Nilpotent invariance of semi-topological K-theory of dg-algebras and the lattice conjecture
aa r X i v : . [ m a t h . K T ] F e b NILPOTENT INVARIANCE OF SEMI-TOPOLOGICAL K-THEORY OFDG-ALGEBRAS AND THE LATTICE CONJECTURE
ANDREI KONOVALOV
Abstract.
We show existence of a natural rational structure on periodic cyclic homology, con-jectured by L. Katzarkov, M. Kontsevich, T. Pantev, for several classes of dg-categories, includingproper connective C -dg-algebras and dg-categories of local systems. The main ingredient is de-rived nilpotent invariance of A. Blanc’s semi-topological K-theory, which we establish along theway. Introduction
One of the features of algebraic geometry over the field of complex numbers is existence ofa pure Hodge structure on de Rham cohomology of smooth proper varieties. Noncommutativealgebraic geometry studies k -dg-categories, aka noncommutative schemes. A fruitful idea is inthe realm of dg-categories one can still define many properties and invariants of schemes, such assmoothness, properness, (direct sums of) Dolbeault cohomology groups and de Rham cohomology,– the correspondence between the commutative and noncommutative worlds being the functorassigning to a scheme X/k the dg-category of perfect complexes
Perf( X ) . The last two invariantsare presented by Hochschild homology and periodic cyclic homology and they come with the Hodge-de Rham spectral sequence, which degenerates for smooth proper dg-categories when k is a fieldof characteristic 0 ([Kal]). In [KKP], the authors consider noncommutative schemes over C andsuggest to look for a counterpart for Hodge structures, which are already partially presented thanksto the aforementioned degeneration of the spectral sequence. One of the missing parts is a naturalrational structure, i.e. a functor F : dgCat C −→ M od Q and a natural transformation F → HP( · / C ) ,such that, for a smooth proper dg-category T , the induced morphism F ( T ) ⊗ Q C → HP(
T / C ) isan equivalence (i.e. a quasi-isomorphism).In this paper, we consider the topological K-theory of dg-categories functor K top , defined byA. Blanc as a promising candidate for the role of integral structure. We prove a statement aboutrational structure, which we later will refer to as the "lattice conjecture", in several cases, whichare listed below (cf. Corollary 5.6, Theorem 6.1, Corollary 6.6, Theorem 6.16). Theorem 1.1.
Let LC ⊂ dgCat C be the full subcategory of dg-categories on which the naturaltransformation of functors K top ( T ) ⊗ C → HP(
T / C ) is an equivalence. Then LC contains thefollowing classes of dg-categories: a) T = Perf( B ) where B is a connected proper dg-algebra; b) T = Perf( B ) where B is a connected dg-algebra, such that H B is a nilpotent extension of acommutative C -algebra of finite type; c) T = Loc( M, C ) where M is a connected locally contractiblespace with some condition on its fundamental group (see Theorem 6.16); d) T = Perf( X ) where X is a derived C -scheme, such that its classical part is a separated scheme of finite type. LC The author was partially supported by Basis Foundation grant 18-1-6-95-1, Leader (Math), by the HSE UniversityBasic Research Program, and by Simons-IUM fellowship. atisfies 2-out-of-3 property with respect to exact triples of dg-categories and is closed under Morita-equivalences and taking retracts. In section 4.7, [Bla], the author considered finite-dimensional classical algebras and used a variantof K top , called pseudo-connective topological K-theory, to provide periodic cyclic homology of suchalgebras with a rational structure. Since these algebras lie in the class a) of Theorem 1.1, it followsthat better-behaving topological K-theory works just as well, which can also be seen directly (seeProposition 6.2). We also explain in Proposition 6.4 that Orlov’s result (Theorem 2.19, [Orl])implies that finite-dimensional smooth C -dg-algebras also lie in LC .Topological K-theory of dg-categories is defined using a topological realization functor, whichgeneralizes the procedure of analytification from C -varieties to arbitrary (spectrum-valued) invari-ants of schemes. To prove Theorem 1.1, we study the behaviour of the realization functor, focusingon the case of Hochschild-type invariants. It allows us to establish the following result, crucial forproving the main theorem. Theorem 1.2.
Let v : B → A be a nilpotent extension of connective C -dg-algebras. Then theinduced map K st ( B ) → K st ( A ) is an equivalence. After derived nil-invariance is established, proving most of the cases does not require muchwork, but for the case of local systems on M we need to understand topological K-theory of groupalgebras. This we can do only under some assumptions on the group, which corresponds to puttingassumptions on the fundamental group of M . Concretely, we ask the group to satisfy the Burgheleaconjecture and the rational Farrell-Jones conjecture, which are both established for a large classof groups, – and under these assumptions we prove the lattice conjecture. We also suggest a newapproach to constructing a counterexample to the Farrell-Jones conjecture. Structure of the paper.
In the first section, we state the two main theorems and recall some mo-tivation behind the lattice conjecture. We also fix some conventions that will be used throughoutthe paper and sketch the structure of the exposition.The second section is devoted to considering two realizations functors, which allow one to extendthe functor of taking the space of complex points with analytic topology from schemes to spectralpresheaves. We show that these two functors coincide, which allows us later to use properties ofboth.The realization formalism was used in [Bla] to define semi-topological K-theory, which after in-verting the Bott element becomes topological K-theory. In the third section, we recall the necessarydefinitions and statements from [Bla].Semi-topological K-theory of a dg-category T is built from algebraic K-theories of different base-changes of T . And, while algebraic K-theory itself is a very complicated invariant, to some extent,it can be approximated by Hochshild and (variants of) cyclic homology. In the fourth section, weconsider realizations of Hochschild-type invariants. In particular, we show that the realizations of HH and HC vanish.In the fifth section, we recall the definition of derived nilpotent invariance and prove the Theo-rem 1.2 using the computations from the previous section.The last section is devoted to considering consequences of Theorem 1.2. In particular, we proveTheorem 1.1 and sketch some other possible applications of our ideas. onventions. Most of the time, we work over the field of complex numbers C . This is because thevery first constructions are using existence of an analytification functor, assigning to a separatedscheme of finite type over C its space of complex points with analytic topology.We freely use the language of ∞ -categories as in the works of J. Lurie. We do not use heavymachinery though; the main benefit is to avoid a lot of cofibrant replacements when working withlocalizations and to work conveniently with categories enriched in homotopy types and spectra.By a left Kan extension we mean the ∞ -categorical version, see [LurHTT], Definition 4.3.3.2,and similarly other colimits in ∞ -categories (which may be presented as homotopy colimits inappropriate model structures).By M od A , we denote the dg-category of (right) modules over a dg-algebra A . By Perf( A ) wedenote the full subcategory of compact objects in M od A .The dg-category of A -modules has a forgetful (aka generalised Eilenberg-MacLane) functor tothe stable ∞ -category Sp of spectra. Since we are working with K-theory, it would be convenientoften to consider such invariants as HH( · /k ) , HP( · /k ) as spectra forgetting the k -linear structure.The (semi-)topological K-theory functor will as well take values in the category of spectra, though,for the purposes of the lattice conjecture, it can be rationalized at any moment.Small triangulated (=idempotent-complete pretriangulated) C -dg-categories form an ∞ -category dgCat C . To any small dg-category T one can assign its triangulated envelope Perf( T ) , – this canbe thought of as the fibrant replacement functor in the Tabuada’s model category of small C -dg-categories and Morita-equivalences ([Tab]). This theory is equivalent to the ∞ -category ofidempotent-complete C -linear small stable ∞ -categories (Corollary 5.5, [Cohn]).By a weakly localizing invariant, we mean a (Morita-invariant) functor F : dgCat C −→ C to a stable ∞ -category C , which sends exact triangles of triangulated dg-categories to fiber sequencesin C . Note that some authors call such functors "localizing"; in this paper, this term will be reservedfor functors that also commute with filtered colimits ( HP and HC − are not localizing in this strongsense). Acknowledgements.
I would like to thank Chris Brav, my advisor, for sharing his ideas and a lotof helpful discussions. I also would like to thank D. Kaledin and A. Prikhodko for many fruitfulconversations we had while I was working on this paper, and S. Gorchinskiy for his attention to myMaster’s thesis where a special case of the main result was proved.2.
Realization functors
In this section, we introduce two ways of assigning the so-called topological realization spectrumto a spectral presheaf on the category of affine schemes of finite type over C and prove that, infact, these two procedures produce the same spectrum. The realization formalism will later beused to define semi-topological K-theory of dg-categories following A. Blanc [Bla]. Each of the twoprocedures has its own advantages, which will be used throughout the paper.The results of this section are not new, they appear in a similar form in [AH] with spaces insteadof spectra as a target category. Besides introducing notation, this chapter is intended to clarify theproof of the comparison theorem (see also Remark 2.1). e denote by Aff f . t . C ⊂ Sch f . t . C the category of affine schemes of finite type and the categoryof separated schemes of finite type over the field of complex numbers and by Aff sm C ⊂ Sch sm C thecorresponding full subcategories of smooth schemes. There is a standard functorial way to definea topology on the set of complex points of a separated scheme X of finite type over C . We denoteby X an this space and by X han the corresponding homotopy type. So we have the functors Sch f . t . C ( · ) han / / ( · ) an (cid:15) (cid:15) Spc Σ ∞ ( · ) + / / SpTop cl , where we denote by Spc the ∞ -category of homotopy types, by Sp the ∞ -category of spectra andby Top cl the 1-category of Hausdorff compactly generated topological spaces, where we use thesubscript cl to emphasize that the functor ( · ) an takes values in honest topological spaces instead ofhomotopy types. Note that this functor commutes with gluing of schemes, so ( · ) an is determinedby its restriction to the subcategory of affine schemes. By abuse of notation, we will use ( · ) an and ( · ) han to denote the corresponding restrictions.Now we want to extend the functor ( · ) han to simplicial presheaves on Aff f . t . C . The first, straight-forward way to do this is via left Kan extension along the Yoneda embedding: Aff f . t . C ( · ) han / / Y (cid:15) (cid:15) Spc Σ ∞ ( · ) + / / SpPr
Spc (Aff f . t . C ) , Y ∗ ( · ) han ssssssssss The value of Y ∗ ( · ) han on a presheaf F can be expressed via the usual colimit formula for left Kanextensions: Y ∗ ( F ) han ≃ colim Y ( X ) → F ∈ Y/F X han . The functors Y ∗ ( · ) han and Σ ∞ ( · ) + commute with colimits, so by the universal property of sta-bilization ([LurHA], Corollary 1.4.4.5), we define Re : Aff f . t . C ( · ) han / / Y (cid:15) (cid:15) Spc Σ ∞ ( · ) + / / SpPr
Spc (Aff f . t . C ) Y ∗ ( · ) han ssssssssss (cid:15) (cid:15) Pr Sp Aff f . t . C , Re ; ; ①①①①①①①①①①①①①①①①①①①①①①① using that the stabilization of Pr Spc (Aff f . t . C ) is the stable ∞ -category Sp(Pr
Spc (Aff f . t . C )) ≃ Pr Sp Aff f . t . C ([LurHA], Remark 1.4.2.9). The functor Re has a right adjoint given by E
7→ { X Hom(Σ ∞ + X han , E ) } . emark 2.1. If instead of doing this two-step procedure, we take simply the left Kan extensionof Σ ∞ ( · ) han+ to Pr Sp Aff f . t . C , the resulting functor will not commute with the loops functor and itsvalue on a presheaf F will actually coincide with its value on the connective cover ˜ F of F : Lan Σ ∞ ( Y ) + (Σ ∞ ( · ) han+ )( F ) ≃ colim Y ( X ) → F (Σ ∞ ( X ) han+ ) ≃ colim Y ( X ) → ˜ F (Σ ∞ ( X ) han+ ) ≃ Lan Σ ∞ ( Y ) + (Σ ∞ ( · ) han+ )( ˜ F ) . Remark 2.2.
Note that we could do the same procedure, starting from the category Aff f . t . C sm . Wedenote the resulting functor by Re sm .Now we consider the other way to extend the analytification functor to Pr Sp Aff f . t . C .Take some F ∈ Pr Sp Aff f . t . C . We can construct its left Kan extension along the complex pointsfunctor: Aff f . t . C op F / / ( · ) an (cid:15) (cid:15) SpTop op cl , an ∗ F < < ①①①①①①①①① and then take its value on the standard cosimplicial object ∆ • top : ∆ −→ Top cl . Now we define Re ′ F to be the realization of the resulting simplicial spectrum: Re ′ F := | (an ∗ F )(∆ • top ) | . So weobtain a functor Re ′ as the following composition: Re ′ : Pr Sp (Aff f . t . C ) an ∗ −−→ Pr Sp Top cl (∆ • top ) ∗ −−−−−→ Pr Sp (∆) |·| −→ Sp . This construction was considered in [FW01] and was used to define semi-topological K-theoryfor quasi-projective varieties (see also [FW05]).Now we prove a version of Antieau-Heller Theorem 2.3 [AH]:
Theorem 2.3.
There is a natural equivalence between the functors Re and Re ′ .Proof. To prove that Re ≃ Re ′ , note that both functors commute with colimits and shifts, henceit is sufficient to provide an equivalence on their restrictions to the subcategory of representablepresheaves.For X ∈ Aff f . t . C we want to compute Re ′ (Aff f . t . C ( · , X )) . Since Σ ∞ + preserves left Kan extensions,by applying Yoneda lemma twice: ∀ H ∈ Pr Set (Top cl ) Pr Set (Aff f . t . C )(Aff f . t . C ( · , X ) , H ( · ) an ) ≃ H ( X an ) ≃ Pr Set (Top cl )(Top cl ( · , X an ) , H ) , and using the adjunction between left Kan extension and precomposition, we deduce ( an ∗ (Aff f . t . C ( · , X )))( M ) ≃ Σ ∞ (Top cl ( M, X an )) + . and, by functoriality, we get Re ′ (Aff f . t . C ( · , X )) ≃ | [ n ] Σ ∞ ( Sing n ( X an )) + | ≃ Σ ∞ ( X han ) + ≃ Re (Aff f . t . C ( · , X )) and Re ′ ≃ Re . (cid:3) Unravelling the definitions from the next section, we get the following corollary. orollary 2.4. There are natural isomorphisms: K st ( T ) ≃ | [ n ] colim ∆ n top → (Spec R ) an ∈ ( · ) an / ∆ n top K( T ⊗ C R ) | , ˜K st ( T ) ≃ | [ n ] colim ∆ n top → (Spec R ) an ∈ ( · ) an / ∆ n top ˜K( T ⊗ C R ) | . These formulae have the advantage as being colimits of K-theory spectra instead of havingcolimits over some K-theory functor.3.
K-theories of dg-categories and topological Chern character
In this section we give a brief survey of results in [Bla].Consider the functors K( T ⊗ C · ) , ˜K( T ⊗ C · ) : Aff f . t . C op −→ Sp , assigning to an affine scheme S = Spec R the spectra of algebraic K-theory and connective algebraicK-theory of the dg-category T ⊗ C R respectively. Note, that, due to Morita-invariance of K-theory,it essentially does not matter if one replaces the dg-category T ⊗ C R by T ⊗ C Perf( S ) .Using topological realization, one defines semi-topological K-theory: Definition 3.1. ([Bla], Definition 4.1) The nonconnective/connective semi-topological K-theory ofa C -dg-category T is the spectrum K st ( T ) := Re (K( T ⊗ C · )) / ˜K st ( T ) := Re ( ˜K( T ⊗ C · )) . Bymonoidality of Re ([Bla], Proposition 3.12), K st can be considered here as a functor from K( · ) -modules to K st ( C ) -modules and analogously for the connective version.The coming definition of topological K-theory is based on the following calculation. Theorem 3.2. ( [Bla] , Theorems 4.5, 4.6) There are canonical equivalences of ring spectra K st ( C ) ≃ ˜K st ( C ) ≃ ku , where ku is the connective cover of the topological K-theory spectrum KU . Here the second equivalence is due to a direct computation and the first one comes from vanishingof negative algebraic K-theory for regular affine schemes and the following theorem:
Theorem 3.3. ( [FW03] , Corollary 2.7 or [Bla] , Theorem 3.18) For any spectral presheaf F ∈ Pr Sp (Aff f . t . C ) , the functor l ∗ : Pr Sp (Aff f . t . C ) −→ Pr Sp (Aff f . t . C sm ) , which restricts a given presheaf to the subcategory of affine smooth schemes, produces an equiv-alence: Re sm ( l ∗ F ) ≃ Re ( F ) . Remark 3.4.
Due to Theorem 2.3, for X a projective weakly normal variety, K st ( X ) is equivalentto K semi ( X ) of Friedlander-Walker, which was defined in terms of analytification of the ind-scheme M or ( X, Gr ) . In particular, if X = Spec C , K semi ( C ) is the connective spectrum corresponding tothe group-completion of Gr an = G n ≥ Gr ( n, ∞ ) an , which is BU × Z , so the statement of Theorem 3.2 ([Bla], Theorems 4.5) follows. And similarly forthe statement of Proposition 3.6 ([Bla], Theorems 4.5) below. ote that KU ≃ ku [ β − ] ≃ K st ( C )[ β − ] for an appropriate choice of the Bott generator β in π (K st ( C )) . Now define topological K-theory Definition 3.5. ([Bla], Definition 4.13) The topological K-theory of a C -dg-category T is the KU -module K st ( T )[ β − ] .This definition is compatible with the classical one: Proposition 3.6. ( [Bla] , Proposition 4.32) Let X be a separated C -scheme of finite type. Thenthere exists a canonical isomorphism: K top ( P erf ( X )) ≃ KU ( X han ) . So, at least in the case of decent schemes, (semi-)topological K-theory, unlike algebraic K-theory, tends to have not very large homotopy groups. There is a topological Chern character Ch top : K top → HP , and K top ( T ) ⊗ Q can be considered as a candidate for the role of a rationalstructure in the periodic cyclic homology HP( T ) .By establishing compatibility with the classical topological Chern character Ch utop : KU ( X han ) → H C [ u ± ]( X han ) , Blanc proves the lattice conjecture in the case of schemes ([Bla], Proposition 4.32).Blanc also considered the simplest noncommutative case, the case of finite-dimensional dg-algebras, in which the lattice conjecture was proved after replacing K top by ˜K top , which in generalhas worse properties (in particular, there is no simple reason for ˜K top to be a localizing invariant).In our Theorem 6.1 and Proposition 6.4, we consider two different generalizations of the aforemen-tioned case and prove the lattice conjecture for each of them. In particular, it follows that there isno need for replacing K top by ˜K top in [Bla], Proposition 4.37, either by dimension comparison orby our Proposition 6.2, whcih directly shows that the natural map ˜K top → K top is an equivalencein this case. 4. Topological realization of Hochschild-type invariants
In this section, we consider behaviour of the topological realization applied to Hochschild-typeinvariants. The main statement here is that the topological realization of negative cyclic homologycoincides with the topological realization of periodic cyclic homology, thus is nil-invariant in thesense that we will make precise in the next section.From now on, k is any subfield of C , so one can consider C -dg-categories as k -dg-categories anddefine Hochschild-type invariants in the k -linear setting. By forgetting the H Z -module structure,we consider these invariants as functors to the category of spectra.We start by computing the topological realization of the relative version of Hochschild homology. Proposition 4.1.
Let T be a C -dg-category. Then the topological realization of the functor HH( T ⊗ C · /k ) : Aff f . t . C op −→ Sp R HH( T ⊗ C R/k ) is trivial. roof. We start with noticing that, since Hochschild homology is a monoidal functor (this seemsto be a folklore statement, see Proposition 2.1 of [AV] for a proof and §4.2.3 of [Lod] for an earlierincarnation of this phenomenon) and topological realization is monoidal (by Proposition 3.12 [Bla]), Re HH( · /k ) is a connective ring spectrum and Re (HH( T ⊗ C · /k )) is a Re HH( · /k ) -module. So itsuffices to prove that Re HH( · /k ) is trivial. We show that its π -group is trivial, hence the identitymorphisms on Re HH( · /k ) and on Re (HH( T ⊗ C · /k )) are trivial as well.By Proposition 3.10 [Bla], it suffices to prove that every two elements a, b ∈ HH ( C /k ) ≃ C arealgebraically equivalent, i.e., for any two morphisms h C ⇒ HH( · /k ) , there exists an algebraic curve C and a commutative diagram h C a ❍❍❍❍❍❍❍❍❍ (cid:15) (cid:15) h C / / HH( · /k ) h C b ; ; ✈✈✈✈✈✈✈✈✈ O O Take C = Spec C [ t ] , the morphisms Spec C → C given by the ideals ( t − a ) and ( t − b ) and themorphism h C → HH( · /k ) given by t . The diagram is commutative by definition, so we are done. (cid:3) Now we can deduce the following important statement.
Proposition 4.2.
Let T be a (smooth proper) C -dg-category. Thena) the topological realization of the functor HC( T ⊗ C · /k ) : Aff f . t . C op −→ Sp is zero;b) the topological realization of the morphism of the functors HC − ( T ⊗ C · /k ) −→ HP( T ⊗ C · /k ) : Aff f . t . C op −→ Sp is an equivalence. Here by HC , HC − and HP we denoted the cyclic homology, the negative cyclic homology andthe periodic cyclic homology respectively. Proof.
The cofiber of the map HC − ( T ⊗ C · /k ) −→ HP( T ⊗ C · /k ) is HC( T ⊗ C · /k )[2] , so the claimb) follows from a), since topological realization commutes with colimits and shifts.To prove a), we proceed by Re (HC( T ⊗ C · /k )) ≃ Re (colim BS HH( T ⊗ C · /k )) ≃ colim BS Re HH( T ⊗ C · /k ) ≃ , using commutation with colimits and the previous proposition. (cid:3) Since the realization of negative cyclic homology coincides with the realization of periodic cyclichomology, it, in a sense, accumulates good properties of both theories. This idea will be used inthe next section. . Nilpotent extensions of connective dg-algebras
In this section, we remind the definition of (derived) nil-invariants and theorems about nil-invariance of the fiber of the cyclotomic trace K inv and of periodic cyclic homology. We use thesetheorems to prove nil-invariance of semi-topological K-theory.Suppose that v : B → A is a map of connective dg-algebras or connective ring spectra, such thatthe induced map π v is a surjection with a nilpotent kernel I : π A ≃ π B/I . Then we call themap v a nilpotent extension of connective ring spectra. We say that the functor from the categoryof (connective) dg-algebras is nil-invariant if it induces equivalences on nilpotent extensions. Thefollowing two theorems provide two important examples of nil-invariants: Theorem 5.1. ( [G85] ) Let k be a field of characteristic 0 and v : B → A be a nilpotent extensionof connective k -dg-algebras. Then the induced map HP(
B/k ) → HP(
A/k ) is an equivalence. Theorem 5.2. ( [DGM] , see also [Ras] for a modern presentation) Let v : B → A be a nilpotentextension of connective ring spectra. Then the induced map fib( ˜K( B ) tr ( B ) −−−→ TC( B )) → fib( ˜K( A ) tr ( A ) −−−→ TC( A )) is an equivalence. Here tr : ˜K → TC is the cyclotomic trace map Remark 5.3.
Using the stable structure, the last theorem can be reformulated as a statementabout equivalence of coker( ˜K( B ) → ˜K( A )) and coker(TC( B ) → TC( A )) . The former cokernel isequivalent to coker(K( B ) → K( A ) ) due to nil-invariance of non-positive K-theory. Proposition 5.4.
Let v : B → A to be a nilpotent extension of connective S -algebras. Thenthe induced map on the coconnective truncations of algebraic K-theory τ ≤ K( B ) → τ ≤ K( A ) is anequivalence. This seems to be a folklore statement. We learned it in this generality from S. Raskin; theexperts probably knew it for a long time.
Sketch of the proof.
We proceed in two steps.The first step is to show nil-invariance of K . For a connective ring R , K , ( R ) ≃ K , ( π R ) ([DGM], 3.1.2 or [LurKM], Lecture 20, Corollaries 3, 4), so we can reduce to classical rings and thento Noetherian classical rings using that K-theory commutes with filtered colimits. For a classicalNoetherian nilpotent ring extension A = B/I , the natural correspondence
P roj fg ( B ) −→ P roj fg ( A ) P P ′ ≃ P ⊗ B A is a bijection on the sets of isomorphism classes, which can be shown by reduction to the square-zero case and application of an explicit process of lifting idempotents.The second step is to use Bass’s formula to deduce nil-invariance for K i − from nil-invariancefor K i and to proceed by induction on − i . (cid:3) ow we use the theorems above and the statements from the previous sections to prove the mainresult: Theorem 5.5.
Let v : B → A be a nilpotent extension of connective C -dg-algebras. Then theinduced map K st ( B ) → K st ( A ) is an equivalence.Proof. We want to use the Re ′ presentation of semi-topological K-theory to express it as a colimitof algebraic K-theories, then to apply Theorem 5.2 termwise, and then, after commuting somecolimits, use our comparison of the realizations of negative cyclic homology and periodic homology,together with nil-invariance of the latter.Remind that, by Corollary 2.4, the map K st ( B ) → K st ( A ) can be expressed as | colim ∆ n top → (Spec R ) an K( B ⊗ C R ) | K st ( v ) −−−−→ | colim ∆ n top → (Spec R ) an K( A ⊗ C R ) | , where the map is determined by the map of functors K( B ⊗ · ) → K( A ⊗ · ) .Consider the cokernel of this map. Commuting the colimits we can express it as coker(K st ( v )) ≃ | colim ∆ n top → (Spec R ) an (coker(K( B ⊗ C R ) → K( A ⊗ C R ))) | . Now, since R in the diagram is taking values in classical regular rings, the map v ⊗ R is a nilpotentextension, so, using functoriality of the cyclotomic trace map, we can apply Theorem 5.2 termwise: coker(K st ( v )) ≃ | colim ∆ n top → (Spec R ) an (coker(TC( B ⊗ C R ) → TC( A ⊗ C R ))) | . For a connective Q -algebra S , TC( S ) ≃ TC − ( S ) ≃ THH( S ) hS ≃ HH( S/ Q ) hS ≃ HC − ( S/ Q ) ,so all the finite coefficient type data vanishes and we get coker(K st ( v )) ≃ | colim ∆ n top → (Spec R ) an (coker(HC − ( B ⊗ C R ) → HC − ( A ⊗ C R ))) | . We can commute the colimits in the formula again, to get back the cokernel of realizations: coker(K st ( v )) ⊗ Q ≃ coker( Re ( HC − ( B ⊗ C · )) → Re ( HC − ( A ⊗ C · ))) . By Proposition 4.2, functoriality of u -inversion in HC − and one more iteration of commuting thecolimits, we get coker(K st ( v )) ⊗ Q ≃ coker( Re ( HP ( B ⊗ C · )) → Re ( HP ( A ⊗ C · ))) ≃≃ Re (coker( HP ( B ⊗ C · ) → HP ( A ⊗ C · ))) ≃ Re (0) ≃ , where for the second to the last equivalence we apply Theorem 5.1. (cid:3) orollary 5.6. Let v : B → A be a nilpotent extension of connective C -dg-algebras. Then Ch top A ⊗ C : K top ( A ) ⊗ C → HP( A/ C ) is an equivalence if and only if so is the map Ch top B ⊗ C : K top ( B ) ⊗ C → HP( B/ C ) . Proof.
We have a natural commutative diagram K top ( B ) ⊗ C Ch top B / / (cid:15) (cid:15) HP( B/ C ) (cid:15) (cid:15) K top ( A ) ⊗ C Ch top A / / HP( A/ C ) , where the right column map is an equivalence by Theorem 5.1. By Theorem 5.5, the left columnis an equivalence as well, since there is a natural equivalence K top ( · ) ⊗ C ≃ K st ( · ) ⊗ ku KU ⊗ C ≃ (K st ( · ) ⊗ Q ) ⊗ ku Q ( KU Q ) ⊗ Q C and this is a product of a nil-invariant and a constant functor. Sothe upper map is an equivalence iff so is the bottom. (cid:3) Some examples and applications
Connective dg-algebras.
We remind that any smooth and proper C -dg-category is Morita-equivalent to a smooth proper C -dg-algebra, so the question about existence of a natural (Morita-invariant) rational structure on periodic cyclic homology of smooth proper C -dg-categories es-sentially boils down to existence of a natural rational structure on periodic cyclic homology ofdg-categories of the form Perf( B ) for B a smooth and proper C -dg-algebra. As an applicationof Theorem 5.5, we prove that K top provides such a structure in the case of connective proper C -dg-algebras. The smoothness assumption is not necessary in this case. Theorem 6.1.
Let B be a connective proper C -dg-algebra. Then the complexified Chern charactermap K top ( B ) ⊗ C → HP( B ) is an equivalence.Proof. We use Corollary 5.6 to reduce to the case of classical finite-dimensional algebra B red withoutnilpotents, hence semi-simple. Now, by Wedderburn’s theorem, we deduce that B red is Morita-equivalent to a finite product of C , so our statement follows from the corresponding equivalence for Spec C and additivity. (cid:3) This theorem generalizes Proposition 4.37 [Bla] because of the following simple observation.
Proposition 6.2.
Let v : B → A be a nilpotent extension of connective C -dg-algebras and A be aclassical Noetherian C -algebra of finite global dimension. Then the natural maps ˜K st ( B ) −→ K st ( B ) nd ˜K top ( B ) −→ K top ( B ) are equivalences of spectra. This shows that in the setting of Proposition 4.37 [Bla], ˜K st ( B ) ≃ K st ( B ) and the pseudo-topological K-theory can safely be replaced by K top even without proving Theorem 6.1 (from whichit would follow by dimension counting). Remark 6.3.
There are simple examples when ˜K st ( T ) = K st ( T ) , e. g. taking T := C [ t , t ] / ( t t (1 − t − t )) works: ˜K st ( C [ t , t ] / ( t t (1 − t − t ))) ≃ ku ⊕ Σ ku , while K st ( C [ t , t ] / ( t t (1 − t − t ))) ≃ Ω ku ⊕ ku ,so K st − ( C [ t , t ] / ( t t (1 − t − t ))) ≃ Z . However, it seems a tricky task to find a decent dg-category T , such that ˜K top ( T ) = K top ( T ) . Proof of Proposition 6.2.
For every regular classical commutative C -algebra D , if B → A is a nil-extension, B ⊗ D → A ⊗ D is a nil-extension as well. Also Noetherian C -algebras of finite globaldimension have no negative K-theory, so ˜K( A ⊗ D ) ≃ K( A ⊗ D ) under these assumptions. Usingnilinvariance of non-positive algebraic K-theory groups Proposition 5.4 and Corollary 2.4, we getequivalences: K st ( B ) ≃ | [ n ] colim ∆ n top → D an K( B ⊗ C D ) | ≃ | [ n ] colim ∆ n top → D an ˜K( B ⊗ C D ) | ≃ ˜K st ( B ) and the equality K top ( B ) ≃ ˜K top ( B ) follows. (cid:3) Using Proposition 6.1 (and nil-invariance of periodic cyclic homology, in order to get a simpleranswer), one can also write formulas expressing HP in terms of the realization of the stack of perfect(or projective) modules, analogous to [Bla], section 4.7 in this context.We also want to mention that Proposition 4.37 [Bla] can be generalized in another way.
Proposition 6.4.
Let B be a smooth C -dg-algebra, which is finite dimensional, i.e. B is Morita-equivalent to a C -dg-algebra A , such that after forgetting the multiplication, the differential andthe grading on A one gets a finite-dimensional C -vector space: dim( L i ∈ Z A i ) < ∞ . Then thecomplexified topological Chern character Ch top : K top ( B ) ⊗ C → HP( B/ C ) is an equivalence. Note that the property of being finite-dimensional is stronger than properness: dg-algebrasare rarely formal (or have a finite-dimensional model). In particular, by [Orl], Corollary 2.21, K ( T ) ≃ Z ⊕ n for T Morita-equivalent to a smooth finite dimensional B , but even for a propersmooth variety X K ( X ) is usually not finitely generated. Proof.
In a recent paper, Orlov generalizes the Auslander construction and shows that, for a finitedimensional C -dg-algebra A , there is a fully faithful embedding of Perf( A ) into the category ofperfect complexes over a dg-algebra E constructed from A such that the category Perf( E ) comeswith a full exceptional collection ([Orl], Theorem 2.19). he equivalence K top ( E ) ⊗ C ≃ HP( E / C ) follows from existence of a full exceptional collection on E and the case of Spec C , since both invariants are weakly localizing. Since we assumed that A/ C issmooth, the embedding Perf( A ) −→ Perf( E ) is admissible, hence, by functoriality, the equivalence K top ( A ) ⊗ C ≃ HP( A/ C ) follows from the equivalence K top ( E ) ⊗ C ≃ HP( E / C ) , since the former isa retract in the latter. (cid:3) Derived schemes.
We can use our nilinvariance result to generalize [Bla], Proposition 4.32to the derived setting. Restricted to the class of commutative dg-algebras, Theorem 5.5 statesthat semi-topological K-theory of a derived affine scheme X / C only depends on the correspondingreduced classical subscheme X/ C . This observation stays true globally. Proposition 6.5.
Let X be a quasi-compact quasi-separated derived scheme over C . Denote X :=( π X ) red → X the reduction of the classical truncation of X . Let F be a nil-invariant weakly localizinginvariant. Then the corresponding morphism F ( X ) → F ( X ) is an equivalence. Applying this proposition to F := K st and F := HP( · / C ) together with Proposition 3.6, we getthe following corollary. Corollary 6.6.
Let X / C be a derived scheme, such that its classical part π X is a separated schemeof finite type (this includes laft schemes of [GR] ). Then there are canonical equivalences: K top (Perf( X )) ≃ K top (Perf( π X )) ≃ KU (( π X ) han ) , K top (Perf( X )) ⊗ C ≃ HP(Perf( X )) ≃ HP(Perf( π X )) . As the proof of Proposition 6.5 below suggests, this equivalences can be understood by covering X by affines and proceeding inductively. The reason behind the finiteness assumption is that, unlike(semi-)topological K-theory, periodic cyclic homology does not commute with filtered colimits. Proof of Proposition 6.5.
The basic idea is to prove that weakly localizing invariants satisfy ‘derivedZariski descent’, by mimicking the standard argument from [TT]. In the qcqs case, this allows usto reduce the question to derived affine case, which is secured by Theorem 5.5 (and in the caseof π X separated of finite type allows to reduce to derived affine schemes with finite type classicalpart where one has comparison between two versions of topological K-theory and periodic cyclichomology).So the statement follows from the following observation (basically, contained in [CMNN], Ap-pendix A). Proposition 6.7.
Let X be a qcqs derived scheme together with two open quasicompact subschemes U ⊂ X ⊃ U and let F be a weakly localizing invariant. Then the triple F ( X ) → F ( U ) ⊕ F ( U ) → F ( U × X U ) is a fiber sequence. roof of Proposition 6.7. The statement follows from two facts. One is the existence of an exacttriple of triangulated dg-categories
Perf Z ( X ) −→ Perf( X ) −→ Perf( U ) for j : U ⊂ X as before, Z a closed complement in the underlying topological space of X to U and Perf Z ( X ) the sub-dg-category of perfect complexes supported on Z , i.e., such F ∈
Perf( X ) that j ∗ F = 0 ([CMNN], Corollary A.10). The other observation is the independence of Perf Z ofthe scheme Z is contained in: j ∗ : Perf Z ( X ) → Perf Z ( U ) is an equivalence ([CMNN], PropositionA.14). This allows us to identify the fibers of F ( X ) → F ( U ) and F ( U ) → F ( U × X U ) , finishingthe proof. (cid:3)(cid:3) Corollary 6.6 can be formulated slightly more generally.
Corollary 6.8.
Let Y / C be a derived algebraic space, together with a surjective etale map X → Y ,such that X is as in Corollary 6.6. Then the topological Chern character K top (Perf( Y )) ⊗ C → HP(Perf( Y )) . is an equivalence.Proof. By Theorem A.4 [CMNN], the topological Chern character, being a map of weakly localizinginvariants valued in Q -modules, is a map of etale sheaves, so we can compute both invariants byCech diagrams, which by Corollary 6.6 are equivalent. (cid:3) This minor generalization comes for free, but we also are optimistic about lattice conjecture forstacky quotients of derived schemes.6.3.
Group algebras.
Let G be a group. We are interested in understanding the topologicalChern character map for the corresponding group algebra C [ G ] , at least after complexification. Inthe next subsection, this will be applied to prove the lattice conjecture for ∞ -categories of localsystems on reasonable topological spaces.We start by considering two simple classes of groups for which all the necessary work has alreadybeen done in previous sections. Proposition 6.9.
Let G be a finite group or a finitely generated abelian group. Then the topologicalChern character induces an equivalence K top ( C [ G ]) ⊗ C ≃ HP( C [ G ]) . Proof.
In the finite group case the statement follows from Theorem 6.1; in the finitely generatedabelian group case the equivalence follows from Proposition 3.6 . (cid:3) emark 6.10. Of course, in these cases one can always write down an explicit formula for semi-topological K-theory: the semi-topological K-theory of C [ Z ⊕ n ] is just the connective part of thetopological K-theory of n -torus, adding cyclic groups corresponds to taking several copies of it; inthe case of a finite group, one can reduce to the semi-topological K-theory of several copies of C : K st ( C [ G ]) ≃ ku ⊕ m .For a general group, one cannot expect such a result to be true. Nevertheless, the latticeconjecture can be proved for a surprisingly large class of groups via reducing to their finite subgroups.In [Bur], it was shown that the target of the map K top ( C [ G ]) ⊗ C → HP( C [ G ]) can always be understood via homology of the cyclic subgroups of G , namely the isomorphism(1) HP ∗ ( C [ G ]) ≃ L [ g ] ∈h G i fin H [ ∗ ] ( C g ; C ) ⊕ L [ g ] ∈h G i ∞ T G ∗ ( g ; C ) , was proved, where h G i fin / h G i ∞ stand for the conjugacy classes of the finite / infinite orderelements in G , C g is the centralizer of g ∈ G , H [ ∗ ] ( C g ; C ) := Q i ∈ Z H ∗ +2 i ( C g ; C ) , T G ∗ ( g ; C ) :=lim ←− (cid:0) · · · S −→ H ∗ +2 n ( C g / h g i ; C ) S −→ H ∗ +2 n − ( C g / h g i ; C ) S −→ · · · (cid:1) and S are Gysin maps.As we remind below, the image of the topological Chern character projects as zero on the secondterm in the formula above, hence vanishing of this term is a necessary condition for the latticeconjecture to be true for C [ G ] . The condition of such vanishing for a group G is sometimes called(generalized) Burghelea conjecture. The [EM] paper is a good source on the current state of theBurghelea conjecture, in particular it has been shown there that the conjecture holds for a largeclass of groups. Remark 6.11.
In Section 5, [EM], there were also given counterexamples to the Burghelea con-jecture with relatively good finiteness properties. Since the topological Chern character map knowsnothing about T G term, those group algebras provide (very non-proper) counterexamples to thelattice conjecture. Note also that Burghelea did not in any way expect vanishing of T G -term forall groups: he only asked if it is true for the groups admitting a finite classifying space and he alsogave a (not finitely generated) example of a group where the T G -term is nonzero.The left hand side of the map K top ( C [ G ]) ⊗ C → HP( C [ G ]) is built via colimits from the spectra of the form K( R [ G ]) ⊗ C where R runs through the categoryof finitely generated regular C -algebras. K-theory of group algebras has been studied a lot for thelast decades in relation to the Farrell-Jones’s conjecture. Rationally for C -algebras this conjecturebecomes similar to the Burghelea’s computation: roughly speaking it states that the K-theory of C [ G ] can be computed in terms of the K-theories of C [ H ] for finite cyclic subgroups H ⊂ G .In [LR06], the authors show that an additive invariant I can be promoted to an Or G -specrum I = I R (i.e. a functor from the subcategory of G -sets of the form G/H to spectra), such that I R ( G/H ) ≃ I ( R [ H ]) . Then one can left Kan extend an Or G -spectrum I to a functor ( − ) ∧ Or G I ∈ Fun(Top G , Sp) – homotopy groups of its values are usually denoted by H G ∗ ( − , I ) . Restricting to = C , applying this construction to the Chern character and substituting the map of G -spaces E F G → G/G = pt , one obtains a commutative square: E F G ∧ Or G K ( C [ − ]) / / (cid:15) (cid:15) G/G ∧ Or G K ( C [ − ]) ≃ K( C [ G ]) (cid:15) (cid:15) E F G ∧ Or G HP ( C [ − ]) / / G/G ∧ Or G HP ( C [ − ]) ≃ HP( C [ G ]) . Here E F G ∈ Top G is the classifying space for a family F of subgroups in G , which can becharacterized by requiring its subspace of H -invariants ( E F G ) H to be contractible if H ∈ F , andempty otherwise. It follows that the usual colimit diagram for the left Kan extension E F G ∧ Or G I does not have I ( G/H ) ≃ I ( R [ H ]) -terms for H / ∈ F , and for F = h ( e ) i E F G ∧ Or G I ≃ B G + ∧ I ( R ) .We refer to [LR06] for details and to [RV], [LR05], [L¨uck] for more information on the Farrell-Jones conjecture and relevant equivariant homotopy theory; in the case of a finite type regular C -algebra R and a group G , the rational Farrell-Jones conjecture states that for a family of finite cyclicgroups F = F Cyc G =: F Cyc , the assembly map E F G ∧ Or G K ( R [ − ]) Q → G/G ∧ Or G K ( R [ − ]) Q ≃ K( R [ G ]) Q is an equivalence (here Propositions 2.14 and 2.20 of [LR05] are used to reduce to F Cyc instead of
V Cyc ). As before, we use that the realization, Bott-inverting and complexificationfunctors commute with colimits to get a commutative square:(2) E F Cyc ( G ) ∧ Or G K top ( C [ − ]) / / (cid:15) (cid:15) G/G ∧ Or G K top ( C [ − ]) ≃ K top ( C [ G ]) (cid:15) (cid:15) E F Cyc ( G ) ∧ Or G HP ( C [ − ]) / / G/G ∧ Or G HP ( C [ − ]) ≃ HP( C [ G ]); whenever the Farrel-Jones conjecture is true for G , the top map is an equivalence of C [ β ] -modules.The bottom map is another way to write the embedding of the first term in (1), in particular, it isan equivalence iff the Burghelea conjecture is satisfied by G . Now we formulate the main result ofthis subsection. Proposition 6.12.
The complexified topological Chern character is an equivalence for C [ G ] if thegroup G belongs to one of the following classes:(i) hyperbolic groups;(ii) finite-dimensional CAT(0) groups;(iii) mapping class groups of compact orientable surfaces with punctures;(iv) systolic groups;(v) compact 3-manifold groups;(vi) Coxeter groups;(vii) right-angled Artin groups;(viii) solvable groups of finite Hirsch lenghth.Proof. The Farrell-Jones is true for these classes of groups ([BLR], [BL], [BB], [Rou], [DJ], [W]), sothe top map is an equivalence. The Burghelea conjecture is true for all these groups ([Ji], [EM]), sothe bottom map is an equivalence. The left map is an equivalence as well, since the complexifiedtopological Chern character becomes an equivalence of functors after restriction to the subcategoryof algebras of the form C [ H ] for H any finite (cyclic) group (Proposition 6.9). Therefore the rightvertical map is an equivalence. Remark 6.13.
When the group G has no non-trivial finite cyclic subgroups, i.e. G is torsion-free,the square (2) takes the simpler form(3) B G + ∧ K top ( C ) / / (cid:15) (cid:15) K top ( C [ G ]) (cid:15) (cid:15) B G + ∧ HP( C ) / / HP( C [ G ]); equivariant theory is no longer necessary here. Remark 6.14.
Let us say that the group G satisfies the semi-topological / topological / rationaltopological Farrel-Jones conjecture if the assembly map E V Cyc ( G ) ∧ Or G I → I ( C [ G ]) is an equiv-alence for the corresponding invariant. Of course, the usual Farrel-Jones conjecture for regular C -algebras = ⇒ F J st = ⇒ F J top = ⇒ F J top Q . By the proof of Proposition 6.12, in the triple h theBurghelea conjecture for G , the lattice conjecture for C [ G ] , F J top Q i , any two statements imply thethird. This might be helpful, for instance, for constructing a counterexample to the Farrell-Jonesconjecture: it is enough to provide a counterexample to the lattice conjecture of a form C [ G ] where G is a group for which Burghelea’s conjecture is secured. Actually, instead of C [ G ] one can considerthe algebra C ∗ (Ω M ) for any M , s.t. for π M Burghelea conjecture is true, see Corollary 6.15 andRemark 6.17.6.4.
Local systems.
We finish with one more application of Theorem 5.5. Nilinvariance of semi-topological K-theory implies that, applied to the ∞ -category of local systems of chain complexeson a decent space M , semi-topological K-theory depends only on the fundamental group π M .Let M be a pointed connected topological space which is locally of singular shape in the senseof [LurHA], Definition A.4.15. Then there is a fully faithful functor Spc
Sing( M ) −→ Shv( M, Spc) from Kan fibrations over
Sing( M ) to the category of sheaves on M with the essential image thesubcategory of locally constant sheaves ([LurHA], Theorem A.4.19, see also [To¨en], Theoreme 2.13),so, roughly speaking, the theory of locally constant sheaves on nice enough spaces can be definedon the level of homotopy types.Now we define the ∞ -category of local systems of C -complexes on a homotopy type. We considerthe functor LOC ∗ : Spc −→ DGCAT op C to the category of presentable dg-categories given by E Fun(
E, M od C ) (cf. [LurKM], Lecture21), which can also be defined as the left Kan extension from its value on pt . The image of the map f : E → E is given by the precomposition functor f ∗ , which has both left and right adjoints f ! and f ∗ . The ! -pushforwards preserve compact objects, so we can define a functor Loc( − , C ) : Spc −→ dgCat C E (Fun( E, M od C )) c ; f f ! . he functor Loc( − , C ) is equivalent to E Perf( C ∗ (Ω E )) . This seems to be a folklore statement,which is partially proved in [LurKM], Lecture 21; we omit the details here, since in the end we areonly interested in the categories of the form Perf( C ∗ (Ω E )) , which can be taken as the definition of Loc( E, C ) . By precomposing with M Sing • ( M ) , Loc( − , C ) can be made into a functor fromthe 1-category of topological spaces, which we, by abuse of notation, will again denote Loc( − , C ) .Applying Theorem 5.5, we get the following corollary. Corollary 6.15.
Let M be a pointed connected locally contractible topological space. Then there isa natural equivalence K st (Loc( M, C )) ≃ K st ( C [ π M ]) . In particular, when the fundamental group of M is nice enough, we deduce the lattice conjecturefor the category of local systems. Theorem 6.16.
Let M be a pointed connected locally contractible topological space such that itsfundamental group belongs to one of the following classes:(i) hyperbolic groups;(ii) finite-dimensional CAT(0) groups;(iii) mapping class groups of compact orientable surfaces with punctures;(iv) systolic groups;(v) compact 3-manifold groups;(vi) Coxeter groups;(vii) right-angled Artin groups;(viii) solvable groups of finite Hirsch lenghth.Then the topological Chern character K top (Loc( M, C )) ⊗ C → HP(Loc( M, C )) is an equivalence.Proof. Apply Corollary 5.6 and Proposition 6.12. (cid:3)
Remark 6.17.
The Corollary 6.15 shows that extending the lattice conjecture from group alge-bras to local systems does not bring new objects, but adds flexibility. Larger supply of maps inthe category of spaces should make semi-topological K-theory of local systems more computable.Remark 6.14 suggests to search for a space M with a fundamental group G , satisfying the Burghe-lea conjecture, such that the complexified topological Chern character has a nontrivial kernel –the fundamental group of such a space would provide a counterexample to the usual Farrell-Jonesconjecture. It might be easier sometimes to find a model with nice geometrical properties for such M than for B G . Remark 6.18.
The category
Perf( C ∗ (Ω M )) has one more incarnation. When M is a smoothconnected manifold, Perf( C ∗ (Ω M )) is Morita-equivalent to the wrapped Fukaya category of thecotangent bundle T ∗ M (see Theorem 1.1, Corollary 6.1 [GPS], or Theorem 1.1 [Ab] for the earliertreatment of closed oriented case). Theorem 6.16 guarantees that, under certain conditions on π M , HP( W ( T ∗ M )) comes with an integral structure. Since the topological Chern character is a map ofweakly localizing invariants, Lemma 6.2 and Corollary 6.3 [GPS] allow to extend this structure tocertain plumbings – it is enough to check joint fully-faithfullness of the maps in the diagrams (6.2),(6.3). The next logical step would be to try and generalize Theorem 6.16 to certain categories ofconstructible sheaves, which are related to more complicated wrapped Fukaya categories. eferences [Ab] M. Abouzaid, A cotangent fibre generates the Fukaya category , Adv. Math. 228 (2011), no. 2, 894–939.[AH] B. Antieau, J. Heller,
Some remarks on topological K-theory of dg categories , arXiv preprint 1709.01587.[AV] B. Antieau, G. Vezzosi,
A remark on the Hochschild-Kostant-Rosenberg theorem in characteristic p , arXivpreprint 1710.06039.[BB] A. Bartels, M. Bestvina,
The Farrell-Jones conjecture for mapping class groups . Preprint, available atarXiv:1606.02844, 2016.[Bla] A. Blanc,
Topological K-theory of complex noncommutative spaces , Compositio Math. 152 (2016), 489–555.[BL] A. Bartels, W. L¨uck,
The Borel Conjecture for hyperbolic and CAT(0)-groups . Ann. of Math. (2),175(2):631–689, 2012.[BLR] A. Bartels, W. L¨uck, H. Reich,
The K-theoretic Farrell-Jones Conjecture for hyperbolic groups . Invent. Math.,172(1):29–70, 2008.[Bur] D. Burghelea,
The cyclic homology of the group rings . Commentarii Mathematici Helvetici. 60. 354-365.10.1007/BF02567420 (1985).[CMNN] D. Clausen, A. Mathew, N. Naumann, J. Noel,
Descent in algebraic K-theory and a conjecture of Ausoni-Rognes , arXiv preprint 1606.03328 (2017).[Cohn] L. Cohn,
Differential graded categories are k-linear stable infinity categories , arXiv: 1308.2587, 2013.[DGM] Bj ø rn Ian Dundas, Thomas Goodwillie, and Randy McCarthy. The local structure of algebraic K-theory ,volume 18. Springer Science & Business Media, 2012.[DJ] M. W. Davis, T. Januszkiewicz,
Right-angled Artin groups are commensurable with right-angled Coxeter groups .J. Pure Appl. Algebra, 153(3):229–235, 2000.[EM] A. Engel, M. Marcinkowski.
Burghelea conjecture and asymptotic dimension of groups , Journal of Topologyand Analysis 12 (02), 321-356[FW01] E. Friedlander, M. Walker.
Comparing K-theories for complex varieties . Amer. J. Math., 123(5):779–810,2001.[FW03] E. Friedlander, M. Walker.
Rational isomorphisms between K-theories and cohomology theories , Inventionesmathematicae 154 (2003), no. 1, 1–61.[FW05] E. Friedlander, M. Walker,
Semi-topological K-theory , Handbook of K-theory (2005), 877–924.[G85] T.G. Goodwillie,
Cyclic homology, derivations, and the free loopspace . Topology 24 (1985), no. 2, 187215.[GPS] S. Ganatra, J. Pardon, V. Shende,
Microlocal Morse theory of wrapped Fukaya categories , arXiv:1809.08807,2020.[GR] D. Gaitsgory, N. Rozenblyum,
A Study in Derived Algebraic Geometry Vol. I. Correspondences and duality ,volume 221 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2017.pp. 1–26.[Ji] R. Ji,
Nilpotency of Connes’ Periodicity Operator and the Idempotent Conjectures , K-Theory 9 (1995), 59–76.[Kal] D. Kaledin,
Spectral sequences for cyclic homology . In Algebra, geometry, and physics in the 21st century,volume 324 of Progr. Math., pages 99–129. Birkhauser/Springer, Cham, 2017.[KKP] L. Katzarkov, M. Kontsevich, and T. Pantev,
Hodge theoretic aspects of mirror symmetry , arXiv preprintarXiv:0806.0107 (2008).[Lod] J.-L. Loday,
Cyclic homology . Grundlehren der Mathematischen Wissenschaften 301. Springer-Verlag, Berlin,1998.[LR05] W. L¨uck, H. Reich.
The Baum-Connes and the Farrell-Jones Conjectures in K- and L-theory . In Handbookof K-theory. Vol. 2, pages 703–842. Springer, Berlin, 2005.[LR06] W. L¨uck and H. Reich.
Detecting K-theory by cyclic homology . Proc. London Math. Soc. (3), 93(3):593–634,2006.[L¨uck] Wolfgang L¨uck.
Isomorphism Conjectures in K- and L-Theory . In preparation, preliminary version availableat him.uni-bonn.de/lueck/.[LurHTT] J. Lurie,
Higher topos theory [LurHA] J. Lurie,
Higher algebra , 2017.[LurKM] J. Lurie,
Lecture 20, 21, Algebraic K-Theory and Manifold Topology (Math 281)
Finite-dimensional differential graded algebras and their geometric realizations , arXiv preprintarXiv:1907.08162 (2019) Ras] S. Raskin,
On the Dundas-Goodwillie-McCarthy theorem , arXiv preprint arXiv:1807.06709 (2018)[Rou] S. K. Roushon,
The Farrell-Jones isomorphism conjecture for 3-manifold groups . J. K-Theory, 1(1):49–82,2008.[RV] H. Reich, M. Varisco.
Algebraic K-theory, assembly maps, controlled algebra, and trace methods . InSpace—time—matter, pages 1–50. De Gruyter, Berlin, 2018.[SP] The Stacks Project Authors.
Stacks Project . http://stacks.math.columbia.edu, 2020.[Tab] G. Tabuada,
Invariants additifs de dg-catgories . Internat. Math. Res. Notices 53 (2005), 33093339.[To¨en] B. To¨en,
Vers une interpretation Galoisienne de la theorie de l’homotopie , Cahiers de topologie et geometriedifferentielle categoriques, Volume XLIII (2002), 257-312.[TT] R. W. Thomason and Thomas Trobaugh.
Higher algebraic K-theory of schemes and of derived categories . InThe Grothendieck Festschrift, Vol. III, volume 88 of Progr. Math., pages 247–435. Birkhauser Boston, Boston, MA,1990[W] Christian Wegner. The Farrell-Jones conjecture for virtually solvable groups. J. Topol., 8(4):975–1016, 2015.
National Research University Higher School of Economics, Russian Federation
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