U -controlled. The chain c is locally finite if, for every bounded set B , the set of points in supp( c ) which meet B isfinite. An n -chain c : X n +1 → Z is controlled if it is locally finite and U -controlled for someentourage U of X . Definition 1.6.
Let X be a bornological coarse space. Then, for n ∈ N , X C n ( X ) denotes thefree abelian group generated by the locally finite controlled n -chains on X .We will also represent n -chains as formal sums (cid:88) x ∈ X n +1 c ( x ) x, that are locally finite and U -controlled.The boundary map ∂ : X C n ( X ) → X C n − ( X ) is defined as the alternating sum ∂ := (cid:80) i ( − i ∂ i of the face maps ∂ i ( x , . . . , x n ) := ( x , . . . , ˆ x i , . . . , x n ) . The graded abelian group X C ∗ ( X ) , endowed with the boundary operator ∂ extended linearly to X C ∗ ( X ) , is a chaincomplex [BE16, Sec. 6.3]. When X is a G -bornological coarse space, we let X C Gn ( X ) bethe subgroup of X C n ( X ) given by the locally finite controlled n -chains that are also G -invariant. The boundary operator restricts to X C G ∗ ( X ) , and ( X C G ∗ ( X ) , ∂ ) is a subcomplex of ( X C ∗ ( X ) , ∂ ) .If f : X → Y is a morphism of G -bornological coarse spaces, then we consider the map onthe products X n → Y n sending ( x , . . . , x n ) to ( f ( x ) , . . . , f ( x n )) . It extends linearly to a map X C G ( f ) : X C Gn ( X ) → X C Gn ( Y ) that involves sums over the pre-images by f . This describes afunctor X C G : G BornCoarse → Ch with values in the category Ch of chain complexes over YCLIC HOMOLOGY FOR BORNOLOGICAL COARSE SPACES 7 the integers Z . The ∞ -category Ch ∞ of chain complexes is defined as the localization (in therealm of ∞ -categories [Lur14, Sec. 1.3.4]) of the nerve of the category Ch at the class W ofquasi-isomorphism of chain complexes Ch ∞ := N( Ch )[ W − ] . By post-composing the functor X C G with the functor(1.3) EM : Ch loc −−→ Ch ∞ (cid:39) −→ H Z − Mod → Sp (the Eilenberg-MacLane correspondence between chain complexes ans spectra [Shi07, Thm.1.1] or [BE16, Sec. 6.3]), we get a functor to the ∞ -category of spectra(1.4) X H G := EM ◦ X C G : G BornCoarse → Sp called equivariant coarse ordinary homology : Theorem 1.7. [BEKW17, Thm. 7.3]
The functor X H G is a G -equivariant Sp -valued coarsehomology theory. The same can be done for a base ring k instead of Z . Example 1.8.
Assume that X is the one point space and the base ring of coefficients is k .Then, the chain complex X C ∗ ( X ) has one free generator in each dimension, and the boundarymaps are either the null map or the identity, depending on the degree. The coarse homologygroups are in positive and negative degree and the base ring k in degree .1.4. The category of controlled objects.
The goal of this subsection is to recall thedefinition of the additive category V G A ( X ) of G -equivariant X -controlled A -objects [BEKW17,Def. 8.3] and of the functor V G A : G BornCoarse → Add sending a G -bornological coarsespace to the category V G A ( X ) . Our main reference is [BEKW17, Sec. 8.2] where the mainresults of this paragraph are given.Let G be a group and let X be a G -bornological coarse space. Remark 1.9.
The bornology B ( X ) on X defines a poset with the partial order induced bysubset inclusion; hence, B ( X ) can be seen as a category.Recall that an additive category is a category enriched on abelian groups, with a zero objectand all finite biproducts. Let A be an additive category with strict G -action. For every element g in G and every functor F : B ( X ) → A , let gF : B ( X ) → A denote the functor sending abounded set B in B ( X ) to the A -object g ( F ( g − ( B ))) (and defined on morphisms B ⊆ B (cid:48) asthe induced morphism of A ( gF )( B ⊆ B (cid:48) ) : gF ( g − ( B )) → gF ( g − ( B (cid:48) )) ).If η : F → F (cid:48) is a natural transformation between two functors F, F (cid:48) : B ( X ) → A , we denoteby gη : gF → gF (cid:48) the induced natural transformation between gF and gF (cid:48) . Definition 1.10. [BEKW17, Def. 8.3] A G -equivariant X -controlled A -object is a pair ( M, ρ ) consisting of a functor M : B ( X ) → A and a family ρ = ( ρ ( g )) g ∈ G of natural isomorphisms ρ ( g ) : M → gM , satisfying the following conditions:(1) M ( ∅ ) ∼ = 0 ;(2) for all B, B (cid:48) in B ( X ) , the commutative diagram M ( B ∩ B (cid:48) ) M ( B ) M ( B (cid:48) ) M ( B ∪ B (cid:48) ) is a push-out; LUIGI CAPUTI (3) for all B in B ( X ) there exists a finite subset F of B such that the inclusion induces anisomorphism M ( F ) ∼ = −→ M ( B ) ;(4) for all elements g, g (cid:48) in G we have the relation ρ ( gg (cid:48) ) = gρ ( g (cid:48) ) ◦ ρ ( g ) , where gρ ( g (cid:48) ) isthe natural transformation from gM to gg (cid:48) M induced by ρ ( g (cid:48) ) . Notation 1.11. If ( M, ρ ) is an X -controlled A -object and x is an element of X , we will oftenwrite M ( x ) istead of M ( { x } ) for the value of the functor M at the bounded set { x } of X .Let X be a G -bornological coarse space and let ( M, ρ ) be an equivariant X -controlled A -object. Let B be a bounded set of X and let x be a point in B . The inclusion { x } → B induces a morphism M ( { x } ) → M ( B ) of A . The conditions of Definition 1.10 imply that M ( { x } ) = 0 for all but finitely many points of B and that the canonical morphism (inducedby the universal property of the coproduct in A )(1.5) (cid:77) x ∈ B M ( { x } ) ∼ = −→ M ( B ) is an isomorphism.Let X be a G -bornological coarse space. The U -thickening U [ B ] (1.1) of a bounded subset B of X is bounded and U -thickenings preserve the inclusions of bounded sets; we get a functor U [ − ] : B ( X ) → B ( X ) . Definition 1.12. [BEKW17, Def. 8.6] Let ( M, ρ ) and ( M (cid:48) , ρ (cid:48) ) be G -equivariant X -controlled A -objects and let U ∈ C G ( X ) be a G -invariant entourage of X . A G -equivariant U -controlledmorphism ϕ : ( M, ρ ) → ( M (cid:48) , ρ (cid:48) ) is a natural transformation ϕ : M ( − ) → M (cid:48) ◦ U [ − ] such that ρ (cid:48) ( g ) ◦ ϕ = ( gϕ ) ◦ ρ ( g ) for all g in G .The set of G -equivariant U -controlled morphisms ϕ : ( M, ρ ) → ( M (cid:48) , ρ (cid:48) ) is denoted by Mor U (( M, ρ ) , ( M (cid:48) , ρ (cid:48) )) . For every bounded set B of X , the inclusion U ⊆ U (cid:48) induces aninclusion U [ B ] ⊆ U (cid:48) [ B ] ; this yields a natural transformation of functors M (cid:48) ◦ U [ − ] → M (cid:48) ◦ U (cid:48) [ − ] ,hence a map Mor U (( M, ρ ) , ( M (cid:48) , ρ (cid:48) )) → Mor U (cid:48) (( M, ρ ) , ( M (cid:48) , ρ (cid:48) )) by post-composition.By using these structure maps we define the abelian group of G -equivariant controlledmorphisms from ( M, ρ ) to ( M (cid:48) , ρ (cid:48) ) as the colimit Hom V G A ( X ) (( M, ρ ) , ( M (cid:48) , ρ (cid:48) )) := colim U ∈C G Mor U (( M, ρ ) , ( M (cid:48) , ρ (cid:48) )) . Definition 1.13. [BEKW17] Let X be a G -bornological coarse space and let A be an additivecategory with strict G -action. The category V G A ( X ) is the category of G -equivariant X -controlled A -objects and G -equivariant controlled morphisms.Let k be a field. When A is the category of finite dimensional k -vector spaces, then wedenote by V Gk ( X ) the associated category of G -equivariant X -controlled (finite dimensional) k -modules. Lemma 1.14. [BEKW17, Lemma 8.7]
The category of equivariant X -controlled A -objects V G A ( X ) is additive. YCLIC HOMOLOGY FOR BORNOLOGICAL COARSE SPACES 9
Let f : ( X, C , B ) → ( X (cid:48) , C (cid:48) , B (cid:48) ) be a morphism of G -bornological coarse spaces. If ( M, ρ ) is a G -equivariant X -controlled A -object, we consider the functor f ∗ M : B (cid:48) → A defined by f ∗ M ( B (cid:48) ) := M ( f − ( B (cid:48) )) for every bounded set B (cid:48) in B (cid:48) and defined on morphisms in thecanonical way. For every g in G , the family of transformation f ∗ ρ = (( f ∗ ρ )( g )) g ∈ G is givenby the natural isomorphisms ( f ∗ ρ )( g ) : f ∗ M → g ( f ∗ M ) with (( f ∗ ρ )( g ))( B (cid:48) ) := ρ ( g )( f − ( B (cid:48) )) . The pair f ∗ ( M, ρ ) := ( f ∗ M, f ∗ ρ ) defined in this way is a G -equivariant X (cid:48) -controlled A -object [BEKW17, Sec. 8.2]. Assume also that U is an invariant entourage of X and that ϕ : ( M, ρ ) → ( M (cid:48) , ρ (cid:48) ) is an equivariant U -controlled morphism. Then, the set V := ( f × f )( U ) is a G -invariant entourage of X (cid:48) and the morphism:(1.6) f ∗ ϕ := (cid:18) f ∗ M ( B (cid:48) ) ϕ f − B (cid:48) ) −−−−−→ M ( U [ f − ( B (cid:48) )]) → f ∗ M ( V [ B (cid:48) ]) (cid:19) B (cid:48) ∈B (cid:48) is V -controlled. We have just described a functor f ∗ := V G A ( f ) : V G A ( X ) → V G A ( X (cid:48) ) . We denote by(1.7) V G A : G BornCoarse → Add . the functor from the category of G -bornological coarse spaces to the category of small additivecategories obtained in this way. Remark 1.15. If A is a k -linear category, then the functor V G A : G BornCoarse → Add refines to a functor V G A : G BornCoarse → Cat k from the category of G -bornological coarsespaces to the category of small k -linear categories.The following properties of the functor V G A are shown in [BEKW17]: Remark 1.16.
Let ( X, C , B ) be a G -bornological coarse space and U ∈ C G be a G -invariantentourage of X . Then, X U := ( X, C U , B ) is a G -bornological coarse space by restriction of thestructures. The natural map X U → X induces an additive functor Φ U : V G A ( X U ) → V G A ( X ) which is the identity on objects as the definition of equivariant X controlled A -objects does notdepend on the coarse structure. Moreover, the category V G A ( X U ) can be seen as a subcategoryof V G A ( X ) . On the other hand, every controlled morphism in V G A ( X ) is U -controlled for someentourage U in C G . Therefore, the category V G A ( X ) is the filtered colimit V G A ( X ) ∼ = colim U ∈C G V G A ( X U ) indexed on the poset of G -invariant entourages of X . Lemma 1.17. [BEKW17, Lemma 8.11]
Let f, g : X → X (cid:48) be two morphisms of G -bornologicalcoarse spaces. If f and g are close to each other, then they induce naturally isomorphic functors f ∗ ∼ = g ∗ : V G A ( X ) → V G A ( X (cid:48) ) . Let A be an additive category and denote by ⊕ its biproduct. Definition 1.18.
An additive category A is called flasque if it admits an endofunctor S : A →A and a natural isomorphism id A ⊕ S ∼ = S . Lemma 1.19. [BEKW17, Lemma 8.13] If X is a flasque G -bornological coarse space, thenthe category V G A ( X ) of G -equivariant X -controlled A -objects is a flasque category. We conclude with the definition of coarse algebraic K -homology: Definition 1.20. [BEKW17, Def. 8.8] Let G be a group and let A be an additive categorywith strict G -action. The G -equivariant coarse algebraic K -homology associated to A is the K -theory of the additive category of A -controlled objects: K A X G := K ◦ V G A : G BornCoarse → Sp . When A is the category of finite dimensional k -vector spaces, we denote by K X Gk theassociated K -theory functor. The properties of the functor V G A reviewed above are used inorder to prove the following: Theorem 1.21. [BEKW17, Thm. 8.9]
Let G be a group and let A be an additive categorywith strict G -action. Then, the functor K A X G is a G -equivariant Sp -valued coarse homologytheory. Keller’s cyclic homology for dg-categories
In this section we recall Keller’s construction of cyclic homology for dg-categories [Kel99].As Keller’s construction uses also Kassel’s mixed complexes [Kas87], we first recall someproperties of dg-categories and mixed complexes and then we define Keller’s Hochschild andcyclic homology for dg-categories.2.1.
Dg-categories.
Our main references in this subsection are [Kel06, Toë11]. We let k be acommutative ring and ⊗ the tensor product over k . We start with the definition of a dg-algebra: Definition 2.1. A differential graded k -algebra A (shortly, a dg-algebra), is a Z -graded k -algebra A = (cid:76) p ∈ Z A p endowed with a differential d either of degree − (chain complexconvention) or of degree (cochain complex convection) that satisfies the Leibniz rule d ( ab ) = d ( a ) b + ( − p ad ( b ) for all a ∈ A p and p ∈ Z , and all b ∈ A .A left dg-module M over a dg-algebra A is a left graded module M = ⊕ p ∈ Z M p endowed with adifferential d (of the same degree as the differential of A ) such that d ( ma ) = d ( m ) a +( − p md ( a ) for every m ∈ M p and a ∈ A . A morphism of dg-modules is a homogeneous morphism ofdegree of the underlying graded modules commuting with the differentials. The category ofdg-modules over the dg-algebra A and morphisms of dg-modules is denoted by A - Mod .A dg-category over k is a category enriched on (the category of) chain complexes of k -modules.We spell it out: Definition 2.2.
A small differential graded category A (shortly, a dg-category) consists of thefollowing data: • a small set of objects obj( A ) (denoted A as well); • for each pair of objects A, B of A , a chain complex of k -modules Hom A ( A, B ) ; • for each triple of objects A, B, C in A compositions Hom A ( A, B ) ⊗ Hom A ( B, C ) → Hom A ( A, C ) satisfying the associativity relations; • for each object A of A , a morphism k → Hom A ( A, A ) satisfying the usual unit conditionwith respect to the compositions.For example, every dg-algebra is a dg-category with a single object. Let A and A (cid:48) be smalldg-categories. A dg-functor F : A → A (cid:48) consists of a map F : obj( A ) → obj( A (cid:48) ) of sets and, foreach pair of objects A, B in A , of a morphism F ( A, B ) : Hom A ( A, B ) → Hom A (cid:48) ( F ( A ) , F ( B )) YCLIC HOMOLOGY FOR BORNOLOGICAL COARSE SPACES 11 of chain complexes satisfying the usual unit and composition conditions. We denote by dgcat k the category of small dg-categories (over k ) and dg-functors. Example 2.3.
An additive category A is a dg-category in a canonical way: for every object A in A , Hom A ( A, A ) is a chain complex concentrated in degree . We denote by(2.1) ι : Add → dgcat the functor from the category of small additive categories to the category of small dg-categoriesthat sends an additive category to the corresponding dg-category.If A is a dg-category, then the dg-category A op defined as the category with the same objectsas A and morphisms Hom A op ( A, B ) := Hom A ( B, A ) , is a dg-category. Let Ch dg ( k ) be thedg-category of chain complexes over the dg-algebra k . Definition 2.4.
A left dg-module over A is a dg-functor A op → Ch dg ( k ) . There is a suitable category of dg-modules over a dg-category A , whose objects are thedg-modules over A and whose morphisms are the natural transformations of dg-functors F : M → N such that F ( A ) : M ( A ) → N ( A ) is a morphism of complexes for every object A of A . Observe that this is coherent with the definition of the category of dg-modules (over adg-algebra) defined in the text after Definition 2.1. Definition 2.5.
Let A be a dg-category and let M, N be two dg-modules over A . A morphismof dg-modules M → N is a quasi-isomorphism if it induces a quasi-isomorphism of chaincomplexes M ( A ) → N ( A ) for every object A in A . Remark 2.6.
The category of dg-modules (over a dg-algebra or a dg-category) admits twoQuillen model structures where the weak equivalences are the object-wise quasi-isomorphismsof dg-modules; these are the injective and the projective model structure induced from theinjective and projective model structure on chain complexes, respectively. We remark that thecategory of dg-modules over a dg-algebra, equipped with the projective model structure (hencethe fibrations are the object-wise epimorphisms), is a combinatorial model category; see, forexample [Coh13, Rem. 2.14].If A is a dg-category, we can define an associated derived category: Definition 2.7. [Kel06, Sec. 3.2] The derived category D ( A ) of a dg-category A is thelocalization of the category of dg-modules over A at the class of quasi-isomorphisms.The objects of D ( A ) are the dg-modules over A and the morphisms are obtained frommorphisms of dg-modules by inverting the quasi-isomorphisms. It is a triangulated categorywith shift functor induced by the -translation and triangles coming from short exact sequencesof complexes.Let A and B be two small dg-categories. A dg-functor F : A → B is called a
Moritaequivalence if it induces an equivalence of derived categories. For a precise definition of Moritaequivalences we refer to [Kel06, Sec. 3.8], or [Coh13, Def. 2.29].
Theorem 2.8. [Tab05, Thm. 5.1]
The category dgcat k of small dg-categories over k admits thestructure of a combinatorial model category whose weak equivalences are the Morita equivalences. For a description of fibrations and cofibrations we refer to [Tab05, Thm. 5.1], or [Kel06, Thm.4.1]. We conclude with the definition of short exact sequences of dg-categories:
Definition 2.9. [Kel06, § 4.6] A short exact sequence of dg-categories is a sequence ofmorphisms
A → B → C inducing an exact sequence of triangulated categories D b ( A ) → D b ( B ) → D b ( C ) in the sense of Verdier.2.2. The ∞ -category of mixed complexes. In this subsection we describe the (cocompletestable ∞ -)category of unbounded mixed complexes. Definition 2.10. [Kas87, §1] A mixed complex ( C, b, B ) is a triple consisting of a Z -graded k -module C = { C p } p ∈ Z together with differentials b and Bb = ( b p : C p → C p − ) p ∈ Z and B = ( B p : C p → C p +1 ) p ∈ Z of degree − and respectively, satisfying the following identities: b = 0 , B = 0 , bB + Bb = 0 . Morphisms of mixed complexes are given by maps commuting with both the differentials b and B . The category of mixed complexes and morphisms of mixed complexes is denoted by Mix .When the differentials are clear from the context, we refer to a mixed complex ( C, b, B ) byits underlying k -module C .Let Λ be the dg-algebra over the field k (2.2) Λ := · · · → → k(cid:15) −→ k → → · · · generated by an indeterminate (cid:15) of degree , with (cid:15) = 0 and differential (of degree − ) d ( (cid:15) ) = 0 . Mixed complexes are nothing but dg-modules over the dg-algebra Λ : Remark 2.11. [Kas87] The category
Mix of mixed complexes is equivalent (in fact, isomorphic)to the category of left dg Λ -modules, which we denote by Λ - Mod . In fact, a mixed complex ( C, b, B ) yields a differential graded left Λ -module whose underlying differential graded moduleis ( C, b ) and where the multiplication (cid:15) · c is defined by (cid:15) · c := B ( c ) . Morphisms of mixedcomplexes correspond to morphisms of dg- Λ -modules. We denote by L : Mix → Λ - Mod thefunctor sending a mixed complex to the associated Λ -dg-module and by R : Λ - Mod → Mix its inverse functor.The category of dg- k -modules admits a combinatorial model structure (the projective modelstructure, see Remark 2.6), whose weak equivalences are the objects-wise quasi-isomorphismsof dg-modules (Definition 2.5). In the language of mixed complexes this translates as follows: Definition 2.12.
A morphism ( C, b, B ) → ( C (cid:48) , b (cid:48) , B (cid:48) ) of mixed complexes is called a quasi-isomorphism if the underlying b -complexes are quasi-isomorphic via the induced chain map ( C, b ) → ( C (cid:48) , b (cid:48) ) . Remark 2.13.
Quasi-isomorphisms of mixed complexes correspond to quasi-isomorphisms of Λ -dg-modules, i.e., the functors L and R of Remark 2.11 preserve quasi-isomorphisms.We now introduce the ∞ -category Mix ∞ of mixed complexes. We recall that, if C is anordinary category and W denotes a collection of morphisms of C , then N( C )[ W − ] is the ∞ -category obtained by the nerve N( C ) of C by inverting the set of morphisms W [Cis, Def.7.1.2 & Prop. 7.1.3], [Lur14, Def. 1.3.4.1]. YCLIC HOMOLOGY FOR BORNOLOGICAL COARSE SPACES 13
Definition 2.14.
The ∞ -category Mix ∞ := N( Mix )[ W − mix ] of mixed complexes is defined as the localization of the (nerve of the) category Mix at theclass W mix of quasi-isomorphisms of mixed complexes.Analogously, the ∞ -category Λ - Mod ∞ is defined as the localization of the category Λ - Mod of dg- Λ -modules at the class W of quasi-isomorphisms of dg- Λ -modules:(2.3) Λ - Mod ∞ := N(Λ - Mod )[ W − ] . Proposition 2.15.
The ∞ -category Mix ∞ is a cocomplete stable ∞ -category.Proof. The category Λ - Mod is a (pre-triangulated) dg-category. By applying the dg-nervefunctor N dg [Lur14, Constr. 1.3.1.6] we obtain an ∞ -category N dg (Λ - Mod ) [Lur14, Prop.1.3.1.10]. The dg-nerve functor sends pre-triangulated dg-categories to stable ∞ -categories[Fao17, Thm. 4.3.1], [Lur14, Prop. 1.3.1.10]. The ∞ -category N dg (Λ - Mod ) is stable and itshomotopy category can be identified (as triangulated category) with the derived category D (Λ) associated to the dg-algebra Λ .The category Λ - Mod is equipped with a combinatorial simplicial model structure byRemark 2.6. By [Lur14, Prop. 1.3.1.17] and by the fact that the simplicial nerve of thesimplicial category associated to Λ - Mod is equivalent to the localization
N(Λ - Mod )[ W − ] (by[Lur14, Rem. 1.3.4.16 & Thm. 1.3.4.20] where we also use that the model category Λ - Mod iscombinatorial, hence admits functorial factorizations), the two constructions
N(Λ - Mod )[ W − ] and N dg (Λ - Mod ) present equivalent ∞ -categories. Hence, the ∞ -category Λ - Mod ∞ is astable ∞ -category. The ∞ -category Λ - Mod ∞ is also cocomplete by [Lur14, Prop. 1.3.4.22]because the model category Λ - Mod is combinatorial.The categories
Mix and Λ - Mod are isomorphic by Remark 2.11 and the functor L : Mix → Λ - Mod and its inverse R : Λ - Mod → Mix preserve quasi-isomorphisms by Remark 2.13. Thisyields an equivalence of ∞ -categories N( Mix )[ W − mix ] → N(Λ - Mod )[ W − ] that proves the statement. (cid:3) Remark 2.16.
The homotopy category of the stable ∞ -category Mix ∞ is canonically equiva-lent to the derived category D (Λ) of the dg-algebra Λ .We conclude the subsection with the definition of Hochschild and cyclic homology of mixedcomplexes.A mixed complex ( C, b, B ) functorially determines a double chain complex B C [Lod98, §2.5.10] by means of the differentials b and B :(2.4) B C := (cid:16) . . . ←− ( C, b ) B ←− ( C [ − , b C [ − ) B ←− . . . B ←− ( C [ − n ] , b C [ − n ] ) B ←− . . . (cid:17) ; here, the chain complex ( C, b ) is placed in bi-degree (0 , ∗ ) , i.e., B C (0 , ∗ ) = ( C ∗ , b ) , and thechain complex ( C [ − n ] , b C [ − n ] ) , placed in bi-degree ( n, ∗ ) , is the chain complex ( C, b ) shiftedby − n , hence B C ( p,q ) = C q − p for p ≥ and B C ( p,q ) = 0 for p < . The total chaincomplex Tot( B C ) , functorially associated to the double chain complex B C , is the chaincomplex defined in degree n by Tot n ( B C ) = (cid:76) i ≥ C n − i with differential d acting as follows: d ( c n , c n − , . . . ) := ( bc n + Bc n − , . . . ) . Let Ch be the category of chain complexes over k . Consider the forgetful functor(2.5) forget : Mix → Ch sending a mixed complex ( M, b, B ) to its underlying chain complex ( M, b ) , and the functor(2.6) Tot( B− ) : Mix → Ch just described above. Definition 2.17. [Kas87, Sec. 1] Let ( C, b, B ) be a mixed complex. The Hochschild homology HH ∗ ( C ) of ( C, b, B ) is the homology of the underlying chain complex ( C, b ) . Its cyclic homology HC ∗ ( C ) is the homology of the associated chain complex Tot( B C ) .We remark that this definition agrees with the usual definition of Hochschild and cyclichomology of algebras [Kas87].2.3. Keller’s cyclic homology.
In this subsection we recall Keller’s Hochschild and cyclichomology for dg-categories [Kel99]. Let k be a commutative ring with identity and let A be a k -algebra. Then, one can associate to A a cyclic module Z ∗ ( A ) [Goo85] ( i.e., a cyclic object inthe category of k -modules) defined in degree n as the ( n + 1) -th tensor product of A over k . Inthe same way, one can construct a cyclic module out of an additive category A [McC94, Def.2.1.1]. We present these constructions in the more general setting of dg-categories. Definition 2.18. [Kel99] Let C be a small dg-category over k . The additive cyclic nerve of C is the cyclic k -module defined by: CN n ( C ) := (cid:77) Hom C ( C , C ) ⊗ Hom C ( C , C ) ⊗ · · · ⊗ Hom C ( C , C n ) where the sum runs over all the objects ( C , C , . . . , C n ) in C n +1 . The face and degeneracymaps, and the cyclic action, are defined as follows: d i ( f ⊗ · · · ⊗ f n ) = (cid:40) f ⊗ f ⊗ · · · ⊗ f i ◦ f i +1 ⊗ · · · ⊗ f n if ≤ i ≤ n − − n + σ f n ◦ f ⊗ f ⊗ · · · ⊗ f n if i = ns i ( f ⊗ · · · ⊗ f n ) = (cid:40) f ⊗ f ⊗ · · · ⊗ f i ⊗ id C i +1 ⊗ f i +1 ⊗ · · · ⊗ f n if ≤ i ≤ n − f ⊗ f ⊗ · · · ⊗ f n ⊗ id C if i = nt ( f ⊗ · · · ⊗ f n ) = ( − n + σ ( f n ⊗ f ⊗ · · · ⊗ f n − ) where σ = (deg f n )(deg f n − + · · · + deg f ) .We get a covariant functor from the category of small dg-categories over k to the categoryof cyclic k -modules. To every cyclic k -module, we can then associate a mixed complex as wenow describe.Let M be a cyclic k -module. If d i and s i denote the i -th face and i -th degeneracy maps of M respectively and t n +1 denotes the cyclic operator in degree n , then we let b : M n → M n − be the alternating sum(2.7) b := n (cid:88) i =0 ( − i d i of face maps, we set N := (cid:80) ni =0 t in +1 be the sum of the powers of the cyclic operator t n +1 anddefine the cochain map B : M n → M n +1 as the composition(2.8) B := ( − n +1 (1 − t n +1 ) sN. YCLIC HOMOLOGY FOR BORNOLOGICAL COARSE SPACES 15
Here, s denotes the extra degeneracy s = ( − n +1 t n +1 s n : M n → M n +1 . Remark 2.19.
Let M be a cyclic module. Then, the triple ( M, b, B ) , where b and B are thedifferentials (2.7) and (2.8) respectively, is a mixed complex. Morphisms of cyclic modulescommute with the face and the degeneracy maps and with the cyclic operators; they yield inthis way morphisms of mixed complexes and a functor from the category of cyclic modules tothe category of mixed complexes. Definition 2.20. [Kel99, Def. 1.3] We denote by
Mix : dgcat k → Mix the functor from the category of small dg-categories over k to the category of mixed complexesdefined as composition of the additive cyclic nerve functor of Definition 2.18 with the functorof Remark 2.19.Thanks to the work of Keller, we know that this functor enjoys many useful properties,among others agreement, additivity and localization [Kel99]. As we work in the context of ∞ -categories, we will spell them out in this language.From now on we assume that k is a field. The ∞ -category of small of dg-categories dgcat k, ∞ := N( dgcat k )[ W − Morita ] is the localization at the class W Morita of Morita equivalences.By [Kel99, Thm 1.5], the functor
Mix of Definition 2.20 sends Morita equivalences of dg-categories to quasi-isomorphisms of mixed complexes and descends to a functor dgcat k Mixdgcat k, ∞ Mix ∞ loc Mix locMix between the localizations. Keller’s Localization Theorem [Kel99, Thm. 1.5] can be thensummarized as follows: Theorem 2.21. [Kel99, Thm. 1.5]
Let k be a field. The functor Mix : dgcat k → Mix ∞ satisfies the following:(1) it sends equivalences of small dg-categories to equivalences of mixed complexes;(2) it commutes with filtered colimits;(3) it sends short exact sequences A → B → C of dg-categories to cofiber sequences of
Mix ∞ .Moreover, if A is a k -algebra, there is an equivalence of mixed complexes Mix( A ) → Mix(proj A ) where proj A is the additive category of finitely generated projective modules. Observe that the functor
Mix preserves filtered colimits, hence the functor loc ◦ Mix preservesfiltered colimits because the localization preserves filtered colimits as well, and by commutativityof the diagram also
Mix ◦ loc . By Proposition 2.15, the ∞ -category Mix ∞ is stable andcocomplete and cofiber sequences of Mix ∞ [Lur14, Def. 1.1.1.6] are detected in its homotopycategory, i.e., in D (Λ) . We observe here that Keller’s theorem holds in a more general setting(for more general rings and for exact categories). However, we only need these properties in thecontext of additive categories. Moreover, in such context, Keller’s functor Mix is equivalent tothe cyclic homology functor constructed by McCarthy [McC94] (see also [Cap19, Lemma 3.4.4& Rmk. 3.4.5]). Equivariant coarse Hochschild and cyclic homology
For a fixed base field k and group G , we define equivariant coarse Hochschild homology X HH Gk and cyclic homology X HC Gk versions of the classical Hochschild and cyclic homology of k -algebras. This will be achieved by first studying an intermediate equivariant coarse homologytheory X Mix Gk with values in the ∞ -category of mixed complexes. We then provide somecomparison results.3.1. The equivariant coarse homology theory X Mix Gk . Let k be a field, Cat k the categoryof small k -linear categories, V Gk : G BornCoarse → Cat k the functor of Remark 1.15, let Mix : dgcat k → Mix be the functor of Definition 2.20, ι : Cat k → dgcat k the functor ofExample 2.3 and loc the localization functor loc : Mix → Mix ∞ . Definition 3.1.
We denote by X Mix Gk the following functor X Mix Gk : G BornCoarse Cat k dgcat k Mix Mix ∞ V Gk ι Mix loc to the ∞ -category of mixed complexes.The main result of the section is the following theorem: Theorem 3.2.
The functor X Mix Gk : G BornCoarse Mix ∞ is a G -equivariant Mix ∞ -valued coarse homology theory.Proof. The category
Mix ∞ is stable and cocomplete by Proposition 2.15. The functor X Mix Gk satisfies coarse invariance (see Proposition 3.3), vanishing on flasque spaces (see Proposition3.4), u-continuity (see Proposition 3.5) and coarse excision (see Theorem 3.6), i.e., the axiomsdescribing an equivariant coarse homology theory of Definition 1.5. (cid:3) We now prove that the functor X Mix Gk satisfies the axioms of Definition 1.5. Proposition 3.3.
The functor X Mix Gk : G BornCoarse → Mix ∞ satisfies coarse invariance.Proof. If f : X → Y is a coarse equivalence of G -bornological coarse spaces, then it inducesa natural equivalence f ∗ : V Gk ( X ) → V Gk ( Y ) by Lemma 1.17. Keller’s functor Mix sendsequivalences of dg-categories to equivalences of mixed complexes by Theorem 2.21 (1). Hence,the functor f ∗ induces the equivalence X Mix Gk ( X ) ∼ −→ X Mix Gk ( Y ) in Mix ∞ , i.e., the functor X Mix Gk is coarse invariant. (cid:3) Recall the definition of flasque spaces Definition 1.3.
Proposition 3.4.
The functor X Mix Gk : G BornCoarse → Mix ∞ vanishes on flasque spaces.Proof. By Lemma 1.19, the category V Gk ( X ) is a flasque category, hence there exists anendofunctor S : V Gk ( X ) → V Gk ( X ) such that id V Gk ( X ) ⊕ S ∼ = S . By [Sch11, Theorem 2.3.11](see also [Cap19, Thm. 3.3.5] for a complete proof of Schlichting’s theorem in this setting),the morphisms Mix(id) ⊕ Mix( S ) and Mix(id ⊕ S ) ∼ = Mix( S ) are equivalent in Mix ∞ . Thismeans that the morphism X Mix Gk (id) : X Mix Gk ( X ) → X Mix Gk ( X ) is equivalent to the -morphism and that X Mix Gk ( X ) (cid:39) . (cid:3) YCLIC HOMOLOGY FOR BORNOLOGICAL COARSE SPACES 17
Proposition 3.5.
The functor X Mix Gk : G BornCoarse → Mix ∞ is u -continuous.Proof. Let X be a G -bornological coarse space, and let C G be the poset of G -invariantcontrolled sets. By Remark 1.16, there is an equivalence V Gk ( X ) (cid:39) colim U ∈C G V Gk ( X U ) of k -linear categories, hence of dg-categories. The functor Mix commutes with filtered colimits,and we get the equivalence X Mix Gk ( X ) (cid:39) colim U ∈C G X Mix Gk ( X U ) in Mix ∞ , which shows that the functor X Mix Gk is u-continuous. (cid:3) Theorem 3.6.
The functor X Mix Gk : G BornCoarse → Mix ∞ satisfies coarse excision. Before giving the proof of this theorem we first need some more terminology.
Definition 3.7. [Kas15] A full additive subcategory A of an additive category U is a Karoubi-filtration if every diagram X → Y → Z in U , with X, Z ∈ A , admits an extension
X Y ZU U ⊕ U ⊥ U ∼ = i p with U ∈ A .By [Kas15, Lemma 5.6], this definition is equivalent to the classical one [Kar70, CP97]. If A is a Karoubi-filtration of U , we can construct a quotient category U / A . Its objects are theobjects of U , and the morphisms sets are defined as follows: Hom U / A ( U, V ) := Hom U ( U, V ) / ∼ where the relation identifies pairs of maps U → V whose difference factorizes through an objectof A .Let X be a G -bornological coarse space and let Y = ( Y i ) i ∈ I be an equivariant big family on X (see Definition 1.4). The bornological coarse space Y i is a subspace of X with the inducedbornology and coarse structure. The inclusion Y i (cid:44) → X induces a functor V Gk ( Y i ) → V Gk ( X ) which is injective on objects. The categories V Gk ( Y i ) and V Gk ( Y ) := colim i ∈ I V Gk ( Y i ) are fullsubcategories of V Gk ( X ) . Lemma 3.8. [BEKW17, Lemma 8.14]
Let Y be an equivariant big family on the G -bornologicalcoarse space X . Then, the full additive subcategory V Gk ( Y ) of V Gk ( X ) is a Karoubi filtration. Let X be a G -bornological coarse space, and ( Z, Y ) be an equivariant complementary pair.Consider the functor(3.1) a : V Gk ( Z ) /V Gk ( Z ∩ Y ) → V Gk ( X ) /V Gk ( Y ) induced by the inclusion i : Z → X ; on objects, it coincides with i ∗ : V Gk ( Z ) → V Gk ( X ) , but onmorphisms it sends an equivalence class [ A ] of A in the equivalence [ i ∗ ( A )] of i ∗ ( A ) . Lemma 3.9. [BEKW17, Prop. 8.15]
The functor a (3.1) is an equivalence of categories.Proof of Theorem 3.6. Let X be a G -bornological coarse space, and let ( Z, Y ) be an equivariantcomplementary pair on X . By Lemma 3.8, V Gk ( Z ∩ Y ) ⊆ V Gk ( Z ) and V Gk ( Y ) ⊆ V Gk ( X ) areKaroubi filtrations and yield the following sequences of k -linear categories: V Gk ( Z ∩ Y ) → V Gk ( Z ) → V Gk ( Z ) /V Gk ( Z ∩ Y ) and V Gk ( Y ) → V Gk ( X ) → V Gk ( X ) /V Gk ( X ∩ Y ) . By [Sch04, Ex. 1.8, Prop. 2.6] (see also [Cap19, Rem. 3.3.12]), Karoubi filtrations induce shortexact sequences of dg-categories, hence by Theorem 2.21 we get cofiber sequences of mixedcomplexes. The inclusion
Z (cid:44) → X induces a commutative diagram (where the rows are theobtained cofiber sequences) Mix( V Gk ( Z ∩ Y )) Mix( V Gk ( Z )) Mix( V Gk ( Z ) /V Gk ( Z ∩ Y ))Mix( V Gk ( Y )) Mix( V Gk ( X )) Mix( V Gk ( X ) /V Gk ( X ∩ Y )) a ∗ where a ∗ is the map induced by a : V Gk ( Z ) /V Gk ( Z ∩ Y ) → V Gk ( X ) /V Gk ( Y ) (3.1). By Lemma 3.9,the functor a yields an equivalence of categories, hence of mixed complexes and the left squareis a co-Cartesian square in Mix ∞ .In order to conclude the proof, we recall that X Mix Gk ( Y ) is defined as the filtered colimit X Mix Gk ( Y ) = colim i X Mix Gk ( Y i ) and that V Gk ( Y ) := colim i ∈ I V Gk ( Y i ) . The functor
Mix com-mutes with filtered colimits of dg-categories, hence we have the equivalence
Mix( V Gk ( Y )) =Mix(colim i V Gk ( Y i )) (cid:39) colim i Mix( V Gk ( Y i )) and the same holds for Z ∩ Y . By using theseidentifications, we obtain the co-Cartesian square in Mix ∞ X Mix Gk ( Z ∩ Y ) X Mix Gk ( Z ) X Mix Gk ( Y ) X Mix Gk ( X ) meaning that X Mix Gk satisfies coarse excision. (cid:3) Coarse Hochschild and cyclic homology.
In this subsection we define equivariantcoarse Hochschild and cyclic homology. The functors forget :
Mix → Ch (2.5), sending amixed complex to the underlying chain complex, and Tot( B− ) : Mix → Ch (2.6), sending amixed complex to the total complex of its associated bicomplex, send quasi-isomorphisms ofmixed complexes to quasi-isomorphisms of chain complexes, hence they descend to functorsbetween the localizations. Definition 3.10.
Let k be a field, G a group and Ch ∞ the ∞ -category of chain complexes.The G -equivariant coarse Hochschild homology X HH Gk (with k -coefficients) is the G -equivariant Ch ∞ -valued coarse homology theory X HH Gk : G BornCoarse Mix ∞ Ch ∞ . X Mix Gk forgetdefined as composition of the functor X Mix Gk of Definition 3.1 and of the functor forget (2.5).The composition X HC Gk : G BornCoarse Mix ∞ Ch ∞ . X Mix Gk Tot( B− ) involving composition with the functor Tot( B− ) (2.6) is G -equivariant coarse cyclic homology .The definitions are justified by the following: YCLIC HOMOLOGY FOR BORNOLOGICAL COARSE SPACES 19
Theorem 3.11.
The functors X HH Gk : G BornCoarse → Ch ∞ and X HC Gk : G BornCoarse → Ch ∞ are G -equivariant Ch ∞ -valued coarse homology theories.Proof. By Theorem 3.2, the functor X Mix Gk : G BornCoarse → Mix ∞ is an equivariant coarsehomology theory and satisfies coarse invariance, coarse excision, u -continuity, and vanishingon flasques. The functors forget : Mix ∞ → Ch ∞ and the functor Tot( B− ) : Mix ∞ → Ch ∞ commute with filtered colimits and send cofiber sequences to cofiber sequences. The twocompositions with X Mix Gk satisfy coarse invariance, coarse excision, u -continuity, and vanishingon flasques, hence the functors X HH Gk and X HC Gk are equivariant coarse homology theories. (cid:3) The category of mixed complexes has a natural symmetric monoidal structure inducedby tensor products between the underlying chain complexes [Kas87]. As tensor products ofmixed complexes preserve equivalences, by [Hin16, Prop. 3.2.2] we get a symmetric monoidal ∞ -category Mix ⊗∞ := N( Mix ⊗ )[ W ⊗ , − mix ] → N( Fin ∗ ) . By [Kas87, Thm. 2.4], the functor Mix has a lax symmetric monoidal refinement (see also [CT12]). This implies that coarse Hochschildand cyclic homologies are lax symmetric monoidal functors:
Proposition 3.12.
The functors X HH Gk and X HC Gk admit lax symmetric monoidal refine-ments: X HH G, ⊗ k : N( G BornCoarse ⊗ ) → Ch ⊗∞ and X HC G, ⊗ k : N( G BornCoarse ⊗ ) → Ch ⊗∞ . where Ch ⊗∞ is the ∞ -category of chain complexes with its standard symmetric monoidalstructure.Proof. By [BC19, Thm. 3.26] and [Kas87, Thm. 2.4], the functor X Mix Gk of Definition 3.1admits a lax symmetric monoidal refinement X Mix G, ⊗ k : N( G BornCoarse ⊗ ) → Mix ⊗∞ . As the functors forget (2.5) and
Tot( B− ) (2.6) are lax symmetric monoidal, coarse Hochschildand cyclic homology are lax symmetric monoidal refinements as well. (cid:3) Comparison results.
In this subsection, we compare equivariant coarse Hochschildhomology with the classical version of Hochschild homology for k -algebras. Furthermore, weshow that the forget-control map for coarse Hochschild homology is equivalent to the associatedgeneralized assembly map. Notation 3.13.
Let A be a k -algebra. We denote by C HH ∗ ( A ; k ) and C HC ∗ ( A ; k ) the chain complexes computing the Hochschild and cyclic homology of the mixed complex Mix( A ) associated to the cyclic object Z ∗ ( A ) associated to A [Goo85, Lod98].Let {∗} be the one point bornological coarse space, endowed with a trivial G -action. Proposition 3.14.
There are equivalences of chain complexes X HH k ( ∗ ) (cid:39) C HH ∗ ( k ; k ) and X HC k ( ∗ ) (cid:39) C HC ∗ ( k ; k ) between the coarse Hochschild (cyclic) homology of the point and the classical Hochschild (cyclic)homology of k . Proof.
By Theorem 2.21, the mixed complex
Mix( A ) associated to a k -algebra A is equivalentto the mixed complex associated to the k -linear category of finitely generated projective A -modules. When X is a point endowed with a trivial G -action and k is a field, the k -linearcategory V k ( X ) is isomorphic to the category Vect f.d.k of finite dimensional k -vector spaces, i.e., Mix( V k ( {∗} )) (cid:39) Mix (cid:0)
Vect f.d.k (cid:1) (cid:39)
Mix( k ) , which is enough to prove the statement. (cid:3) Let G be a group. By Example 1.2, there is a canonical G -bornological coarse space G can , min = ( G, C can , B min ) associated to it. Proposition 3.15.
There are equivalences of chain complexes: X HH Gk ( G can , min ) (cid:39) C HH ∗ ( k [ G ]; k ) and X HC Gk ( G can , min ) (cid:39) C HC ∗ ( k [ G ]; k ) between the G -equivariant coarse Hochschild and cyclic homologies of G can , min and the classicalHochschild and cyclic homologies of the group algebra k [ G ] .Proof. The category V Gk ( G can , min ) of G -equivariant G can , min -controlled finite dimensional k -vector spaces is equivalent to the category Mod fg , free ( k [ G ]) of finitely generated free k [ G ] -modules [BEKW17, Proposition 8.24]. By Theorem 2.21, Keller’s mixed complex Mix(
Mod fg , free ( k [ G ])) of the category of finitely generated free k [ G ] -modules is equivalent tothe mixed complex associated to the category Mod fg , proj ( k [ G ]) of finitely generated projectivemodules (because Morita equivalent dg-categories). Therefore, the result follows from the chainof equivalences of mixed complexes Mix( V Gk ( G )) (cid:39) Mix(
Mod fg , free ( k [ G ])) (cid:39) Mix(
Mod fg , proj ( k [ G ])) (cid:39) Mix( k [ G ]) , where the last equivalence is again true by Theorem 2.21. (cid:3) Let X be a G -set and let X min , max denote the G -bornological coarse space with minimal coarsestructure and maximal bornology. Let G be a group, H a subgroup of G and endow the set G/H with the minimal coarse structure and the maximal bornology. The previous calculationcan be extended to G -bornological coarse spaces of the form ( G/H ) min , max ⊗ G can , min : Remark 3.16.
Let G be a group, H a subgroup of G and endow the set G/H with theminimal coarse structure and the maximal bornology; then, we get an equivalence of chaincomplexes: X HH Gk (( G/H ) min , max ⊗ G can , min ) (cid:39) C HH ∗ ( k [ H ]; k ); the same holds for equivariant coarse cyclic homology.One of the main applications of coarse homotopy theory is within the studying of assemblymap conjectures. We conclude this subsection with a comparison result between the forget-control maps for equivariant coarse Hochschild and cyclic homology and the associated assemblymaps. Recall the definitions of the cone functor O ∞ hlg [BEKW17, Def. 10.10], of the forget-control map β [BEKW17, Def. 11.10] and of the coarse assembly map α [BEKW17, Def. 10.24].By [BEKW17, Thm 11.16], the forget-control map for a G -equivariant coarse homology theory E can be compared with the classical assembly map for the associated G -equivariant homologytheory E ◦ O ∞ hlg : G Top → C .By applying the Eilenberg-MacLane correspondence (1.3), we can assume that the equivariantcoarse homology theories X HH Gk and X HC Gk are equivariant spectra-valued coarse homologytheories. YCLIC HOMOLOGY FOR BORNOLOGICAL COARSE SPACES 21
Definition 3.17.
Let HH Gk := EM ◦ X HH Gk ◦O ∞ hlg : G Top → Sp be the G -equivariant homol-ogy theory associated to equivariant coarse Hochschild homology.Let Fin be the family of finite subgroups of G . The following is a consequence of [BEKW17,Thm 11.16]: Proposition 3.18. [Cap19, Prop. 4.2.7]
The forget-control map β G can , min ,G max , max for X HH Gk is equivalent to the assembly map α E Fin
G,G can , min for the G -homology theory HH Gk . Furthermore, the assembly map α E Fin
G,G can , min for the G -homology theory HH Gk (hence,the forget-control map β G can , min ,G max , max for X HH Gk ) is split injective by [LR06, Thm 1.7].4. From coarse algebraic K -theory to coarse ordinary homology In this section is to define a natural transformation Φ X HH Gk : X HH Gk −→ X C G from equivariant coarse Hochschild homology X HH Gk to the Ch ∞ -valued equivariant coarseordinary homology X Mix G and, analogously, a natural transformation Φ X HC Gk from equivariantcoarse cyclic homology. The construction of the transformation Φ X HH Gk divides in the followingsteps: • For every G -bornological coarse space X , we consider its associated k -linear category V Gk ( X ) of controlled objects, hence the associated additive cyclic nerve CN( V Gk ( X )) .For every tensor element A ⊗ . . . ⊗ A n in the additive cyclic nerve of V Gk ( X ) and every n + 1 points x , . . . , x n of X , we define a trace-like map, which gives an element of k (see Notation 4.2); • by letting x , . . . , x n vary, this yields a G -equivariant locally finite controlled chainon X , i.e., an element of X C Gn ( X ) (see Definition 4.3 and Lemma 4.4); by letting A ⊗ . . . ⊗ A n vary we get a map ϕ : CN ∗ ( V Gk ( X )) → X C G ∗ ( X ) that is a chain mapwith respect to the differential d = (cid:80) ( − i d i of CN( V Gk ( X )) (see Proposition 4.6); • the additive cyclic nerve CN( V Gk ( X )) yields a mixed complex with the differentials b and B as in Remark 2.19; the chain map ϕ extends to a map of mixed complexes ˜ ϕ (see Lemma 4.7) and yields a natural transformation of equivariant coarse homologytheories Φ X HH Gk : X HH Gk −→ X C G (see Theorem 4.8).We now proceed with the precise construction.Let V Gk ( X ) be the k -linear category of X -controlled finite dimensional k -vector spaces ofDefinition 1.10. The additive cyclic nerve associated to V Gk ( X ) (see Definition 2.18) is given indegree n by CN n ( V Gk ( X )) = (cid:77) (( M ,ρ ) ,..., ( M n ,ρ n )) (cid:32) n (cid:79) i =0 Hom(( M i +1 , ρ i +1 ) , ( M i , ρ i )) (cid:33) where the index i runs cyclically in the set { , . . . , n } and the sum ranges over all the tuples (( M , ρ ) , . . . , ( M n , ρ n )) of objects of V Gk ( X ) . Remark 4.1.
For every controlled morphism A i : ( M i +1 , ρ i +1 ) → ( M i , ρ i ) (see Definition 1.12)in Hom(( M i +1 , ρ i +1 ) , ( M i , ρ i )) and for every pair of points x and y of X , there is a well-defined k -linear map A x,yi : M i +1 ( x ) → M i ( y ) induced by A i .We use the following notation: Notation 4.2.
Let A ⊗ . . . ⊗ A n be an element of (cid:78) ni =0 Hom(( M i +1 , ρ i +1 ) , ( M i , ρ i )) with A i : ( M i +1 , ρ i +1 ) → ( M i , ρ i ) and (( M , ρ ) , . . . , ( M n , ρ n )) a tuple of objects of V Gk ( X ) . Let ( x , . . . , x n ) be a point of X n +1 . The symbol ( A ◦ · · · ◦ A n ) | ( x , . . . , x n ) denotes the linear operator ( A ◦ · · · ◦ A n ) | ( x , . . . , x n ) : M ( x n ) → M ( x n ) defined as thecomposition ( A ◦ . . . ◦ A n ) | ( x , . . . , x n ) := M ( x n ) A xn,xn − n −−−−−−→ M n ( x n − ) A xn − ,xn − n − −−−−−−−→ . . . A x ,x −−−−→ M ( x ) A x ,xn −−−−→ M ( x n ) of the induced operators A x i ,x i +1 i : M i ( x i ) → M i +1 ( x i +1 ) . It is an endomorphism of M ( x n ) ,which is a finite dimensional k -vector space.Let X be a G -bornological coarse space and let X C n ( X ) be the k -linear vector spacegenerated by the locally finite controlled n -chains on X (see Definition 1.6). Definition 4.3.
We let ϕ n : CN n ( V Gk ( X )) → X C n ( X ) be the map defined on elementarytensors as ϕ n : A ⊗ . . . ⊗ A n (cid:55)−→ (cid:88) ( x ,...,x n ) ∈ X n +1 tr (( A ◦ · · · ◦ A n ) | ( x , . . . , x n ) : M ( x n ) → M ( x n )) · ( x , . . . , x n ) and extended linearly. Lemma 4.4.
The n -chain ϕ n ( A ⊗ . . . ⊗ A n ) is locally finite and controlled.Proof. In order to prove that ϕ n ( A ⊗ . . . ⊗ A n ) is locally finite and controlled we show thatits support supp( ϕ n ( A ⊗ . . . ⊗ A n )) (1.2) is locally finite and that there exists an entourage U of X such that every x = ( x , . . . , x n ) in supp( ϕ n ( A ⊗ . . . ⊗ A n )) is U -controlled.We first observe that the operators A i : ( M i +1 , ρ i +1 ) → ( M i , ρ i ) are U i -controlled for someentourage U i of X . By Definition 1.12, A i is given by a natural transformation of functors M i +1 → M i ◦ U i [ − ] satisfying an equivariance condition. For every point x in X , A i restrictsto a morphism M i +1 ( x ) → M i ( U i [ x ]) ∼ = (cid:77) x (cid:48) ∈ U i [ x ] M i ( x (cid:48) ) where the direct sum has only finitely many non-zero summands.Let K be a bounded set of X . The set of points x n ∈ K for which M ( x n ) is non-zero is finite(as a consequence of Definition 1.13). For such a fixed x n , there are only finitely many points x n − ∈ U n [ K ] such that the corresponding map A x n ,x n − n : M ( x n ) → M n ( x n − ) is non-zero.The set U n [ K ] is a bounded set of X , the morphism A n − : M n → M n − is U n − -controlledand we can repeat the same argument for A n − , hence for each A i . This implies that the n -chain is locally finite because, for the given bounded set K , we have found only finitelytuples ( x , . . . , x n ) in the support of ϕ n ( A ⊗ . . . ⊗ A n ) that meet K .The chain is also U -controlled, where U is the entourage U := U ◦ · · · ◦ U n of X . (cid:3) YCLIC HOMOLOGY FOR BORNOLOGICAL COARSE SPACES 23
Remark 4.5.
Let X be a G -bornological coarse space. Let ( M, ρ ) be a G -equivariant X -controlled finite dimensional k -vector space and let g be an element of the group G . Then, ρ ( g ) (Definition 1.10) is a natural isomorphism between the functors M and gM . The morphisms inthe category V Gk ( X ) satisfy a G -equivariant condition (see Definition 1.12). These observationsimply that the following diagram is commutative M ( gx n ) M n ( gx n − ) . . . M ( gx n ) gM ( x n ) gM n ( x n − ) . . . gM ( x n ) ∼ = A gxn,gxn − n ∼ = A gx ,gxn ∼ = ∼ = gA xn,xn − n gA x ,xn for A ⊗ . . . ⊗ A n in CN n ( V Gk ( X )) with A i : ( M i +1 , ρ i +1 ) → ( M i , ρ i ) , where the isomorphismsare induced by ρ i ( g ) . Hence, the image of ϕ n is a G -invariant locally finite controlled n -chainon X .Let ∂ i : X C Gn ( X ) → X C Gn − ( X ) be the i -th differential of the chain complex X C G ( X ) andlet d i : CN n ( V Gk ( X )) → CN n − ( V Gk ( X )) be the i -th face map of CN( V Gk ( X )) . Consider thechain complex underlying the additive cyclic nerve CN( V Gk ( X )) . Proposition 4.6.
The maps ϕ n : CN n ( V Gk ( X )) → X C Gn ( X ) of Definition 4.3 extend to achain map ϕ : (CN( V Gk ( X )) , d ) → ( X C G ( X ) , ∂ ) . Proof.
It is easy to see that the following square is commutative: CN n ( V Gk ( X )) X C Gn ( X )CN n − ( V Gk ( X )) X C Gn − ( X ) d i ϕ n ∂ i ϕ n − This is achieved by using that the trace map is additive, invariant under cyclic permutationsand that the morphisms A i ◦ A i +1 factor through all the points x (cid:48) i of X (that give contributionzero up to finitely many). For a complete proof we refer to [Cap19, Prop. 4.3.6]. (cid:3) If M is a cyclic module, by Remark 2.19, we get a mixed complex. The chain complex X C G ( X ) is also a mixed complex with the differential B = 0 . Lemma 4.7.
The chain map ϕ : CN( V Gk ( X )) → X C G ( X ) of Definition 4.3 extends to a map ˜ ϕ : Mix( V Gk ( X )) → X C G ( X ) that is a morphism of mixed complexes.Proof. The proof is a simple computation and uses the definition of the operator B (2.8) andthat the trace is invariant under cyclic permutations. See also [Cap19, Lemma 4.3.7]. (cid:3) We can now construct the natural transformation Φ X HH Gk : X HH Gk → X C G : Theorem 4.8.
The map ϕ extends to natural transformations Φ X HH Gk : X HH Gk → X C G . and Φ X HC Gk : X HC Gk → (cid:77) n ∈ N X C G of G -equivariant Ch ∞ -valued coarse homology theories. Proof.
The map ϕ : (CN( V Gk ( X )) , d ) → ( X C G ( X ) , ∂ ) of Definition 4.3 is a chain map byProposition 4.6. Let f : X → Y be a morphism of G -equivariant bornological coarse spaces.Consider the induced chain map X C G ( f ) : X C G ( X ) → X C G ( Y ) and the induced functor f ∗ = V Gk ( f ) : V Gk ( X ) → V Gk ( X (cid:48) ) . By functoriality of the additive cyclic nerve, f ∗ inducesa morphism CN( f ∗ ) : CN ∗ ( V Gk ( X )) → CN ∗ ( V Gk ( Y )) of cyclic modules (hence, a chain mapbetween the underlying chain complexes as well).The diagram CN n ( V Gk ( X )) X C Gn ( X )CN n ( V Gk ( Y )) X C Gn ( Y ) ϕ n CN( f ∗ ) X C G ( f ) ϕ n is commutative. The map ϕ extends to the associated mixed complexes by Lemma 4.7and this extension preserves the commutative diagram (of associated mixed complexes).After localization and application of the forgetful functor (recall the definition of equivariantcoarse Hochschild homology in terms of X Mix Gk , Definition 3.10), the map ϕ yields a naturaltransformation of equivariant coarse homology theories Φ X HH Gk : X HH Gk → X C G . To every mixed complex C , we associate the chain complex Tot( B C ) (2.4) defined by Tot n ( B C ) = (cid:76) i ≥ C n − i with differential d ( c n , c n − , . . . ) = ( bc n + Bc n − , . . . ) . By Lemma4.7, we conclude that the map ϕ extends to a chain map on the total complex as well, and toa natural transformation of coarse homology theories Φ X HC Gk : X HC Gk → (cid:77) n ∈ N X C G where the sum is indexed on the natural numbers because the (mixed complex associated to)the additive cyclic nerve of V Gk ( X ) is positively graded. (cid:3) The following result implies that the transformation Φ X HH k : X HH k → X C is non-trivial: Proposition 4.9. If X is the one point space {∗} , then the transformation Φ X HH k : X HH k ( ∗ ) → X C ( ∗ ) induces an equivalence of chain complexes.Proof. Let c : {∗} n +1 → k be an n -chain in X C n ( ∗ ) ; we identify this chain with the element c ∈ k that is its image. Let ι n : X C n ( ∗ ) → CN n ( V k ( ∗ ))) be the map sending c to the element ( · c ) ⊗ ( · k ) ⊗ . . . ⊗ ( · k ) . This extends to a chain map that gives a section of the trace map, i.e., ϕ ◦ ι = id .As coarse Hochschild homology and coarse ordinary homology of the point are both isomor-phic to the Hochschild homology of the ground field k (by Example 1.8 and Proposition 3.14),we get equivalences of chain complexes X HH k ( ∗ ) (cid:39) C HH ∗ ( k ) (cid:39) X C k ( ∗ ) . By using these equivalences and the section ϕ ◦ ι = id , we obtain that, when X is the onepoint space, the transformation Φ X HH k induces an equivalence of chain complexes. (cid:3) By applying the Eilenberg-MacLane correspondence EM (1.3), we now assume that equivari-ant coarse Hochschild and cyclic homology take values in the ∞ -category Sp of spectra. Theclassical trace map constructed by McCarthy [McC94, Sec. 4.4] extends to a transformation YCLIC HOMOLOGY FOR BORNOLOGICAL COARSE SPACES 25 from equivariant coarse algebraic K -homology to equivariant coarse Hochschild homology[Cap19, Prop. 4.4.1]: Proposition 4.10.
There are natural transformations K X Gk → X HH Gk and K X Gk → X HC Gk induced by the Dennis trace maps from algebraic K-theory to Hochschild homology. In particular, when X is the G -bornological coarse space G can , min , the induced map K X Gk ( G can , min ) → X HH Gk ( G can , min ) is the classical Dennis trace map K ( k [ G ]) → HH( k [ G ]) by McCarthy’s agreement result[McC94, Sec. 4.5], by [BEKW17, Prop. 8.24] and by Proposition 3.15.By composition of the transformations of Proposition 4.10 and of Theorem 4.8 we get anatural transformation K X Gk → X HH Gk EM◦ Φ X HH Gk −−−−−−−−→ X H Gk from equivariant coarse algebraic K -homology to coarse ordinary homology. For example, when X is the G -bornological coarse space G can , min , we get a map K ( k [ G ]) → H ( G ; k ) from thealgebraic K -theory of the group ring k [ G ] to the ordinary homology of G with k -coefficients. Acknowledgements.
This work formed part of the author’s PhD at Regensburg University.It is a pleasure to again acknowledge Ulrich Bunke, for which this work would not exist without.The author also thanks Clara Löh, Denis-Charles Cisinski and Alexander Engel for helpfuldiscussions and/or comments. The author is supported by the DFG Research Training GroupGRK 1692 “Curvature, Cycles, and Cohomology”.
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