Cyclotomic complexes

Cyclotomi c complexes D. Kaledin Con ten ts 1 Cyclic cate gories. 4 1.1 Connes’ c yclic catego r y . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Cyclotomic catego ry . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Extended categories. . . . . . . . . . . . . . . . . . . . . . . . 9 1.4 More on extended cyclic category . . . . . . . . . . . . . . . . 11 1.5 Homologi cal pr op erties. . . . . . . . . . . . . . . . . . . . . . 1 3 1.6 Homologi cal v anishing. . . . . . . . . . . . . . . . . . . . . . . 15 2 Cyclic Mack ey functors. 17 2.1 The quotien t category d escription. . . . . . . . . . . . . . . . 17 2.2 Geometric ﬁxed p oints. . . . . . . . . . . . . . . . . . . . . . 20 2.3 Coalgebra description. . . . . . . . . . . . . . . . . . . . . . . 23 2.3.1 A ∞ -coalg ebr a. . . . . . . . . . . . . . . . . . . . . . . 23 2.3.2 The comparison theorem. . . . . . . . . . . . . . . . . 26 2.4 Restriction and corestriction. . . . . . . . . . . . . . . . . . . 30 3 Cyclotomic co mplexes. 32 3.1 Normalized Λ R -graded coalgebras. . . . . . . . . . . . . . . . 32 3.2 Reduced Λ R -graded coalgebras. . . . . . . . . . . . . . . . . . 33 3.3 Como dules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4 Equiv arian t homology . 40 4.1 Generalities on equiv arian t homotop y . . . . . . . . . . . . . . 40 4.2 Cyclic sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.3 Cyclic Mac key fu n ctors. . . . . . . . . . . . . . . . . . . . . . 48 4.4 Cyclotomic complexes. . . . . . . . . . . . . . . . . . . . . . . 53 1 5 Filtered Dieudo nn´ e modules. 57 5.1 Deﬁnitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 5.2 Filtered ob jects. . . . . . . . . . . . . . . . . . . . . . . . . . 59 5.3 Cyclic expansion and sub division. . . . . . . . . . . . . . . . . 61 5.4 Stabilization. . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 5.5 Comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 6 T op ological cyclic homology . 68 7 App endix. 71 In tro du ction. This pap er is a sequel to [Ka2]; the goal of b oth pap ers is to try to unders tand ho w some purely h omologic al notions used in [Ka1] are related to T op ological Cyclic Homology of [BHM ]. This turned out to b e a r ather length y p ro ject, since o n e has to construct appropr iate homological coun terparts of sev eral notions from stable homotop y th eory . The p ap er [Ka2] dea lt with deriv ed Mac key fu nctors, homological ana- logues of genuine G -equiv ariant s p ectra of [LMS]. While th e ab elian category of Mac key functors is very well kno wn in stable homotop y theory , and do es pla y an imp ortan t role, its most naiv e derived generalization turned to b e not quite well-behav ed. Thus a s lightly diﬀerent d eriv ed v ersion of Mac key functors w as constr u cted and studied in [Ka2]. The p resen t pap er deals w ith cyclot omic sp ectra of [BM], esp ecially as present ed in [HM]. T o ﬁn d a homological counte r p art for those, one ﬁr st h as to go beyo n d [Ka2]: b y deﬁnition, cyclotomic sp ectra are equiv arian t with resp ect to the circle group S 1 , an d [Ka2 ] only dealt w ith ﬁnite groups G . W e cannot really construct a go o d homological analogue of all S 1 -sp ectra sp ectra, but w e do construct a category D M Λ( Z ) of “cyclic Mac k ey func- tors” whic h captures the part of the equiv arian t stable category relev an t to T op ologi cal Cyclic Homology . W e then int r o duces th e triangulated ca tegory D ΛR ( Z ) of “cyclotomic complexes”. Ideally , the relation b et ween cyclot omic complexes and cyclotomic sp ec- tra should b e expressed by a comm u tativ e diagram (0.1) D ΛR( Z ) − − − − → Cycl   y   y D ( Z ) − − − − → StHom 2 of “bra ve new schemes” , understo od for example as tensor triangulated cat- egories with so me en h ancemen t. Here StHom is t h e stable homotop y cate - gory , D ( Z ) is the derived catego ry of ab elian grou p s, and Cycl is the catego r y of cyclotomic s p ectra. The diagram should b e “almost Cartesian”. More precisely , it should b ecome Cartesian if we restrict our attension to the s u b- catego ry Cycl o ⊂ Cycl of cyclotomic sp ectra T with trivial geometric ﬁxed p oints Φ S 1 T with resp ect to the wh ole group S 1 . A t p resen t, such a nice picture seems wa y b ey ond r eac h ; b esides the ob vious diﬃculties w ith making al l the “bra v e n ew” n otions precise, it seems that up to n o w, n o-one constructed Cycl even as a triangulated category . Th u s in practice, w e restrict our atten tion to the follo win g t wo things: (i) w e co n struct an equiv arian t homology cyclotomic complex C q ( T ) fo r ev ery cyclotomic s p ectrum T ; th is ough t to corresp ond t o the top arro w in (0.1), (ii) w e construct a top ological cyclic homology fun ctor TC on the categ ory D ΛR( Z ) in suc h a wa y th at for any cyclo tomic sp ectrum T , TC( C q ( T )) is naturally iden tiﬁed with the h omology of the s p ectrum TC( T ). W e then adopt a diﬀerent p ersp ectiv e and giv e a completely diﬀeren t and v ery simp le description of the catego r y D ΛR( Z ). As it h ap p ens, cyclotomic complexes are essen tially equiv alen t to “ﬁltered Dieudonn´ e mo dules” of [FL]. Filtered Dieudon n´ e mo dules are rather simple linear-algebraic gadgets with a deep meaning — they give a p -adic coun terpart of D eligne’s notion of a mixed Ho d ge structure, and the wh ole story acquires a distinctly motivic ﬂa vo u r. A more detailed discussion of this is a v ailable in [Ka3]. Filtered Dieudonn ´ e mo d ules arise natur ally as the cristalline cohomology of algebraic v arieties o ver Z p , while cyclotomic sp ectra app ear as T op olog ical Ho c hschild Homology sp ectra of ring sp ectra A . In view of the equiv alence w e established, there are many area s of int ersection wh ere one can compare the t wo constru ctions. W e did not attempt to do so in this pap er; w e stic k to pure linear algebra, and lea v e the geo m etric applications for futur e researc h. The only comparison r esult that we pro v e says that for proﬁn itely complete cyclotomic complexes, the top ological cyclic homology TC in fact coincides with the syntomic cohomolog y o f [FM] familiar in th e theory of Dieudonn´ e mo dules. The p ap er is organized as follo w s . T o b egin the story , we need some mo del for S 1 -equiv ariant sp aces and their homology; for b etter or for w orse, w e ha ve c h osen to use the com binatorial approac h usin g A. Connes’s cat- egory Λ. Section 1 con tains necessary fact ab out the cate gory Λ and its 3 v arious cousin s . In Section 2, we constru ct cyclic Mac ke y functors. Section 3 d eals with cyclot omic complexes. Sectio n 4 is the tecnical heart of the pap er: here we constru ct equiv ariant h omology f u nctors from S 1 -sp ectra to D M Λ( Z ) an d from cyclotomic sp ectra to D ΛR( Z ). Then in Section 5, w e forget all ab out top ology and prov e the comparison theorem b et ween cyclotomic complexes and ﬁ ltered Dieudonn ´ e mo du les. Finally , in Section 6 w e brieﬂy discuss top ological cyclic h omology , and pro ve th e c omparison theorems for TC. App endix conta in s some tec hnicalities, mostly from [Ka2]. Ac kno wledgements. Many discussions on the sub ject with G. Merzon w ere very inspirational and h elpful. I o we a lot to L. Hesselholt for his pa- tien t explanation ab out top ology (this go es b oth for this pap er in particular, and f or the w hole pro ject in general). I am grate f u l for V. V olog o d sky for explanations ab out Dieudonn´ e mo dules. It is a pleasur e to thank A. Beilin- son, V. Drinfeld, V. Ginzburg, D. Kazhdan, and J.P . Ma y for their int erest in this work. A part of the pap er wa s ﬁ nished at the Hebrew Unive r sit y of Jerus alem, a n d another part wa s done while visiting the Univ ers ity of Chicago; the hospitalit y of b oth places is gratefully ac knowledged. 1 C yc lic categories. 1.1 Connes’ cyclic category . Recall that A. Connes’ cyclic c ate gory Λ is a small catego ry whose ob jects [ n ] are indexed by p ositiv e in tegers n , n ≥ 1. Maps b et wee n [ n ] and [ m ] can b e deﬁned in v arious equiv alen t w a ys; for the con ve n ience of the r eader, we r ecall tw o of these descriptions. T op olo gic al description . T h e ob j ect [ n ] is though t of as a “wh eel” – a cellular decomp osition of the c ircle S 1 with n 0-cells, c alled vertic es , and n 1-cells, called e dges . A con tinuous map f : S 1 → S 1 induces a map e f : R → R of the unive rs al co ve r s ; sa y t h at f is monotonous if e f ( a ) ≥ e f ( b ) for any a, b ∈ R , a ≥ b . Then morph isms from [ n ] to [ m ] in the catego r y Λ are homotopy classes of monotonous con tinuous maps f : [ n ] → [ m ] whic h ha ve degree 1 and send v ertices to v ertices. Combinatoria l description . Cons id er the category Λ big of totally ordered sets equipp ed with an order-preservin g endomorph ism τ . Let [ n ] ∈ Λ big b e the set Z w ith the natural l in ear order and endomorphism τ : Z → Z , τ ( a ) = a + n . Let Λ ∞ ⊂ Λ big b e t h e f ull sub category spann ed by [ n ], n ≥ 1. F or any [ n ] , [ m ] ∈ Λ ∞ , the set Λ ∞ ([ n ] , [ m ]) is acted up on by the endomorphism τ (on the left, or on the right, by d eﬁnition it does not 4 matter). W e deﬁn e the set of maps Λ([ n ] , [ m ]) in the catego r y Λ by (1.1) Λ([ n ] , [ m ]) = Λ ∞ ([ n ] , [ m ]) /τ . Other descriptions are p ossib le, see [L, C hapter 6] and App en d ix to [FT]. F or an y [ n ] ∈ Λ, the set V ([ n ]) of v ertices of the corresp ond ing decomp o- sition of the circle can b e naturally identiﬁed with the s et Λ([1] , [ n ]) of maps from [1] to [ n ], and the set E ([ n ]) of edges can b e iden tiﬁed with th e set Λ([ n ] , [1]) – in particular, E ( − ) is a con tra v ariant fu nctor (geo metrically , the preimage of a n e d ge is cont ained in exactly one edge). The automorphism group Aut([ n ]) is the cyclic group Z /n Z generated b y the clo c kw ise r ota- tion; we will d enote the generator by σ . In the com b inatorial description, σ corresp onds to the map Z → Z , a 7→ a + 1. Giv en an int eger p ≥ 2, one can d eﬁne a cat egory Λ p b y taking the same set of ob jects [ n ], n ≥ 1, and setting Λ p ([ n ] , [ m ]) = Λ ∞ ([ n ] , [ m ]) /τ p . The catego ry Λ p is in termediate b et w een Λ ∞ and Λ; in particular, the ob vi- ous pro jections Λ ∞ ([ n ] , [ m ]) /τ p → Λ ∞ ([ n ] , [ m ]) /τ together d eﬁne a functor π p : Λ p → Λ . The functor π p is a biﬁb ration with ﬁ b er pt p = [1 / ( Z /p Z )], the group oid with one ob ject and automorphism group Z /p Z . On the other hand, Λ p ([ n ] , [ m ]) can b e identiﬁed with the s et of maps f : [ np ] → [ mp ] in Λ suc h that f ◦ σ n = σ m ◦ f ; this giv es a canonical fun ctor i p : Λ p → Λ suc h that on ob jects, w e hav e i p ([ n ]) = [ np ]. Denote by (1.2) Λ I = a p ≥ 1 Λ p the disj oin t union of all the categories Λ p , p ≥ 1. Th en the fu nctors i p and π p can b e considered together as t wo functors (1.3) i, π : Λ I → Λ . The catego ry Λ is self-du al: an equiv alence Λ ∼ = Λ opp sends every ob ject to itself, and a morphism [ n ] → [ m ] represen ted by a map f : Z → Z go es to the map represente d by f ♯ : Z → Z , (1.4) f ♯ ( a ) = max { b ∈ Z | f ( b ) ≤ a } . 5 In the t op ologic al description, the dualit y in terchanged ed ges and v ertices and corresp onds to taking the dual cellular d ecomp osition. F or any [ n ] , [ m ] ∈ Λ, the set of maps Λ([ n ] , [ m ]) is ﬁnite. T h e groups Aut([ n ]) and Aut([ m ]) act on Λ([ n ] , [ m ]) b y comp ositions, and b oth these actions are stabilizer-free. W e will need the f ollo w ing slightl y mo r e general fact. Lemma 1.1. Assume given thr e e inte gers m , n , l such tha t m, l ≥ 1 , n ≥ 2 , and a map f : [ nl ] → [ m ] in Λ such that f ◦ σ l = σ l 1 ◦ f for some inte ge r l 1 , 0 ≤ l 1 < m . Then m = nl 1 . Pr o of. Use the com binatorial description of Λ. Then f is represent ed by an order-preserving map e f : Z → Z su c h that (1.5) e f ( a + nl ) = e f ( a ) + m, e f ( a + l ) = e f ( a ) + l 1 + bm for an y a ∈ Z , wh ere b is a ﬁxed int eger indep endent of a . Since a ≤ a + l ≤ a + nl , th is implies 0 ≤ l 1 + bm ≤ m , so that either l 1 = 0 and b = 1, or b = 0. The ﬁrst case is imp ossible since σ l acts on Λ([ nl ] , [ m ]) without ﬁxed p oints. Thus b = 0, and (1.5) immediately implies the claim.  The catego ry Λ / [1] of ob j ects [ n ] ∈ Λ equipp ed with a map [ n ] → [1] is naturally equiv alen t to the category ∆ of non-empt y ﬁnite totally ordered sets: geometrically , Λ / [1] is the category of w heels with a ﬁxed edge, and remo ving this edge creates a canonical clo c kwise total ord er on the set of v ertices of the wheel. W e thus ha v e a n atural discrete ﬁbration ∆ ∼ = Λ / [1] → Λ indu cing a coﬁbration j o : ∆ opp → Λ opp . Du ally , the category [1] \ Λ of ob jects [ n ] ∈ Λ equipp ed w ith a map [1] → [ n ] is equiv alen t to ∆ opp , so that w e get a n atural discrete ﬁbration j : ∆ → Λ opp (geometrica lly , ∆ opp is the catego ry of wheels with a ﬁxed v ertex). The same constru ctions work for the categories Λ n , n ≥ 2. In particular, we obtain a canonical fun ctor j n : ∆ → Λ opp n , and we hav e π ◦ j n = j for any n ≥ 2. Let ∆ n → ∆ b e the biﬁ b ration obtained by the Cartesian squ are ∆ n − − − − → Λ opp n   y   y π n ∆ j n − − − − → Λ opp . 6 Then the f unctor j n giv es a splitting ∆ → ∆ n of this b iﬁbration, so that w e ha ve ∆ n ∼ = ∆ × pt n . In more down-to -earth terms, this means that the group Z /n Z acts on the f unctor j n . W e can also comp ose j n with th e embedd ing i n ; this results in a comm utativ e diagram ∆ j n − − − − → Λ opp n r n   y   y i n ∆ j − − − − → Λ opp , where r n : ∆ → ∆ is the e dgewise sub division functor giv en by (1.6) r n ([ m ]) = [ n ] × [ m ] , where [ m ], [ n ] are totally ordered s ets with m resp . n element s, and [ n ] × [ m ] is giv en the left-to-righ t lexicographical order. 1.2 Cyclotomic category . W e no w in tro du ce the follo wing deﬁnition based on the top ological description of the category Λ. Deﬁnition 1.2. The cycloto mic c ate g ory Λ R is the sm all ca tegory with the same ob jects [ n ], n ≥ 1, as the category Λ. If we th ink of [ n ] as a conﬁguration of n mark ed p oin ts on a circle S 1 , then m orp hisms from [ n ] to [ m ] in th e category Λ R are homotop y classes of monotonous conti nuous maps f : [ n ] → [ m ] w hic h sen d marke d p oints to marke d p oin ts and ha ve p ositiv e degree, deg f ≥ 1. The only diﬀerence w ith the category Λ is that th e maps are allo w ed to ha ve degree bigger th an 1. A t ypical n ew map is obtained as follo ws . F or ev ery conﬁguration of n p oints on a circle and any p ositiv e in teger l ≥ 1, consider the l -fold ´ etale co ve r π l : S 1 → S 1 , and the conﬁguration of nl preimages of n mark ed p oint s. Then π giv es a w ell-deﬁned map π n,l : [ n l ] → [ n ] in the category Λ R . Moreo ver, ev ery map f : [ m ] → [ m ] of degree l in the category Λ R factors as f = π n,l ◦ f ′ for some f ′ : [ m ] → [ nl ] of d egree 1, and su c h a factoriza tion is unique up to the action of the group Z /l Z of d ec k transformations of the co vering π l : S 1 → S 1 . T h u s the set Λ R l ([ m ] , [ n ]) of degree- l maps from [ m ] to [ n ] is naturally identi ﬁ ed with the quotien t (1.7) Λ R l ([ m ] , [ n ]) = Λ([ m ] , [ nl ]) / ( Z /l Z ) b y the action of the group Z /l Z generated b y σ n : [ n l ] → [ nl ]. In particular, Λ R l ([ m ] , [ n ]) is ﬁn ite f or ev ery [ m ], [ n ] and l . 7 Deﬁnition 1.3. A map f : [ m ] → [ n ] in the catego ry Λ R is horizontal if it is of d egree 1. A m ap f : [ m ] → [ n ] of some degree l ≥ 1 is vertic al if the map f ′ : [ m ] → [ nl ] in the decomp osition f = π n,l ◦ f is in vertible. It follo ws f rom the discussion ab ov e that v ertical an d h orizon tal m aps form a factorization system on Λ R in the s en se of Deﬁnition A.2. The sub category Λ R h formed b y horizon tal maps is by d eﬁ nition equiv alen t to Λ. Moreo ver, for any group G , let O G b e the catego ry of ﬁnite G -orbits – that is, ﬁnite sets equipp ed with a transitive G -action. Then the s u b category Λ R v ⊂ Λ R form ed b y v ertical maps is o bviously equiv alen t to th e orb it catego ry O Z – the equiv alence sen ds [ n ] ∈ Λ R to the orbit Z /n Z (all ﬁn ite Z -orbits are of this form). Lemma 1.4. F or any p air of a horizonta l map h : [ m 1 ] → [ m ] and a vertic al map v : [ m 2 ] → [ m ] in Λ R , ther e exists a Cartesian squar e [ m 12 ] h 1 − − − − → [ m 2 ] v 1   y   y v [ m 1 ] h − − − − → [ m ] with horizontal h 1 and vertic al v 1 . Pr o of. C lear.  Comp osing the fu nctors i and π of (1.3) with the natural emb edding Λ ∼ = Λ R h ֒ → Λ R , w e obtain fun ctors e i, e π : Λ I → Λ R. Moreo ver, the quotien t maps π n,l , n, l ≥ 1 tak en together deﬁne a v ertical map (1.8) e v : e i → e π . Let f Λ I b e the category of vertic al maps v : [ m ] → [ m ′ ] in Λ R , with maps from v 1 : [ m 1 ] → [ m ′ 1 ] to v 2 : [ m 2 ] → [ m ′ 2 ] giv en by commutati ve squares (1.9) [ m 1 ] f − − − − → [ m 2 ] v 1   y   y v 2 [ m ′ 1 ] f ′ − − − − → [ m ′ 2 ] 8 with h orizon tal f , f ′ . Then sending a ∈ Λ I to e v : e i ( a ) → e π ( a ) deﬁ n es a functor (1.10) Λ I → f Λ I . Lemma 1.5. The functor (1.10) is an e quivalenc e of c ate gories. Pr o of. C lear.  Sending a w heel [ n ] ∈ Λ R to the set V ([ n ]) of its vertic es deﬁnes a functor Λ R → Sets. W e let (1.11) ∆ R e j − − − − → Λ R opp b e the discrete ﬁbration corresp ond ing to V b y the Grothendiec k construc- tion. Lemma 1.6. The functor δ = deg ◦ e j : ∆ R → [1 / N ∗ ] is a c oﬁbr ation, with ﬁb er ∆ , and tr ansition functor r m c orr esp onding to m ∈ N ∗ given by the e dgewise sub division f u nctor. Pr o of. By deﬁn ition, ∆ R is op p osite to the full s u b category in the slice catego ry [1] \ Λ R sp anned b y horizon tal maps h : [1] → [ n ], [ n ] ∈ Λ R . More- o v er, it in herits f r om Λ R th e ve r tical/horizo ntal factorization system. One no w imm ediately deduces that vertic al maps are Cartesian with resp ect to δ , w h ile th e ﬁb er of δ is sp anned b y horizont al maps.  1.3 Extended cat e gories. Let N ∗ b e the monoid of p ositiv e i ntegers l ≥ 1 with resp ect to m ultiplication, and let [1 / N ∗ ] b e the category with one ob ject 1 and Hom [1 / N ∗ ] (1 , 1) = N ∗ . Sending a map to its degree giv es then a fu nctor (1.12) deg : Λ R → [1 / N ∗ ] . This functor has a section, the fully faithfu l embedd ing α : [1 / N ∗ ] → Λ R whic h send s 1 to [1] ∈ Λ R (any map [1] → [1] is uniquely determined by its degree). Moreo ver, let I = 1 \ [1 / N ∗ ] b e the catego ry of ob jects a ∈ [1 / N ∗ ] equipp ed with a map 1 → a (the slice catego ry ). Equiv alen tly , I is N ∗ considered as a p artially ordered set with 9 order giv en by divisibilit y , and turn ed in to a category in the s tand ard w ay . Then we hav e a n atural coﬁbration (1.13) I → [1 / N ∗ ] whose ﬁb er is the set N ∗ considered as a discrete category . By the Grothen- diec k construction, this corresp onds to a f u nctor [1 / N ∗ ] → Sets sending 1 to N ∗ , in other w ords, to an action of the monoid N ∗ on itself; the action is by righ t multiplicat ion. Let no w N ∗ act on it self b oth on th e right a n d on th e left, and l et I b e corresp onding category coﬁb ered ov er [1 / N ∗ ] × [1 / N ∗ ] w ith ﬁb er N ∗ . Equiv alently , I is obtained by the C artesian s quare I − − − − → I   y   y [1 / N ∗ ] × [1 / N ∗ ] − − − − → [1 / N ∗ ] , where the b ottom map is induced b y the pro duct map N ∗ × N ∗ → N ∗ . Comp osing the coﬁbration I → [1 / N ∗ ] × [1 / N ∗ ] with the pro jection on to the righ t multiple [1 / N ∗ ], w e obtain a coﬁbration (1.14) I → [1 / N ∗ ] , with the ﬁb er I . W e also h av e a natural Cartesian functor I → I ; on ﬁb ers, it is giv en b y the inclusion of th e discrete cate gory N ∗ in to I (wh ic h is nothing but N ∗ considered as a partially ordered set). Explicitly , ob jects in I are p ositiv e in tegers n ≥ 1, and morph isms are generated b y morph isms (1.15) F l , R l : n → nl for an y n, l ≥ 1, su b ject to relations F n ◦ F m = F nm , R n ◦ R m = R nm , F n ◦ R m = R m ◦ F n for any n, m ≥ 1. This is opp osite to the category int r o duced by T. Go o d- willie in [Go o d]. Deﬁnition 1.7. The extende d c yc lic c ate gory Λ Z is giv en by Λ Z = Λ R × [1 / N ∗ ] I , and the extende d cyclotomic c ate gory g Λ R is giv en b y g Λ R = Λ R × [1 / N ∗ ] I , where in b oth cases, Λ R → [1 / N ∗ ] is the d egree fu nctor d eg of (1.12), and I → [1 / N ∗ ] resp. I → [1 / N ∗ ] are the coﬁbrations (1.13) resp. (1.14). 10 The s ection α : [1 / N ∗ ] → Λ R of the d egree fu n ctor deg : Λ R → [1 / N ∗ ] induces a functor (1.16) e α : I → g Λ R. By deﬁnition, w e h a v e coﬁbrations (1.17) λ : Λ Z → Λ R , e λ : g Λ R → Λ R, with ﬁb ers iden tiﬁed with N ∗ resp. I . Explicitly , ob jects in either Λ Z or g Λ R are giv en by pairs h [ n ] ∈ Λ R, l ∈ N ∗ i ; we w ill denote sich a pair b y [ n | m ]. A m ap from [ n | m ] to [ n ′ , m ′ ] in Λ Z resp. g Λ R is a map f : [ n ] → [ n ′ ] in Λ R su c h that m ′ = m deg f , resp. m ′ = l m deg f for some in teger l ≥ 1. The v ertical/horizon tal factorizatio n system then indu ces an analogous v ertical/horizon tal factorizati on systems on Λ Z and g Λ R . Denote b y Λ Z v , Λ Z h ⊂ Λ Z , g Λ R h , g Λ R h ⊂ g Λ R the su b categories sp anned by vertica l, resp. h orizon tal maps. Then Λ Z h and g Λ R h decomp ose as (1.18) Λ Z h = a m ≥ 1 Λ Z m h ∼ = N ∗ × Λ , g Λ R h = I × Λ , where Λ Z m h is the fu ll sub category spanned by the ob jects [ n | m ], n ≥ 1, and for every m , the category Λ Z m h is naturally equ iv alen t to Λ. On the other hand, the catego ry Λ Z v decomp oses as (1.19) Λ Z v = a n ≥ 1 Λ Z n v ∼ = a n ≥ 1 O Z /n Z , where Λ Z n h is the full sub catego r y spanned b y ob jects [ n ′ | m ′ ] with n = n ′ m ′ , and for ev ery n , th e category Λ Z n h is naturally equiv alent to th e category O Z /n Z . These decomp ositions are ind u ced by th e identiﬁca tions Λ R h ∼ = Λ, Λ R v ∼ = O Z . 1.4 More on extended cyclic category . Here are some more simple prop erties of the category Λ Z . By Y oneda, the category Λ Z is fully and faithfully em b edded in to the category F un(Λ Z opp , Sets). Restricting to Λ = Λ Z 1 h ⊂ Λ Z , we obtain a f unctor Y : Λ Z → F un (Λ opp , Sets) . This giv es an alternativ e p u rely com binatorial description of the category Λ Z , since w e h av e the follo wing result. 11 Lemma 1.8. The functor Y i s ful ly faithful. Pr o of. By deﬁnition, Y ([ n | 1]) is the fu nctor h [ n ] : Λ opp → Sets represente d b y [ n ] ∈ Λ, and by (1.7), Y ([ n | m ]) for m ≥ 2 is the quotien t h [ nm ] / ( Z /m Z ) of h [ nm ] b y the action of the cyclic group Z /m Z ⊂ Aut([ nm ]). W e then hav e Hom( Y ([ n | m ]) , Y ([ n ′ | m ′ ])) = Hom( h [ nm ] , h [ n ′ m ′ ] / ( Z /m ′ Z )) Z /m Z =  Λ([ nm ] , [ n ′ m ′ ]) / ( Z/m ′ Z )  Z /m Z , and by Lemm a 1.1, this is non-empt y only if m ′ = l m for some l , and coincides with Λ([ n, n ′ l ]) / ( Z /l Z ) . Again b y (1.7) , this coincides with Λ Z ([ n | m ] , [ n ′ | m ′ ]) = Λ R l ([ n ] , [ n ′ l ]).  Lemma 1.9. F or any p air of a horizontal map h : [ n 1 | m ] → [ n | m ] and a vertic al map v : h : [ n ′ | m ′ ] → [ n | m ] , ther e exists a c artesian squar e (1.20) [ n ′ 1 | m ′ ] v ′ − − − − → [ n 1 | m ] h ′   y   y h [ n ′ | m ′ ] v − − − − → [ n | m ] in Λ Z with horizontal h ′ and vertic al v ′ . Pr o of. Use Lemma 1.4 and notice that deg v 1 = deg v .  By virtue of this Lemma, w e can d eﬁne a n ew category c Λ Z as follo ws: the ob jects are the same as in Λ Z , morphisms from c to c ′ are give n by isomorphism classes of diagrams (1.21) c v ← − − − − c 1 h − − − − → c ′ with ve rtical v and horizonta l h . Comp osition is give n by pullb acks. Note that a diagram (1.21) has no non-trivial automorph ism s. Th er efore c Λ Z h as a natural factorizati on system, with horizon tal resp . ve r tical maps repre- sen ted by a diagram with inv ertible v resp. h . As b efore, w e d en ote the corresp ondin g s u b categories b y c Λ Z h , c Λ Z v ⊂ c Λ Z . W e ha ve decomp osition c Λ Z v = a n ≥ 1 O opp Z /n Z , Λ Z h = N ∗ × Λ 12 induced by (1.1 9 ) and (1.18) , so th at w e ha ve c Λ Z v ∼ = Λ Z opp v and c Λ Z h ∼ = Λ Z h . Note that the equiv alence Λ ∼ = Λ opp of (1.4) giv es an equiv alence (1.22) Λ Z h ∼ = N ∗ × Λ → Λ Z opp h ∼ = N ∗ × Λ opp . Lemma 1.10. Ther e exists en e quivalenc e of c ate gories c Λ Z ∼ = Λ Z opp which r estricts to the identity func tor c Λ Z v ∼ = Λ Z opp v → Λ Z opp v ⊂ Λ Z opp and to the e q uivalenc e c Λ Z h ∼ = Λ Z h → Λ Z opp h ⊂ Λ Z opp of (1.22) . Pr o of. T he shap e of the equiv alence c Λ Z ∼ = Λ Z opp is p r escrib ed by the con- ditions: it is identica l on ob j ects and on vertica l morp hisms, and it send s a horizon tal morp hism h in some Λ Z m h ⊂ Λ Z , Λ Z m h ∼ = Λ to h ♯ . T o see that this is consisten t, we hav e to c h ec k that for an y Cartesian square (1.20), the diagram [ n ′ 1 | m ′ ] v ′ − − − − → [ n 1 | m ] h ′ ♯ x   x   h ♯ [ n ′ | m ′ ] v − − − − → [ n | m ] is comm utativ e. This imm ediately follo ws fr om the construction: b oth maps h and h ′ can b e represen ted by the same m ap [ n 1 m ] → [ nm ].  1.5 Homological prop ert ies. The geometric reali zation | Λ | of the nerv e of the category Λ is homotop y equiv alen t to B U (1), the classifying space of the unit circle group U (1). In particular, the cohomology of the category Λ ∼ = Λ opp with co eﬃcien ts in the constan t functor Z is giv en by H q (Λ opp , Z ) = H q (Λ , Z ) ∼ = Z [ u ] , where u is a generator of degree 2. T o see u ∈ H 2 (Λ , Z ) exp licitly , asso- ciate to a wheel [ n ] ∈ Λ its cellular cohomology complex C q ([ n ] , Z ). S ince top ological ly , ev ery wh eel is the circle S 1 ∼ = U (1), w e ha ve an exact sequen ce 0 − − − − → Z − − − − → C 0 ([ n ] , Z ) − − − − → C 1 ([ n ] , Z ) − − − − → Z − − − − → 0 for an y [ n ] ∈ Λ. Th is sequence dep ends fun ctorially on [ n ], so that w e obtain an exact sequence (1.23) 0 − − − − → Z b 0 − − − − → j ∗ Z B − − − − → j o ! Z b 1 − − − − → Z − − − − → 0 13 of functors in F u n(Λ opp , Z ), where we hav e identiﬁed C 1 ([ n ] , Z ) ∼ = Z [ E ([ n ])] ∼ = ( j o ! Z )([ n ]) , C 0 ([ n ] , Z ) ∼ = Z [ V ([ n ])] ∗ ∼ = ( j ∗ Z )([ n ]) , with j : ∆ → Λ opp , j o : ∆ opp → Λ opp as in Sub section 1.1. The exact sequence (1.23) r epresen ts by Y on ed a an element u ∈ Ex t 2 ( Z , Z ); this is the generator of the p olynomial algebra Ext q ( Z , Z ) = H q (Λ opp , Z ). The cellular cohomology complexes are also functorial with r esp ect to maps of h igher degrees, so that the exact sequence (1.23) extends to the catego ry F un (Λ R, Z ). The extended sequence tak es the form (1.24) 0 − − − − → Z b 0 − − − − → e j ∗ Z B − − − − → E b 1 − − − − → deg ∗ Z (1) − − − − → 0 , where e j is as in (1.11), E ∈ F un(Λ R opp , Z ) denotes the fun ctor [ n ] 7→ C 1 ([ n ] , Z ), and Z (1) ∈ F un([1 / N ∗ ] , Z ) is the functor corresp onding to Z with ev ery n ∈ N ∗ acting b y multiplicat ion by n . Using the functors i , π of (1.3), we can pull bac k the sequence (1.2 3 ) to the category Λ I opp in tw o wa ys, and the map (1.8 ) ind uces a m ap b etw een these pullb ac ks. After ﬁxing a p ositiv e in teger n ≥ 2 and r estricting to Λ opp n ⊂ Λ I opp , what we obtain is a comm utativ e diagram (1.25) 0 − − − − → Z − − − − → i ∗ n j ∗ Z i ∗ n B − − − − → i ∗ n j opp ! Z − − − − → Z − − − − → 0 id   y   y v n   y v n   y n id 0 − − − − → Z − − − − → π ∗ n j ∗ Z π ∗ n B − − − − → π ∗ n j opp ! Z − − − − → Z − − − − → 0 , where the vertica l maps η n are isomorp h isms. T he geomet r ic realization | Λ n | of the cat egory Λ n has the same homotop y t yp e as | Λ | , and the fu n ctor i n : Λ n → Λ indu ces a h omotop y equiv alence of the realizations. The ﬁrst line in (1.25) represen ts by Y oneda the generator u ∈ H 2 (Λ opp n , Z ) of the cohomology algebra H q (Λ opp n , Z ) ∼ = H q (Λ opp , Z ) ∼ = Z [ u ] . On the other hand, th e functor π n : Λ n → Λ d o es not induce a homotop y equiv alence of realizations: o n the lev el of r ealizat ions | Λ n | ∼ = | Λ | ∼ = B U (1), the map ind uced by π n corresp onds to the n -fold co v ering U (1) → U (1). the second line in (1.25) represents the elemend π ∗ n = nu ∈ H 2 (Λ opp n , Z ). 14 1.6 Homological v anishing. W e w ill also need some cohomologica l v an- ishing results on small categories ∆ R and Λ R opp . Firs t recall that if w e take an y p ≥ 1 and let r p : ∆ → ∆ b e the edgewise su b divison f unctor (1.6), then for an y M ∈ D (∆ , k ), the natural map H q (∆ , r ∗ p M ) → H q (∆ , M ) is an isomorphism (see e.g. [Ka1, L emm a 1.14], but in fact the statement is v ery well-kno wn ). By adjunction, this means that the natural map (1.26) r p ! k → k is a quasiisomorphism for any p ≥ 1 (here k denotes the constant functors). Lemma 1.11. L et h : ∆ ∼ = ∆ R h → ∆ R b e the natur al emb e dding, and assume gi ven an obje ct M ∈ D (∆ R opp , Z ) such that H q (∆ , M ) = 0 . Then H q (∆ R, M ) = 0 . Pr o of. L et δ : ∆ R → [1 / N ∗ ] b e the coﬁbr ation of Lemma 1.6. It su ﬃces to pro ve that R q δ ∗ M = 0. Equiv alen tly , H q ( g ∆ R, κ ∗ M ) = 0 , where g ∆ R is the category of ob jects [ n ] ∈ ∆ R equipp ed with a map δ ([ n ]) → 1 — in other w ords, a n u mb er m ∈ N ∗ — and κ : g ∆ R → ∆ R is the forgetful functor. F or any l ≥ 1, let ι l : ∆ → g ∆ R b e the em b edding sen ding [ n ] to itself equipp ed with the num b er l . W e also hav e the forgetful functor δ : g ∆ R → I induced by δ : ∆ R → [1 / N ∗ ], and by Lemma 1.6, this is a coﬁbration with transition functors r n . Therefore the fu nctor i ∗ l i m ! : D (∆ , k ) → D (∆ , k ) is t r ivial un less m = lp for some p , and isomorphic to r p ! if this is the case. T hen b y (1.26), the adjun ction map i l ! k → k is an isomorph ism at ι p (∆) ⊂ g ∆ R for all p dividing l . Therefore k ∼ = lim → i l ! k , so that it suﬃces to pro ve th at RHom q ( i l ! k , κ ∗ M ) = 0 for an y l . By adjunction, this means H q (∆ , ι ∗ l κ ∗ M ) = 0, and w e ha ve κ ◦ ι l = id .  15 Corollary 1.12. F or any M ∈ F un(Λ R opp , k ) , we have H q (Λ R opp , M ⊗ E ) = 0 , wher e E ∈ F u n (Λ R opp , k ) is as in (1.24) . Pr o of. As in [Ka1, Lemma 1.10], (1.24) implies by devissage that it suﬃces to pro ve that H q (∆ R, e j ∗ ( M ⊗ E )) = 0 , and b y Lemma 1. 11 , it suﬃces to pr o v e that H q (∆ , j ∗ h ∗ ( M ⊗ E )) = 0. This is dual to [Ka1, Lemma 1.10].  By Corollary 1.12, (1.24) indu ces a long exact sequence of cohomology (1.27) H q (Λ R opp , M ) − − − − → H q (∆ R, e j ∗ M ) − − − − → − − − − → H q − 1 (Λ R opp , M ⊗ deg ∗ Z (1)) − − − − → for an y M ∈ F u n(Λ R opp , Z ). Prop osition 1.13. Assume g i ven a functor M ∈ F un(Λ R opp , Z ) , and as- sume further that M is pr oﬁnitely c omplete. Then the natur al map H q (Λ R opp , M ) → H q (∆ R, e j ∗ M ) is an isomorphism. Pr o of. By the pro jection form ula, we ha v e H q (Λ R opp , M ⊗ deg ∗ Z (1)) ∼ = H q ([1 / N ∗ ] , Z (1) ⊗ R q deg ∗ M ) , and by (1.27), we ha ve to pr o v e that th is is tautologically 0. S ince R q deg ∗ comm utes with proﬁnite completion, it suﬃces to prov e th e f ollo w ing. Lemma 1.14. F or any pr oﬁnitely c omplete M ∈ F un([1 / N ∗ ] , Z ) , we have H q ([1 / N ∗ ] , M ⊗ Z (1)) = 0 . Pr o of. E v ery p roﬁnitely complete ab elian group M decomp oses as M = Y p M p , where the prod uct is ov er all primes p , and M p is pro- p co mp lete. Therefore w e may assu me that M is p ro- p complete for some prime p . Decomp ose th e 16 monoid N ∗ in to the pr o duct N ∗ = N × N ∗ { p } , where N p ⊂ N consists of all p ositiv e in tegers pr im e to p , and N ⊂ N ∗ is the monoid of all p o w er p n , n ≥ 0. Then by the K ¨ un neth form ula, it suﬃ ces to p ro ve that H q ([1 / N ] , Z (1) ⊗ M ) = 0 . But for any M ′ ∈ F un([1 / N ] , Z ), H q ([1 / N ] , M ′ ) can ob viously b e compu ted b y the t w o-term complex M ′ id − t − − − − → M ′ , where t : M ′ → M ′ is the actio n of the generator 1 ∈ N . In o u r case M ′ = M ⊗ Z (1), M ′ is pr o- p complete, and t is divisib le by p . Therefore id − t is in vertible.  2 C yc lic Mac k ey functors. 2.1 The quotient category description. Consider the wreath prod uct Λ Z ≀ Γ of the enh anced cyclic category Λ Z w ith the catego ry Γ of ﬁnite sets; explicitly , Λ Z ≀ Γ can b e id entiﬁed with the full sub catego ry in F un(Λ opp , Sets) spanned by ﬁ nite disjoin t unions of ob jects [ n | m ] ∈ Λ Z ⊂ F un(Λ opp , Sets) (in particular, we h av e a n atural full em b edd ing Λ Z ⊂ Λ Z ≀ Γ ). A m ap a s ∈ S [ n s | m s ] → a s ′ ∈ S ′ [ n s ′ | m s ′ ] b et ween t wo s uc h unions indexed by ﬁn ite sets S , S ′ consists of a map f : S → S ′ and a map f s : [ n s | m s ] → [ n f ( s ) | m f ( s ) ] for any s ∈ S . Sa y that a map h f , { f s }i is ve rtic al if f s is v ertical for an y s . Say that the m ap is horizonta l if f is inv ertible, and f s is horizonta l for an y s . Th en v ertical and h oziron tal maps ob viously form a factorizatio n system on Λ Z ≀ Γ, and w e hav e the follo wing. Lemma 2.1. F or any vertic al map v : [ a ] → [ b ] in Λ Z ≀ Γ and any map f : [ b ′ ] → [ b ] , ther e exists a Cartesian squar e [ a ′ ] v ′ − − − − → [ b ′ ] f ′   y   y f [ a ] v − − − − → [ b ] with vertic al v ′ . 17 Pr o of. Since ve rtical and horizon tal maps form a factorization system on Λ Z ≀ Γ, w e may assume that f is either horizon tal or vertica l. In the ﬁrst case, the claim follo ws fr om Lemma 1.9. In th e second case, it suﬃces to p ro ve that the catego ry (Λ Z ≀ Γ) v = Λ Z v ≀ Γ has pu llbac ks. This folo ws from (1.19), since for every m , the category O Z /m Z ≀ Γ h as pu llbac ks (it is equiv alent to the category of ﬁnite sets equipp ed with an action of the group Z /m Z ).  Our deﬁnition of (deriv ed) cyclic Mac k ey fu nctors m im icks the deﬁtion of deriv ed Mac k ey fun ctors giv en in [Ka2, Section 3] (and more sp eciﬁcally , in S ubsection 3.4). F or any tw o ob jects c, c ′ ∈ Λ Z , w e let Q ≀ Λ Z ( c, c ′ ) b e the catego ry of diagrams (2.1) c v ← − − − − c 1 f − − − − → c ′ in Λ Z ≀ Γ w ith v ertical v . Maps fr om a diagram c ← c 1 → c ′ to a d iagram c ← c 2 → c ′ are giv en by s uc h maps g = h g , { g s }i : c 1 → c 2 that g comm u tes with f and w ith v , and eac h of the comp onen t m ap s g s is in vertible. W e ob viously hav e Q ≀ Λ Z ( c, c ′ ) = Q (Λ Z ( c, c ′ )) ≀ Γ, where Q Λ Z ( c, c ′ ) ⊂ Q ≀ Λ Z ( c, c ′ ) is th e sub catego r y of diagrams (2.1) with c 1 ∈ Λ Z ⊂ Λ Z ≀ Γ, and in vertible maps b et we en them. This identi ﬁ cation giv es the pro jection functor ρ c,c ′ : Q ≀ ( c, c ′ ) → Γ send ing a diagram c ← c 1 → c ′ to th e ﬁn ite set S of comp onents of the ob ject c 1 = ` s ∈ S [ n s | m s ] ∈ Λ Z ≀ Γ. W e let T c,c ′ = ρ ∗ T ∈ F un( Q ≀ ( c, c ′ ) , Z ) b e the f u nctor from Q ≀ ( c, c ′ ) to the category of ab elian groups obtained by pullbac k f rom the functor T ∈ F u n(Γ , Z ), T ( S ) = Z [ S ]. By Lemma 2.1, for an y c, c ′ , c ′′ ∈ Λ Z we h a v e a natural functor m c,c ′ ,c ′′ : Q ≀ ( c, c ′ ) × Q ≀ ( c ′ , c ′′ ) → Q ≀ ( c, c ′′ ) giv en b y pu llbac k. T h is op eration is asso ciativ e, so that we h av e a 2-category Q Λ Z with the same ob j ects as Λ Z , and with morphism categories Q ≀ ( − , − ). Analogously to [Ka2, (3.7)], w e h a ve natural m ap s µ c,c ′ ,c ′′ : T c,c ′ ⊠ T c ′ ,c ′′ → m ∗ c,c ′ ,c ′′ T c,c ′′ , and these maps are asso ciativ e on triple pro du cts. Therefore b y [Ka2, Sub- section 1.6], we ha ve an A ∞ -categ ory B q with th e same ob j ects as Λ Z , with morphisms giv en by B q ( c, c ′ ) = C q ( Q ≀ ( c, c ′ ) , T c,c ′ ) , 18 the bar complex of the category Q ≀ ( c, c ′ ) with co eﬃcient s in th e functor T c,c ′ , and w ith the compositions indu ced by the f unctors m c,c ′ and the maps µ c,c ′ ,c ′′ . F or an y ab elian category Ab, consider the deriv ed catego ry D ( B opp q , Ab) of A ∞ -functors from the opp osite category B opp q to the category of complexes of ob jects in Ab. By d eﬁnition, the category Λ Z is emb edded in to the 2- catego ry Q Λ Z — the em b edd ing f unctor q : Λ Z → Q Λ Z is id en tical on ob jects, and sends a m orp hism in to a diagram (2.1) with v = id . F or an y c, c ′ ∈ Λ Z , the restriction q ∗ T q ( c ) ,q ( c ′ ) is the constant fu nctor Z , s o th at by restriction, we obtain a natural fu nctor (2.2) q ∗ : D ( B opp q , Ab) → D (Λ Z opp , Ab) . Let h : Λ Z h → Λ Z b e the natural emb edding; comp osing q ∗ with h ∗ , we obtain a restriction functor D ( B opp q , Ab) → D (Λ Z opp h , Ab) . Deﬁnition 2.2. A cyclic Mackey functor with v alues in an ab elian cate- gory Ab is an A ∞ -functor M ∈ D ( B opp q , Ab) whose r estriction h ∗ q ∗ M ∈ D (Λ Z opp h , Ab) is lo cally constan t in the sense of Deﬁnition A.1. Cyclic Mac k ey f unctors form a fu ll triangulated sub category in the cat- egory D ( B opp q , Ab) whic h we will den ote by D M Λ(Ab ) ⊂ D ( B opp q , Ab). In fact, in th is pap er we will only need the case Ab = k -mo d, the category of mo d ules o v er a comm utativ e ring k ; to simp lify notation, we will denote D M Λ( k -mod ) = D M Λ( k ). By deﬁn ition, for ev ery p ositiv e in teger m ≥ 1, the emb edding O Z /m Z ∼ = Λ Z m v ⊂ Λ Z of (1.19) extends to a 2-functor (2.3) i m : Q ≀ O Z /m Z → Q Λ Z , where Q ≀ O Z /m Z is the 2-category of [Ka2, Sub s ection 3.4], and Q ≀ O Z /m Z ⊂ Q ≀ O Z /m Z is the full sub catego ry spanned by Z /m Z -orbits. This 2-functor is compatible with the co eﬃcien ts T c,c ′ ∈ F un( Q ≀ ( c, c ′ ) , Z ), so it extends to an A ∞ -functor e B O Z /m Z q → B q , where e B O Z /m Z q is as in [Ka2, Subsection 3.5]. The category e B O Z /m Z q is b y deﬁnition self-dual; thus b y restriction, for any comm u tativ e ring k w e ob tain a natural fun ctor i ∗ m : D ( B opp q , k ) → D M ( Z /m Z , k ) , 19 where D M ( Z /m Z , k ) is the cate gory of deriv ed Z /m Z -Mac key functors con- structed in [Ka2]. W e also ha ve the left-adjoin t f unctor i m ! : D M ( Z /m Z , k ) → D ( B opp q , k ) . 2.2 Geometric ﬁxed p oin ts. Consider no w the category c Λ Z of Subsec- tion 1.4. This category c Λ Z is also embed d ed into Q Λ Z — th e em b edding functor b q : c Λ Z → Q Λ Z is identica l on ob jects, and sends a diagram (1.21) to the corresp ond in g diagram (2.1) (we recall th at diagrams (1.21) ha v e n o automorphisms, so there are no choice s in vol ved in this constru ction). As in the case of the functor q of (2.2 ), th e f unctor b q is compatible w ith the co eﬃcien ts T c,c ′ , so that w e hav e the restriction functor b q ∗ : D ( B opp q , k ) → D ( c Λ Z opp , k ) . F or any p ositiv e inte ger m ≥ 1, the 2-functor i m restricts to the em b edding O Z /m Z → c Λ Z , so that w e h av e a fun ctorial isomorphism (2.4) i ∗ m ◦ b q ∗ ∼ = b q ∗ m ◦ i ∗ m , where b q ∗ m is the restriction with resp ect to the 2-functor b q m : O Z /m Z → Q ≀ ( O Z /m Z ) considered in [Ka2, Sub section 5.3] (there denoted b y q ), Recall n o w that we h a ve the d ecomp ositions (1.19) , (1. 18 ), and the iden- tiﬁcation Λ Z h ∼ = c Λ Z h ∼ = N ∗ × Λ. Deﬁne a functor ν : F un(Λ Z opp , k ) = F un( c Λ Z opp h , k ) → F u n ( c Λ Z opp , k ) b y setting ν ( M )([ n | m ]) = M ([ n | m ]) f or any M ∈ F un( c Λ Z opp h , k ), [ n | m ] ∈ c Λ Z , with (2.5) v ( M )( v ◦ h ) = ( M ( v ◦ h ) , v inv er tib le , 0 , v not in vertible , where f = v ◦ h is the horizon tal/v ertical factorization of a m ap f in c Λ Z h . It is easy to see that this is w ell-deﬁned, and giv es an exact functor ν . By abuse of notation, w e will d enote by the s ame letter ν its extension to the deriv ed catego ries. W e ha ve a left-adjoin t functor ϕ : F un( c Λ Z opp , k ) → F un(Λ Z opp h , k ), and its d eriv ed fun ctor L q ϕ : D ( c Λ Z opp , k ) → D (Λ Z opp h , k ) is left-adjoin t to ν : D (Λ Z opp h , k ) → D ( c Λ Z opp , k ). 20 Deﬁnition 2.3. The (ge ometric) ﬁxe d p oint fu nctor Φ : D ( B opp q , k ) → D (Λ Z opp h , k ) is giv en by Φ = L q ϕ ◦ b q ∗ . W e note that b y deﬁn ition, the ﬁxed p oint s functor Φ actually splits into comp onent s Φ m : D ( B opp q , k ) → D ((Λ Z m h ) opp , k ) ∼ = D (Λ opp , k ) n umb ered b y p ositiv e in tegers m ≥ 1. W e will need some compatibilit y results b et we en Φ and geomet r ic ﬁxed p oint f u nctors for d eriv ed Mac key functors constructed in [Ka2]. Namely , ﬁx a p ositiv e integ er m ≥ 1, and consider the em b edd ing i m of (2.3) and the corresp onding r estriction functor i ∗ m . L et O Z /m Z b e th e category of Z /m Z -orbits an d in vertible maps b et we en them, and deﬁne the functor ν m : F un( O Z /m Z , k ) → F un( O Z /m Z , k ) b y the s ame formula (2.5) as the functor ν ( ν m ( M )( v ) = M ( v ) for in ve rtib le v , and 0 otherwise). Let ϕ m : F un( O Z /m Z , k ) → F u n( O Z /m Z , k ) b e its left- adjoin t fu n ctor, with the d eriv ed fu nctor L q ϕ m . W e then ha ve an obvio u s isomorphism (2.6) i ∗ m ◦ ν ∼ = ν m ◦ i ∗ m . By adjunction, the isomorphisms (2.4) and (2.6) indu ce b ase c hange maps (2.7) i ∗ m ◦ L q ϕ → L q ϕ m ◦ i ∗ m , i m ! ◦ b q ∗ m → b q ∗ ◦ i m ! , where by abu se of n otation, i m denotes b oth the 2-fun ctor (2.3), and its restrictions i m : O Z /m Z → c Λ Z opp , i m : O Z /m Z → c Λ Z opp h to b q m ( O Z /m Z ) and b q m ( O Z /m Z ). Lemma 2.4. F or any p ositive inte ger m ≥ 1 , the b ase change maps of (2.7) ar e i somorph isms. 21 Pr o of. Let b h : c Λ Z opp h → c Λ Z opp , h m : O Z /m Z → O Z /m Z b e the tautologic al em b eddings. Then we ob viously hav e b h ∗ ◦ ν ∼ = id , h ∗ m ◦ ν ∗ m ∼ = id , so that by adjunction, L q ϕ ◦ b h ! ∼ = id , L q ϕ m ◦ h m ! ∼ = id . On the other hand, w e hav e the horizon tal/v ertical factorizati on system on c Λ Z opp ∼ = Λ Z , and Lemma A.4 y ields i ∗ m ◦ b h ! ∼ = h m ! ◦ i ∗ m . Therefore the map i ∗ m ◦ L q ϕ ◦ b h ! → L q ϕ m ◦ i ∗ m ◦ b h ! is an isomorp h ism, so th at the ﬁ rst of th e maps (2.7) b ecomes an isomor- phism after ev aluating at an y ob ject E ∈ D ( c Λ Z opp , k ) of the form E = b h ! E ′ , E ′ ∈ D ( c Λ Z opp h , k ). S ince the categ ory D ( c Λ Z opp , k ) is generated by ob jects of this form, the map i ∗ m ◦ L q ϕ → L q ϕ m ◦ i ∗ m is itself an isomorphism. F or the second of th e m aps (2.7), let b h ′ = b q ◦ b h , h ′ m = b q m ◦ h m . Then again, the h orizon tal/v ertical fact orization system on c Λ Z ≀ Γ sho ws that every diagram (2.1) decomp oses as c v ← − − − − c 1 v 1 − − − − → c 2 h − − − − → c ′ with v ertical v , v 1 and horizon tal h , and this implies that i ∗ m ◦ b h ′ ∗ ∼ = h ′ m ∗ ◦ i ∗ m . The pro of is the same as in Lemma A.4, except that in (A.3), one replaces the Hom-sets c Λ Z opp ( − , − ), c Λ Z opp v ( − , − ) with the Hom-categ ories Q ≀ c Λ Z ( − , − ), resp. Q ≀ ( O Z /m Z )( − , − ). Therefore the base c hange map i ∗ m ( b q ∗ E ) → b q ∗ m ( i ∗ m E ) is an isomorphism for an y E ∈ D ( c Λ Z opp , k ) of the form E = b h ′ ∗ E ′ , E ′ ∈ D ( c Λ Z opp h , k ). O b jects of this f orm generate the d er ived category , so that i ∗ m ◦ b q ∗ ∼ = b q ∗ m ∼ = i ∗ m ; b y adju nction, this giv es the claim.  Corollary 2.5. F or any M ∈ D ( B opp q , k ) and any p ositive inte gers m, n ≥ 1 , we have a natur al isomorphism Φ m ( M )([ n ]) = Φ( M )([ n | m ]) ∼ = Φ [( Z /nm Z ) / ( Z /m Z )] i ∗ mn M , 22 wher e [( Z /nm Z ) / ( Z /m Z )] ∈ O Z /nm Z is understo o d as a Z /nm Z -orbit, and Φ c , c ∈ O Z /nm Z is the ge ometric ﬁxe d p oints functor of [ K a2, Subse ction 5.1]. Mor e over, f or any derive d Mackey f unctor M ∈ D M ( Z /mn Z , k ) , we have a natur al i somorph ism (2.8) Φ m ( i mn ! M ) ∼ = i mn ! (Φ [( Z /nm Z ) / ( Z /m Z )] M ) . Pr o of. I mmediately follo w s f rom Lemma 2.4 and [Ka2, Prop osition 6.5].  2.3 Coalgebra description. W e will now adapt the constructions of [Ka2, Section 6] and use th e ﬁxed p oint f unctor Φ to obtain another d e- scription of the category D M Λ( k ) of cyclic Mac key fun ctors. 2.3.1 A ∞ -coalgebra. C onsider th e cyclotomic category Λ R of Sub s ec- tion 1.2 with its vertic al/horizon tal factorizati on system. F or an y ob ject [ m ] ∈ Λ R , let [ m ] \ Λ R b e the category of ob jects [ m ′ ] ∈ Λ R equip p ed with a map [ m ] → [ m ′ ]. T he factorization system on Λ R induces a f actorizat ion system on [ m ] \ Λ R for any [ m ]. Lemma 2.6. F or any vertic al map v : [ a ] → [ b ] in [ m ] \ Λ R and any map f : [ b ′ ] → [ b ] , ther e exists a Cartesian squar e [ a ′ ] v ′ − − − − → [ b ′ ] f ′   y   y f [ a ] v − − − − → [ b ] with vertic al v ′ . Pr o of. The same argumen t as in Lemma 2.1 sh o ws that the w reath pro duct catego ry Λ R ≀ Γ h as ﬁb ered pr o ducts. Therefore so do es the slice category [ m ] \ (Λ R ≀ Γ). It remains to notice that the em b edding [ m ] \ Λ R → [ m ] \ (Λ R ≀ Γ) has a left-adjoint (an y S ∈ Λ R ≀ Γ is a disjoint union of ob jects in Λ R , and an y m ap [ m ] → S factors u niquely through one of these ob jects).  No w, for an y morphism f : [ m ′ ] → [ m ] in Λ R and for any int eger n ≥ 0, consider all diagrams (2.9) [ m ′ ] g − − − − → [ m 0 ] v 0 − − − − → . . . v n − 1 − − − − → [ m n ] v n − − − − → [ m ] in Λ R su c h v i , 0 ≤ i ≤ n is v ertical, v n is not in vertible, and f = v n ◦· · · ◦ v 0 ◦ g . Let V n ( f ) b e the group oid of all such diagrams and isomorphisms b et we en them. Since Λ R is small, V n ( f ) is small for an y f , n . 23 Assume th at f = f (1) ◦ f (2) for some [ m ′′ ] ∈ Λ R and some maps f (1) : [ m ′′ ] → [ m ], f (2) : [ m ′ ] → [ m ′′ ]. Then for any l ≥ 1 and any diagram α ∈ V n ( f ) of the form (2.9), w e can app ly Lemma 2.6 and form a comm utative diagram [ m ′ ] g − − − − → [ m 0 ] v 0 − − − − → . . . v n − 1 − − − − → [ m n ] v n − − − − → [ m ]    f (1) 0 x   f (1) n x   f (1) x   [ m ′ ] g ′ − − − − → [ m ′ 0 ] v ′ 0 − − − − → . . . v ′ n − 1 − − − − → [ m ′ n ] v ′ n − − − − → [ m ′′ ] , where f (2) = v ′ n ◦ · · · ◦ v ′ 0 ◦ g ′ , and all the commutativ e squares are Cartesian squares in the category [ m ′ ] \ Λ R . F or any i , 0 ≤ i ≤ n , w e h a v e a natur al v ertical map ν i = v ′ n ◦ · · · ◦ v i : [ m ′ i ] → [ m ′′ ]. T ak e th e minimal i su c h that ν i is an isomorphism, let α (2) b e the diagram [ m ′ ] g ′ − − − − → [ m ′ 0 ] v ′ 0 − − − − → . . . v ′ i − 2 − − − − → [ m ′ i − 1 ] ν i ◦ v ′ i − 1 − − − − − → [ m ′′ ] in V i ( f (2) ), an d let α (1) b e the diagram [ m ′′ ] f (1) i ◦ ν − 1 i − − − − − → [ m i ] v i − − − − → . . . v n − − − − → [ m ] in V n − i ( f (1) ). Th en sending α to α (1) × α (2) giv es a well-deﬁned functor (2.10) V n ( f ) → a 0 ≤ i ≤ n V i ( f (1) ) × V n − i ( f (2) ) . This construction is obvio u sly asso ciativ e: for an y l -tuple f 1 , . . . , f l of com- p osable maps in Λ R , we can comp ose the functors (2.10) and obtain a fun ctor (2.11) V n ( f 1 ◦ · · · ◦ f l ) × I l → a n 1 + ··· + n l = n V n 1 ( f 1 ) × · · · × V n l ( f l ) , where I l is the l -th group oid of the monoidal category op erad of Deﬁni- tion A.8. Th ese fun ctors are compatible with the n atural op erad structur e on I q in the ob vious sense. F or an y i , 1 ≤ i ≤ n , forgetting the ob j ect [ m i ] in a diagram (2.9) giv es a fun ctor δ i : V n ( f ) → V n − 1 ( f ), and these fu n ctors s atisfy the relations b et ween simplicial face maps (n ot only up to an isomorphism , but on the nose). Therefore w e can deﬁne a bicomplex T q , q ( f ) by setting (2.12) T q , q ( f ) = C q ( V q ( f ) , Z ) , 24 the bar complex of the group oid V q ( f ) with coeﬃcient s in the co n stan t functor Z . One d iﬀerential in the bicomplex (2.1 2 ) comes fr om the bar complex, and the other one is given by d = d 1 − d 2 + · · · ± d n , where d i is the map in duced by the fun ctor δ i . T he coprod u ct op erations (2.11) strictly comm ute with the functors δ i , so that w e hav e canonical op erations C q ( I l , Z ) ⊗ T q ( f 1 ◦ · · · ◦ f l ) → T q ( f 1 ) ⊗ · · · ⊗ T q ( f n ) , again compatible with the asymmetric op erad structure on C q ( I q , Z ). Fix- ing a map A ss ∞ → C q ( I q , Z ), as in the Ap p end ix, we equip the collectio n T q ( − ) with a structur e of a Λ R -graded A ∞ -coalg ebr a in th e sense of [Ka2, Subsection 1.5.4]. F or future u se, w e r ecord right a wa y some elemen tary prop erties of the Λ R -graded A ∞ -coalg ebr a T q . Lemma 2.7. (i) The A ∞ -c o algebr a T q is augmente d, and for any hori- zontal map f , we have T l ( f ) = 0 for l ≥ 1 . (ii) F or any c omp osable maps f 1 , f 2 such that f 1 is horizontal, the c opr o d- uct map b 2 : T q ( f 2 ◦ f 1 ) → T q ( f 1 ) ⊗ T q ( f 2 ) ∼ = T q ( f 2 ) is an isomorphism. (iii) Assume given an n -tuple f 1 , . . . , f n of c omp osable maps in Λ R , and assume that f i is horizontal f or i ≥ 2 , and n ≥ 3 . Then the c orr e- sp onding A ∞ -op er ation b n : T q ( f n ◦ f 1 ) → T q ( f 1 ) ⊗ · · · ⊗ T q ( f n ) ∼ = T q ( f n ) is e qual to 0 . Pr o of. (i) is ob vious: for an y f , V 0 ( f ) is by deﬁnition a single p oin t, and if f is horizon tal, V l ( f ) is empt y for any l ≥ 1. T o see (ii) , note that since f 1 is horizon tal, th e only non-trivial term in the copro duct (2.10) is a map (2.13) V l ( f 2 ◦ f 1 ) → V l ( f 2 ) × V 0 ( f 1 ) = V l ( f 2 ) , and sending a diagram α ∈ V n ( f 2 ) of the form (2.9) to the diagram [ m ′ ] g ◦ f 1 − − − − → [ m 0 ] v 0 − − − − → . . . v n − 1 − − − − → [ m n ] v n − − − − → [ m ] deﬁnes a map V n ( f 2 ) → V n ( f 2 ◦ f 1 ) whic h is strictly in verse to (2.13). More- o v er, th is in verse map construction is ob viously s tr ictly asso ciativ e, so that 25 for an y n -tup le f 1 , . . . , f n of comp osable maps with horizonta l f 2 , . . . , f n and n ≥ 3, w e obtain a single map V l ( f n ) → V n ( f n ◦ · · · ◦ f 1 ) . Therefore the copro d uct map (2.10) is also strictly asso ciativ e, so that th e map (2.11) factors through the natural map I n → pt . Th is by deﬁnition means that the higher A ∞ -op eration b n v anishes, w hic h pr o v es (iii).  Lemma 2.8. L et h : Λ R h ∼ = Λ → Λ Z , i : Λ R v ∼ = O Z → Λ Z b e the tautolo gic al emb e ddings. Then h ∗ T q is the trivial Λ - gr ade d A ∞ -c o algebr a, h ∗ T q ( f ) ∼ = Z f or any map f in Λ , while i ∗ m T q is isomorphic to the O Z -gr ade d A ∞ -c o algebr a of [Ka2, Subse ction 6.3.3]. Pr o of. The ﬁrs t claim immediatel y follo ws from Lemma 2.7. F or the second claim, note that if f is v ertical, all th e diagrams (2.9) consist of v ertical maps, th us coincide with the diagrams used in [Ka2, Subsection 6.3.3].  2.3.2 The comparison theorem. Consider no w the natural cobifration λ : Λ Z → Λ R of (1.17), an d let λ ∗ T q b e the Λ Z -graded coalgebra obtained b y pu llb ac k. F or any ring k , consider the deriv ed categ ory D (Λ Z , λ ∗ T q , k ) of A ∞ -comod ules ov er λ ∗ T q . By Lemma 2.8, the pu llbac k h ∗ λ ∗ T q with r esp ect to the tautolog ical embed ding h : Λ Z h → Λ Z is the trivial Λ Z h -graded A ∞ -coalg ebr a, so that w e hav e a natural pullbac k functor h ∗ : D (Λ Z , λ ∗ T q , k ) → D (Λ Z opp h , k ) . Let D M Λ T ( k ) ⊂ D (Λ Z , λ ∗ T q , k ) b e the full sub category spanned b y ob jects M w hose restriction h ∗ M is lo cally constan t in the sense of Deﬁnition A.1. W e w ant to sho w that the category D M Λ T ( k ) is n aturally equiv alen t to the catego ry D M Λ( k ) of k -mo d-v alued cyclic Mac key functors. T o construct a comparison fu nctor b et we en these t wo catego ries, let V l ([ n | m ]), [ n | m ] ∈ Λ Z , l ≥ 0 b e the group oid of d iagrams [ n 1 | m 1 ] v 1 − − − − → . . . v n − 1 − − − − → [ n l | m l ] v n − − − − → [ n | m ] in Λ Z with v ertical v 1 , . . . , v n and non-in vertible v n . Let σ l : V l ([ n | m ]) → c Λ Z b e the functor w hic h sends su ch a d iagram to [ n 1 | m 1 ] ∈ c Λ Z , or to [ n | m ] is l = 0. F or any A ∞ -functor E q from B q to the category of complexes of k -mo dules, let Φ [ n | m ] q ( E q ) b e the total complex of the triple complex C q ( V q ([ n | m ]) , σ ∗ l b q ∗ E q ) , 26 where tw o d iﬀeren tials are induced by the d iﬀeren tials in E q and in the bar complex, and the third one is as in [Ka2, S u bsection 6.3.1]. Then the same construction as in [Ka2, Sub section 6.3.2] sho w s that th e collection Φ [ n | m ] q ( E q ) has a natural structure of an A ∞ -comod ule o v er λ ∗ T q , so that w e obtain a functor (2.14) Φ q : D ( B opp q , k ) → D (Λ Z , λ ∗ T q , k ) . Cho ose an integer m ≥ 1, and consider the em b edd ing i m : O Z /m Z ∼ = Λ Z m v → Λ Z for some m ≥ 1. Th e comp osition λ ◦ i m : O Z /m Z → Λ R factors th rough the n atur al em b edding O Z /m Z ⊂ O Z = Λ R v → Λ R . Th ere- fore by Lemm a 2.8, the O Z /m Z -graded A ∞ -coalg ebr a i ∗ m λ ∗ T q is isomorphic to the A ∞ -coalg ebr a of [Ka2, Su bsection 6.3.3], and w e hav e a natural pu llbac k functor i ∗ m : D (Λ Z , λ ∗ T q , k ) → D M ( Z /m Z , k ) , where D M ( Z /m Z , k ) is the category of k -v alued derived Mac k ey fun ctors of [Ka2] for the group Z /m Z . W e hav e an obvio u s isomorphism (2.15) i ∗ m ◦ h ∗ ∼ = h ∗ m ◦ i ∗ m , where h m : O Z /m Z → O Z /m Z is the tautologic al em b edding, an d h ∗ m : D M ( Z /m Z , k ) → D ( O Z /m Z , k ) is the corresp onding pullbac k f unctor. W e note that by [Ka2, Lemma 6. 18], the fun ctor h ∗ m admits a r igh t-adjoin t func- tor h m ∗ . Moreo ve r, by co n struction, the comparison functor Φ q of (2.1 4 ) re- stricts to the corresp onding functor for O Z /m Z — that is, w e h a v e a canonical isomorphism (2.16) Φ [ m ] q ◦ i ∗ m E ∼ = i ∗ m ◦ Φ q , where on th e left-hand side, Φ [ m ] q is the fun ctor Φ q of [Ka2, Theorem 6.17], for C = O Z /m Z . Lemma 2.9. The functor h ∗ : D (Λ Z , λ ∗ T q , k ) → D (Λ Z opp h , k ) admits a right-adjoint h ∗ : D (Λ Z opp h , k ) → D (Λ Z , λ ∗ T q , k ) , and for every m ≥ 1 , the b ase change map i ∗ m ◦ h ∗ → h m ∗ ◦ i ∗ m induc e d by (2.15) is an isomorphism. Pr o of. I mmediately follo w s f rom Lemma A.6 and L emma A.7.  27 Lemma 2.10. Assume gi ven a triangulate d sub c ate gory D ′ in the c ate gory D (Λ Z , λ ∗ T q , k ) which is c lose d with r esp e cts to arbitr ary pr o ducts and c on- tains al l the obje cts h ∗ M , M ∈ D (Λ Z opp h , k ) . Then D ′ = D (Λ Z , λ ∗ T q , k ) . Pr o of. F or any inte ger n ≥ 1, let h n : Λ ∼ = Λ Z n h ⊂ Λ Z h b e the em b edding of the n -th comp onen t of the decomp osition (1.18). F or an y M ∈ D (Λ Z opp h , k ), let the supp ort Supp ( M ) ⊂ N ∗ b e the set of all suc h n ≥ 1 that h ∗ l M is non-zero for s ome l divid in g n . F or any M ∈ D (Λ Z , λ ∗ T q , k ), let Supp M = Supp h ∗ M . Note th at for any tw o integ ers n, n ′ ≥ 1, h ∗ n ′ ◦ h ∗ ◦ h n ∗ = 0 unless n divides n ′ , and h ∗ n ◦ h ∗ ◦ h n ∗ ∼ = id (by Lemma 2.9, we ma y c h ec k b oth statemen ts after app lyin g the functors i ∗ m , m ≥ 1, and then they immediately follo w from [Ka2, Lemma 6.18]). Therefore in p articular, for an y n ≥ 1 and an y M ∈ D (Λ Z , λ ∗ T q , k ), w e h a v e Supp ( h ∗ h n ∗ h ∗ n h ∗ M ) ⊂ S upp ( M ) . Moreo ver, give n an ob ject M ∈ D (Λ Z , λ ∗ T q , k ), let M [1] b e the cone of th e adjunction map M → h ∗ ( h ∗ M ) m , where m is the smallest int eger in Supp ( M ). T hen Supp ( M [1] ) = Supp ( M ) \ { m } . By indu ction, let M [ n ] = ( M [ n − 1] ) [1] for an y n > 1. Then w e ha ve natural maps M [ n ] → M , and their cones f M [ n ] lie in D ′ and form an in verse system. W e hav e a compatible system of maps η n : M → f M [ n ] , n ≥ 1. By induction, for ev ery n > n ′ ≥ 1, the transition map f M [ n +1] → f M [ n ] b ecomes an isomorphism after applying h ∗ n ′ h ∗ , and so d o es the map η n . L et f M = holim f M [ n ] , where holi m is deﬁned b y the telescope construction, and let η : M → f M b e the natural map. T hen for every n ≥ 1, th e inv erse system h ∗ n h ∗ f M [ n ′ ] stabilizes for n ′ > n , an d h ∗ n h ∗ ( η ) is an isomorphism. Th u s η itself is an isomorphism. But b y construction, f M lies in th e category D ′ .  Lemma 2.11. The c omp osition h ∗ ◦ Φ q : D ( B opp q , k ) → D (Λ Z opp h , k ) is isomorphic to the ﬁxe d p oints functor Φ of Deﬁnition 2.3. 28 Pr o of. By construction, h ∗ Φ q ( E ) for an y E ∈ D ( B opp q , k ) only d ep ends on the restriction b q ∗ E ∈ D ( c Λ Z opp , k ) — the construction of the functor Φ q also giv es a functor ϕ q : D ( c Λ Z opp , k ) → D (Λ Z opp h , k ), and we ha ve h ∗ ◦ Φ q ∼ = ϕ q ◦ b q ∗ . F or an ob ject E ∈ F un( c Λ Z opp , k ), the d egree-0 homology of the complex ϕ q ( E ) is easily seen to b e isomorphic to ϕ ( E ), an d th is isomorph ism is functorial in E . Th u s by the unive r s al prop ert y of the deriv ed functor, it extends to a map e : ϕ q → L q ϕ. W e ha ve to pro ve that e : ϕ q ( E ) → L q ϕ ( E ) is a n isomorphism for a ny E ∈ D ( c Λ Z opp , k ) of the form E = b q ∗ E ′ , E ′ ∈ D ( B opp q , k ). T o do this, it suﬃces to prov e that i ∗ m ( e ) : i ∗ m ◦ h ∗ ◦ Φ q → i m ◦ Φ is an isomorp hism for any m ≥ 1. By (2.16) and (2.15), the left-hand side is isomorph ic to h ∗ m ◦ Φ [ m ] q ◦ i ∗ m , and b y [Ka2, Lemma 6.15] , h ∗ m ◦ Φ [ m ] is iso- morphic to the direct sum of the f unctors Φ [( Z /nm Z ) / ( Z /m Z )] of C orollary 2.5. T o ﬁ nish the pro of, it suﬃces to in vok e Corollary 2.5.  Prop osition 2.12. The functor Φ q of (2.14) is an e quivalenc e of c ate gories, and it i dentiﬁes D M Λ( k ) ⊂ D ( B opp q , k ) with D M Λ T ⊂ D (Λ Z , λ ∗ T q , k ) Pr o of. As in Lemma A.6, let D ′ ⊂ D (Λ Z , λ ∗ T q , k ) b e the sub catego ry of ob jects M in th e cate gory D (Λ Z , λ ∗ T q , k ) su c h that the functor Hom(Φ q ( − ) , M ) from D ( B opp q , k ) to D ( k ) is representable. The geometric ﬁxed p oin ts functor Φ of Sub section obvio u sly has a righ t-adjoin t. Therefore b y Lemma 2.11, D ′ satisﬁes all the assumptions of Lemma 2.10. Thus D ′ is the whole category D (Λ Z , λ ∗ T q , k ), and Φ q admits a righ t-adjoin t fun ctor Φ − 1 q : D (Λ Z , λ ∗ T q , k ) → D ( B opp q , k ) . By Lemma 2.11, the comp osition Φ − 1 q ◦ h ∗ is righ t-adjoin t to the geometric ﬁxed p oin ts fu nctor Φ of Deﬁn ition 2.3. Moreo ver, for ev ery m ≥ 1, the functor Φ [ m ] q of (2.1 6 ) is a n equiv alence of ca tegories by [Ka 2 , Theorem 6.17]. Let Φ − 1 m b e the inv erse equiv alence. Then b y Corollary 2.5, the base c hange m ap i ∗ m ◦ Φ − 1 q ◦ h ∗ → Φ − 1 m ◦ h m ∗ ◦ i ∗ m , 29 b eing adjoint to the direct pro du ct of isomorphism s (2.8), is itself an iso- morphism. Then b y Lemma 2.9, the base c h ange map i ∗ m (Φ − 1 q ( M )) → Φ − 1 m ( i ∗ m ( M )) is an isomorphism for any M ∈ D (Λ Z , λ ∗ T q , k ) of the form M = h ∗ M ′ , M ′ ∈ D (Λ Z opp , k ), and by Lemma 2.10, this implies that it is an isomorphism for an y M . Thus we hav e i ∗ m ◦ Φ q ◦ Φ − 1 ∼ = Φ [ m ] ◦ Φ − 1 m ◦ i ∗ m , i ∗ m ◦ Φ − 1 ◦ Φ q ∼ = Φ − 1 m ◦ Φ [ m ] ◦ i ∗ m for an y m ≥ 1. Since Φ [ m ] q and Φ − 1 m are mutually inv erse equiv alences of catego ries, this means that the adjunction maps Id → Φ − 1 ◦ Φ q , Φ − 1 ◦ Φ q → Id b ecome isomorphisms after restricting to Λ Z v . S ince this restriction is ob viously a conserv ativ e functor, w e conclude that Φ q and Φ − 1 are mutually in verse equiv alences of categories. T o pr o v e that Φ q iden tiﬁes D M Λ( k ) and D M Λ T ( k ), note that h ∗ ◦ Φ q ∼ = h ∗ .  2.4 Restriction and corestriction. W e will no w mak e some add itional observ ations on cyclic Mac ke y functors for fu ture u se. First of all, r ecall that we ha ve 2-functors q : Λ Z → Q Λ Z , b q : c Λ Z → Q Λ Z , and th e agree on horizon tal maps, q ◦ h ∼ = b q ◦ b h , so that we hav e an isomorphism h ∗ ◦ q ∗ ∼ = b h ∗ ◦ b q ∗ . Lemma 2.13. The b ase change map b h ! ◦ h ∗ → b q ∗ ◦ q ! induc e d by the isomorphism h ∗ ◦ q ∗ ∼ = b h ∗ ◦ b q ∗ is itself an i somorph ism. Pr o of. As in the pro of of Lemma 2.4, this follo ws by the s ame argumen t as in the pro of of Lemma A.4.  In particular, this Lemma shows that we h a v e n atural ident iﬁcations h ∗ ◦ Φ q ◦ q ! ∼ = Φ ◦ q ! ∼ = L q ϕ ◦ b q ∗ ◦ q ! ∼ = L q ϕ ◦ b h ! ◦ h ∗ , and since L q ϕ ◦ b h ! is adjoin t to b h ∗ ◦ ν = Id , the r igh t-hand side is just h ∗ . In fact more is true: th e comp osition Φ q ◦ q ! : D (Λ Z opp , k ) → D (Λ Z , λ ∗ T q , k ) is naturally isomorphic to the corestriction f u nctor ξ ∗ with r esp ect to the augmen tation m ap of the augmen ted Λ Z -graded A ∞ -coalg ebr a λ ∗ T q (to con- struct an isomorphism, resolv e a functor E ∈ F un(Λ Z opp , k ) b y functors 30 of the f orm i m ! E m , E m ∈ F un( O opp Z /m Z , k ) and ap p ly [Ka2, Lemma 6.20]). Therefore the corestriction f u nctor ξ ∗ has a r igh t-adjoin t fun ctor ξ ∗ giv en b y ξ ∗ = q ∗ ◦ Φ − 1 q : D (Λ Z , λ ∗ T q , k ) → D (Λ Z opp , k ) . W e also h a v e (2.17) h ∗ ◦ ξ ∗ ◦ ξ ∗ ∼ = h ∗ ◦ q ∗ ◦ q ! ∼ = b h ∗ ◦ b q ∗ ◦ q ! ∼ = b h ∗ ◦ b h ! ◦ h ∗ . T o compu te the right-hand sid e m ore eﬀectiv ely , it is useful to consider the catego ry Λ I ∼ = f Λ I of Lemma 1.5. Consider also the p ro du ct Λ I × N ∗ , and deﬁne the pro jections i, π : Λ I × N ∗ → c Λ Z opp h ∼ = Λ Z h ∼ = Λ × N ∗ b y (2.18) i = i × id , π = π × ρ m on Λ m × N ∗ ⊂ Λ I × N ∗ , where ρ m : N ∗ → N ∗ is the map giv en b y multiplicati on b y the in teger m ≥ 1. Lemma 2.14. We have a natur al isomorphism b h ∗ ◦ b h ! ∼ = π ! ◦ i ∗ . Pr o of. Under th e iden tiﬁcation c Λ Z opp ∼ = Λ Z , b h go es to the tautolo gical em b edding h : Λ Z h → Λ Z . Let Λ I b e the category of vertic al maps v : a → a ′ in Λ Z , with maps b et ween th em giv en by commutati ve squares a 1 f − − − − → a 2 v 1   y   y v 2 a ′ 1 f ′ − − − − → a ′ 2 with horizon tal f (and arbitrary f ′ ). Let s : Λ I → Λ Z h , t : Λ I → Λ Z b e the functors sending a map to its source resp. target. Then t is a coﬁbration, and s h as a left adjoint ι : Λ Z h → Λ I sen d ing a ∈ Λ Z to its identit y map. Then b h = t ◦ ι , so that h ! ∼ = t ! ◦ ι ! ∼ = t ! ◦ s ∗ . It remains to notice that we hav e a n atural Cartesian square f Λ I × N ∗ h − − − − → Λ I π   y   y t LZ h h − − − − → Λ Z , so that h ∗ ◦ t ! ∼ = π ! ◦ h ∗ b y base change, and we h a ve s ◦ h = i .  31 3 C yc lotomic complexes. W e can no w in tro d uce th e main su b ject of the pap er, a cyclotomic complex. Fix a comm utativ e rin g k . Consider the cyclotomic category Λ R and the Λ R -graded A ∞ -coalg ebr a T q of Subsection 2.3. Let e λ : g Λ R → Λ R b e the coﬁbration of (1.17), and let h : g Λ R h ∼ = Λ × I → g Λ R b e the n atural em- b eddin g. By Lemma 2.7, the restriction h ∗ e λ ∗ T q is the trivial g Λ R h -graded A ∞ -coalg ebr a, h ∗ e λ ∗ T q ( f ) = Z for an y map f in g Λ R h , so that we h a v e a restriction fun ctor h ∗ : D ( g Λ R, e λ ∗ T q , k ) → D (Λ R h , k ) . Deﬁnition 3.1. A cyclotomic c omplex o ver k is an A ∞ -comod ule M q o v er e λ ∗ T q with v alues in the category k -mo d such that h ∗ M q ∈ D ( g Λ R h , k ) is lo cally constan t in the sense of Deﬁnition A.1. The deriv ed category of cyclotomic complexes o ver k will b e denoted denoted by D ΛR( k ). 3.1 Normalized Λ R -graded coalgebras. Deﬁnition 3.1 is sh ort enough, but it is only as exp licit and amenable to computations as the A ∞ -coalg ebr a T q (that is, not particularly). In this Section, we will p ro vide more explicit descriptions of the categories D ΛR( k ). In the pro cess of doing so, w e will also obtain a more con venien t description of the ca tegories D M Λ( k ) of S ection 2. W e s tart with the follo win g redu ction similar to [K a2, Sub s ection 7.5,7.6]. Deﬁnition 3.2. Assume given a ﬁnite group G . A complex E q of Z [ G ]- mo dules is str ongly acyclic with r esp ect to G if for any su bgroup H ⊂ G , H 6 = G , and an y Z [ H ]-mo d u le V we hav e lim n → H q ( H , V ⊗ F n E q ) = 0 , where F q E q is the s tu pid ﬁltration on the complex E q . A map f : E q → E ′ q b et ween Z [ G ]-mo du les is a str ong quasiisomorphism with resp ect to G if its cone is strongly acyclic. Deﬁnition 3.3. A Λ R -graded A ∞ -coalg ebr a R q is called normalize d if has the prop erties (i)-(iii) of Lemma 2.7. In particular, the coalgebra T q is norm alized (by Lemma 2.7). F or any normalized Λ R -graded A ∞ -colag ebr a R q and an y map f : [ m ] → [ m ′ ] is the catego ry Λ R , the complex R q ( f ) is equipp ed, b y deﬁn ition, with an action of the cyclic group Aut([ m ]). 32 Deﬁnition 3.4. An A ∞ -map ξ : R q → R ′ q b et ween normalized Λ R -graded A ∞ -coalg ebr as is a str ong quasiisomorphism if f or any map f : [ m ] → [ m ′ ], the corresp onding map ξ : R q ( f ) → R ′ q ( f ) is a strong quasiisomorphism with resp ect to the sub group Aut( f ) ⊂ Aut([ m ]) c onsistin g of su c h g ∈ Aut([ m ]) that f ◦ g = f . Prop osition 3.5. F or any str ong A ∞ -quasiisomorphism ξ : R q → R ′ q b e- twe en no rmalize d Λ R -gr ade d A ∞ -c o algebr as and a ny c ommutative ring k , the c or estriction functors ξ ∗ : D ( g Λ R, e λ ∗ R q , k ) → D ( g Λ R, e λ ∗ R ′ q , k ) , ξ ∗ : D (Λ Z , λ ∗ R q , k ) → D (Λ Z , λ ∗ R ′ q , k ) b etwe en the derive d c ate gories of A ∞ -c omo dules is an e quivalenc e of c ate- gories. Pr o of. Both Lemma 2.9 and Lemma 2.10 hold for an y n ormalized Λ R -graded A ∞ -coalg ebr a instead of T q , with the same p ro of. Moreo ver, they also hold for g Λ R instead of Λ Z (again with the same p ro of ). Thus as in the p ro of of Prop osition 2.12, the functors ξ ∗ admits a righ t-adjoin t fun ctors ξ ∗ : D ( g Λ R, e λ ∗ R ′ q , k ) → D ( g Λ R, e λ ∗ R q , k ) , ξ ∗ : D (Λ Z , λ ∗ R ′ q , k ) → D (Λ Z , λ ∗ R q , k ) . Moreo ver, for an y in teger m ≥ 1, w e ha v e a natural em b edding i m : O Z /m Z ∼ = Λ R m v → Λ R and th e corresp ondin g restriction fun ctors i ∗ m : D ( g Λ R, e λ ∗ R q , k ) → D ( O Z /m Z , i ∗ m e λ ∗ R q , k ) , i ∗ m : D ( g Λ R, e λ ∗ R ′ q , k ) → D ( O Z /m Z , i ∗ m e λ ∗ R ′ q , k ) . F or Λ Z , these fu n ctors w ere already considered in Section 2. F or either Λ Z or g Λ R , we hav e an obvious isomorphism ξ ∗ m ◦ i ∗ m ∼ = i ∗ m ◦ ξ ∗ , where ξ m = i ∗ m ( ξ ), and th e corresp onding base c han ge map i ∗ m ◦ ξ ∗ → ξ m ∗ ◦ i ∗ m is also an isomorphism, as in Lemma 2.9. Thus it su ﬃces to p ro ve that for ev ery m , ξ ∗ m and ξ m ∗ are mutually inv erse equiv alences of categories. This is [Ka2, Lemma 7.14].  3.2 Reduced Λ R -graded coalgebras. The ﬁ rst corollary of Prop osi- tion 3.5 is analogous to the reduction done in [Ka2, Subsection 7.5]. 33 Deﬁnition 3.6. A normaliz ed Λ R -graded A ∞ -coalg ebr a R q is called r e- duc e d if f or any map f in Λ R of degree n > 1, R ( f ) = 0 unless n is prime. F or a normalized Λ R -graded A ∞ -coalg ebr a R q , its r e duction R r ed q is deﬁned b y setting R r ed q ( f ) = ( R q ( f ) if the degree of f is 1 or a prime n umb er , 0 , otherwise , with the A ∞ -op erations b eing the same as in R q when it makes sense, and 0 otherwise. F or any n ormalized Λ R -graded A ∞ -coalg ebr a R q , w e ob viously h a v e a canonical map R r ed q → R q . Lemma 3.7. F or any normalize d Λ R -gr ade d A ∞ -c o algebr a R q , the c anon- ic al map R r ed q → R q induc es e quivalenc es of c ate g ories D ( g Λ R, e λ ∗ R r ed q , k ) ∼ = D ( g Λ R, e λ ∗ R q , k ) , D (Λ Z , λ ∗ R r ed q , k ) ∼ = D (Λ Z , λ ∗ R q , k ) . Pr o of. By [Ka2, Lemma 7.15 (ii)], the map R r ed q → R q is a strong qu asi- isomorphism in the sense of Deﬁnition 3.4; the claim then follo ws f rom Prop osition 3.5.  W e now observ e that a reduced Λ R -graded A ∞ -coalg ebr a is essen tially a linear ob ject: all the p oten tially n on-linear com ultiplication maps are 0 b e deﬁnition. T o make this pr ecise, let Λ I r ed = a p prime Λ p ⊂ Λ I , where Λ I is as in (1.2). Deﬁnition 3.8. An A ∞ -functor fr om a small catego r y C to some ab elian catego ry Ab is normalize d if for any n -tup le f 1 , . . . , f n of comp osable in v ert- ible m aps f 1 , . . . , f n in C , n ≥ 3, the corresp ondin g A ∞ op eration b n is equal to 0. Lemma 3.9. The c ate gory of r e duc e d normalize d Λ R -gr ade d A ∞ -c o algeb- r as and A ∞ -maps b etwe en them is e qui v alent to the c ate gory of normalize d A ∞ -functors fr om Λ I opp r ed to Z -mo d and A ∞ -maps b e twe en them. 34 Pr o of. Assume giv en a redu ced normalized Λ R -graded A ∞ -coalg ebr a R q . F or any ob ject [ m ] ∈ Λ R w e hav e the slice catego r y Λ R h / [ m ] of horizonta l maps f : [ n ] → [ m ], [ n ] ∈ Λ R . Since R q is normalized, f or an y map g : [ m ] → [ m ′ ] w e can d eﬁne a fun ctor R g q from (Λ R h / [ m ]) opp to complexes of ab elian groups by setting R g q ( f ) = R q ( g ◦ f ) . This functor is constan t (all transition maps are isomorphisms). Let P q ( g ) = lim ← R g q ( f ) , where the limit is tak en ov er the category (Λ R h / [ m ]) opp . Moreo ver, an y horizon tal map h : [ m ] → [ m ′ ] in duces a functor Λ R h / [ m ] → Λ R h / [ m ′ ], f 7→ h ◦ f ; for any map g : [ m ′ ] → [ m ′′ ], r estriction with r esp ect to th is functors giv es a natural map h ∗ : P q ( g ) → P q ( g ◦ h ) , and this is asso ciativ e in the ob vious sens e. Recall that we ha v e the equiv alence (1.1 0 ), and restrict it to Λ I r ed ⊂ Λ I . F or eve r y ob ject a ∈ Λ I r ed , let P q ( a ) = P q ( v ( a )) . Then P q ( − ) has a natural structure of a normalized A ∞ -functor from Λ I opp r ed to complexes of ab elian groups: for ev ery n -tuple i p ( a 0 ) f 1 − − − − → i p ( a 1 ) f 2 − − − − → . . . f n − − − − → i p ( a n ) v 0   y v 1   y   y v n π p ( a 0 ) f ′ 1 − − − − → π p ( a 1 ) f ′ 2 − − − − → . . . f ′ n − − − − → π p ( a n ) of maps of the form (1.9), the A ∞ op eration P q ( a n ) → P q ( a 0 ) is the comp osition of the map ( f n ◦ · · · ◦ f 1 ) ∗ : P q ( a n ) → P q ( v n ◦ f n ◦ · · · ◦ f 1 ) = P q ( f ′ n ◦ · · · ◦ f ′ 1 ◦ v 0 ) and the map P q ( f ′ n ◦ · · · ◦ f ′ 1 ◦ v 0 ) → P q ( v 0 ) = P q ( a 0 ) 35 induced by A ∞ -op eration on R q . Con versely , assume g iven a normalized A ∞ -functor P q from Λ I opp r ed to complexes of ab elian groups. F or any m ap f : [ m ] → [ n ] of p rime degree in Λ R , let C ( f ) b e the catego ry of d iagrams [ m ] h − − − − → [ m ′ ] v − − − − → [ n ] with v ertical v , horizon tal h , and f = v ◦ h . Then sending su c h a diagram to P q ( v ) give s a fun ctor P q ( f ) from C ( f ) to complexes of ab elian group s , and w e set R q ( f ) = lim ← P q ( f ) , where the limit is tak en ov er C f . W e lea v e it to the reader to c h eck that the A ∞ -functor stru cture on P q induces a reduced normalized A ∞ -coalg ebr a structure on R q ( − ), and that b oth constructions are mutually inv erse.  No w we can make our ﬁ nal reduction. S a y that a normalized A ∞ -functor M q from Λ I opp r ed to Z -mo d is admissible if for an y ob ject [ a ] ∈ Λ p ⊂ Λ I r ed , we ha ve M i ([ a ]) = ( 0 , i < 0 , Z , i = 0 , and M i ([ a ]) is a free Z [ Z /pZ ]-mo dule for i ≥ 1. W e will say th at a reduced normalized Λ R -graded A ∞ -coalg ebr a R q is adm issib le if so is the A ∞ functor E q corresp ondin g to R q under the equiv alence of Lemma 3.9. Note that the A ∞ -coalg ebr a T q is admissible by [Ka2, Prop osition 7.8]. Lemma 3.10. F or any two admissible r e duc e d normalize d Λ R - gr ade d A ∞ - c o algebr as R q , R ′ q , and ring k , we have c anonic al e qu ivalenc es D ( g Λ R, e λ ∗ R ′ q , k ) ∼ = D ( g Λ R, e λ ∗ R q , k ) , D (Λ Z , λ ∗ R ′ q , k ) ∼ = D (Λ Z , λ ∗ R q , k ) . Pr o of. Cho ose a pr o jectiv e r esolution e P q of the constan t functor Z ∈ F un(Λ I opp r ed , Z ), and let P 0 = Z , P i = e P i − 1 for i ≥ 1. Then P q is an admissi- ble normalized A ∞ -functor from Λ I opp r ed to Z -mo d, and e P q is h -pro jectiv e by Lemma A.5. Therefore for any other admissib le normalized A ∞ -functor P ′ q , w e hav e an A ∞ -map ξ : P q → P ′ q . Moreo ver, for every [ m ] ∈ Λ p ⊂ Λ I r ed , th e map ξ ([ m ]) is a strong quasi- isomorphism with resp ect to Z /p Z ⊂ Aut([ m ]). Let P ′ q b e the A ∞ -functor 36 corresp ondin g to t h e A ∞ -coalg ebr a R q , and let b R q b e the A ∞ -coalg ebr a corresp ondin g to the A ∞ -functor P q . T hen ξ in duces a map ξ : b R q → R q , and this map is a strong quasiisomorph ism in the sense of Deﬁnition 3.4. Therefore by Pr op osition 3.5, the corestrictio n functors corresp onding to e λ ∗ ξ , resp . λ ∗ ξ are equ iv alences of categories. Analogously for R ′ q .  3.3 Como dules. Consider now the category g Λ R h ∼ = Λ × I and the pro d - ucts Λ I r ed × N ∗ ⊂ Λ I r ed × I . Deﬁne the functors i, π : Λ I r ed × I → g Λ R h b y the same form ula as in (2.18). Moreo ver, let τ : Λ I r ed × I → Λ I r ed b e the tautologic al pro jection. An y complex P q of fun ctors in F un(Λ I opp r ed , Z ) is in p articular a normalized A ∞ -functor from Λ I opp r ed to Z -mod. F ix such a complex P q so th at it is admissible in th e sense of Lemm a 3.10. F or any comm utativ e ring k , consider the category of pairs h V q , ϕ i of complexes V q in F un( g Λ R opp h , k ) equipp ed with a map ϕ : π ∗ V q → i ∗ V q ⊗ τ ∗ P q . In verting quasiisomorphisms in this catego ry , we obtain a triangulated cat- egory denoted by D ( g Λ R h , P q , k ). F orgetting the map ϕ giv es a functor e h ∗ : D ( g Λ R h , P q , k ) → D ( g Λ R opp h , k ) . Analogously , let D (Λ Z , P q , k ) b e the category of pairs h V q , ϕ i of complexes V q in F un(Λ Z opp h , k ) equip p ed w ith a map ϕ : π ∗ V q → i ∗ V q ⊗ τ ∗ P q , with inv erted quasiisomorph ism s, wh er e i and π are as in (2.18). Restricting from g Λ R h to Λ Z h giv es a f orgetful functor (3.1) D ( g Λ R h , P q , k ) → D (Λ Z h , P q , k ) , and w e h a v e the restriction functor h ∗ : D (Λ Z h , P q , k ) → D (Λ Z opp h , k ) . Let R q b e the r educed normalized g Λ R -graded A ∞ -coalg ebr a corresp ondin g to P q under the equiv alence of Lemma 3.9, and consid er the d eriv ed cate- gories D (Λ Z , e λ ∗ R q , k ), D ( g Λ R, e λ ∗ R q , k ). 37 Lemma 3.11. Ther e exists a c anonic al e q uivalenc es of c ate g ories D (Λ Z h , P q , k ) ∼ = D (Λ Z , λ ∗ R q , k ) , D ( g Λ R h , P q , k ) ∼ = D ( g Λ R, e λ ∗ R q , k ) c ommuting with the r estriction functors h ∗ , r esp. e h ∗ .. Pr o of. By Lemma A.5, w e ma y mo d ify the d eﬁ nition of the derived categ ory D ( g Λ R h , P q , k ) by replacing co mp lexes in F un( g Λ R opp h , k ) with A ∞ -comod ules o v er the trivial g Λ R opp h -graded A ∞ -coalg ebr a, and analog ous ly for Λ Z . Th en the resulting catego r y of complexes is tautologic ally equiv alen t the cat egory of A ∞ -comod ules ov er e λ ∗ R q , resp. λ ∗ R q , and the equiv alence comm utes with e h ∗ , r esp. h ∗ . In particular, it preserves quasiisomorphisms, hence descends to the derive d categories.  Let no w D c ( g Λ R h , P q , k ) ⊂ D ( g Λ R h , P q , k ) b e the full su b category sp anned b y such M ∈ D ( g Λ R h , P q , k ) that h ∗ M ∈ D ( g Λ R opp h , k ) is lo cally constan t in the sense of Deﬁn ition A.1, and let D c (Λ Z h , P q , k ) ⊂ D (Λ Z h , P q , k ) b e th e full s u b category spanned b y M ∈ D (Λ Z h , P q , k ) with lo cally constan t h ∗ M . Then combining Lemma 3.11 with Lemma 3.7 and Lemma 3.10, we obtain the follo wing eﬀectiv e description of the categories D M Λ( k ), D ΛR( k ) of k - v allued cyclic Mac key fun ctors and k -v alued cyclotomic complexes. Prop osition 3.12. F or any admissible c omplex P q of functors f r om Λ I r ed to Z -mo d , ther e exist c anonic al e quivalenc es D M Λ( k ) ∼ = D c (Λ Z h , P q , k ) , D ΛR( k ) ∼ = D c ( g Λ R h , P q , k ) of triangulate d c ate gories.  Moreo ver, restricting f rom g Λ R to Λ Z as in (3.1), we obtain restriction functors h ∗ : D ( g Λ R opp , k ) → D (Λ Z opp , k ) , h ∗ : D ( g Λ R h , P q , k ) → D (Λ Z opp , k ) . Let D w ( g Λ R opp h , k ) ⊂ D ( g Λ R opp h , k ), D w ( g Λ R h , P q , k ) ⊂ D ( g Λ R h , P q , k ) b e the full su b categories spanned by such M th at h ∗ M ∈ D (Λ Z opp , k ) is locally constan t. Prop osition 3.13. (i) The r estriction functor e h ∗ has a right-adjoint e h ∗ : D ( g Λ R opp h , k ) → D ( g Λ R h , P q , k ) , and it sends the sub c ate gory D w ( g Λ R opp h , k ) ⊂ D ( g Λ R opp h , k ) into the sub- c ate g ory D w ( g Λ R h , P q , k ) ⊂ D ( g Λ R h , P q , k ) . 38 (ii) Assume g i ven a triangulate d sub c ate gory D ′ ⊂ D w ( g Λ R h , P q , k ) which is close d with r esp e cts to arbitr ary pr o ducts and c ontains al l the obje cts h ∗ M , M ∈ D (Λ Z opp h , k ) . Then D ′ = D w ( g Λ R h , P q , k ) . Pr o of. O n the lev el of catego r ies of complexes, th e forgetful fu nctor e h ∗ has an ob vious adjoint given b y e h ∗ V q = π ∗ ( i ∗ V q ⊗ τ ∗ P q ) , with th e tautological map ϕ . Since ev ery complex in F un(Λ R opp h , k ) has an h -injectiv e replacemen t, th is d escend s to the deriv ed categories. Lemma 3.14. F or a ny l ≥ 0 , let F l P q ⊂ P q b e the l -term of the stupid ﬁltr ation on P q (that is, ( F l P q ) m = P m for m ≤ l , and 0 otherwise). Then for every h -inje ctive c omplex V q in F un( g Λ R opp h , k ) , the natur al map (3.2) π ∗ ( i ∗ V q ⊗ τ ∗ P q ) → lim l → R q π ∗ ( i ∗ V q ⊗ τ ∗ F l P q ) is a quasiisomorphism. Pr o of. W e obvio u s ly ha ve i ∗ V q ⊗ τ ∗ P q ∼ = lim l → i ∗ V q ⊗ τ ∗ F l P q , so that it suﬃces to pro ve th at the map π ∗ ( i ∗ V q ⊗ τ ∗ F l P q ) → R q π ∗ ( i ∗ V q ⊗ τ ∗ F l P q ) is a qu asiisomorph ism for any l ≥ 0. By induction, this redu ces to proving that for an y integ ers m , l ≥ 0, n ≥ 1, we h av e R n π ∗ ( i ∗ V m ⊗ τ ∗ P l ) = 0 . This can b e c hec ked after ev aluating at any ob ject a ∈ g Λ R h ; b y base change, w e hav e to show that H n ( Z /p Z , i ∗ V m ( b ) ⊗ P l ( τ ( b ))) = 0 for ev ery b ∈ Λ p × I ⊂ Λ I r ed × I . But since V q is h -injectiv e, V q ( c ) is an h -injectiv e complex of Z [Aut( c )]-mo d ules for an y c ∈ g Λ R h , so that i ∗ V m ( b ) is a free Z /p Z -mo du le. Hence so is the pro d uct i ∗ V m ( b ) ⊗ τ ∗ P l ( b ).  39 No w, by P rop osition 3.12, the category D ( g Λ R h , P q , k ) do es not dep end on the c hoice of an admissible co mp lex P q . Cho ose P q so that F l P q is quasiisomorphic to the s h ift Z [ l ] of the constant functor Z ∈ F un(Λ I opp r ed , k ) for ev ery ev en intege r l ≥ 0. Let π : Λ I r ed × N ∗ → Λ Z h b e the r estriction of the fu n ctor π to Λ I r ed × N ∗ . T hen h ∗ ◦ R q π ∗ ∼ = R q π ∗ ◦ h ∗ b y base c hange, and R q π ∗ : D (Λ I opp r ed , k ) → D (Λ Z opp h , k ) ob viously send s D c (Λ I opp r ed , k ) into D c (Λ Z opp h , k ). Th us r estricting to ev en l in the righ t-hand sid e of (3.2), w e see that for ev ery V q ∈ D w ( g Λ R opp h , k ), e h ∗ e h ∗ V q ∼ = lim l → R q π ∗ i ∗ V q [2 l ] indeed lies in D w ( g Λ R opp h , k ). T his ﬁnishes the pro of of (i) . (ii) now follo ws b y exactly the same argumen t as in Lemm a 2.10.  4 E quiv arian t homology . 4.1 Generalities on equiv arian t homotop y . T o ﬁx notation, we start b y recalling some general fact f r om equiv arian t stable homotop y theory . The standard reference here is [LMS]; we mostly follo w the exp osition in [HM] whic h cont ains in a concise form ev eryth ing we will need. Let G b e a compact Lie group. A G -CW c omplex X is a p ointe d CW complex X equipp ed with a con tin uous ﬁxed-p oint- p reserving G -act ion suc h that for an y g ∈ G , the ﬁxed-p oin ts sub set X g ⊂ X is a sub complex. Con- sider the category of p oin ted G -toplogical spaces and G -equiv ariant maps b et ween them considered up to a G -equiv arian t homotopy , and let G -T op b e the fu ll sub catego r y sp anned by spaces homotopy equiv alent to G -CW complexes. W e note that for ev ery closed Lie sub group H ⊂ G , send ing X to the ﬁxed p oints sub set X H ⊂ X giv es a well -d eﬁ ned functor G -T op → W H -T op , where W H = N H /H , and N H ⊂ G is the n orm alizer of the subgroup H . Giv en a ﬁnite-dimensional representat ion V of the group G ov er R , w e will denote b y S V the one-p oin t compactiﬁcation of V , with inﬁnity b eing the ﬁxed p oint. F or any X ∈ G -T op , w e will d en ote Σ V X = X ∧ S V , and w e will denote b y Ω V X th e space of based con tin uous map s from S V to X . The functors Σ V , Ω V : G -T op → G -T op are ob viously adjoint. 40 A G -universe is an R -v ector space U equipp ed with a con tinous linear G - action and a G - inv arian t p ositiv e-deﬁnite inner pro d uct. F or any G -univ erse U , a G -pr esp e ctrum X indexe d on U is a collectio n of G -CW complexes X ( V ), one for eac h ﬁnite-dimensional G -in v ariant subs p ace V ⊂ U , and G -equiv arian t conti nuous maps (4.1) X ( V ) → Ω W X ( V ⊕ W ) , one for every pair of transversal mutually orthogonal ﬁn ite-dimensional G - in v arian t subspaces V , W ⊂ U , sub ject to an obvious asso ciativit y condition. The ca tegory of G -sp ectra indexed on U and homoto py cla sses of maps b et ween them will b e denoted G -sp( U ). A G -presp ectrum is a sp e ctrum if the maps (4.1 ) are homeomorph isms. The ca tegory of G -sp ectra indexed on U and homoto py cla sses of maps b et ween them will b e denoted G -Sp( U ). W e ha ve the tautological embedd ing G -Sp( U ) → G -sp ( U ), and it adm its a left-adjoin t sp e ctriﬁc ation functor L giv en b y (4.2) Lt ( V ) = lim W → Ω W Σ W X ( V ⊕ W ) , where the limit is tak en o v er all ﬁnite dimensional G -inv arian t subsp aces W ⊂ U orthogonal to V . F or an y inclusion u : U 1 ⊂ U 2 of G -un iv erses, w e ha v e an ob vious restric- tion fun ctor ρ # ( u ) : G -sp( U 2 ) → G -sp( U 1 ), called change of u niverse , and this fun ctor h as a left-adjoin t ρ # ( u ) : G -sp( U 1 ) → G -sp( U 2 ) giv en by e ρ # ( u )( t )( V ) = Σ V − ( V ∩ u ( U 1 )) t ( u − 1 ( V )) , where V − ( V ∩ u ( U 1 )) ⊂ V is the orthogonal to the intersec tion V ∩ u ( U 1 ) ⊂ V . T he functor ρ # ( u ) sen ds sp ectra to sp ectra; th e corr esp onding fu nctor ρ # ( u ) : G -Sp( U 2 ) → G -Sp( U 1 ) has a left-adjoint ρ # : G -Sp( U 1 ) → G -Sp( U 2 ) giv en b y ρ # ( u ) = L e ρ # ( u ) , where L is the sp ectriﬁcation functor (4.2). In particular, sp ectra indexed on a trivial un iv erse U = 0 are ju st G - spaces, and the r estriction G -Sp → G -T op with resp ect to the em b edding 0 ֒ → U is the forgetful f u nctor sending a G -sp ectrum to its v alue at 0. Its righ t-adjoint is called the susp ension sp e c trum fu nctor and denoted Σ ∞ : G -T op → G -Sp( U ). Explicitly , Σ ∞ X = L e Σ ∞ X , w here e Σ ∞ X ∈ G -sp( U ) is giv en b y e Σ ∞ X ( V ) = Σ V X. Of the n on -trivial G -univ erses, those of tw o p articular t yp es are im p ortan t. 41 (i) U = R ∞ with th e trivial G -action (where ∞ is assumed to b e count- able). Then G -sp ectra indexed on U are called naive equiv ariant G - sp ectra; w e den ote the category G -Sp( U ) b y G -Sp naive . (ii) A G -univ erse is c omplete if ev ery ﬁnite-dimensional rep r esen tation V of the compact Lie group G app ears in U a count able num b er of times. All complete G -univ erses are obvio u sly isomorphic. A G -sp ectrum indexed on a complete G -univ erse U is kno w n as a genuine equiv arian t G -sp ectrum; w e w ill d enote G -Sp( U ) simply by G -Sp. W e note that b oth G -Sp naive and G -Sp are triangulated catego ries, with shifts giv en X [ n ]( W ) = X ( W ⊕ R n ). In addition, G -Sp h as an auto equ iv- alence Σ V : G -Sp → G -Sp for ev ery ﬁn ite-dimensional rep r esen tation V , giv en by Σ V X ( W ) = X ( W ⊕ V ) (to make this p recise, one has to ﬁ x an isomorphism U ⊕ V ∼ = U , or U ⊕ R n ∼ = U in the naive case, and apply the c hange of un iv erses fu nctor). Assume giv en a closed Lie subgroup H ⊂ G . Then for an y G -universe U , U H is a W H -univ erse, complete if U was complete. F or any G -sp ectrum X ∈ G -Sp( U ), the L ewis-May ﬁxe d p oints sp e ctrum X H ∈ W H -Sp( U H ) is giv en b y X H ( V ) = X ( V ) H for an y ﬁ n ite-dimensional V ⊂ U H ⊂ U . There is a second ﬁxed p oin ts fu nc- tor Φ H : G -Sp( U ) → H -Sp( U H ) called the g e ometric ﬁxe d p oints functor . T o deﬁne it, one chooses a ﬁn ite-dimensional G -in v arian t su bspace W ( V ) ⊂ U for every ﬁnite-dimensional W H -in v arian t V ⊂ U H suc h that V = W ( V ) H and [ V ⊂ U H W ( V ) = U, and sets (4.3) ϕ H t ( V ) = t ( W ( V )) H for an y t ∈ G -sp( U ), and Φ H X = Lϕ H X for any G -sp ectrum X ∈ G -Sp( U ). Here ϕ H dep end s of the choice of the subspaces W ( V ), bu t the sp ectriﬁcation Φ H do es not (for a more inv arian t description of the fun ctor Φ H whic h make th is explicit, see [HM , Lemma 1.1]). F or any X ∈ G -Sp, there is a natur al map X H → Φ H X, 42 and this map is functorial in X . F or naiv e G -sp ectra, b oth ﬁ xed p oints fun ctors obvio us ly coincide. In the gen uine ca se, let us ﬁx a complete G -un iverse U . Th en U G ⊂ U is isomorphic to R ∞ , so that G -Sp ( U G ) is G -Sp naive , and the in clusion u : U G → U induces a pair of adjoin t fun ctors ρ # ( u ) : G -Sp → G - S p naive , ρ # ( u ) : G -Sp naive → G -Sp. W e then h av e comm u tative diagrams G -T op Σ ∞ − − − − → G -Sp naive ρ # ( u ) − − − − → G -Sp ( − ) H   y ( − ) H   y   y Φ H W H -T op Σ ∞ − − − − → W H -Sp naive ρ # ( u ′ ) − − − − → W H -Sp and (4.4) G -Sp naive ρ # ( u ) ← − − − − G -Sp ( − ) H   y   y ( − ) H W H -Sp naive ρ # ( u ′ ) ← − − − − W H -Sp , where u ′ is the em b edding U G ⊂ U H (and the W H -univ erse U H is ob viously complete). 4.2 Cyclic sets. F rom now on, w e let G = S 1 = U (1), the unit circle. Then it is w ell-kno w n that G -spaces are related to cyclic sets. Let u s r ecall the relation (for details and references, see e.g. [L]). F or an y ob ject [ n ] ∈ Λ, let | [ n ] | b e its top ological realization: the union of p oin ts num b ered b y vertic es v ∈ V ([ n ]) and op en interv als I e n umb ered b y edges e ∈ E ([ n ]), with the natur al top ology making | [ n ] | into a circle. F or an y fun ction a : E ([ n ]) → R such that a > 0, we can make | [ n ] | into a metric space | [ n ] | ( a ) by assigning length a ( e ) to the int erv al I e . L et R ([ n ]) o b e the space of pairs h a, b i of a function a : E ([ n ]) → R , a > 0, and a metric-preserving mon otonous con tin u ous m ap b : | [ n ] | ( a ) → S 1 to the un it circle S 1 ⊂ § . Suc h a map b exists if and only if a 1 + · · · + a n = 2 π , and the space of all su c h maps is n on-canonically iden tiﬁed with S 1 , so that w e hav e a non-canonical homeomorphism R ([ n ]) o ∼ = S 1 × T o n − 1 , where for any m ≥ 0, T o m ⊂ T m is the int erior of the standard m -simplex T m . E m b ed R ([ n ]) o in to a compact space R ([ n ]) by allo w ing a to take zero 43 v alues (and degenerate m etrics on | [ n ] | ). W e then ha ve R ([ n ]) ∼ = S 1 × T n − 1 . This decomp osition is n ot canonical, but the space R ([ n ]) itself is completely canonical, and b y construction, it carries a con tin u os G -action. Moreo ver, for any m ap f : [ n ] → [ m ] and a pair h a, b i ∈ R ([ m ]), let f ( a )( e ) = X f ( e ′ )= e s ( e ′ ) , so th at we ha ve an obvious metric-preservin g map | [ n ] | ( f ( a )) → | [ m ] | ( a ), and let f ( b ) b e the comp osition of this map w ith b . This makes R in to a con tra v arian t fu nctor from Λ to G -T op . T urn it in to a co v ariant fun ctor by applying the dualit y Λ opp ∼ = Λ. T he r esu lt is a functor (4.5) R : Λ → G -T op . By the standard Kan extension pr o cedure, it extends un iqu ely to a colimit- preserving r e alization functor Real : Λ opp Sets → G -T op suc h that Real ◦ Y ∼ = R , w h ere Y : Λ → Λ opp Sets is the Y oneda emb edding. W e also the righ t-adjoint functor S : G -T op → Λ opp Sets suc h that S ( X )([ n ]) = Maps G ( R ([ n ]) , X ) , for any X ∈ G -T op , where Maps G ( − , − ) stands for the set of G -equiv arian t unbased maps. It i s w ell kno wn that for an y A ∈ Λ opp Sets, the realiza tion Real( A ) is homeomorphic to t h e usual geometric realization of the simplicial set j ∗ A ∈ ∆ opp Sets, and for an y X ∈ G -T op, the adjun ction m ap Real( S ( X )) → X is a h omotop y equiv alence for every X ∈ G -T op (see [L], or [Dri] for a mo dern treatmen t). I n particular, if for any set A we denote by Z [ A ] the free ab elian group spanned by A , and apply this p oin t wise to S ( X ) : Λ → Sets, then we ha ve (4.6) H q (Λ opp , Z [ S ( X )]) ∼ = H q ( X hG , Z ) , 44 where X hG is the homotop y quotient of X by the G -action. Ho w ever, for our applications the fu nctors Z [ S ( X )] ∈ F un (Λ opp , Z ) are inco v enient b ecause they are not lo cally constant in t h e s ense of D eﬁn i- tion A.1. T o remedy this, w e note that the s ets S ( X )([ n ]) all carry a natural top ology . Deﬁnition 4.1. F or an y X ∈ G -T op, its chain c omplex C q ( X ) is a complex in F un(Λ opp , Z ) giv en by C q ( X )([ n ]) = C q ( S ( X )([ n ]) , Z ) , where C q ( − , Z ) is the n orm alized singular chain homology complex with co eﬃcien ts in Z . Since all the maps R ([ n ]) → R ([ n ′ ]) are homotop y equiv alences, C q ( X ), unlik e Z [ S ( X )], is locally co ns tan t in the sense of Deﬁnition A.1. More explicitly , C q ( X ) is obtained as follo ws. W e consider the functor e R : Λ × ∆ → G -T op giv en by e R ([ n ] × [ m ]) = R ([ n ]) × T m − 1 , where [ m ] ∈ ∆ is th e totally ordered set with m elemen ts. The f unctor e R also extends to a pair of adjoin t functors g Real : F un(Λ opp × ∆ opp , Sets) → G -T op , e S : G -T op → F un(Λ opp × ∆ opp , Sets) . to obtain C q ( S ( X )), we take Z [ e S ( X )] ∈ F un(Λ opp × ∆ opp , Z ), and app ly the normaliz ed c hain complex construction ﬁb erwise with resp ect to the pro jection τ : Λ × ∆ → Λ. T o explain the relation b et ween Z [ S ( X )] and C q ( X ), let ι m : Λ → Λ × ∆ b e the em b edding give n by ι ([ n ]) = [ n ] × [ m ], for any m ≥ 1. Then ι 1 is adjoin t to τ , so that for any A ∈ F un(Λ opp × ∆ opp , Sets), we ha v e a natural adjunction map (4.7) τ ∗ ι ∗ 1 A → A. Apply this to e S ( X ) for some X ∈ G -T op. By deﬁn ition, we ha v e ι ∗ 1 e S ( X ) ∼ = S ( X ); taking free ab elian groups and apssing to normalized c h ain complexes, w e obtain a n atural map (4.8) Z [ S ( X )] → C q ( X ) . 45 Lemma 4.2. Assme given A ∈ F un(Λ opp × ∆ opp , Sets) such that f or any map f : [ m ] → [ n ] i n ∆ , the c orr esp onding map ι ∗ n A → ι ∗ m A induc es a homotopy e qui valenc e of ge ometric r e alizations. Then the map Real( ι ∗ A ) ∼ = g Real( τ ∗ ι ∗ A ) → g Real( A ) induc e d by (4.7) is a homotopy e quivalenc e. Pr o of. L et Real Λ : F u n(Λ opp × ∆ opp , Sets) → ∆ opp -T op b e the ﬁb erwise geometric realiza tion fu nctor with r esp ect to the pro jec- tion Λ × ∆ → ∆. Then the assumption on A means that (4.7) b ecomes a homotop y equiv alence already after applyin g Real Λ .  F or any X ∈ G -T op, we ha ve ι ∗ m e S ( X ) ∼ = S (Maps(∆ m − 1 , X )), s o th at e S ( X ) automatically satisﬁes the conditions of Lemma 4.2. Th u s in p articu- lar, the natural map Real( S ( X )) → g Real( e S ( X )) is a homotop y equiv alence, and the map (4.9) H q (Λ , Z [ S ( X )]) → H q (Λ , C q ( X )) induced by (4.8) is an isomorph ism. More generally , if we d enote by C : D (Λ , Z ) → D c (Λ , Z ) the left-adjoin t fu nctor to the em b edding D c (Λ , Z ) ⊂ D (Λ , Z ), then C ( Z [ S ( X )]) ∼ = C q ( X ). The complexes C q ( X ) are usually rather big. W e will need smaller mo d - els f or the standard G -orbit spaces G/C n , where C n = Z /n Z ⊂ G is the group of n -th ro ots of u nit y . T o obtain s uc h m o dels, consid er the fu nctors j n : ∆ → Λ opp n of Subsection 1.1, with the natural C n -actions on them. Deﬁne functors J n : Λ opp × ∆ opp → Sets by s etting (4.10) J n ([ m ] × [ l ]) = Λ( i n j n ([ l ]) , [ m ]) /C n , where C n acts through its action on j n . Lemma 4.3. F or any n ≥ 1 , we have a natur al homotop y e quivalenc e g Real( J n ) ∼ = G/C n . 46 Pr o of. By Lemma 4.2, it suﬃces to construct equiv alences Real( ι ∗ l J n ) ∼ = G/C n for an y [ l ] ∈ ∆. By deﬁnition, w e hav e ι ∗ l J n ∼ = Y ( j n ([ l ])) /C n = Y ([ nl ]) /C n , where Y ([ nl ]) ⊂ Λ opp Sets is the Y oneda image of [ n l ] ∈ Λ. Sin ce Real is colimit-preserving, we hav e homeomorphisms Real( Y ([ nl ]) /C n ) ∼ = Real( Y ([ nl ])) /C n ∼ = R ([ nl ]) /C n , and R ([ nl ]) is indeed homotop y-equiv alent to G .  W e will no w in tro du ce the ﬁ xed p oin ts subsets in to t h e picture. By Lemma 1.8, the functor R of (4.5) extends to a functor R = Real ◦ Y : Λ Z → G -T op , so that w e obtain an adjoin t pair of fu nctors Real : F un(Λ Z opp , Sets) → G -T op , S : G -T op → F un(Λ Z opp , Sets) . W e k eep the same n otation b ecause the functors are direct extensions of the functors Real and S of Sub s ection 4.2. In eﬀect, the restriction f u nctor Λ opp Sets → F u n(Λ Z opp , Sets) admits a fully faithful left-adjoin t L : Λ opp Sets → F un(Λ Z opp , Sets) , and we hav e S ∼ = L ◦ S , Real ∼ = Real ◦ L . Explicitly , if we restrict L ( A ) to Λ ∼ = Λ Z m h ⊂ Λ Z , we h a v e (4.11) L ( A ) | Λ Z m h = π m ∗ i ∗ m A, where the adjoin t π m ∗ : Λ opp m Sets → Λ opp Sets to the pullb ac k fun ctor π ∗ m is obtained b y taking Z /m Z -ﬁxed p oin ts ﬁb erwise, and the vertica l map s in Λ Z act b y natur al inclusions of ﬁxed p oint s. As in Su bsection 4.2, we extend the fun ctors S and Real to the pro du ct catego ry Λ Z × ∆. F or any X ∈ G -T op, we deﬁn e its e xtende d chain c omplex e C q ( X ) of functors in F un(Λ Z opp , Z ) b y setting e C q ( X ) = C q ( S ( X )) = N ( Z [ e S ( X )]) , 47 where N is the normalized c hain complex fun ctor applied ﬁb erwise to the pro jection Λ Z × ∆ → Λ Z . Moreo ver, deﬁne the r e duc e d chain c omplex C q ( X ) as C q ( X )([ n | m ]) = e C q ( X )([ n | m ]) / Z [ o ] , where o ∈ Maps S 1 ( R ([ m | n ]) , X ) is the d istinguished p oin t. Th en the stan- dard shuﬄe map ind u ces a natur al quasiisomorphism (4.12) C q ( X ) ⊗ C q ( Y ) → C q ( X ∧ Y ) for any X , Y ∈ G -T op , where X ∧ Y is the smash pr o duct, and the tensor pro du ct in the r igh t-hand sid e is the p oin t wise tensor p r o duct in the category F un(Λ Z opp , Z ). The r ed uced c hain complex functor C q ( − ) easily ext end s to the category G -Sp naive of naive G -equiv arian t sp ectra. Namely , for any X ∈ G -Sp naive and in tegers i, j ≥ 0, the tr an s ition maps (4.1) induce b y adju nction maps Σ i X ( R ⊕ j ) → X ( R ⊕ i + j ) , whic h giv e rise to maps (4.13) C q ( X ( R ⊕ i ))[ − i ] ∼ = C q (Σ i X ( R ⊕ j )) → C q ( X ( R ⊕ i + j )) . Deﬁnition 4.4. The e quivariant homolo gy c omplex C naive q ( X ) of a naive G -sp ectrum X ∈ G - S p naive is giv en by C naive q ( X ) = lim i → C q ( X ( R ⊕ i ))[ i ] ∈ D (Λ Z opp , Z ) , where the limit is tak en with resp ect to the maps (4.13). 4.3 Cyclic Mac k ey f unctors. T o extend th e equ iv arian t homolog y com- plex of Deﬁnition 4.4 to gen uine G -sp ectra, w e n eed to p ass to the catego ry D M Λ( Z ) of cyclic Mac k ey f unctors of Section 2. W e will use the mo del of Prop osition 3.12, with an appropr iately c h osen complex P q . W e b egin with the follo w in g observ ation. F or any n ≥ 1, let P n q b e the complex in F un(Λ Z opp , Z ) obtained as the cone of the natural augmen tation map N ( L ( J n )) → Z , where Z ∈ F u n(Λ Z opp , Z ) is the constan t functor, and J n : Λ opp × ∆ opp → Sets is as in (4.10). W e note that for any [ m | l ] ∈ Λ Z , P n q ([ m | l ]) is a ﬁnite- length complex of ﬁn itely generated free ab elian groups. No w, let C (1) b e 48 the s tandard complex represent ation of the group G = U (1), and let C ( n ) = C (1) ⊗ n . T reat C ( n ) as a 2-dimensional real representa tion by restriction of scalars, and denote Σ n = Σ C ( n ) : G -T op → G -T op . Lemma 4.5. F or any X ∈ G -T op , we have a natur al functorial quasiiso- morphism P n q ⊗ C q ( X ) → C q (Σ n X ) . Pr o of. W e hav e S C ( n ) ∼ = Σ( G/C n ), the n on-equiv ariant susp ension of the standard orbit G/C n . The homotop y equiv alence of Lemma 4.3 indu ces b y adjunction a map J n → e S ( G/C n ) , and this map induces a quasiisomorphism P n q → C q ( S C ( n ) ) . Com binin g this with (4.12), w e get the claim.  No w let us ﬁx a complete G -univ erse U , b y taking (4.14) U = M n ≥ 0 C ( n ) ∞ , where ∞ means the sum of a coun table n umb er of copies (this is complete, since C ( − n ) ∼ = C ( n ) as real representati ons). Sa y that a map ν : N → N is adm iss ible if ν ( i ) 6 = 0 for at most a ﬁnite n umb er of i . F or an y admissib le s equence ν , let V ( ν ) = M i C ( i ) ⊕ ν ( i ) ⊂ U, and let Σ ν = Σ V ( ν ) = Σ i 1 ) ν ( i 1 ) ◦ · · · ◦ (Σ i n ) ν ( i n ) , P ν q = O 1 ≤ l ≤ n  P i l q  ⊗ ν ( i l ) , where i 1 ≤ · · · ≤ i n are all in tegers suc h that ν ( i l ) 6 = 0. Then Lemma 4.5 immediately giv es canonical quasiisomorphisms (4.15) P ν q ⊗ C q ( X ) → C q (Σ ν ( X )) . 49 W e will call th e sub spaces V ( ν ) ⊂ U c el lular ; the collectio n of all cellular subspaces is ob viously coﬁn al in the collectio n of all ﬁnite-dimensional G - in v arian t subsp aces V ⊂ U . W e are now ready to deﬁne our complex P q in F u n(Λ I opp r ed , Z ). Note that for any m, l ≥ 1, the natural C m -action on i ∗ m J n ([ l ]) = J n ([ ml ]) fac tors through a free action of the qu otient C m / ( C m ∩ C n ). Th erefore for an y v ertical map v : a → b in Λ Z , the corresp onding map v : P n q ( b ) → P n q ( a ) is an isomorphism if deg ( v ) divides n , and the natural inclusion Z = P n 0 ( b ) → P n q ( a ) otherwise. Moreo ver, for any p rime p not dividing n , the restriction i ∗ p ( P n q ) with resp ect to the fu nctor i p : Λ p → Λ ∼ = Λ Z 1 h ⊂ Λ Z is the constan t functor Z in d egree 0, wh ile i ∗ p P n i ([ m ]) is a free Z /p Z -mo d ule for an y [ m ] ∈ Λ p and an y i ≥ 1. W e now deﬁne a complex P q of fun ctors in F un(Λ I opp r ed , Z ) by setting P q = lim → i ∗ p P ν q on Λ p ⊂ Λ I r ed , where the limit with resp ect to the natural inclusions Z → P n q is tak en ov er all admissible sequen ces ν suc h that ν ( pl ) = 0, l ∈ N . Th en P q is admissible in the sense of Prop osition 3.12, so that we hav e a canonical equiv alence D M Λ( Z ) ∼ = D (Λ Z h , P q , Z ) . Remark 4.6. T o h elp the reader visualize the complex P q , w e note that on Λ p ⊂ Λ I r ed , it is essen tially giv en by i ∗ p lim V → C q ( S V ) , where the limit is tak en o ver all cellular subspaces in U orthogonal to U C p ⊂ U . The only d iﬀerence is in that we use more economical simplicial mo d els for the spheres. By d eﬁnition, P ν q for an y n giv es an ob ject in the categ ory D (Λ Z h , P q , Z ). The corresp ondin g complex is h ∗ P ν q , 50 the restriction with r esp ect to the em b edding Λ Z h → Λ Z , and the map ϕ : π ∗ h ∗ P ν q → i ∗ h ∗ P ν q is induced by the action of the vertica l m aps on P nu q . Explicitly , denote b y ν p the map giv en by ν p ( m ) = ( ν ( m ) , m = pl , l ≥ 1 , 0 , otherwise . Then w e h a v e P ν q = P ν p q ⊗ P ν − ν p q , and the Λ p -comp onen t ϕ p of the map ϕ is the pro du ct of an isomorphism π ∗ p h ∗ P ν p q ∼ = i ∗ p h ∗ P ν p q and the natural inclusion Z ∼ = π ∗ p h ∗ P ν − ν p q → i ∗ p h ∗ P ν − ν p q . If w e treat the complexes P n q up to a qu asiisomorph ism, then on Λ Z m h ⊂ Λ Z h , the restriction h ∗ P n q is giv en by ( Z [2] , n = ml, l ≥ 1 , Z , otherwise . Let us no w ﬁx a pro jectiv e resolution Q q of the constant functor Z ∈ F un(Λ Z opp h , Z ), and let us deﬁne a complex Q n q b y setting Q n q = ( Z [ − 2] , n = ml, l ≥ 1 , Z , otherwise on Λ Z m h ⊂ Λ Z h . T hen Q n q ⊗ h ∗ P n q is quasiisomorph ic to the constant fu nctor Z , and sin ce Q q is pro jectiv e, this can b e realized b y an actual quasiisomor- phism ε n : Q q → Q n q ⊗ h ∗ P n q . Moreo ver, for any ad m issible ν : N → N and any m ≥ 1, let d ( m, ν ) = X l ≥ 1 ν ( ml ) , and let Q ν q b e a complex in F u n(Λ Z opp h , Z ) giv en by Q q [ − 2 d ( m, ν )] on Λ Z m h ⊂ Λ Z h . T hen taking tensor pro duct of the maps ε n , we obtain a system of quasiisomorphisms (4.16) ε ν : Z → Q ν q ⊗ h ∗ P ν q 51 for all admissible ν . F or an y prime p , we hav e a natural map ε ν − ν p : π ∗ p Q ν q → i ∗ p ( Q ν q ⊗ P ν − ν p q ) , and comp osing these maps with the n atural em b eddings i ∗ p P ν − ν p q → P q , we equip Q ν q with a n atural structure of an ob ject in D (Λ Z h , P q , Z ). W e are now ready to deﬁne our equiv arian t h omology functor. Assume giv en a pr esp ectrum t ∈ G -sp( U ). F or any t wo admissible ν, ν ′ : N → N , we ha ve a transition map Σ ν ′ t ( V ( ν )) → t ( V ( ν + ν ′ )) adjoin t to the map (4.1). By (4.15) , these maps in d uce canonical maps P ν ′ q ⊗ C q ( t ( V ( ν ))) → C ( V ( ν + ν ′ )) . Let no w ξ ∗ : D (Λ Z opp , Z ) → D (Λ Z , λ ∗ T q , Z ) ∼ = D M Λ( Z ) b e the corestriction functor with r esp ect to the augmenta tion map of the A ∞ -coalg ebr a λ ∗ T q . Then these maps comp osed with the maps (4.16) ind u ce tr ansition maps (4.17) ξ ∗ C q ( t ( V ( ν )) → Q ν q ⊗ ξ ∗ P ν q ⊗ ξ ∗ C q ( t ( V ( ν ))) → → Q ν q ⊗ ξ ∗ C q ( t ( V ( ν + ν ′ ))) Deﬁnition 4.7. The e quivariant chain c omplex C q ( X ) ∈ D M Λ( Z ) of a G - presp ectrum t ∈ G -sp( U ) is giv en b y (4.18) C q ( t ) = lim ν → Q ν q ⊗ ξ ∗ C q ( X ( V ( ν ))) , where the limit is tak en o ver all the admissible maps ν : N → N with resp ect to the transition maps (4.17). Lemma 4.8. F or any t ∈ G -sp( U ) with sp e ctriﬁc ation L t , the adjunction map t → Lt i nduc es an isomorphism C q ( t ) → C q ( Lt ) . Pr o of. Since cellular subsp aces are coﬁnal, in it suﬃces to tak e th e limit o v er cellular su bspaces in (4.2). Then sub stituting (4.2) into (4.18), w e see that C q ( Lt ) is giv en by lim ν ≤ ν ′ → Q ν q ⊗ ξ ∗ C q (Ω V ( ν ′ − ν ) t ( V ( ν ′ ))) , where the limit is tak en o v er all pairs ν ≤ ν ′ of admissible maps ν, ν ′ : N → N . The su bset of all p airs with ν = ν ′ is coﬁnal in this set, s o it suﬃces to tak e the limit ov er such p airs. This is exactly C q ( t ).  52 Corollary 4.9. F or any X ∈ G -T op , we have C q (Σ ∞ X ) ∼ = ξ ∗ C ( X ) , and for any X ∈ G -Sp naive , we have (4.19) C q ( ρ # ( u ) X ) ∼ = C q ( X ) , wher e ρ # ( u ) is the change-of-universe f u nctor asso ciate d to the emb e dding R ⊕∞ = U G ⊂ U . Pr o of. By Lemma 4.8, we can replace Σ ∞ with e Σ ∞ and ρ # ( u ) with e ρ # ( u ). Then for t = e Σ ∞ X , all the transition maps in the ﬁ ltered limit of (4.18) are quasiisomorphism s, and for t = e ρ # ( u ) X , the only p ossibly n on -trivial transition maps are the those of (4.13).  4.4 Cyclotomic co mplexes. Now for any m ≥ 1, let C m = Z /m Z ⊂ G = U (1) b e the group of th e m -th ro ots of un ity . T he m -p o wer map giv es an isomorph ism p m : G/C m → G , and w e ha ve p m ◦ p n = p nm , m, n ≥ 1. W e h a ve obvio u s canonical G -equiv ariant isomorphisms u m : U C m ∼ = U, where U is the complete G -universe (4.14), and w e h av e u m ◦ u n = u nm , n, m ≥ 1. Deﬁnition 4.10. F or any X ∈ G -Sp = G -Sp ( U ) a n d an y m ≥ 1, the extende d ge ometric ﬁxe d p oints sp e ctrum b Φ m ( X ) ∈ G -Sp is giv en by b Φ m ( X ) = ρ # ( u m )Φ C m ( X ) , and the extende d L ewis-May ﬁxe d p oints sp e ctrum b Ψ m ( X ) ∈ G -Sp is giv en b y b Ψ m ( X ) = ρ # ( u m ) X C m . F or any m ≥ 1, let ρ m : I → I b e th e m ultiplication my m , as in (2.18). It comm utes w ith the N ∗ -action, hence induces an endofun ctor e ρ m : Λ Z → Λ Z comm uting with the pro jection λ : Λ Z → Λ R . By (4.11 ) , w e hav e e ρ ∗ m LA ∼ = LA C m 53 for an y A ∈ Λ opp Sets. I n p articular, for ev ery X ∈ G -T op we hav e (4.20) e ρ ∗ m C q ( X ) ∼ = C q ( X C m ) , and b y (4.13) , these quasiisomorphisms indu ce quasiisomorphisms (4.21) e ρ ∗ m C q ( X ) ∼ = C q ( X C m ) for ev ery naiv e G -sp ectrum X ∈ G -Sp naive . W e wa nt to obtain a v ersion of this for gen u ine G -sp ectra. Since λ ◦ e ρ m = λ , we tautologically ha ve e ρ ∗ λ ∗ T q ∼ = λ ∗ T q , so that w e hav e a p ullbac k fu nctor e ρ ∗ m : D M Λ( Z ) → D M Λ( Z ) . Lemma 4.11. F or any m ≥ 1 and any X ∈ G -Sp , we have a natur al functorial isomorphism C q ( b Φ m ( X )) ∼ = e ρ ∗ m C q ( X ) . Pr o of. F or an y map ν : Z → Z , let r ( ν ) : Z → Z b e giv en b y r ( ν )( ma + b ) = ν ( a ), a ∈ Z , 0 ≤ b < n . If ν is admissible, then so is r ( ν ). By Lemma 4.8, w e m a y replace b Φ m with the fu nctor ϕ m = e ρ # ( u m ) ◦ ϕ C m . Cho ose the subspaces W ( V ) ⊂ U so that W ( V ( ν )) = V ( r ( ν )) for an y admissible ν : Z → Z . Then C q ( ϕ m X ) is give n by (4.22) lim ν → Q ν q ⊗ ξ ∗ C q ( X ( V ( r ν )) C m ) , and since the sequences r ( ν ) are coﬁn al in the set of all admissible sequences, e ρ ∗ m C q ( X ) is giv en b y (4.23) lim ν → e ρ ∗ m Q r ( ν ) q ξ ∗ C q ( X ( V ( r ν ))) . By deﬁnition, the pullbac k fu nctor commutes with corestriction, e ρ ∗ m ◦ ξ ∗ ∼ = ξ ∗ ◦ e ρ ∗ m , and w e hav e the isomorp hisms (4.20). It r emains to notice th at b y deﬁnition, we ha ve e ρ ∗ m Q r ( ν ) q ∼ = Q ν q for an y admissib le ν .  W e n o w consider the righ t-adjoint ξ ∗ : D M Λ( Z ) → D (Λ Z opp , Z ) to the corestriction ξ ∗ , as in Su bsection 2.4. The isomorphism (4.19) than induces a base c hange map (4.24) C q ( ρ # ( u )( X )) → ξ ∗ C q ( X ) 54 for an y X ∈ G -Sp. By virtue of (4.21) and (4.4), we ha ve a natural isomor- phism e ρ ∗ m C q ( ρ # ( u )( X )) ∼ = C q ( b Ψ m ( X )) for an y genuine G -sp ectrum X ∈ G -Sp and an y m ≥ 1. Lemma 4.12. The b ase change map (4.24) is an isomorphism for any X ∈ G -Sp , so that C q ( b Ψ m ( X )) ∼ = e ρ ∗ m ξ ∗ C q ( X ) for any m ≥ 1 . Pr o of. It s uﬃces to prov e that the map (4.24) b ecomes an isomorphism after ev aluating at an y ob j ect [ n | m ] ∈ Λ Z . By (4.4 ) and the d eﬁ nition of the equiv arian t homology complex C q ( − ) of a naiv e G -sp ectrum, w e hav e a natural quasiisomorphism C q ( ρ # ( u )( X ))([ n | m ]) ∼ = C q ( X C m ) , where X C m is treate d as a non-equiv arian t sp ectrum, and C q ( − ) in the righ t-hand side is obtained by applying the limit of Deﬁnition 4.4 to the usual non-equiv arian t reduced singular c hain complex fu nctor C q ( − , Z ) ( n is irrelev an t sin ce the c hain complexes of Deﬁnition 4.1 are lo cally constant ). Th u s w e ha ve to pro ve that for any n , m ≥ 1, th e natural map C q ( X C m ) → ( ξ ∗ C q ( X ))([ n | m ]) induced by (4.24) is an isomorphism. On the other h and, b y the deﬁnition of the fu nctor C q : G -Sp → D M Λ( Z ), we hav e an isomorphism Σ V ◦ C q ∼ = C q ◦ Σ V for an y V = V ( ν ) ⊂ U , and the base change map C q ◦ (Σ V ) − 1 → (Σ V ) − 1 ◦ C q is also an isomorphism. Therefore (4.24) is an isomorph ism f or some X ∈ G -Sp if and only if it is an isomorp hism for Σ V X . Since ev ery X ∈ G -Sp is a ﬁltered colimit of sp ectra of the form (Σ V ) − 1 Σ ∞ Y , Y ∈ G -T op , it suﬃces to pr o v e that (4.24) is an isomorp hism for su sp ension sp ectra Σ ∞ Y . By Corollary 4.9, w e h a ve a quasiisomorph ism C q (Σ ∞ Y ) ∼ = ξ ∗ C q ( Y ); 55 th us what we ha ve to p ro ve is that for an y Y ∈ G - T op and an y [ n | m ] ∈ Λ Z , the natural map C q ((Σ ∞ Y ) C m ) → ( ξ ∗ ξ ∗ C q ( Y ))([ n | m ]) induced by (4.24) is a quasiisomorphism . The right- h and side can b e com- puted by (2.17) and Lemma 2.14; it is giv en b y M l,p ≥ 1 , lp = m C q ( Z /l Z , C q ( Y )([ nl | p ]) , where b y deﬁn ition, w e h a v e C q ( Y )([ nl | p ]) ∼ = C q ( Y C p , Z ) , the redu ced c hain complex of the ﬁ xed-p oints set Y C p . The d esired isomor- phism then b ecomes the tom Diec k-Segal splitting [HM, Section 1].  W e no w return to top ology , and recall the follo wing fun damen tal notion (w e again follo w [HM], and r efer to that pap er for further information and references). Deﬁnition 4.13. A cyclotomic structur e on a gen uine G -sp ectrum T ∈ G -Sp is giv en by a collection of homotop y equiv alences r m : b Φ m T ∼ = T , one for eac h in teger m ≥ 1, su c h that r 1 = id and r n ◦ r m = r nm for an y t w o in teger n , m > 1. Example 4.14. Assume given a p ointed CW complex X , and let LX = Maps( S 1 , X ) b e its fr ee lo op space. Then for any ﬁnite subgroup C ⊂ S 1 , the isomorphism S 1 ∼ = S 1 /C in duces a homeomorphism Maps( S 1 , X ) C = Maps( S 1 /C, X ) ∼ = Maps( S 1 , X ) , and these homeomorph ism pro vide a canonical cyclotomic stru cture on the susp ension sp ectrum Σ ∞ LX . Then by Lemma 4.11, a cyclotomic str u cture { r m } on T induces a col- lection of quasiisomorphisms r m : e ρ ∗ m C q ( T ) → C q ( T ) , m ≥ 1 56 suc h that r m ◦ r n = r nm . If we treat the equ iv ariant c hain complex C q ( T ) as an ob ject h h ∗ C q ( T ) , ϕ i in the category D (Λ Z h , P q , Z ), th en the maps r m extend h ∗ C q ( T ) to a complex in the category F un( g Λ R opp , Z ), an d this extension is compatible with the map ϕ . Therefore C q ( T ) canonically deﬁnes an ob ject e C q ( T ) in the d eriv ed ca tegory D ( g Λ R h , P q , Z ). By Prop ositon 3.1 2 , this tur ns C q ( T ) into a cyclot omic complex. Thus w e can ﬁnally j u stify our terminology by int r o ducting the follo wing d eﬁnition. Deﬁnition 4.15. The cyclotomic complex e C q ( T ) ∈ D ΛR( Z ) is called the e quiv ariant chain c omplex of th e cyclotomic sp ectrum T . 5 Fil tered Dieudonn´ e mo dules. 5.1 Deﬁnitions. W e n o w wan t to compare cyclotomic complexes to a diﬀeren t and m u ch simp ler alge b raic notion whic h app eared ea rlier in a diﬀeren t context – the notion of a ﬁltered Dieudonn ´ e mo dule. Deﬁnition 5.1. Let k b e a ﬁnite ﬁeld of charact eristic p , with it F rob enius map, and let W b e its ring of Witt v ectors, with its canonical lifting of the F rob enius map. A ﬁlter e d Dieudonn´ e mo dule ov er W is a ﬁnitely generated W -mo dule M equipp ed with a decreasing ﬁltration F q M , ind exed by all in tegers, and a collect ion of F rob eniu s-semilinear maps ϕ i : F i M → M , one for eac h intege r i , su c h that (i) ϕ i | F i +1 M = pϕ i +1 , and (ii) the map X ϕ i : M i F i M → M is surjectiv e. This deﬁnition was in tro d uced b y F on taine and Lafaille [FL] as a p -adic analog of the n otion of a Ho dge stru cture. Under certain assu m ptions, the de Rham cohomology H q D R ( X ) of a smo oth compact algebraic v ariet y X/W has a natural ﬁltered Dieudonn ´ e mo dule structure, with F q b eing the Ho d ge ﬁltration and the maps ϕ q induced by the F rob enius end omorphism of the sp ecial ﬁb er X k = X ⊗ W k . The category of ﬁltered Dieudonn´ e mo d ules is obviously additive, but there is more: j u st as f or mixed Ho dge stru ctures, a small m iracle happ ens, and the category is actually ab elian. F or this, the n ormalization condition 57 (ii) of Deﬁn ition 5.1 plays the crucial role. If one is pr ep ared to w ork with non-ab elian additiv e categ ories, this cond ition ca n b e dropp ed . F or the purp oses of present p ap er, the follo w ing notion will b e conv enient. Deﬁnition 5.2. A gener alize d ﬁlter e d Dieudonn´ e mo dule (gFDM for sh ort) is an ab elian group M equipp ed with (i) a decreasing ﬁltration F q M , indexed b y all int egers and suc h that M = S F i M , and (ii) for eac h in teger i , eac h p ositiv e in teger j ≥ 1, and eac h prime p , a map ϕ p i,j : F i M → M /p j M , suc h that ϕ p i,j +1 = ϕ p i,j mo d p j , and ϕ p i,j = pϕ p i +1 ,j on F i +1 M ⊂ F i M . This diﬀers from Deﬁnition 5.1 in that w e n o longer requir e the normal- ization condition (ii) , we restrict our atten tion to p rime ﬁelds r ather than ﬁnite ﬁelds, and w e co llect to gether the structures for all primes. Note, ho wev er, that if M is ﬁn itely generated ov er Z p , then all other p rimes act on M b y in vertible maps, and th e extra maps ϕ l q , q for l 6 = p are all 0. In eﬀect, for every prime p and ev ery in teger i , w e can collect all the maps ϕ p i, q in to a s ingle map ϕ p i : F i M → d ( M ) p , where d ( M ) p means the pro- p completion of the ab elian group M . If M is ﬁnitely generated o ve r Z p , w e h a v e d ( M ) p ∼ = M and d ( M ) l = 0 for l 6 = p . Complexes of gFDMs are deﬁned in the ob vious wa y . A map b et w een suc h complexes is a qu asiisomorphism if it ind uces a quasiisomorph ism of the asso ciated graded qu otien ts gr F . I n verting suc h quasiisomorphisms, w e obtain a triangulated “derived ca tegory of gFDMs” whic h w e will denote by F D M . F or every gFDM M and any in teger i , w e will denote by M ( i ) the same M with the ﬁltration F q t wisted by i – that is, w e set F j M ( i ) = F j − i M . Under this conv entio n , we in tro duce th e folo wing “t wisted 2-p erio d ic” v ersion of the category F D M . Deﬁnition 5.3. The triangulated category F D M per is obtained by inv ert- ing quasiisomorphisms in the catego r y of complexes of gFDMs M q equipp ed with an isomorphism M ∼ = M [2](1). W e can no w formulate the m ain resu lt of this Section. 58 Theorem 5.4. Ther e is a natur al e quivalenc e F D M per ∼ = D ΛR( Z ) b etwe en the twiste d 2 -p erio dic derive d c ate gory of gFDMs, on one hand, and the derive d c ate gory of cyclotomic c omplexes of ab elian gr oups in the sense of Deﬁnition 3.1, on the other hand. W e will pro ve this in Sub section 5.5, in the more precise form of Prop o- sition 5.17, after ﬁ nishing the necessary pr eliminaries. W e start with some generalities on ﬁltered ob jects. 5.2 Filtered ob je cts. By a ﬁlter e d obje ct in an ab elian category Ab w e will un d erstand an ob ject E ∈ Ab equipp ed with a decreasing ﬁltrati on F q n umb ered b y all intege res. Maps and complexes of ﬁltered ob jects are deﬁned in the ob vious w ay . Deﬁnition 5.5. A map f : E q → E ′ q b et ween t wo ﬁletred complexes of ob jects in Ab is a ﬁlter e d quasiisomorphism if the induced map f : F i E q /F i +1 E q → F i E ′ q /F i +1 E ′ q is a quasiisomorphism for ev ery intege r i . Remark 5.6. W e d o not require that a ﬁltered quuasiisomorph ism induces a quasiisomorphism of the u nderlying complexes E q , E ′ q of ob jects in Ab, nor of the induvidu al pieces F i E q , F i E ′ q of the ﬁltrations. The ﬁlter e d derive d c ate g ory D F (Ab) is obtained by inv erting ﬁ ltered quasiisomorphisms in the category of ﬁ ltered complexes and ﬁltered maps. The p erio dic ﬁlter e d derive d c ate gory D F per (Ab) is similarly obtained fr om the category of complexes o f ﬁlte r ed ob jects V q in Ab equipp ed w ith a n isomorphism u : V q ∼ = V q − 2 (1) , where V (1) means a t wist of ﬁltration. E xplicitly , such a complex is giv en b y the t w o ﬁltered ob jects V 0 , V 1 , and the t wo ﬁltered m aps d 1 : V 1 → V 0 , d 0 : V 0 → V 1 (1) ∼ = V − 1 suc h that d 1 ◦ d 0 = 0 = d 0 ◦ d 1 ; th e other terms in the complex are then giv en b y V 2 q = V 0 , V 2 q +1 = V 1 with the same diﬀeren tials d 0 , d 1 . 59 If Ab is the category of ab elian groups, we obtain th e p erio dic ﬁltered deriv ed catego r y D F per ( Z ) of ﬁltered ab elian groups. F or any integ er n , let Z ( n ) ∈ D F per b e the ob j ect V giv en by V 0 = Z , v 1 = 0, with the ﬁ ltration F n V 0 = V 0 and F n +1 V 0 = 0. Then the ob jects Z ( n ), n ∈ Z generate the catego ry D F per ( Z ) in the follo w ing sens e. Lemma 5.7. Any triangulate d sub c ate gory D ′ ⊂ D F per ( Z ) close d under arbitr ary sums and pr o ducts and c ontaining Z with the trivial ﬁltr ation F 0 Z = Z , F 1 Z = 0 is e qual to the whole D F per ( Z ) . Pr o of. Since D ′ is closed u nder ta kin g cones, it ob viously con tains any ﬁltered complex h V q , F q i w ith b oun ded ﬁltration F q (that is, F i V q = 0, F j V q = V q for some inte gers i , j ). Assume giv en an abritrary ﬁltered com- plex h V q , F q i . Th en th e natural maps (5.1) V q ← − − − − lim i → F − i V q − − − − → lim i → lim j ← F − i V q /F j V q induce isomorphisms on gr F , th us b ecome isomorp hisms in D F per ( Z ). T h u s w e may replace V q with the d ouble limit in the righ t-hand s id e of (5.1). The direct limit can b e compu ted by the telescop e construction. Moreo v er, for an y i , the in verse system F i V q /F q V q satisﬁes the Mittag-Leﬄer condition, so that the inv er s e limit ca n also b e computed by the telesco p e constru ction. Since D ′ is closed under pro d ucts and sums, it is also closed under telescopes, so that it m u st con tain V q .  W e note that by (5.1), we ma y represent an y ob ject in D F per ( Z ) b y a ﬁltered complex h V q , F q i which is admissible in the follo wing sense: b oth natural maps lim i → F − i V q → V q , V q → lim i ← V q /F i V q are isomorphisms of complexes. T o w ork with ﬁ ltered ab elian groups, it is con venien t to use Rees ob jects. Consider the algebra Z [ t ] of p olynomials in one v ariable t . Say that a mo d u le M ov er Z [ t ] is t -adic al ly c omplete if the natural m ap M → lim i ← M /t i M is an isomorphism (th u s in our terminology , “complete” in cludes “sepa- rated”). W e tu r n Z [ t ] int o a g r ad ed ring by assigning degree − 1 to th e generator t . 60 Lemma 5.8. The ﬁlter e d derive d c ate gory D F ( Z ) is e quivalent to the ful l sub c ate gory in the derive d c ate gory of the ab elian c ate g ory of Z -gr ade d Z [ t ] - mo dules M q sp anne d by t -adic al ly c omplete mo dules. Pr o of. F or an y ﬁltered ab elian group h M , F q i , the corresp onding graded Z [ t ]-mo dule f M q called the R e es obje ct of M q is giv en by f M q = M lim i → F q M /F q + i M , with t induced by the natur al em b eddings F q M → F q − 1 M . W e n ote that f M q is automatically t -adically complete, and a ﬁltered quasiisomorphism of complexes of ab elian groups ind uces a qu asiisomorphism of Rees ob jects. T o get the inv erse corresp ond ence, note that eve r y graded Z [ t ]-mo d ule M q has a ﬁnite resolution by mod ules with no t -torsion, so that it is enough to consider graded mo d ules M q with injectiv e map t . Suc h a mo du le M q is sen t to f M = lim t → M q , with F i f M ⊂ f M b eing the image of the natural em b edd ing M i → f M for an y in teger i .  The equiv alence of Lemma 5.8 has an ob vious p erio dic v ersion: the Rees ob ject mo del of the categ ory D F per ( Z ) is obtained by inv erting quasiiso- morphisms in the category of pairs h f M q , q , u i of a complex f M q , q of graded t -adically complete Z [ t ]-mo du les M q and an isomorphism (5.2) e u : f M q , q ∼ = f M q +1 , q − 2 . 5.3 Cyclic expansion and sub division. No w let Ab = F u n(Λ opp , Z ) be the category of cyclic ab elian groups. Let I 0 = j o ! Z , I 1 = j ∗ Z b e as in (1.23), and let d 1 = B : I 1 → I 0 , d 0 = b 0 ◦ b 1 : I 0 → I 1 , where b 0 , b 1 and B are again as in (1.23) . Th en since (1.23) is exact, we ha ve d 1 ◦ d 0 = 0 = d 0 ◦ d 1 . Moreo ver, if w e deﬁne ﬁltrations F q on I 0 and I 1 b y setting F 0 I l = I l , F 1 I l = 0, l = 0 , 1, then b oth d 0 and d 1 are ﬁltered maps, so that w e hav e a p erio dic ﬁltered complex I q of ob jects in F un (Λ opp , Z ). Deﬁnition 5.9. F or any ob ject in D F per ( Z ) represent ed by a p erio d ic com- plex V q of admissible ﬁltered ab elian groups, its cyclic exp ansion Exp ( V q ) is a complex of cyclic ab elian groups giv en by Exp ( V q ) = F 0 ( V q ⊗ Z [ u ] I q ) , 61 where u is th e p erio d icity map on V q and I q , and F 0 is tak en with resp ect to the pro d u ct ﬁltration. In terms of the corresp ondin g p erido c complex e V q , q of Rees ob jects, cyclic expansion is giv en by (5.3) Exp ( V q ) i = ( e V 0 , q ⊗ I 1 )[1] ⊕ ( e V 0 , q ⊗ I 0 ) , with the diﬀerentia l d = d V ⊗ id + d I , where d V is the diﬀerent ial on e V 0 , q , and d I is equal to id ⊗ d 1 on the ﬁ r st sum mand, and to t e u ⊗ d 0 on the second one, where e u : e V q , q ∼ = e V q +1 , q − 2 is as in (5.2). In p articular, for every ob ject [ n ] ∈ Λ, Exp ( V q )([ n ]) is the sum of a ﬁnite num b er of copies of e V 0 , q and its shift e V 0 , q [1]; therefore cyclic expans ion commute s with arbitrary sums and arbitrary pro du cts. Lemma 5.10. Cyclic exp ansion induc es an e quivalenc e of c ate gories Exp : D F per ( Z ) ∼ = D c (Λ opp , Z ) b etwe en D F per ( Z ) and the fu l l sub c ate gory D c (Λ opp , Z ) ⊂ D (Λ opp , Z ) sp anne d by obje cts which ar e lo c al ly c onstant in the sense of D eﬁnition A.1. Pr o of. Since (1.23) is exact, F 1 I q = I < 0 is a resolution of the constan t fun c- tor Z ∈ F un (Λ , Z ). By ind uction, th is immediately implies that for any p eri- o dic ﬁltered complex V q , the homology functors H q ( Exp ( V q )) ∈ F un(Λ opp , Z ) are giv en by H i ( Exp ( V q )) ∼ = ( F 0 V i /F 1 V i ) ⊗ Z . In particular, a ﬁ ltered quasiisomorph ism of p erio dic ﬁltered complexes in- duces a quasiisomorphism of their cyclic expansions, so th at Exp ind uces a well-deﬁned triangulated f unctor f rom the category D F per ( Z ) to th e full sub category D c (Λ opp , Z ) ⊂ D (Λ opp , Z ). Moreov er, for any in teger n , we h a v e Exp ( Z ( n )) ∼ = Z [2 n ] . Since the fund amen tal group π 1 ( | Λ | ) ∼ = π 1 ( B U (1)) is trivial, ev ery lo cally constan t functor Λ opp → Z -mo d must b e constan t; therefore Z generates the triangulated category D c (Λ opp , Z ) in the same s ense as in L emma 5.7. Since Exp comm utes with arb itrary sum s and arbitrary pro du cts, it therefore suﬃces to p ro ve that Exp is fu lly faithful on the ob jects Z ( n ) – that is, th e natural map Exp : RHom q D F per ( Z ) ( Z ( n ) , Z ( m )) → RHom q D (Λ opp , Z ) ( Z [2 n ] , Z [2 m ]) 62 is a quasiisomorphism for any t wo in tegers n , m . This immediately f ollo w s from the isomorphism H q (Λ opp , Z ) ∼ = Z [ u ].  No w assume giv en a p ositiv e intege r n ≥ 1, and let δ n : Z [ t ] → Z [ t ] b e the map whic h send s t to n t . Sin ce the the ideal ( n t ) ⊂ Z [ t ] lies ins id e ( t ) ⊂ Z [ t ], the d irect im age δ n ∗ M of a t -adically complete Z [ t ]-mo dule M is automatica lly t -adically complete. Deﬁnition 5.11. F or any p ositiv e inte ger n ≥ 1 and a ﬁltered ab elian group M with the corresp onding adm iss ible Rees ob ject M q as in Lemma 5.8, the n -th sub division Div n ( M ) is the ﬁltered ab elian group corr esp onding to the graded ab elian group δ n ∗ M q . In other w ords, M q remains the same as a graded ab elian group, but the map t is rep laced by its multiple nt . W e n ote that the und erlying ﬁltered ab elian group M itself migh t c h ange under sub d ivision: for example, if M = Z (0) = Z , then Div n ( M ) ∼ = Q , with the ﬁltration giv en by F 1 Q = 0, F i Q = n i Z ⊂ Q for i ≤ 0. F or any n , Div n is ob viously an endofunctor of the category of ﬁltered ab elian groups, and it descends to an endofu nctor Div n : D F per ( Z ) → D F per ( Z ) of the p erio dic derived category D F per ( Z ). Fix an in teger n ≥ 1, and recall the tw o functors i n , π n : Λ n → Λ of Subsection 1.1. Lemma 5.12. F or any p erio dic c omplex V q of adm issible ﬁlter e d ab elian gr oups, we have a f unctorial isomorphism π n ∗ i ∗ n Exp ( V q ) ∼ = Exp ( Div n ( V q )) . Pr o of. Let I ′ l = π n ∗ i ∗ n I l , d ′ l = π n ∗ i ∗ n d l , l = 0 , 1. Applying the f u nctor π n ∗ i ∗ n to (5.3), w e s ee that the complex π n ∗ i ∗ n Exp ( V q ) is giv en by π n ∗ i ∗ n Exp ( V q ) i = ( e V 0 , q ⊗ I 1 )[1] ⊕ ( e V 0 , q ⊗ I 0 ) , with the diﬀerent ial d = d V ⊗ id + d I , w here d V is the diﬀeren tial on e V 0 , q , and d I is equal to id ⊗ d ′ 1 on the ﬁrst su mmand, and to t e u ⊗ d ′ 0 on the second one. By (1.25), we ha v e canonica l isomorphisms I ′ 0 ∼ = I 0 , I ′ 1 ∼ = I 1 , and u nder these isomorphisms, we ha v e d ′ 1 = d 1 and d ′ 0 = nd 0 . Thus t e u ⊗ d ′ 0 = nt e u ⊗ d 0 , and π n ∗ i ∗ n Exp ( V q ) is exactly isomorph ic to the expression (5.3) for th e complex Exp ( Div n ( V q )).  63 5.4 Stabilization. F or an y ﬁltered ab elian group M w ith the corresp ond- ing ad m issible Rees ob j ect M q , we hav e a tautolo gical map M → M (1); applying the sub division fu nctor Div n , w e obtain a n atural map (5.4) Div n ( M ) → Div n ( M (1)) . Deﬁnition 5.13. In the situation of Deﬁnition 5.1 1 , the stabilize d n - th sub- division Stab n ( M ) is giv en by Stab n ( M ) = lim l → Div n ( M ( l )) , where the limit is tak en with resp ect to the tautologica l maps (5.4). Lemma 5.14. F or any ﬁlter e d ab elian gr oup M which is admissible in the sense of Subse ction 5.2 and any prime p ≥ 2 , we have Stab p ( M ) ∼ = c M p ⊗ Z p Q p , with the ﬁltr ation given by F l c M p = p l c M p , wher e c M p = lim l ← M /p l M denotes the pr o- p c ompletion of the gr oup M . Pr o of. O n the lev el of Rees ob j ects, M q = lim → Div p ( M ) q is isomorphic to M in every d egree l , M l ∼ = M , with t ∈ Z [ t ] acting by m ultiplication b y p . Ho we ver, this is not t -adically complete. Th u s wh en w e app ly the equ iv alence of Lemma 5.8, the result is lim l ← M /t l M q ∼ = c M p , as required.  As a corollary , w e see that the p eriod ic derived category F D M per of generalized ﬁ ltered Dieudonn ´ e mo du les of Subsection 5.1 is equiv alen t to the catego r y of ﬁltered p eriod ic complexes V q of ab elian group s equip p ed with a map ϕ p : V q → Sta b p ( V q ) for an y pr ime p ≥ 2. 64 No w let I q b e the p erio dic complex of functors in F un(Λ , Z ) of Subsec- tion 5.3, and let P q ⊂ I q b e its canonical trun cation at 0. Explicitly , we hav e P 0 = Z , the constant functor, and P 2 l − 1 = I 1 , P 2 l = I 0 for an y l ≥ 1. The complex P q is acyclic . Moreo v er, th e pullbac k i ∗ P q with resp ect to the fun ctor i : Λ I r ed → Λ of Su b section 3.3 is admissible in the sense of Lemma 3.10. Let η + : Z → F 0 I q , η − : Z ∼ = P 0 → P q b e the n atural em b eddings, and deﬁne a map η : I q → P q ⊗ ( F 0 I q ) b y (5.5) η = ( η − ⊗ i d , on I l , l ≤ 0 , id ⊗ η + , on I l , l ≥ 0 . Let F l P q , l ≥ 0 b e the stu p id ﬁltration on the complex P q . Then for any l ≥ 0, the map η induces a map (5.6) η : F − l I q → F 0 I q ⊗ F 2 l P q , and this map is a qu asiisomorphism (b oth s id es are qu asiisomorphic to Z [2 l ]). Lemma 5.15. F or any n ≥ 1 , l ≥ 0 , and any c omplex V ∈ F un (Λ opp n , Z ) , the natur al map (5.7) η : π n ∗ ( V ⊗ i ∗ n F − l I q ) → π n ∗ ( V ⊗ i ∗ n F 0 I q ⊗ i ∗ n F 2 l P q ) is a quasiisomorphism. Pr o of. Both sides are ﬁ nite-length complexes in F un(Λ opp n , Z ), and after ev al- uating at an ob ject [ m ] ∈ Λ n , b oth sid es giv e complexes of free Z [ Z /n Z ]- mo dules. Therefore w e m a y r eplace π n ∗ with its deriv ed functor R q π n ∗ . Then the claim immediately follo ws from the fact that (5.6) is a quasiiso- morphism.  Corollary 5.16. F or any inte ger n ≥ 1 and any twiste d p erio dic c omplex V q of ﬁlter e d ab elian gr oups, the map η of (5.5) induc es a quasiisomorphism Exp ( Stab n ( V q )) ∼ − − − − → lim l → π n ∗ i ∗ n ( Exp ( V q ) ⊗ F l P q ) . 65 Pr o of. By Lemma 5.12 and Deﬁnition 5.13, w e h av e Exp ( Stab n ( V q )) ∼ = lim l → π n ∗ i ∗ n Exp ( V q ( l )) , and b y deﬁn ition, w e ha ve Exp ( V q ( l )) = F 0 ( V q ⊗ F − l I q ) for an y integ er l ≥ 0.  5.5 Comparison. Consider no w the category D (Λ R h , i ∗ P q , k ) of Subsec- tion 3.3, with the sp eciﬁc c h oice of the complex P q made in Su bsection 5.4. Let τ : Λ R h ∼ = Λ × I → Λ b e the tautol ogical p ro jection. Then for an y t wisted per io dic complex V q of ab elian group s, a c ollection of m aps ϕ p : V q → Sta b p ( V q ) for all prime p indu ces a map η ◦ τ ∗ Exp ( ϕ ) : τ ∗ Exp ( V q ) → π ∗ i ∗ τ ∗ ( Exp ( V q ) ⊗ P q ) , whic h giv es by adju n ction a map e ϕ : π ∗ τ ∗ Exp ( V q ) → i ∗ ( τ ∗ Exp ( V q ) ⊗ i ∗ P q ) . Sending h V q , { ϕ p }i to h τ ∗ Exp ( V q ) , e ϕ i then deﬁnes a comparison fun ctor (5.8) Exp : F D M per → D c (Λ R h , P q , Z ) . By Prop osition 3.12, the follo wing result immediately yields Theorem 5.4. Prop osition 5.17. The functor Exp of (5. 8 ) i s an e quivalenc e of c ate gories. T o pro ve this result, we n eed to generalize it. Consider the t wisted p erio dic ﬁltered derive d category D F per ( I opp , Z ) of fun ctors f rom I opp to ab elian groups. The cyclic expansion functor giv es a functor (5.9) Exp : D F per ( I opp , Z ) → D w (Λ R opp h , Z ) , where D w (Λ R opp h , Z ) ⊂ D (Λ R opp h , Z ) is as in Pr op osition 3.13. Lemma 5.10 immediately sho w s that this functor Exp is an equiv alence of cat egories. Moreo ver, for any n ≥ 1, let ν n : I → I b e the fun ctor giv en by multi- plication by n , as in (2.18), and f or an y twisted p erio dic ﬁltered complex M ∈ DF per ( I opp , Z ), let ^ Stab n ( M ) = ν ∗ n Stab n ( M ) . 66 Let ^ F D M per b e the d eriv ed category of suc h complexes M equipp ed with a collect ion of maps ϕ p : M → g Stab p ( M ) for all p rimes p . Denote by e h ∗ : ^ F D M per → D F per ( I opp , Z ) the forgetful functor. It has an obvious righ t-adjoin t e h ∗ giv en b y (5.10) e h ∗ ( M )( n ) = M ⊕ Y p | n Stab p ( M ) , where the pro duct is tak en o ver all p rime divisors of the intege r n ∈ I . Moreo ver, the constr u ction of the functor Exp of (5.8) also giv es a fu nctor (5.11) g Exp : ^ F D M per → D w (Λ R h , i ∗ P q , Z ) , and b y construction, w e hav e (5.12) e h ∗ ◦ g Exp ∼ = Exp ◦ e h ∗ , where e h ∗ in the left-hand side is the restriction functor of Pr op osition 3.13. Prop osition 5.18. The functor (5.11) i s an e quivalenc e of c ate gories. Pr o of. As in th e pro of of Prop osition 2.12, Prop osition 3.13 (ii ) im p lies th at the functor g Exp has a righ t-adjoin t functor K : D w (Λ R h , i ∗ P q , Z ) → ^ F D M per . Moreo ver, by (5.10) , Corollary 5.16 and Lemma 3.14, the base c hange map g Exp ◦ e h ∗ → e h ∗ ◦ Exp induced by (5.12) is an isomorphism. Therefore b y ad j unction, e h ∗ ◦ K ∼ = Exp − 1 ◦ b t ∗ , where Exp − 1 is the equiv alence inv erse to =eqrefexp.w. W e conclude th at e h ∗ ◦ K ◦ g Exp ∼ = Exp − 1 ◦ Exp ◦ e h ∗ , e h ∗ ◦ g Exp ◦ K ∼ = Exp − 1 ◦ Exp ◦ e h ∗ . As in the pro of of Prop osition 2.12, the functor e h ∗ is conserv ative , and Exp and Exp − 1 are m utually inv erse equiv alences of categories. Therefore so are the functors g Exp and K .  67 Pr o of of Pr op osition 5.17 . The tautological pro jection τ : I → pt induces a functor τ ∗ : D F per ( Z ) → D F per ( I opp , Z ) . This is a full emb edding on to the f u ll sub category spann ed by the lo cally constan t f unctors; the adjoin t f unctor is giv en by τ ∗ = R q lim I opp ← . By Lemma 5.14, s tabilized su b division commutes with arbitrary pro du cts, so that τ ∗ ◦ g Stab p ∼ = Stab p ◦ τ ∗ . Therefore τ ∗ and τ ∗ extend to an adjoin t pair of f unctors b et we en F D M per and ^ F D M per , s o that F D M per is identiﬁed with the full sub categ ory in ^ F D M per spanned by ob jects M with lo cally constan t e h ∗ M . By (5.12), the equiv alence of Prop osition 5.18 iden tiﬁes this sub category with D ΛR( Z ) ∼ = D c (Λ R h , i ∗ P q , Z ) ⊂ D w (Λ R h , i ∗ P q , Z ).  6 T op ological cyclic homology . W e ﬁ nish the p ap er with a b rief d iscussion of top ologic al cyclic homology (w e again follo w [HM]). Fix the G -un iverse U as in Su bsection 4.4, and recall that for any m ≥ 1, w e ha v e canonical fun ctors b Ψ m , b Φ m endofunctors of the category G -Sp = G -Sp( U ). W e also h av e natural maps can : b Ψ m → b Φ m . F or an y T ∈ G -Sp and a pair of in tegers r, s > 1, one has a natural non-equiv arian t map F r,s : T C r s → T C r . On the other hand, assume that T is equipp ed w ith a cycloto mic s tr ucture. Then we hav e a n atural map R r,s : T C r s ∼ = ( b Ψ s ( T )) C r can − − − − → ( b Φ s ( T )) C r r s − − − − → T C r , where r s comes fr om the cyclotomic structure on T . T ak en together, the maps F r,s and R r,s deﬁne a functor I ( T ) fr om the category I opp of Subsec- tion 1.3 to th e category of n on-equiv ariant sp ectra: w e let I ( T )( n ) = T C n for an y n ∈ I , and we let the morph ism s F r , R r : s → r s act by the maps F r,s resp. R r,s . Deﬁnition 6.1. The top olo gic al cyclic homolo gy TC( T ) of a cyclotomic sp ectrum T is a non-equiv arian t sp ectrum giv en by TC( T ) = holim I opp I ( T ) . 68 Assume no w give n a cycloto mic complex M ∈ D ΛR ( Z ), and consider the functor e α : I → g Λ R of (1.16). Deﬁnition 6.2. The top olo g i c al c yc lic homo lo gy TC q ( M ) of a cyclotomic complex M ∈ D ΛR( Z ) is giv en by TC q = H q ( I opp , e α ∗ ξ ∗ M ) , where ξ ∗ : D ΛR( Z ) ∼ = D c ( g Λ R, e λ ∗ T q , Z ) → D c ( g Λ R opp , Z ) is the r igh t-adjoin t to the corestriction functor. In particular, assume giv en a G -sp ectrum T , and co n sider its equiv arian t c hain complex C q ( T ) ∈ D M Λ( Z ) of Subsection 4.3. Prop osition 6.3. F or any cyclotomic sp e ctrum T , we have a natur al iso- morphism TC q ( C q ( T )) ∼ = H q (TC( T ) , Z ) , wher e H q ( − , Z ) denotes the homolo gy of a non-e qui variant sp e c trum with c o e ﬃcients in Z . Pr o of. By Lemma 4.12, w e hav e e α ∗ ξ ∗ C q ( T )( m ) ∼ = C q ( T C m ) for an y m ∈ I , so th at e α ∗ ξ ∗ C q ( T ) ∈ D ( I opp , Z ) is isomorp hic to C q ( I ( T ) , Z ), the non -equiv arian t c h ain homology complex of the system of sp ectra I ( T ) used in Deﬁnition 6.1.  In view of the equiv alence of Theorem 5.4, it w ould b e desirable to express th e top ologic al cyclic h omology f unctor TC q of Deﬁnition 6.2 in terms of ﬁltered Dieudonn´ e mo dules. A natural notio n of homology for ﬁltered Dieudonn´ e mo dules is the follo w in g. Deﬁnition 6.4. Syntomic c ohomolo gy of a generalized ﬁltered Dieud onn ´ e mo dule M ∈ F D M per ( Z ) is giv en b y RHom q ( Z , M ) , where Z ∈ F D M per is the trivial ﬁltered Dieudonn´ e mo du le. In general, syn tomic cohomology and top ological cyclic homology are diﬀeren t. Ho wev er, to hav e the follo wing comparison result. 69 Theorem 6.5. Assume that M ∈ D ΛR( Z ) ∼ = F D M per ( Z ) is pr oﬁnitely c omplete. Then we have a natur al i somorphism TC q ( M ) ∼ = RHom q ( Z , M ) . Pr o of. In term of cyclotomic complexes, the trivial Dieudonn ´ e mo d ule Z cor- resp ond s to the corestriction e ξ ∗ Z of the constan t f unctor Z ∈ F un( g Λ R opp , Z ). Then by adjun ction, we h a v e RHom q ( Z , M ) ∼ = H q ( g Λ R opp , ξ ∗ M ) . Th u s it suﬃces to pro ve that for an y proﬁnitely complete M ∈ D ( g Λ R opp , Z ), the natural map H q ( I opp , e α ∗ M ) → H q ( g Λ R opp , M ) is an isomorphism. Equiv alen tly , we hav e to prov e that the map H q ([1 / N ∗ ] , e λ ∗ e α ∗ M ) → H q (Λ R opp , e λ ∗ M ) is an isomorphism. By base change, e λ ∗ e α ∗ ∼ = α ∗ e λ ∗ , and since righ t-derived Kan extensions comm ute with proﬁn ite completions, w e apply Prop osi- tion 1.13 and replace Λ R with ∆ R . But we know th at e j ∗ e λ ∗ M b ecomes lo cally constan t after restricting to ∆ ⊂ ∆ R , and we claim that for any M ′ ∈ D (∆ R, Z ) with lo cally constan t h ∗ M ′ ∈ D (∆ , k ), the map H q ([1 / N ∗ ] , α ∗ M ′ ) → H q (∆ R, M ′ ) is an isomorphism. In deed, b y Lemm a 1.6, we can compute the d irect image δ ! ﬁb erw ise, and since h ∗ M ′ is locally constan t, th e adjunction map δ ∗ δ ! M ′ → M ′ is an isomorphism. Then H q (∆ R, M ′ ) ∼ = H q (∆ R, δ ∗ δ ! M ′ ) ∼ = H q ([1 / N ∗ ] , δ ! M ′ ) , and since δ ◦ α = id , the r igh t-hand side is exactly H q ([1 / N ∗ ] , α ∗ M ′ ).  In conclusion, let me say that to obtain an analogous comparison iso- morphism f or an arb itrary cyclotomic sp ectrum T , one h as to mo dify the deﬁnition of top ologica l cyclic homology T C( T ) b y r eplacing the ﬁxed-p oint s functors T C m in the system I ( T ) with their homotop y ﬁxed p oints  T C m  hG 70 with resp ect to the residual action of th e group G = U (1) (this s hould b e related to the usual TC b y a coﬁb er sequence analogous to (1.27)). I do not kno w whether it makes sense to do it from the top ological p oin t of view. I am very grateful to L. Hesselholt for explaining to me why it do es not matter in the proﬁnitely complete case. 7 App endix. In this App end ix, w e collect some facts used thr oughout the p ap er; most of the facts are w ell-known, but our terminology ma y b e non-stand ard. First of all, a piece of general nonsense. Assume giv en a square A b − − − − → B a   y   y c C d − − − − → D , and assume that a and c admit left-adjoin t fun ctors a ! , c ! . Then an isomor- phism d ◦ a ∼ = c ◦ b induces by adjun ction a m ap c ! ◦ b → a 1 ◦ b. W e call th is map and its v arious adjoints the b ase change map indu ced by the isomorphism d ◦ a ∼ = c ◦ b . F or any category C with ob jects c, c ′ ∈ C , w e denote b y C ( c, c ′ ) the set of maps from c to c ′ , an d w e denote b y C opp the opp osite category , C opp ( c, c ′ ) = C ( c ′ , c ). F or a small categ ory C and a category A , we denote b y F un( C , A ) the category of cuntors fr om C to A . F or a fun ctor f : C → C ′ , f ∗ : F u n( C ′ , A ) → F un( C , A ) is the pullb ack, and f ! , f ∗ are the left and the righ t Kan e xtensions (when they exists). If A is ab elian, w e den ote b y D ( C , A ) the d er ived category of the category F u n( C , A ), and by abuse of notation, w e use f ! and f ∗ to denote the deriv ed functors of the Kan extensions. If A = k -mo d , the categ ory of mo d ules ov er a ring k , we shorten F un( C , k -mod ), D ( C , k -mo d ) to F un( C , k ), D ( C , k ). The homology H q ( C , k ) resp. cohomology H q ( C , k ) of a small category C is deﬁn ed b y taking derv ied Kan extensions with resp ect to the pro jection C → pt . As usual, H q ( C , k ) is an algebra, and H q ( C , k ) is a mo du le o ver H q ( C , k ). T h e homology can b e computed by an explicit b ar c omplex C q ( C , k ). W e also introd uce the follo w ing slightly non-standard deﬁnition. 71 Deﬁnition A.1. Assume giv en a small category C . An ob j ect E q ∈ D ( C , k ) is lo c al ly c onstant if f or any map f : a → b in C , th e map E q ( f ) : E q ( a ) → E q ( a ) is a quasiisomorphism. W e freely u se the notions from [S GA] (Cartesian map, ﬁ bration, coﬁ- bration, biﬁ b ration); for a brief o verview with exactly the same notation as in this pap er, s ee [Ka1, S ection 1]. W e also f reely use the base change isomorphism and pro jection f orm ula of [K a1, Lemma 1.7]. W e also use the follo wing n otion from [Bou]. Deﬁnition A.2. A factorization system on a category C is giv en by t wo sub categories C v , C h ⊂ C suc h that all isomorph isms in C lie b oth in C v and in C h , and any m orphism f in C decomp oses as f = v ◦ h , v ∈ C v , h ∈ C h , and suc h a d ecomp osition is unique up to a unique isomorphism. Example A.3. If γ : C → C ′ is a ﬁb r ation, then ﬁb erwise and Cartesian maps form a factorizatio n system on C . F actorizatio n systems are actually v ery useful gadgets, although they are traditionally relegated to app endices and introd uctions (and we follo w the tradition). Deﬁnition A.2 has sev eral corollaries an d /or equiv alen t re- form ulations. F or example, one can s h o w that C v ∩ C h exactly consists of all the isomorph isms in C ; moreo ver, maps in C v ha ve a un ique lifting prop erty with resp ect to maps in C h , an d vice versa. W e r efer the reader to [Bou] for d iscussion. W e will need one r esult wh ic h is n ot in [Bou]. Assum e that the categ ory C is small, an d let C b e th e category of all ob jects in C and all isomorphisms b et ween them, so that w e hav e a C artesian s quare C v − − − − → C h h   y   y h C v v − − − − → C , where v , h , v , h are the em b eddin g fun ctors. Th en the isomorphism v ∗ ◦ h ∗ ∼ = h ∗ ∼ = v ∗ induces a base c hange m ap (A.1) h ! ◦ v ∗ → v ∗ ◦ h ! of functors from D ( C h , k ) to D ( C v , k ). Lemma A.4. The b ase c hange map (A.1) is an isomorphism. 72 Pr o of. Since the catego r y D ( C h , k ) is generated by represen table fu nctors of the form k c ( c ′ ) = k [ C h ( c, c ′ )] , c, c ′ ∈ C h , it suﬃces to p ro ve that (A.1) b ecomes a n isomorphism after applying to some such f unctor k c ∈ F un ( C h , k ), c ∈ C h . In deed, h ! sends represent able f u nctors in to repr esen table ones, so that we h a v e (A.2) v ∗ h ! k c ( c ′ ) = h ! k c ( c ′ ) = k [ C ( c, c ′ )] for any c ′ ∈ C . But by the deﬁnition of a factorization system, we hav e a natural isomorphism (A.3) C ( c, c ′ ) = a c ′′ ∈C ( C h ( c, c ′′ ) × C v ( c ′′ , c ′ )) / Aut( c ′′ ) , and the actions of the group s Aut( c ′′ ) are free. Therefore (A.2) is isomorphic to h ! M c ′′ ∈ C k [ C h ( c, c ′′ )] , and the right -hand side is exactly v ∗ k c .  W e also use the A ∞ -tec h nology; a brief in tro duction to the relev an t part of it is co ntained in [Ka2, Section 1.5]. Here w e just recall that an A ∞ - algebra is an algebra o ver a certain asymmetric op erad Ass ∞ of complexes of ab elian group s, a coﬁbrant r esolution of the asso ciativ e asymmetric op erad Ass . An A ∞ -coalg ebr a is an A ∞ -algebra in the opp osite cate gory . Since the op erads are asymmetric, one can d eﬁne algebras in an arbitrary tensor catego ry . In particular, for an y set S , one can consider the ca tegory of ab elian groups graded by S × S , with tensor pro duct giv en by ( V ⊗ V ′ ) s,s ′ = M s ′′ ∈ S V s,s ′′ ⊗ V ′ s ′′ ,s . An A ∞ -algebra in this category is a small A ∞ -categ ory with the set of ob jects S . More generally , give n a small category C with the of ob j ects C 0 and the set of morp hisms C 1 , one can consid er the catego r y of C 1 -graded v ector sp aces, with the tensor pro duct giv en by ( V ⊗ V ′ ) f = M f = f ′ ◦ f ′′ V f ′ ⊗ V ′ f ′′ . A C -gr ade d A ∞ -c o algebr a is an A ∞ -coalg ebr a in this category . 73 The derived category D ( B q , Ab) of A ∞ -functors from a s m all A ∞ -cate- gory B q to the cate gory of complexes of ob jects in an ab elian category Ab is deﬁned by consid ering the DG category of all such fu n ctors and A ∞ -maps b et ween them, and in verting quasiisomorphism s . T he inv ertion p ro cedure present s no problems, since the DG category is w ell-b eh av ed (at least if Ab is large enough, f or example Ab = k -mo d for some rin g k ). Ev ery ob j ect has an h -pro jectiv e and an h -injectiv e r eplacemen t. F or ev ery A ∞ -morphism f : B q → B ′ q , w e h a v e the pullbac k fun ctor f ∗ : D ( B ′ q , Ab) → D ( B q , Ab), and it has the righ t and left-adjoin t fu nctors f ∗ , f ! : D ( B q , Ab) → D ( B ′ q , Ab). Giv en a C -graded A ∞ -coalg ebr a R q , one considers the DG category of C -graded k -v alued A ∞ -comod ules o v er R q . O ne can still in vert quasiiso- morphisms to obtain the triangulated deriv ed cate gory D ( C , R q , k ). F or an y A ∞ -map f : R q → R ′ q , w e hav e the corestriction functor f ∗ : D ( C , R q , k ) → D ( C , R ′ q , k ). F or any fu n ctor f : C ′ → C , w e ha v e the pullback C ′ -graded A ∞ - coalge b ra f ∗ R q and a p ullbac k fu n ctor f ∗ : D ( C , R q , k ) → D ( C ′ , f ∗ R q , k ). Ho w ever, h -pro jectiv e replacemen ts usu ally d o not exist at all, and there is no general pro cedure for constructing h -injectiv e ones. Therefore the exis- tence of adjoint s is n on -trivial One case wh ere an ad j oin t do es exist is the em b edding functor i : pt → C of the p oint category on to an ob ject c ∈ C . I n this case, an adjoin t to th e pullbac k functor i ∗ : D ( C , T q , k ) → D ( k ) is giv en b y th e cofree A ∞ -mo dule fun ctor, sendin g M ∈ D ( k ) to an A ∞ -comod ule M c with M c ( c ′ ) ∼ = M ⊗ M f : c ′ → c T q ( f ) . Another situation where things are easy is the trivial C - gr ade d A ∞ -c o algebr a R q giv en by R 0 ( f ) = Z for an y f ∈ C 1 , and R i = 0 for i 6 = 0. Lemma A.5. Assume given a smal l c ate gory C , and let R b e the trivial C -gr ade d A ∞ -c o algebr a. Then e v ery h -pr oje ctive c omplex of functors f r om C opp to k -mod is h -pr oje ctive as an A ∞ -c omo dule over R , and the natur al emb e dding D ( C opp , k ) → D ( C , R , k ) is an e qu i valenc e of c ate gories. Pr o of. L et B q ( c, c ′ ) = Z [ C ( c, c ′ )] b e the free additivie category generated by C , and treat it as an A ∞ -categ ory in th e ob vious wa y . Then by deﬁnition, an A ∞ -comod ule ov er the trivial A ∞ -coalg ebr a R is the s ame thin g as an A ∞ -functor from B opp q to complexes of k -mo dules. Thus it suﬃces to pr o v e the claim for A ∞ -categ ories, we h r e it is w ell-kno w.  74 Lemma A.6. Assume given a smal l c ate gory C , a C -gr ade d A ∞ -c o algebr a T q , a nd a functor ρ : C ′ → C fr om a smal l c ate gory C ′ such that ρ ∗ T q is isomorph ic to the trivial C ′ -gr ade d A ∞ -c o algebr a. Th en the pul lb ack functor ρ ∗ : D ( C , T q , k ) → D ( C ′ , k ) admits a right-adjoint fu nctor ρ ∗ : D ( C ′ , k ) → D ( C , T q , k ) . Pr o of. By deﬁnition, w e ha ve to prov e that for any ob ject M ∈ D ( C ′ , k ), the functor (A.4) N 7→ Hom( ρ ∗ N , M ) from D ( C , T q , k ) to th e catego r y of k -mo dules is represen table. By the cobar construction, eve ry ob ject M ∈ D ( C ′ , k ) is the cone of an end omorp hism of an ob ject M ′ ∈ D ( C ′ , k ) of the form M ′ = Y M i , where eac h M i = M ′ a is the corepresent able fu nctor from C ′ to k -mod cor- resp ond ing to an ob ject a ∈ C ′ and s ome M ′ ∈ k -mo d. Thus it su ﬃces to pro ve that the functor (A.4) is r epresent able for M = M ′ a . The represent in g ob ject is giv en by the cofree A ∞ -comod ule M ′ ρ ( a ) ∈ D ( C , T q , k ).  Lemma A.7. Assume given a smal l c ate g ory C e quipp e d with a f actor- ization system, as in L emma A.4. Mor e over, assume given a C -gr ade d A ∞ -c o algebr a T q such that h ∗ T q is a trivial C h -gr ade d A ∞ -c o algebr a, as in L emma A .6. Then the b ase change map v ∗ ◦ h ∗ → h ∗ ◦ v ∗ of functors fr om D ( C opp h , k ) to D ( C v , v ∗ T q , k ) induc e d by the obvious i somor- phism v ∗ ◦ h ∗ ∼ = h ∗ ◦ v ∗ is itself an i somorph ism. Pr o of. The same pro of as f or Lemma A.4 wo rk s , w ith the adjoin ts con- structed by Lemma A.6.  All the categories of A ∞ -comod ules that we consider in this pap er ought to b e symmetric tensor categories. Ho we ver, to construct the tensor pro d uct, one w ould need to equip the A ∞ -coalg ebr as with some sort of Hopf algebra structure, and this is to o hea vy tec h n ically . Ther efore in general, we a voi d tensor pro d ucts. W e do n eed them in one easy case. Say that a C -graded A ∞ -coalg ebr a R q is augmente d if R i = 0 f or i < 0, and R 0 ( f ) = Z for an y 75 morphism f in C . W e th em hav e an ob vious augmen tation map ξ : Z C → R q , and the corresp ond ing corestriction f u nctor ξ ∗ : D ( C opp , k ) → D ( C , R q , k ) . Ho w ever, w e ha ve more: for ev ery complex M q in F un( C opp , k ) and ev ery A ∞ -comod ule E q o v er R q , w e can d eﬁne th e tensor pro d uct E q ⊗ ξ ∗ M q b y setting ( E q ⊗ ξ ∗ M q )( c ) = E q ( c ) ⊗ M q ( c ) , c ∈ C , with the A ∞ -op erations giv en by the p ro du cts of the A ∞ -op erations in E q and the structure maps of the fu nctor M q . Th is construction is obiv ously asso ciativ e in M q . Finally , in ord er to constru ct A ∞ -categ ories and A ∞ -coalg ebr as, w e use v arious categorical constructions asso ciativ e “up to an isomorph ism”. Here is the protot yp e example (for more details, see [Ka2, Sub section 1.6]). As- sume giv en a small monoidal category C , with an associativit y isomorph ism satisfying the usual p enta gon equation. Then in eﬀect, C is an algebra o v er the follo wing asymmetric op erad. Deﬁnition A.8. The mono dial c ate gory op er ad I n is an op erad of group oids deﬁned as follo ws: (i) on ob jects, I n is the free op er ad generated by a single binary op eration, (ii) on morp hisms, there exists exactly one morph isms b et wee n an y t wo ob jects of the group oid I n . The bar complex C q ( C , k ) is automaticall y an algebra ov er th e op erad C q ( I q , Z ). Bu t the op erad C q ( I q , Z ) is a resolution of the asso ciativ e op erad Ass , and the A ∞ -op erad A ss ∞ is another su c h resolution, and a coﬁbrant one. Th erefore the augmen tation m ap Ass ∞ → Ass f actors through a map Ass → C q ( I q , Z ). Fixing s u c h a factorization once and for all, we turn the bar complex C q ( C , k ) in to an A ∞ -algebra o ve r k . References [BHM] M. B¨ okstedt, W.C. Hsiang, and I. Madsen, T he cyclotomic tr ac e and algebr ai c K -the ory of sp ac es , Inv ent. Math. 111 (1993), 465 –539. [BM] M. B¨ okstedt and I. Madsen, T op ol o gi c al cyclic homolo gy of the i nte gers , in K -the ory (Str asb our g, 1992) , Ast´ erisque 226 (1994), 7–8 , 57–143. 76 [Bou] A.K. Bousﬁeld, Constructions of factorization systems in c ate gories , J. Pure Appl. Algebra 9 (1976/77), 207–220. [Dri] V. Drinfeld, On the notion of ge ometric r e al ization , Mosco w Math. J. 4 (2004), 617–626 . [FT] B. F eigin and B. Tsygan, A dditive K -The ory , in Lecture Notes in Math. 1289 (1987), 97–209. [FL] J.-M. F ontai n e and G. Lafaille, Construction de r epr ´ esentations p -adiques , A n n. Sci. ´ Ecole N orm. S u p. (4) 15 (1982), 547 –608 ( 1983). [FM] J.-M. F ontaine and W. Messing, p -adic p erio ds and p -adic ´ etale c ohomolo gy , in Curr ent tr ends in arithmetic al algebr aic ge ometry (A r c ata, Calif., 1985) , Contemp. Math. 67 , AMS, Providence, R I, 1987; 179–207 . [Goo d] T.G. Goo dwillie, R elative algebr aic K -the ory and cyc l ic homolo gy , Ann. of Math. 124 (1986), 347–40 2. [SGA] A. Grothend iec k, SGA I, Exp os ´ e VI . [HM] L. Hesselholt and I. Madsen, On the K -the ory of ﬁni te algebr as over Witt ve ctors of p erfe ct ﬁelds , T op ology 36 (1997), 29–101 . [Ka1] D. Kaledin, Non-c ommutative Ho dge-to-de Rham de gener ation via the metho d of Deligne-Il lusie , Pure Appl. Math. Q. 4 (2008), 785–875. [Ka2] D. Kaledin, Derive d M ackey functors , to appear in Mosco w Math. J. [Ka3] D. K aledin, Motivic structur es in non-c ommutative ge ometry , to app ear in Pro c. ICM 2010. [KS] M. Kashiwara and P . Schapira, Cate gories and she aves , Grundlehren der Mathe- matisc hen Wissensc haften, 332 , Springer-V erlag, Berlin, 2006 . [LMS] L.G. Lew is, J.P . May , and M. Steinberger, Equivariant stable homotopy the- ory , with contributions by J. E. McClure, Lecture Notes in Mathematics, 1213 , Springer-V erlag, Berl in, 1986. [L] J.-L. Lo day , Cyclic Homolo gy , second ed., Springer, 1998. Steklov Ma th Institute Moscow, USSR E-mail addr ess : kaled in@mccme .ru 77

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