Isomorphism conjectures with proper coefficients

ISOMORPHISM CONJECTURES WITH PR OPER COEFFICIENTS GUILLERMO COR TI ˜ NAS AND EUGENIA ELLI S Abstract. Let G b e a group and l et E be a functor fr om small Z -l inear categories to spectra. Also let A be a ring wi th a G - action. Under mild conditions on E and A one can deﬁne an equiv arian t homology theory of G - simplicial sets H G ( − , E ( A )) with the prop erty that if H ⊂ G i s a subgroup, then H G ∗ ( G/H, E ( A )) = E ∗ ( A ⋊ H ) If now F is a nonempt y fami ly of subgroups of G , closed under conjugation and under subgroups, then the re is a mo del category structure on G -s implicial sets such that a map X → Y is a weak equiv alence (resp. a ﬁbration) if and only if X H → Y H is an equiv alence (resp. a ﬁbration) for al l H ∈ F . The strong isomorphism conjecture f or the quadruple ( G, F , E , A ) asser ts that if cX → X is the ( G, F )-coﬁbran t replacemen t then H G ( cX, E ( A )) → H G ( X, E ( A )) is an equiv alence. The isomorphism conjecture says that this holds when X is the one point space, i n which case c X is the classifyi ng space E ( G, F ). In this pap er we i n troduce an algebraic notion of ( G, F )-properness for G - rings, modell ed on the analogous notion for G - C ∗ -algebras, and sho w th at the strong ( G, F , E , P ) isomorphis m conjecture for ( G, F )-proper P is t rue in seve ral cases of interest in the algebraic K -theory con text. Thus w e give a purely algebraic, discrete coun terpart to a result of Guent ner, Higson and T rout i n the C ∗ - algebraic case. W e apply this to show that under rather general hypothesis, the assemb ly map H G ∗ ( E ( G, F ) , E ( A )) → E ∗ ( A ⋊ G ) can be identiﬁed with the boundary map in the long exact sequence of E -groups asso ciated to certain exact sequence of rings. Along the wa y we prov e sev eral results on excision in algebraic K -theory and cyclic homology whi c h are of independent interest. 1. Introduction Let G b e a group; a family o f subgro ups of G is a nonempty family F closed under conjugation and under taking subgroups. If F is a family of subgroups of G , then a G -simplicia l set X is called a ( G, F ) -c omplex if the stabilizer o f every simplex of X is in F . The categ ory of G -simplicial s e ts can b e equipp ed with a closed mo del structure where an equiv ariant map X → Y is a weak equiv alence (resp. a ﬁbration) if X H → Y H is a weak equiv alence (resp. a ﬁbratio n) for every H ∈ F (see Section 2); ( G, F )-complexe s are the coﬁbra n t ob jects in this model structure (Remark 2.6). By a g eneral co nstruction of Davis and L ¨ uck (see [6]) any functor E from the categor y Z − Ca t of small Z -linear ca tegories to the category Spt of spe ctra whic h sends category equiv alences to equiv a le nces of spectra gives rise to an equiv ariant homology theory of G -spa c es X 7→ H G ( X, E ( R )) for each Corti ˜ nas was supported by CONICET; b oth authors were partial ly supp orted by the M ath- AmSud netw ork U11MA TH-05 and by grants UBACyT 20020100100386 , and MTM2007-64704 (FEDER funds). 1 2 GUILLERMO COR TI ˜ NAS AND EUGENIA E LLIS unital ring R with a G -action (unital G -ring, for short), such that if H ⊂ G is a subgroup, then (1.1) H G ∗ ( G/H, E ( H )) = E ∗ ( R ⋊ H ) is just E ∗ ev aluated a t the cro ssed pro duct. The str ong isomorph ism c onje ctur e for the quadruple ( G, F , E , R ) asserts that H G ( − , E ( R )) sends ( G, F )-eq uiv alences to weak equiv a lences of spectra. The strong isomo rphism c o njecture is e quiv alent to the asser tion that for every G -simplicial set X the map (1.2) H G ( cX , E ( R )) → H G ( X, E ( R )) induced by the ( G, F )-coﬁbrant replacement cX → X is a w eak equiv alence. The weak er isomorphism c onje ctu re is the pa rticular case when X is a po in t; it asserts that if E ( G, F ) ∼ ։ pt is the coﬁbra n t r e placement then the map (1.3) H G ( E ( G, F ) , E ( R )) → H G ( pt, E ( R )) called the assembly map , is an equiv a lence of spe c tr a. This formulation of the conjecture is equiv alent to that o f Davis-L¨ uck, ([6]) which is given in terms of top ological spaces (see P rop osition 2.4 and para graph 2 .7). In this pap er w e are primarily concerned with the strong isomorphism conjec- ture for no nc o nnective algebraic K -theory –denoted K in this paper – homotop y algebraic K -theory K H , and Ho chsc hild and c y clic homolog y H H and H C . Our main results a re outlined in Theorem 1.4 b elow. First we need to expla in the terms “excisive” and “prop er” app earing in the theorem. Let E : Rings → Spt be a func- tor; w e say that a not necessarily unital ring A is E - excisive if whenever A → R is an embedding of A as a tw o sided ideal in a unital ring R , the s e q uence E ( A ) → E ( R ) → E ( R/ A ) is a homotopy ﬁbration. Unital rings are E -excisive for all functors E considered in Theorem 1.4; th us the theorem remains true if “unital” is subs tituted for “ excisive”. By a result of W eibel [30], Homoto p y algebr aic K -theory sa tis ﬁe s excision; this means that every ring is K H -excisive. The rings which a re excisive with r esp ect to cyclic a nd Ho chsc hild ho mology are the same; they were characterized by W o dzicki in [31], where he coined the term H -unital for such rings. By results of Suslin and W odzicki, a ring is exc is iv e for rationa l K -theory if and only if it is H -unital (see [26] for the if part and [31] for the only if part); K -excisive r ings were characterized by Suslin in [25]. Under mild ass umptions on E (the Standing Assumptions 3.3.2), which a re satisﬁed by all the exa mples co nsidered in Theor em 1.4, one can make sense of H G ( − , E ( A )) for not necessar ily unital, E -exc is iv e A (see Section 3). The ring Z ( X ) of p olyno mial functions on a lo cally ﬁnite simplicial set X which are suppo rted on a ﬁnite simplicia l subset, and the r ing C comp ( | X | , F ) of compactly suppo rted contin uous functions with v alues in F = R , C ar e unital if a nd o nly if X is ﬁnite, a nd a re E - excisive for all X and all the functors E of Theor em 1.4; they are ( G, F )-pro per whenever X is a ( G, F )-co mplex. In ge neral if X is a lo cally ﬁnite simplicia l set with a G -ac tion and A is a G -ring, then A is called pr op er ov er X if it carrie s a Z ( X ) -algebra structure whic h is c o mpatible with the actio n of G and satisﬁes Z ( X ) · A = A . W e sa y that A is ( G, F )- pr op er if it is prop er o ver a ( G, F )-co mplex. ISOMORPHISM CONJECTURES WITH PROPER COEFFICIE NTS 3 Theorem 1. 4. L et G b e a gr oup, F a family of sub gr oups, E : Z − Cat → Spt a functor, and P an E -excisive G -ring. The st ro ng isomorphism c onje ctur e for the quadruple ( G, F , E , P ) is satisﬁe d in e ach of t he fo l lowing c ases. i) E = H H or H C and F c ontains al l the cyclic sub gr oups of G . ii) E = K H and P is ( G, F ) -pr op er. iii) E = K and P is p r op er over a 0 - dimensional ( G, F ) -sp ac e. iv) E = K , F c ontains al l the cyclic sub gr oups of G and P is a ( G, F ) -pr op er Q -algebr a. v) E = K ⊗ Q , F c ont ains all the cyclic sub gr oups of G and P is ( G, F ) -pr op er. Part i) of the theorem for unital rings is Prop osition 7.6; that it holds for all H C -excisive rings follows from this by Corolla r y 3 .3.11 and Pr op o sition 6.4. Even for unital rings, pa r t i) g eneralizes a result of L ¨ uck and Reich [18], who proved it under the additional a ssumption tha t G acts tr ivially on A . Theorem 13.1 .1 prov es that part ii) holds for any functor E : Z − Cat → Spt satisfying cer tain pr op e rties, including excision; the fact that K H satisﬁes them is the sub ject of Section 5. W e prov e in Theor em 11.6 tha t part iii) o f the theo rem holds for a n y E satisfying the standing assumptions; that they hold for K -theory is esta blished in Prop osition 4.3.1. Parts iv) and v) ar e the conten t o f Theorem 13.2.1. The co ncept o f prop erness us e d in this a rticle is a discr ete, a lg ebraic transla tion of the analogo us concept of prop er G - C ∗ -algebra . By a re sult of Guentner, Higson and T rout, the full C ∗ -cross e d pro duct version of the Baum-Connes conjecture with co eﬃcients holds whenev er the co eﬃcient algebra is a pro per G - C ∗ -algebra [9]. This result is a basic fact b ehind the Dirac-dual Dira c metho d that was used, for exa mple, in the pr o of of the Baum-Connes co njecture for a- T -menable groups [10]. It is also at the basis of rece nt work of Meyer and Nest ([19],[20],[21]) in which the conjecture a nd the Dira c metho d are recast in ter ms of tria ngulated categorie s. W e exp ect that Theo rem 1.4 can similar ly b e used as a to ol in proving instances of the isomorphism conjecture for (homo to p y) a lgebraic K -theo ry . As a ﬁrst application of Theo rem 1.4 we pro ve the following theorem, which ident iﬁes the assembly ma p (1.3) as the connecting map in an excis ion s equence. Theorem 1.5. L et G b e a gr oup and F a family of sub gr oups. Then t her e is a functor which assigns to e ach G -ring A a G -ring F ∞ A = F ∞ ( F , A ) e quipp e d with an exhaustive ﬁltr ation by G -ide als { F n A : n ≥ 0 } , and a natur al t r ansformation A → F 0 A , which, if E is as in The or em 1.4 and A is E -excisive, have the fol lowing pr op erties. i) The map E ( A ⋊ G ) → E ( F 0 A ⋊ G ) is an e quivalenc e. ii) The fol lowing se quenc e is a homotop y ﬁbr ation E ( F 0 A ⋊ G ) → E ( F ∞ A ⋊ G ) → E (( F ∞ A/ F 0 A ) ⋊ G ) In p articular ther e is a m ap ∂ : Ω E (( F ∞ A/ F 0 A ) ⋊ G ) → E ( F 0 A ⋊ G ) iii) Ther e is an e quivalenc e H G ( E ( G, F ) , E ( A )) ∼ − → Ω E (( F ∞ A/ F 0 A ) ⋊ G ) 4 GUILLERMO COR TI ˜ NAS AND EUGENIA E LLIS which makes the fol lowing diagr am c ommute up to homotopy H G ( E ( G, F ) , E ( A )) ≀   Assembly / / E ( A ⋊ G ) ≀   Ω E (( F ∞ A/ F 0 A ) ⋊ G ) ∂ / / E ( F 0 A ⋊ G ) The theorem above holds more generally for an y functor satisfying certain h y- po thesis, listed in 3 .3.2 and 12.1; see P r op osition 12.2.3 and Theorem 1 2 .3.3. W e also prove a num ber of results ab out K - e x cisive and H - unital r ings which are needed fo r the proo f of the theorems a bove; they are summariz ed in the following theorem. Theorem 1.6. i) If A is a K - ex cisive (r esp. H - unital) G -ring, then A ⋊ G is K - ex cisive (r esp. H - unital). ii) L et { A i } b e a family of rings and let A = L i A i their dir e ct su m , with c o or dinate- wise pr o duct. Then A is K -excisive ( resp . H -u n ital) if and only if e ach A i is. iii) If A and B ar e K -excisive rings, and at le ast one of them is ﬂat as a Z -m o dule, then A ⊗ B is K -excisive. Part i) of Theorem 1.6 r esults by combining Prop ositio ns A.6.3 and A.6.4. Part ii) follows from Pro po sitions A.4.4 and A.4.6. P art iii) is Pr op osition A.5.3. The analogue of part iii) for H - unital rings is true without ﬂatness assumptions, and was prov ed by Suslin and W odzicki in [26, Theor em 7.10]. The rest of this pap er is organized as follo ws. In Section 2 w e for m ulate the isomorphism conjectur e s in terms of closed mo del categor ies. If G is a gr oup, F a family of subgroups and C is either the categor y T op of top o logical spaces or the ca tegory S of simplicial sets, we intro duce clos ed model structures on the equiv ar iant category C G in which a n equiv a riant map X → Y is a weak equiv a lence (resp. a ﬁbration) if X H → Y H is one for every H ∈ F . W e show in Pr op osition 2.4 that the realization and singular functors give a Quillen equiv a lence b etw e e n S G and T op G . In Section 3 we give a list of ﬁve basic conditions for a functor E : Z − Cat → Spt, the Standing Assumptions 3.3.2; all functors E c onsidered in the pap er satisfy them. All but one of these conditions r efer to needed p ermanence prop erties of E -ex cisive ring s; thus they concern only the restriction of E to Rings. The remaining condition is that for all C ∈ Z − Ca t there must b e a n equiv a lence (1.7) E ( A ( C )) ∼ − → E ( C ) Here A ( C ) = M x,y ∈C hom C ( x, y ) is the arrow r ing. The ass ignmen t C → A ( C ) is functorial only for functors which are injectiv e on ob jects; likewise the equiv alence (1.7) is only r equired to be natura l with resp ect to such functors. These conditions imply , for example, that E sends nat- urally equiv alent functors to homotopy equiv alen t maps of sp ectra (Lemma 3.3.6), and that H G ( X, E ( − )) maps extensions o f E -excisive ring s to ho motopy ﬁbrations (Prop osition 3.3.9). W e also discuss a fully functorial construction Z − Ca t → Rings, ISOMORPHISM CONJECTURES WITH PROPER COEFFICIE NTS 5 C → R ( C ) which comes with a ma p p : R ( C ) → A ( C ) and give co ndtions on E un- der whic h E ( p ) is a n equiv alence for all C (Lemma 3.4.3); they apply , for example, when E = K H , but fail fo r E = K (see Example 3.4.2). In Section 4 we presen t the mo del for the (nonconnective) K -theory sp ectrum that we use in this article –essentially b orr ow e d from Pedersen-W eib el’s paper [24]– a nd prov e (Pro pos ition 4.3.1) that it s a tisﬁes the standing a ssumptions. F or this we ne e d several prop erties of K -excisive rings which are pr ov e d in the app endix (including those listed as parts i) and ii) of Theorem 1.6). Section 5 concer ns W eib e l’s homotopy K -theory; the fact tha t it s a tisﬁes the s tanding a ssumptions is prov ed in Pro pos ition 5.5. W e also show (Pr op o sition 5.3) that there is a natur al equiv alence K H ( C ) → K H ( R ( C )) ( C ∈ Z − Ca t). The basic deﬁnitions of Hochsc hild and cyclic ho mo logy for rings and Z -linear categories are reviewed in Section 6, where it is shown (Pr opo sition 6.4) that they sa tisfy the standing assumptions. Part i) o f Theorem 1.4 is prov ed in Section 7 (Pro p ositio n 7.6). In the nex t sectio n we disc uss v ario us Chern characters connecting K -theory with cyclic homolog y . Of these, the relative character ν : K nil ( C ) ⊗ Q = hoﬁb er( K ( C ) → K H ( C )) → Ω − 1 | H C ( C ) | ⊗ Q (deﬁned in (8.2.3))) plays a pr ominent r ole in the article. Here | − | is the sp ectrum asso ciated by the Dold-Kan c o rresp ondence. W e show in Prop ositio n 8.2 .4 that its ﬁber (1.8) K ninf ( C ) = hoﬁb er( ν ) satisﬁes the sta nding a ssumptions, that in addition it is excisive and that K ninf ∗ commutes with ﬁltering colimits. Sectio n 9 reviews some of the prop erties of the ring Z ( X ) of ﬁnitely s upp or ted, integral p olynomial functions on a simplicial set X . F or example, Z ( − ) is fu nctorial for prop er maps, and sends disjoin t unions to direct sums (se e Subsection 9.3). Moreov er, if X is lo cally ﬁnite, and Y ⊂ X is a subo b ject, then the the restriction map Z ( X ) → Z ( Y ) is on to (Corollary 9.4.2). W e also show that if X is lo ca lly ﬁnite, then Z ( X ) is free as an ab elian gr oup (see Lemma 9.3 .7) a nd that if E satisﬁes the sta nding assumptions then the r ing Z ( X ) is E -exc is iv e (Prop osition 9.5.1). Thus by Theo rem 1 .6 iii), the class of K - excisive rings is clo sed under tensoring with Z ( X ) (Prop osition 9.5.3). In Sectio n 10 we consider G -rings which ar e pr oper over a ( G, F )-co mplex X . W e establish discrete analogues of several of the pr o per ties of prop er C ∗ -algebra s disc ussed in [9]. F or a subgroup H ⊂ G we introduce the induction functor Ind G H : H − Rings → G − Rings (Subsection 10.2) and show that it is an equiv alence be t w een H − Rings and the full sub c ategory o f those G -rings which are prop er over the 0-dimensio nal simplicial set G/H (Pro po s ition 10.3 .1). Next we g iv e a discr ete v ariant o f Gre e n’s imprimitivit y theorem; we show in Theor em 10 .4.5 that there is an isomorphis m (1.9) Ind G H ( A ) ⋊ G ∼ = M G/H ( A ⋊ H ) Here M G/H denotes matr ices indexed by G/H × G/H with ﬁnitely many nonzero co eﬃcient s. Also in this section we consider the res triction functor Res H G going from G -rings to H -ring s a nd study the compo sites Ind G H Res H G and Res H G Ind G K for subgroups K, H ⊂ G (Lemmas 10.5.1 and 10.5.4). The materia l in Section 10 is used in the next sectio n to deﬁne, for a gr oup G , a subgro up K ⊂ G , a G -simplicial set X , a functor E : Z − C a t → Spt satisfying the s tanding ass umptions , and a 6 GUILLERMO COR TI ˜ NAS AND EUGENIA E LLIS K -r ing A , an induction map Ind : H K ( X, E ( A )) → H G ( X, E (Ind G K ( A ))) W e show in Prop osition 11.3 that the map a bove is an equiv alence. Then we use this result to pr ov e part iii) of Theor em 1.4 for any functor satisfying a s sumptions 3.3.2; see Theorem 11 .6. The latter theorem is applied in Section 12, where Theorem 1.5 is prov ed for any E satisfying assumptions 3.3.2 and 12.1 (se e Prop osition 12 .2.3 and Theorem 12 .3.3). In Section 13 we b egin by proving par t ii) of Theorem 1.4 for any functor E satsifying excisio n in addition to the hypothes is of 3.3 .2 and 1 2.1 (see Theor em 13.1.1). In particula r, it holds when E is the functor K ninf of (1.8). Parts iv) and v ) of Theorem 1.4 are the cont ent of Theor em 13.2.1). The pro o f uses par t i) of Theor em 1.4, and Theo rem 13.1 .1 applied to K ninf . In the App endix we recall the results of Suslin and W o dzicki on K - excisive a nd H -unital rings, a nd establish Theorem 1 .6 (see Pr o po sitions A.4.4, A.4.6, A.5.3, A.6.3 a nd A.6.4). Notation 1.10 . If C is a (s ma ll) categor y , we write ob C fo r the (sma ll) set o f ob jects and ar C for that o f arrows. W e o ften consider a set X a s a discrete category , who se only arrows a re the identit y maps. I n particular, we do this when X = ob C ; note that there is a faithful functor ob C → C . W e write S for the ca tegory of simplicial sets and T op for that of top olog ical spaces. A family F of subgr oups of a gr oup G is a nonempty family closed under conjugation and under taking subgroups. W e write Or F G for the orbit category relative to the family F ; its ob jects a re the G -sets G/H , H ∈ F ; its homomorphisms are the G -equiv ariant maps. If C a nd D are categor ies, we wr ite C D for the category of functors D → C , where the homomorphisms are the na tural transforma tions. In particular T op G and S G are the categories of G -spaces and G -s implicial s e ts , and T op Or F G op and S Or F G op those of con trav a riant Or F G -spaces and Or F G -simplicial sets. If f : C → C ′ is a functor, we write f ∗ : C D → C ′ D for the functor g 7→ f ◦ g . Thu s for exa mple | | ∗ : S G → T op G is the equiv ar iant geometric realiza tion functor; this notation is used in Section 2. In the r est of the pap er, if C is a chain c o mplex of a belia n groups , | C | is the sp ectrum the Dold-Kan co rresp ondence asso ciates to it. T o p olo gical spaces are consider ed brieﬂy in Section 2 where it is e x plained that we can eq uiv alently work with simplicial sets, which is what we do in the rest of the pap er. In particular –except brieﬂy in Section 2– a sp ectrum is a sequence { n E } of pointed s implicial sets and b onding maps Σ n E → n +1 E . If E , F : C → Spt are functor ial sp ectra, then by a (natura l) map f : E ∼ − → F we mean a zig -zag of natural maps E = Z 0 f 1 − → Z 1 f 2 ← − Z 2 f 3 − → . . . Z n = F such that each r ig h t to left ar row f i is an ob ject-wise weak e q uiv alence. If also the left to right ar rows a re ob ject-wise w eak equiv alences, then w e say that f is a we ak e quivalenc e or simply an e quivalenc e . If { E i } is a family o f sp ectra, w e write L i E i for their wedge or copro duct. Rings in this paper ar e no t assumed unital, unless explicitly stated. W e write Rings for the ca tegory of r ings and ring homomor phisms, a nd Rings 1 for the sub- category of unital rings and unit pr eserving homomorphis ms. W e use the letters A, B for rings, and R , S for unital ring s. If V is an ab elian gr oup, then the tensor algebra o f V is T V = L n ≥ 1 V ⊗ n ; th us fo r us T V is nonunit al. If V is free, then T V is a free nonunital ring. If X is a s et, then M X is the ring of a ll matrices ISOMORPHISM CONJECTURES WITH PROPER COEFFICIE NTS 7 ( z x,y ) x,y ∈ X × X with in teger coe ﬃcie nts, only ﬁnitely ma n y of which are nonzero . If A is a r ing, then M X A = M X ⊗ A ; in particula r M X Z = M X . If { A i } is a family of ring s, then L i A i is their dir e ct sum as ab elian groups, eq uipped with co ordinate-wise m ultiplication. 2. Model ca tegor y structures and assembl y maps W e b egin with some general consider ations on mo del catego ry structures for diagrams of spaces . W e co nsider T op a nd S with their usual, c o ﬁbrantly generated clo sed mo del structures. If C = T op, S , a nd I is a n y small catego ry , then, by [11, Thm. 11 .6.1], C I is aga in a coﬁbrantly ge nerated closed mo del category , with ob ject-wise ﬁbra - tions and weak equiv alences, and wher e generating (trivia l) coﬁbratio ns are of the form a hom I ( α, − ) f : a hom I ( α, − ) dom f → a hom I ( α, − ) co d f with α ∈ I and f : dom f → c od f a generating (trivial) co ﬁbration in C . Recall that the g eometric realiza tion functor | | : S → T op and its r ight adjoint Sing : T op → S form a Quillen equiv alence. Hence b y [ 1 1, T hm. 11.6.5], the induced functors | − | ∗ : S I ⇄ T op I : Sing ∗ are Quillen equiv alences to o. Next ﬁx a gr oup G a nd a fa mily F of subgro ups of G . By the previous discussio n applied to the orbit catego ry Or F G op , we hav e a Quillen equiv alence (2.1) T op Or F G op Sing ∗ - - S Or F G op | | ∗ m m F or C = T op , S , consider the functor R : C G → C Or F G op , R ( X )( G/H ) = map G ( G/H, X ) = X H and its left a djoin t, the co end L : C Or F G op → C G , L ( Y ) = Z Or G Y ( G/ H ) × G/H The Quillen equiv alence (2.1) ﬁts into a diagr am (2.2) T op Or F G op L   Sing ∗ - - S Or F G op L   | | ∗ m m T op G Sing ∗ , , R S S S G | | ∗ l l R S S Lemma 2.3. L et B / / Y A O O i / / X O O 8 GUILLERMO COR TI ˜ NAS AND EUGENIA E LLIS b e a c o c artesian diagr am of G -sets. Assume that i is inje ctive. Then B G / / Y G A G O O i / / X G O O is again c o c artesian. Pr o of. Str a ightf orward.  Prop osition 2.4. L et C = T op , S . i) C G is a close d mo del c ate gory wher e a map f is a ﬁbr ation (r esp. a we ak e qu iv- alenc e) if and only if R ( f ) is. Mor e over C G is c oﬁbr antly gener ate d, wher e the gener ating (trivial) c oﬁbr ations ar e the maps f × id : dom f × G/H → co d f × G/H , with f a gener ating (trivial) c oﬁbr ation and H ∈ F . ii) Each of the p airs of functors of di agr am (2.2) is a Quil len e quivalenc e. Pr o of. O ne ca n give co nditions on tw o sets o f maps a nd a s ubcateg ory of a ca tegory D to be resp ectively the generating coﬁbrations , ge ne r ating trivia l co ﬁbrations and weak e q uiv alences in a clo s ed mo del structure o f D ; see M. Hov ey’s b o ok [12, Thm. 2.1.19]. It is stra ight forward that those conditions are satisﬁed in our c a se, for D = C G . This pro ves i). The top pair of functors in diagram (2.2) is a Quillen equiv alence by the disc us sion ab ov e the pr opo sition. By deﬁnition of ﬁbrations and weak equiv alences in C G , these a re b oth pr eserved and reﬂected by R . In particular ( L, R ) is a Quillen pa ir. T o show that it is an eq uiv alence, it suﬃces, by [12, Cor. 1.3.16], to show that if X ∈ C Or F G op is coﬁbrant, then the unit map (2.5) X → RL X is a w eak equiv alence; in fact we s hall see tha t it is an isomo rphism. Because every coﬁbr a n t ob ject is a retract of a coﬁbrant cell co mplex, it suﬃces to chec k that (2.5) is an isomor phism on cell co mplexes. Because the unit map pr eserves the skeletal ﬁltration, it suﬃces to c heck that X n → R LX n is an is omorphism for all n . By deﬁnition, the generating coﬁbrant c e lls in C Or F G op are of the fo rm ` map G ( − ,G/H ) ∆ n . But for e very T ∈ S , we hav e: RL ( a map G ( − ,G/H ) T )( G/K ) = R ( G/H × T )( G/K ) =( G/H × T ) K = map Or G ( G/K, G/H ) × T = a map G ( − ,G/H ) T Thu s the unit map is a n iso morphism o n cells , and therefore on copr o ducts of cells , since taking ﬁxed po ints under a subgroup preserves copro ducts o f G -simplicial sets. In par ticula r (2.5) is an is omorphism on the zero skeleton o f X . Assume by induction that (2.5 ) is an iso morphism on the n -sk eleton. The n + 1-skeleton is a pushout ISOMORPHISM CONJECTURES WITH PROPER COEFFICIE NTS 9 ` H ∈ I n ` map G ( − ,G/H ) ∆ n / / X n +1 ( − ) ` H ∈ I n ` map G ( − ,G/H ) ∂ ∆ n O O / / X n ( − ) O O Applying L to this diagr am yields a co cartes ian diagra m with injective vertical maps. Hence by Lemma 2.3 and the inductive hypothesis , the diag ram ` H ∈ I n (map G ( − , G/H )) × ∆ n / / RLX n +1 ( − ) ` H ∈ I n (map G ( − , G/H )) × ∂ ∆ n / / O O RLX n ( − ) O O is aga in a pushout. It follows that R LX n +1 ∼ = X n +1 and thus (2.5) is a n iso- morphism on all cell complexe s , as we had to prove. W e ha v e shown that the top horizontal and bo th vertical pair s of functor s are Q uillen equiv alences; by [12, Co r. 1.3.15], this implies that also the b ottom pair is a Quillen equiv alence.  R emark 2.6 . An ob ject o f a coﬁbrantly gener ated category is coﬁbrant if a nd only if it is a retra c t of a cellula r complex built from genera ting coﬁbra n t cells . In the case of S G , every o b ject is built fr om cells of the form ∆ n × G/H for H ⊂ G a subgroup; it is c o ﬁbrant for the mo del structure of Pro po s ition 2 .4 if and o nly if all suc h cells ha ve H ∈ F . Thus the coﬁbrant cell complexes exhaust the class of c oﬁbrant ob jects. Observe also that they ca n b e c haracterized a s those ob jects X ∈ S G such that X H = ∅ for H / ∈ F . Equivariant homolo gy 2.7 . F or the mo del structures o f Prop osition 2.4, the functo- rial coﬁbrant repla cemen t in T op G of the p oint spa c e ∗ is a mo del for the cla ssifying space o f G with r esp ect to F and the coﬁbrant replacement of ∗ in S G is a simplicial version. Moreover be c ause | − | ∗ : S G → T op G is a Quillen equiv alence, it takes the simplicial version to the topo logical one. In par ticular if E is a functor from T op G to sp ectra and π : E ( G, F ) → ∗ is the coﬁbrant replacement in S G , then we hav e a map (2.8) E ( π ) : E ( |E ( G, F ) | ) → E ( ∗ ) If E ( X ) = F % ( X ) = R ( X ) ⊗ Or G F := Z Or G X H + ∧ F ( G /H ) for so me functor F : Or G → S pt , (2.8) is the Davis-L¨ uck assembly map of [6, § 5.1]. In case F = | F ′ | is the geometric rea liz ation of a functoria l sp ectrum in the simplicial set se ns e, w e hav e further | F ′ | % ( | X | ) = | Z Or G X H + ∧ F ′ ( G/H ) | = | F ′ % ( X ) | and the ass em bly map for F is the g eometric realiz a tion of that of F ′ . Hence we can equiv alently work with assem bly maps in the top olo gical o r the simplicial s e tting; we choos e to do the latter. In particular all sp ectra co nsidered henceforth are simplicial. If C is a chain co mplex , we will write | C | for the sp ectrum a sso ciated to it by the Dold-K an cor resp ondence; since top olo g ical spaces will o ccur only rarely 10 GUILLERMO COR TI ˜ NAS AND EUGENIA E LLIS from now on, a nd since we will not use | | to indicate realization, this s ho uld ca use no confusion. 3. Rings a nd ca tegories 3.1. Cros sed p ro ducts and equiv ariant homol ogy . A gr oup oid is a small cat- egory where all arrows are isomorphisms. Let G b e a group oid, a nd let R be a unital ring. An action of G on R is a functor ρ : G → Rings 1 such that ρ ( x ) = R for all x ∈ ob G . F o r example we may take ρ ( g ) = i d R for all a rrows g ∈ ar G ; this is called the trivial action. Whenever ρ is ﬁxed, we omit it from our notation, and write g ( r ) = ρ ( g )( r ) for g ∈ ar G and r ∈ R . Given a tr iple ( G , ρ, R ), we consider a small Z -linear category R ⋊ G . The ob jects of R ⋊ G are those of G , and hom R ⋊ G ( x, y ) = R ⊗ Z [hom G ( x, y )] If s ∈ R and g ∈ hom G ( x, y ), we wr ite s ⋊ g for s ⊗ g . Comp osition is deﬁned by the rule (3.1.1) ( r ⋊ f ) · ( s ⋊ g ) = r f ( s ) ⋊ f g here r, s ∈ R , and f and g are comp osable ar rows in G . In case the ac tio n of G on R is trivia l, we also write R [ G ] for R ⋊ G . Let G b e a group; cons ider the functor G G : G − S ets → G pd which sends a G -set S to its t r ansp ort gr oup oid . By deﬁnition ob G G ( S ) = S , and hom G G ( S ) ( s, t ) = { g ∈ G : g · s = t } . Notation 3.1 .2 . If E is a functor from Z -linear categorie s to spectra, R a unital G -ring, and X a G -space, we put H G ( X, E ( R )) := E ( R ⋊ G G (?)) % ( X ) 3.2. The ring A ( C ) . Le t C be a small Z -linear categor y . Put (3.2.1) A ( C ) = M a,b ∈ ob C hom C ( a, b ) If f ∈ A ( C ) write f a,b for the comp onent in ho m C ( b, a ). The following m ultiplication law (3.2.2) ( f g ) a,b = X c ∈ ob C f a,c g c,b makes A ( C ) int o an asso ciative ring, which is unital if and only if ob C is ﬁnite. Whatever the cardinal of ob C is, A ( C ) is always a ring with lo c al units , i.e. a ﬁltering colimit o f unital rings. A (?) and tensor pr o ducts. The tensor pr o duct of t w o Z -linea r categories C a nd D is the Z -linear c a tegory C ⊗ D with o b( C ⊗ D ) = ob( C ) × ob( D ) and hom C ⊗D (( c 1 , d 1 ) , ( c 2 , d 2 )) = ho m C ( c 1 , c 2 ) ⊗ hom D ( d 1 , d 2 ) W e have A ( C ⊗ D ) = A ( C ) ⊗ A ( D ) Example 3.2.3 . If G is a gr oup o id ac ting trivially on a unital ring R , then A ( R [ G ]) = A ( R ⊗ Z [ G ]) = R ⊗ A ( Z [ G ]) ISOMORPHISM CONJECTURES WITH PROPER COEFFICIE NTS 11 A (?) and cr osse d pr o ducts. If A is any , not necessarily unital ring, a nd G is a group oid acting on A , we put A ( A ⋊ G ) = M x,y ∈ ob G A ⊗ Z [hom G ( x, y )] The rules (3.1.1) and (3.2.2) make A ( A ⋊ G ) into a ring, which in general is non unital and do es not hav e lo cal units. The r ing A ( A ⋊ G ) ma y also b e describ ed in terms of the unitalization ˜ A of A . By deﬁnition, ˜ A = A ⊕ Z equipp ed with the trivial G - action on the Z -summand and the following multiplication (3.2.4) ( a, λ )( b, µ ) = ( ab + λb + aµ, λµ ) W e have (3.2.5) A ( A ⋊ G ) = ker( A ( ˜ A ⋊ G ) → A ( Z [ G ])) Note that A ( A ⋊ G ) is deﬁned, even thoug h A ⋊ G is not. One can actua lly deﬁne A ⋊ G as a nonunital ca tegory , i.e. a ca tegory without identit y morphisms, but we do not go into that in this pa per . Next w e ﬁx a group G and a subgr oup H ⊂ G a nd consider the ring A ( A ⋊ G G ( G/H )) asso ciated to the crossed pro duct by the tra nspo rt g roup oid. Note tha t hom G G ( G/H ) ( H, H ) = H = hom G H ( H/H ) ( H, H ) th us there is a fully faithful functor G H ( H/ H ) → G G ( G/H ). This functor induces a ring homomo rphism  : A ⋊ H = A ( A ⋊ G H ( H/ H )) ⊂ A ( A ⋊ G G ( G/H )) The next lemma compares the map  with the ca nonical inclusion ι : A ⋊ H → M G/H ( A ⋊ H ) , x 7→ e H,H ⊗ x In the following lemma and elsewhere, we make us e of a s ection s : G/H → G of the canonica l pro jection onto the quotien t by a sugroup H ⊂ G . W e sa y that the section s is p ointe d if it is a ma p o f po in ted s ets, that is, if it maps the cla ss of H to the element 1 ∈ G . Lemma 3. 2.6. L et A b e a ring, G a gr oup acting on A , and H ⊂ G a sub gr oup. Then ther e is an isomorphi sm α : A ( A ⋊ G G ( G/H )) ∼ = − → M G/H ( A ⋊ H ) making the fol lowing di agr am c ommute: A ⋊ H  / / ι ' ' O O O O O O O O O O O O A ( A ⋊ G G ( G/H )) ≀ α   M G/H ( A ⋊ H ) The isomorphism α is natur al in A but not in the p air ( G, H ) , as it dep ends on a choic e of p ointe d se ction s : G/H → G of the pr oje ction π : G → G/H . Pr o of. Le t s be as in the lemma; put ˆ g = s ( π ( g )) ( g ∈ G ). The isomorphism α : A ( A ⋊ G G ( G/H )) ∼ = − → M G/H ( A ⋊ H ) is deﬁned as follows. F or b ∈ A , s, t ∈ G , and g ∈ hom G G ( G/H ) ( sH, tH ), put α ( b ⋊ g ) = e tH,sH ⊗ ˆ t − 1 ( b ) ⋊ ( ˆ t − 1 g ˆ s ) It is straig h tforward to chec k that α is an isomor phis m and that α  = ι .  12 GUILLERMO COR TI ˜ NAS AND EUGENIA E LLIS F u nctoriality of A (?) . If F : C 7→ D is a Z -linear functor which is injectiv e on ob jects, then it deﬁnes a ho momorphism A ( F ) : A ( C ) → A ( D ) b y the rule α 7→ F ( α ). Hence we may regar d A a s a functor (3.2.7) A : inj − Z − Cat → Rings from the categ ory of Z -linear categ o ries and functors which are injective on ob jects, to the catego ry of rings. How ev er A ( F ) is not deﬁned fo r general Z -line a r F . R emark 3 .2.8 . The use of the preﬁx inj here diﬀers from that in [6]. Indeed, here inj indicates that functors are injective o n ob jects, whereas in [6], it refers to functors which are injective on ar rows. 3.3. The non unital case. A Milnor squar e is a pullba ck square of rings (3.3.1) R ′   / / R f   S ′ g / / S such tha t either f or g is sur jectiv e. Below w e shall assume f is surjective. Let E : Z − Cat → Spt b e a functor. If A is a not necess arily unital ring, embedded as an ideal in a unital r ing R , w e write E ( R : A ) = ho ﬁber ( E ( R ) → E ( R/ A )). The functor E is said to s atisfy excision for the Milnor squa re (3.3.1) if E ( R ′ )   / / E ( R ) E ( f )   E ( S ′ ) / / E ( S ) is homotopy car tesian. If ker f ∼ = A , then E satisﬁes excisio n on (3.3.1) if and only if E ( R ′ , R : A ) = hoﬁb er( E ( R ′ : A ) → E ( R : A )) is weakly contractible. W e say that the ring A is E -excisive if E satisﬁes excision on every Milnor square (3.3.1) with ker f ∼ = A . Assume unital ring s are E - excisive; if A is any , not necess a rily E -excis iv e ring, we consider its unitalization ˜ A , deﬁned in (3.2.4) ab ov e. Put E ( A ) = hoﬁb er( E ( ˜ A ) → E ( Z )) Because of o ur assumption that unital r ing s are E -excis iv e, if A ha pp ens to b e unital, the t w o deﬁnitions o f E ( A ) are naturally ho motopy equiv a len t. Note that if 0 → A ′ → A → A ” → 0 is an exact sequence of rings and A ′ is E -excisive, then E ( A ′ ) → E ( A ) → E ( A ”) is a ho mo top y ﬁbra tio n. W e say tha t E is excisiv e or that it satisﬁes excision , if every ring is E -excisive. Standing Assumptions 3.3.2 . F r om now on, we sha ll be primar ily concer ned with functors E : Z − Ca t → Spt that satisfy the following: i) Every r ing with lo cal units is E -excisive. ii) If H is a group and A a n E -excisive H -ring, then A ⋊ H is E -excisive. ISOMORPHISM CONJECTURES WITH PROPER COEFFICIE NTS 13 iii) If A is E -excisive, X a set and x ∈ X , then M X A is E -excisive, and E sends the map A → M X A , a 7→ e x,x a to a weak equiv a le nce. iv) Ther e is a natural weak equiv alence E ( A ( C )) ∼ − → E ( C ) of functors inj − Z − Cat → Spt . v) Let { A i : i ∈ I } b e a family of r ings, and let A = L i ∈ I A i be their direct sum, with co ordinate- w is e multiplication. Then A is E -excisive if a nd only if ea ch A i is. Moreov er if these equiv ale n t conditions are satisﬁed, then the map L i E ( A i ) → E ( A ) is an eq uiv alence. R emark 3.3 .3 . Obser ve that standing a ssumptions i)-iii) and v) ar e o nly co ncerned with the re s triction of E to the full sub category Ring s ⊂ Z − Cat, and that assump- tion iv) says that E | Rings determines the whole functor up to weak equiv alence. How e ver the assumptions are enough to prov e for instance that E maps categ ory equiv alences to equiv alences of sp ectra; see 3.3 .7. Note also that the eq uiv alence of iv) is natura l only with r espec t to functors which are injective on ob jects, b ecause A ( − ) is only functorial on inj − Z − Cat. One co uld ask whether it is p ossible to extend a functor E : Rings → Spt s atisfying i)-iii) a nd v) to all of Z − Ca t in such a wa y that iv) is satisﬁed. In the next subsec tion we intro duce a functor R : Z − Cat → Ring s which r estricts to the identit y on Rings a nd a natura l trans- formation p : R → A of functor s inj − Z − Cat → Ring s and discuss c o nditions on E under which E ( p ) is an equiv a lence. R emark 3.3.4 . The exa mples we are primarily interested in, na mely K -theo ry and Ho c hschild and cyclic ho mology , satisfy a s tronger version o f prop erty i). Indeed, they no t only satisfy excision for rings with lo cal units, but also for (ﬂa t) s -unital rings. A ring A is ca lled s - unital if for ev ery ﬁnite co llection a 1 , . . . , a n ∈ A there exists an element e ∈ A such that a i e = ea i = a i . Note that if we add the requirement tha t e be idemp otent w e recov er the notion of ring with lo cal units. As is explained in the App endix (Example A.3.5) ev ery s - unital r ing is excisive fo r bo th Hochschild a nd cyclic homology , and every s -unital ring which is ﬂat as an ab elian group is K -excisive. R emark 3.3.5 . If E satisﬁe s excis ion, then assumptions i) and ii) hold automatica lly , and assumptions iii) a nd v) hold if and only if they hold for unital ring s . Lemma 3.3.6. L et E : Z − Ca t → Spt b e a fun ctor satisfying the standing assump- tions ab ove. If F i : C → D i = 0 , 1 ar e natur al ly isomorph ic line ar functors, then E ( F 0 ) and E ( F 1 ) ar e ho motopic. Pr o of. Le t G [1] = { 0 ⇆ 1 } be the gro upoid with tw o ob jects and exactly one isomorphism b etw een any tw o given (equa l or distinct) ob jects. The linear functors F, G : C → D are equiv alen t if the dotted ar r ow in the following diag ram of Z -linear functors exists and makes it co mm ute C ⊗ Z [ G [1]]   C ⊕ D = C ⊗ Z [ob G [1 ]] ι 0 ⊕ ι 1 5 5 l l l l l l l l l l l l l l F ⊕ G / / D Hence it suﬃces to show that E ( ι 0 ) ∼ = E ( ι 1 ). By a ssumption iv ), we are reduced to showing that E ( A ( ι 0 )) ∼ = E ( A ( ι 1 )). But one chec ks that A ( C ⊗ Z [ G [1 ]]) = M 2 ( A ( C )) 14 GUILLERMO COR TI ˜ NAS AND EUGENIA E LLIS and that the ι i induce the tw o c anonical inclus io ns x 7→ x ⊗ e 1 , 1 , x ⊗ e 2 , 2 , hence we are done by as s umption iii) (see [2, Lemma 2 .2.4], e.g.).  R emark 3.3.7 . It follows from L e mma 3.3.6 that E sends categor y eq uiv alences to equiv alences of sp ectra. Let G b e a group. Assume E satisﬁes the standing a ssumptions abov e. F or A an E -excisive G -ring , consider the O r G -sp ectrum (3.3.8) G/H 7→ E ( A ⋊ G G ( G/H )) = ho ﬁber( E ( ˜ A ⋊ G G ( G/H )) → E ( Z [ G G ( G/H ))]) Applying (?) % to (3.3.8) deﬁnes an equiv ariant homo logy theo ry of G -simplicial sets, which we denote H G ( − , E ( A )). Moreover, for ea c h ﬁxe d G -simplicial set X , H G ( X, E (?)) is a functor of E -ex cisive rings. Obser v e that, for unital A , we hav e t wo deﬁnitions of E ( A ⋊ G G ( − )) and t w o deﬁnitions of H G ( − , E ( A )); the next prop osition says that the tw o deﬁnitions are equiv a len t. Prop osition 3.3.9. L et E : Z − Cat → Spt b e a functor and G a gr oup. Assume that E satisﬁes the st anding assu mptions 3.3.2 ab ove. a) If R is a unital G -ring, then the two deﬁn itions of E ( R ⋊ G G ( − )) and the two deﬁnitions of H G ( − , E ( R )) ar e e qu ivalent. b) If 0 → A ′ → A → A ” → 0 is an exact se quenc e of E -ex cisive G -rings, and X is a G -simplic ial s et , t hen E ( A ′ ⋊ G G ( − )) → E ( A ⋊ G G ( − )) → E ( A ” ⋊ G G ( − )) and H G ( X, E ( A ′ )) → H G ( X, E ( A )) → H G ( X, E ( A ”)) ar e homotopy ﬁ br ations. Pr o of. If A is E -excis iv e and H ⊂ G is a subgroup, then conditions ii) and iii) together with Lemma 3.2.6 imply that A ( A ⋊ G G ( G/H )) is E -excis iv e. Hence, by condition iv), the sp ectrum in (3.3.8) is equiv alent to E ( A ( A ⋊ G G ( G/H ))). In particular, by i), A ( R ⋊ G G ( G/H )) is E -excis iv e for R unital, a nd the map hoﬁb er( E ( ˜ R ⋊ G G ( G/H )) → E ( Z [ G G ( G/H )])) → E ( R ⋊ G G ( G/H )) induced by the pr o jection ˜ R ∼ = R × Z → R is an e q uiv alence. T his proves a ). Moreov er, b ecause A (? ⋊ G G ( G/H )) preserves exa ct sequences, applying (3.3 .8) to the exact sequence of part b) y ields an ob ject-wise homotopy ﬁbration of Or G - sp ectra, which is the ﬁr st homotopy ﬁbr a tion of b). Applying (?) % we obtain the second one.  R emark 3.3.10 . Let E : Z − Cat → Spt and let A b e any , not necessa rily E -excis iv e G -ring, equiv a riantly embedded as a n ideal in a unital G -ring R . Consider the Or G -sp ectrum E ( R ⋊ G G ( − ) : A ⋊ G G ( − )) = hoﬁb er( E ( R ⋊ G G ( − )) → E (( R/ A ) ⋊ G G ( − ))) Put H G ( X, E ( R : A )) = E ( R ⋊ G G ( − ) : A ⋊ G G ( − )) % ( X ) . ISOMORPHISM CONJECTURES WITH PROPER COEFFICIE NTS 15 A ( G, F )-co ﬁbrant replacement cX → X gives rise to a map of homotopy ﬁbrations H G ( cX , E ( R : A )) / /   H G ( cX , E ( R ))   / / H G ( cX , E ( R/ A ))   H G ( X, E ( R : A )) / / H G ( X, E ( R )) / / H G ( X, E ( R/ A )) If H G ( cX , E ( S )) → H G ( X, E ( S )) is an equiv alence for a ll unital S , then b oth the middle a nd right hand side vertical maps are equiv alences; it follows that the sa me is true of the ma p on the left. W e r ecord a pa rticular case of this in the fo llowing corolla r y . Corollary 3.3.11 . L et E : Z − Cat → Spt b e a funct or; assume E satisﬁes t he Standing Assumptions 3.3. 2. F urther let G b e a gr oup, X ∈ S G , F a family of sub gr oups, cX → X and ( G, F ) -c oﬁbr ant r eplac ement. Assume t hat the assembly map H G ( cX , E ( R )) → H G ( X, E ( R )) is an e quivalenc e for every un ital ring R . Then H G ( cX , E ( A )) → H G ( X, E ( A )) is an e quivalenc e for every E -excisive ring A . Prop osition 3 .3.12. L et A ⊳ R b e an ide al in a unital G -ring, close d under the action of G . L et E : Rings → Spt b e a functor satisfying t he st anding assumptions. If A is E -excisive then E ( A ⋊ G G ( − )) → E ( R ⋊ G G ( − ) : A ⋊ G G ( − )) is an obje ct- wise we ak e quivalenc e of O r G -sp e ctr a. Pr o of. The pro of follows fr om Lemma 3 .2.6, using assumptions ii), iii) and iv).  3.4. The ring R ( C ) . Let C be a Z -linear c a tegory . Imitating a co ns truction us ed by M. J oachim ([13 ]) in the C ∗ -algebra context, we s ha ll a sso ciate to C a r ing R ( C ) which is a quo tien t of the tensor alg ebra o f A ( C ); ﬁrs t we need some notation. If M is an abelian group, we write T ( M ) = L n ≥ 1 M ⊗ n for the (unaugmen ted) tenso r algebra. Put R ( C ) = T ( A ( C )) / < { g ⊗ f − g ◦ f : f ∈ hom C ( a, b ) , g ∈ hom C ( b, c ) , a, b, c ∈ ob C } > Note that a n y Z -line a r functor C → D ∈ Z − Cat deﬁnes a homomorphism R ( C ) → R ( D ). Thu s we may regard R as a functor R : Z − Cat → Rings , C 7→ R ( C ) Observe that the canonical sur jection T ( A ( C )) → A ( C ) facto r s through a map (3.4.1) p : R ( C ) ։ A ( C ) whose kernel is the ideal generated by the elements g ⊗ f fo r non-co mpos able g and f . F or example if C ha s only one o b ject, then p is the identit y . In pa rticular any functor E : Rings → Spt can be extended to Z − Ca t via E ( C ) = E ( R ( C )), and p induces a na tural trans formation E ( p ) : E ( C ) → E ( A ( C )) of functors of inj − Z − Ca t. Example 3.4.2 . Let R , S be unital rings, and let C b e the Z -linear category with t wo ob jects a a nd b such that ho m C ( a, b ) = hom C ( b, a ) = 0, ho m C ( a, a ) = R and hom C ( b, b ) = S . Then A ( C ) = R ⊕ S and R ( C ) = R ` S is the no n unital copro duct. W e shall see in P rop osition 4 .3.1 that K -theo ry sa tisﬁes the standing ass umptions; how ev er in g eneral K ∗ ( R ` S ) 6 = K ∗ ( R ) ⊕ K ∗ ( S ). 16 GUILLERMO COR TI ˜ NAS AND EUGENIA E LLIS In Lemma 3.4.3 w e give co nditions on E which guarantee that it sends the map (3.4.1) to a w eak equiv a lence. First w e need some notation. If B is a ring, we write ev i : B [ t ] → B i = 0 , 1 for the ev aluation maps. If f , g : A → B are ring homomor phisms, then a (p olynomia l) elementary homotopy betw een f and g is a ma p H : A → B [ t ] such that ev 0 H = f and ev 1 H = g . A homotopy from f to g is a s equence of homomorphisms f = h 0 , . . . , h n = g and elemen tary homotopies H i : A → B [ t ] from h i to h i +1 . The functor E is invariant under p olynomial homotopy if for every ring A , E sends the inclusio n A ⊂ A [ t ] to a weak equiv alence. B e cause the compo site inc ◦ ev 0 : A [ t ] → A [ t ] is homotopic to the ident it y , if E is inv aria n t under p olynomial homo top y , and f and g are homotopic ring homomorphisms, then E ( f ) and E ( g ) deﬁne the same map in HoSpt . Lemma 3. 4 .3. L et E : Ring s → Spt b e a fun ctor. Assume that E satisﬁes standing assumptions i) and iii). L et C b e a Z -line ar c ate gory such t hat R ( C ) is E -excisive. Then E ∗ sends (3.4.1) to a natu ra l ly split surje ction. Assume in addition that E is invariant under p olynomial homotopy . Then E sends (3.4.1) to a we ak e quivalenc e. Pr o of. Le t ob + C = o b C ` { + } b e the set of ob jects of C with a base p oint added. Consider the homomor phism j : A ( C ) → M ob + C R ( C ) , j ( f ) = f ⊗ e b,a ( f ∈ hom C ( a, b )) W rite p for the map (3 .4 .1). Consider the matr ices V = X a ∈ ob C 1 a ⊗ e a, + W = X a ∈ ob C 1 a ⊗ e + ,a The comp osite q = M ob + C ( p ) ◦ j sends f ∈ A ( C ) to q ( f ) = W f ⊗ e + , + V Observe that left multiplication b y W and rig h t multiplication b y V leave M ob + C A ( C ) stable, and that aV W a ′ = aa ′ for all a, a ′ ∈ M ob + C A ( C ). By the ar g umen t of [2, 2.2.6], all this tog ether with matr ix in v aria nce imply that E ∗ ( q ) = E ∗ (? ⊗ e + , + ) is an iso morphism. This prov es the ﬁrst assertion o f the Lemma. T o prov e the s e cond, it suﬃces to show that r = j ◦ p is ho motopic to the inclusion ι ( a ) = a ⊗ e + , + . If f ∈ hom C ( a, b ), write H ( f ) ∈ M ob + C ( R ( C ))[ t ] for H ( f ) = f ⊗ ( − t ( t 3 − 2 t ) e + , + + t ( t 2 − 1 ) e + ,a + (1 − t 2 )( t 3 − 2 t ) e b, + + (1 − t 2 ) 2 e b,a ) Note that ev 0 H ( f ) = r ( f ), ev 1 H ( f ) = ι ( f ). F urther, one chec ks that if g ∈ hom C ( b, c ), then H ( g f ) = H ( g ) H ( f ). Thus H induces a homomorphism R ( C ) → M ob + C ( R ( C ))[ t ] which is a homotopy fro m r to ι . This concludes the pro of.  Example 3.4 .4 . If E : Rings → Spt is excis ive and homotopy inv a riant and satisﬁes standing assumptions iii) and v), then its extension E ◦ R : Z − Cat → Spt satisﬁes all the standing a ssumptions and ag rees with E on Rings. If F is another extension of E which also sa tisﬁes the s tanding ass umptions, then co mpos ing E ( R ( C )) → E ( A ( C )) with the ma p o f assumption 3 .3.2 iv), we get an e q uiv alence E ( R ( C )) → F ( C ) which is natural with r esp e c t to functors which are injective o n ob jects. ISOMORPHISM CONJECTURES WITH PROPER COEFFICIE NTS 17 4. K -theor y 4.1. The K -theory sp ectrum. Given a Z -linear c ategory C , w e denote by C ⊕ the Z -linear category whose ob jects a re ﬁnite s equences of ob jects o f C , a nd whose morphisms are matrices of morphisms in C with the obvious matrix pr o duct as comp osition. Conc a tenation of se q uences yields a sum ⊕ and hence we obtain, functorially , an additive categor y; write Idem C ⊕ for its idemp otent completion. W e shall also need Kar oubi’s c one Γ( C ) ([15, pp 270 ]). The ob jects of Γ( C ) are the sequences x = ( x 1 , x 2 , . . . ) of ob jects of C suc h that the se t (4.1.1) F ( x ) = { c ∈ C : ( ∃ n ) x n = c } is ﬁnite. A map x → y in Γ( C ) is a matrix f = ( f i,j ) of homo morphisms f i,j : x j → y i such that (1) Ther e exists an N such that every r ow and every column of f has at most N nonzer o entries. (2) The set { f i,j : i, j ∈ N } is ﬁnite. Int ersp ersing of sequences deﬁnes a symmetric monoidal o per ation ⊞ : Γ( C ) × Γ( C ) → Γ( C ) and there is an endofunctor τ such that 1 ⊞ τ ∼ = τ (see [14, § II I]). If C has ﬁnite dire c t sums, e.g. if C = D ⊕ for some Z - line a r categor y D , then the int ersp ersing op eration is naturally e quiv alent to the induced sum ( x ⊕ y ) i = x i ⊕ y i ([14, Lemme 3.3]). In par ticular, if C is additive, then Γ C is a ﬂasqu e additive category ; that is, ther e is an additive endo functor τ : C → C s uch that τ ⊕ 1 ∼ = τ . A mor phism f in Γ( C ) is ﬁnite if f ij = 0 for all but ﬁnitely many ( i, j ). Finite morphisms form an ideal, and we wr ite Σ( C ) for the ca teg ory with the same ob jects as Γ( C ), and morphisms taken mo dulo the ideal o f ﬁnite morphisms . The catego ry Σ( C ) is Karoubi’s su sp ension o f C . By [24, Thm. 5 .3], if C is a dditive, we have a homotopy ﬁbratio n s equence (4.1.2) K Q (Idem C ) → K Q (Γ(Idem C )) → K Q (Σ(Idem C )) Here each of the categor ies is regar ded as a semis imple exact ca tegory , and K Q denotes the ﬁbrant s implicia l set for its algebra ic K -theory . Beca us e Γ(Idem C ) is ﬂasque, K Q (Γ(Idem C )) is contractible, whence K Q (Idem C ) ∼ = Ω K Q (Σ(Idem C )). Now let C be a n y s mall Z -linea r categor y , p oss ibly w itho ut direct sums. C o nsider the sequence of ca tegories (4.1.3) C (0) = Idem( C ⊕ ) , C ( n +1) = Idem(Σ C ( n ) ) Then we have a sp ectrum K ( C ) = { n K ( C ) } , with (4.1.4) n K ( C ) ∼ = K Q ( C ( n ) ) R emark 4.1.5 . If R is a unital ring, then by [15, Pro p. 1.6], we hav e category equiv alences (4.1.6) Idem(Γ(pro j( R ))) ∼ = pro j(Γ( R )) a nd Idem(Σ(pro j( R ))) ∼ = pro j(Σ( R )) Hence th e spectr um K ( R ) deﬁned a b ove is equiv alent to the us ual, Gersten-Ka r oubi- W ago ner spectrum of the ring R . 18 GUILLERMO COR TI ˜ NAS AND EUGENIA E LLIS 4.2. Com paring K ( C ) with K ( A ( C )) . The op er ation ♦ . Let X b e a set a nd let C a nd D be Z -linear c a tegories with ob C = ob D = X . Consider the categor y C ♦ D with set o f ob jects ob( C ♦ D ) = X , homomorphisms hom C ♦ D ( x, y ) = ho m C ( x, y ) ⊕ hom D ( x, y ) and co ordinate-wis e comp osition. If C , D and E are Z -linea r categor ies, we hav e ( C ♦ D ) ⊕ = C ⊕ ♦ D ⊕ Idem(( C ♦ D ) ⊕ ) = Idem C ⊕ × Idem D ⊕ (4.2.1) ( C ♦ D ) ⊗ E = ( C ⊗ E ) ♦ ( D ⊗ E ) (4.2.2) Unitalization. W e hav e already re called the de ﬁnitio n of the unitalization ˜ A o f a not necessa rily unital ring A . Now we need a version of unitalization for Z -linear categorie s; this can be more g enerally deﬁned for no nunital Z -categ ories, but w e will have no o ccasion for that. Let C ∈ Z − Cat; wr ite ˜ C for the category with ob ˜ C = o b C and with homomo rphisms given by hom ˜ C ( x, y ) = ho m C ( x, y ) ⊕ δ x,y Z =  hom C ( x, y ) x 6 = y hom C ( x, x ) ⊕ Z x = y Comp osition b et ween ( f , δ x,y n ) ∈ hom ˜ C ( x, y ) a nd ( g , δ y ,z m ) ∈ hom ˜ C ( y , z ) is deﬁned by the for m ula ( g , δ y ,z m ) ◦ ( f , δ x,y n ) = ( g f + δ y ,z mf + δ x,y g n, δ x,y δ y ,z mn ) Observe that if R is a ring , co ns idered as a Z -linear categor y with one ob ject, then ˜ R → R × Z = R ♦ Z , ( r , n ) 7→ ( r + n · 1 , n ) is an iso morphism. This isomo r phism generalizes to Z -ca tegories as follows. Let Z h ob C i ∈ Z − C a t, be the Z -linear categ ory with the same ob jects as C and homo- morphisms given by hom Z h ob C i ( x, y ) = δ x,y Z W e have an isomor phism of linear catego ries (4.2.3) C ♦ Z h ob C i ∼ = − → ˜ C which is the identit y o n o b jects, as well as on hom C ♦ Z h ob C i ( x, y ) for x 6 = y , and which sends hom C ♦ Z h ob C i ( x, x ) ∋ ( f , n ) 7→ ( f − n 1 x , n ) ∈ hom ˜ C ( x, x ) ISOMORPHISM CONJECTURES WITH PROPER COEFFICIE NTS 19 The m ap K ( C ) → K ( A ( C )) . If C is a Z -linear ca tegory , and x, y ∈ ob C , then by deﬁnition of A ( C ), (4.2.4) hom C ( x, y ) ⊂ A ( C ) and the inclusion is compatible with co mpo s ition. W e also hav e an inclusion (4.2.5) hom ˜ C ( x, x ) ∋ ( f , n ) 7→ ( f , n ) ∈ ] A ( C ) The inclusions (4.2.4) and (4 .2.5) together with the o nly map o b ˜ C → ob ] A ( C ) = { •} deﬁne a functor (4.2.6) φ : ˜ C → ] A ( C ) Observe that Z h ob C i ⊂ ˜ C and that φ ( Z h ob C i ) ⊂ Z ⊂ ] A ( C ). W e have a commut ative diagram ˜ C φ / / π 1   ] A ( C ) π 2   Z h ob C i / / Z Here the v ertical maps are the obvious pro jections. By (4.2 .3) a nd (4.2 .1) we hav e an equiv ale nc e K ( ˜ C ) ∼ − → K ( C ) × K ( Z h ob C i ) Under this equiv alence the map induced b y π 1 bec omes the ca nonical pr o jection; hence its ﬁber is K ( C ). On the other hand, b y deﬁnition, K ( A ( C )) is the ﬁb er of K ( π 2 ). Hence φ induces a map (4.2.7) ϕ : K ( C ) → K ( A ( C )) Prop osition 4.2.8. L et C b e a Z -line ar c ate gory. Then the map (4.2.7) is an e quivalenc e. Pr o of. Be cause b oth the source and the target of (4.2.7) comm ute with ﬁltering colimits, w e may assume that C ha s ﬁnitely ma n y ob jects. Then A ( C ) is unital, and thus we have a n iso morphism ] A ( C ) ∼ = A ( C ) × Z . Recall that the ide mp o- ten t c o mpletion o f an additive catego ry A is the categor y whose o b jects a re the idempo ten t endomorphis ms in A and where a map f : e 1 → e 2 is an element o f hom A (dom e 1 , do m e 2 ) such that f = e 2 f e 1 . One checks that the comp osite C ⊕ → Idem C ⊕ 1 × 0 → Idem C ⊕ × Idem Z h ob C i ∼ = Idem( ˜ C ⊕ ) φ → Idem( ] A ( C ) ⊕ ) ∼ = Idem( A ( C ) ⊕ ) × Idem( Z ⊕ ) → Idem( A ( C ) ⊕ ) is the functor ψ which sends an ob ject ( c 1 , . . . , c n ) to the idemp o ten t diag(1 c 1 , . . . , 1 c n ) and a map f = ( f i,j ) : ( c 1 , . . . , c n ) → ( d 1 , . . . , d m ) to the corresp onding matrix ( f i,j ) ∈ hom A ( C ) ⊕ ( • n , • m ). Because ψ is fully faithful and coﬁna l, it induces an equiv alence K ( C ) → K ( A ( C )). It follows that (4.2 .7) is an equiv alence.  20 GUILLERMO COR TI ˜ NAS AND EUGENIA E LLIS 4.3. K -theory and the standing assumptions. Prop osition 4.3.1. The functor K : Z − Cat → Spt satisﬁes t he standing assump- tions. Pr o of. Ass umption iv) was pro ved in Prop osition 4.2.8 above. The remaining as- sumptions are either prov ed in App endix A or follow fro m results ther ein. By Example A.1.1, rings with lo cal units are K -excisive; hence K -theor y satisﬁes i). Assumption ii) holds by Prop osition A.6.3. If A is K -excisive and X is a set, then M X A is K -excisive, b y Prop osition A.5.3. Assumption iii) follows from this and the fact that K - theory is matrix stable o n unital ring s. Ass umption v) is prov ed in Prop osition A.4.4.  5. Homotopy K -theor y If C is a Z - line a r categor y , then we wr ite C ∆ • for the simplicial Z -linear ca tegory (5.1) C ∆ • : [ n ] 7→ C ∆ n = C ⊗ Z [ t 0 , . . . , t n ] / < t 0 + · · · + t n − 1 > Applying the functor K dimensionwise w e get a simplicial s p ectrum whose total sp ectrum is the homotopy K -the ory sp ectrum K H ( C ). In particula r if R is a unital ring, then K H ( R ) was deﬁned by W eib el in [30]. The following theorem was prov ed in [30]; see also [2, § 5]. Theorem 5.2. (Weib el) The functor K H : Rings → Spt is excisive, matrix in- variant, and invariant under p olynomial homotopy. Prop osition 5.3. Ther e is a natur al we ak e qu ivalenc e K H ( C ) ∼ − → K H ( R ( C )) . Pr o of. W e begin by observing that the inclusions (4.2.4) and ( 4.2.5) lift to inclusions hom C ( x, y ) ⊂ R ( C ) and hom ˜ C ( x, x ) ⊂ ] R ( C ). Thus w e hav e a functor φ ′ : ˜ C → ] R ( C ) Comp osing it with ˜ p : ] R ( C ) → ] A ( C ) we obtain the map φ : ˜ C → ] A ( C ) of (4.2.6) ab ov e. T ensoring with Z ∆ • and applying K ( − ) we obta in a commutativ e diagram K H ( ˜ C ) φ % % K K K K K K K K K K φ ′ / / K H ( ] R ( C )) p   K H ( ] A ( C )) The diagram ab ov e maps to the diagr am K H ( Z h ob C i ) φ / / φ & & N N N N N N N N N N N K H ( Z ) K H ( Z ) ISOMORPHISM CONJECTURES WITH PROPER COEFFICIE NTS 21 T aking ﬁb e rs and using (4.2.1), (4.2.2) and (4 .2.3), w e o bta in a homotopy commu- tative diag r am K H ( C ) ϕ ” / / ϕ ′ & & M M M M M M M M M M K H ( R ( C )) p   K H ( A ( C )) Here ϕ ′ comes from a map of simplicia l spec tr a (5.4) ϕ • : K ( C ⊗ Z ∆ • ) ∼ − → K ( ˜ C ⊗ Z ∆ • : C ⊗ Z ∆ • ) → K ( ] A ( C ) ⊗ Z ∆ • : A ( C ) ⊗ Z ∆ • ) ∼ ← K ( A ( C ) ⊗ Z ∆ • ) , and ϕ 0 = ϕ is the map (4.2.7) , whic h is an equiv alence by P r op osition 4.2 .8. T he same argument o f the pr o of of P rop osition 4.2.8 shows that ϕ n is an equiv alence for every n . On the other hand, by Theorem 5 .2 a nd Lemma 3.4.3, the map p : K H ( R ( C )) → K H ( A ( C )) is an eq uiv alence. It follows that ϕ ” is an equiv a lence to o.  Prop osition 5.5. The functor K H : Z − Cat → Spt satisﬁes the standing assump- tions. Pr o of. All as sumptions except iv) follow from Theorem 5 .2. Assumption iv) follows from the pr o of of Pr op osition 5.3, and a ls o fr om combining the s tatemen t of that prop osition with Lemma 3 .4.3.  6. Cyclic homology Let A b e a ring, and M an A -bimo dule. If a ∈ A a nd m ∈ M , write [ a, m ] = am − ma and [ A, M ] = { X i [ a i , m i ] : a i ∈ A, m i ∈ M } , M ♮ = M / [ A, M ] Let B b e another ring . W e say that B is an alg ebra ov er A if B is eq uipp ed with an A -bimo dule s tr ucture such that the multiplication B ⊗ B → B factors thro ugh an A -bimo dule map B ⊗ A B → B . Consider the gra ded abe lian g roup given in degree n by the n + 1 tenso r p ow er mo dulo A -bimo dule commutators: T ( B / A ) n =  B ⊗ A n +1  ♮ Note T ( B / A ) is a quotient of T ( B / Z ). If B is unital, then T ( B / Z ) car ries a canonical cyclic mo dule structure [29, Section 9.6]; if A is unital a lso, and the A - bimo dule str ucture o n B co mes from a unital homomo rphism A → B , then the structure passe s down to the quo tient; we write C ( B / A ) for T ( B / A ) equipp ed with this cyclic module structure. The cyclic theor y of B / A , which includes Ho chschild, cyclic, neg ative cyclic and p erio dic cyclic homo lo gy , is that of C ( B / A ). If A is unital but B is not, one can unitalize B a s an A -algebr a by ˜ B A = B ⊕ A , ( b, a )( b ′ , a ′ ) = ( bb ′ + ba ′ + ab ′ , aa ′ ); the cyclic theo r y of B / A is that of the cyclic mo dule C ( ˜ B A : B / A ) = ker( C ( ˜ B A / A ) → C ( A/ A )). In the unital ca se, ther e is a natura l quasi- isomorphism C ( B / A ) → C ( ˜ B A : B / A ). In the gener a l ca se, when neither A no r B is assumed to be unital, then B has a canonical ˜ A -alge bra struc tur e, and the cyclic theory of B as an A -a lgebra is that of B as an ˜ A -alge bra; we put M ( B / A ) = C ( ˜ B A : 22 GUILLERMO COR TI ˜ NAS AND EUGENIA E LLIS B / ˜ A ). Note that if A is unital, then M ( B / A ) = C ( ˜ B A : B / A ), whence there is no ambiguit y . W e use the following notation for ho mology; we write H H ( B / A ) = ( M ( B / A ) , b ) for the Ho chsc hild complex , H H ( B / A ) n for its degree n summand, and H H n ( B / A ) for its nth homology group. W e use the same conv en tion with cyclic, neg a tiv e cyclic a nd p erio dic cyclic homolog y , which we denote H C , H N and H P . Let ℓ b e a c o mm utative unital ring and R a unital ℓ -algebra. Recall that R is called sep ar able ov er ℓ if R is pro jective as a n R ⊗ ℓ R op -mo dule. Lemma 6. 1. Le t I b e a ﬁltering p oset, I → ar(Rings) , i 7→ { A i → B i } a functor to t he c ate gory of ring ho momorphisms, and A → B = co lim i ( A i → B i ) . Assume that A i → B i is unital for al l i . Pu t C ( B / A ) = colim i C ( B i / A i ) . Then C ( B / A ) → M ( B / A ) is a quasi-iso morphism. If furthermor e e ach A i is sep ar able over Z , t hen also C ( B ) = C ( B / Z ) → C ( B / A ) is a qu asi-isomorph ism. Pr o of. The ﬁr st assertion follows from the fact that b oth C and M commute with ﬁltering co limits, and that the map is a q uasi-isomor phism in the unital c a se [16, Thm. 1.2.13 ]. The second as sertion follows similarly fro m the unital cas e.  Example 6.2 . Let C b e a small Z -linear catego r y . W e hav e an injective functor Z h ob C i → C , a nd thu s a homo morphism A ( h ob C i ) = Z (ob C ) → A ( C ), whic h is the ﬁltering co limit over the ﬁnite subsets X ⊂ ob C , o f the functor X 7→ ( A ( X ) → A ( C X )). Here C X ⊂ C is the full sub category whose ob jects ar e the elements o f X . Since A ( X ) is sepa rable, the natural maps C ( A ( C )) → C ( A ( C ) / Z (ob C ) ) → M ( A ( C ) / Z (ob C ) ) are quasi-is omorphisms, b y Lemma 6.1. Put C ( C ) = C ( A ( C ) / Z (ob C ) ) Note tha t this cyclic module is functorial on Z − Cat, even though as we have see n in (3.2.7), A ( − ) is only functor ial on inj − Z − Cat. The cyclic mo dule C ( C ) is often called the Z -line ar cyclic nerve of C ([18, § 4.2]). The cyclic theo r y of a Z -linea r category C is that of C ( C ). Note that if R is a unital r ing considered a s a Z -linea r category with one ob ject, then C ( R ) is the same cyclic mo dule that was deﬁned ab ov e. R emark 6.3 . As explained in Example 6.2 a bove, the pro jectio n C ( A ( C )) → C ( A ( C ) / Z (ob C ) ) = C ( C ) is a qua s i-isomorphism. This map has a left inverse C ( C ) → C ( A ( C )); namely the inclusion C ( C ) n = M ( c 0 ,...,c n ) ∈ ob C n +1 hom C ( c 1 , c 0 ) ⊗ · · · ⊗ hom C ( c 0 , c n ) ⊂ A ( C ) ⊗ n +1 = C ( A ( C )) n This inclusion is a quasi-is omorphism, and is compatible with the ma p (4 .2.6); indeed they b oth form part of a map of distinguished triangles: C ( C ) / / inc   C ( ˜ C ) / / φ   C ( Z h ob C i )   C ( A ( C )) / / C ( ] A ( C )) / / C ( Z ) Prop osition 6.4. H o chschild and cyclic homolo gy satisfy t he st anding assump- tions. ISOMORPHISM CONJECTURES WITH PROPER COEFFICIE NTS 23 Pr o of. M. W o dzicki show e d in [31] that the H H -excisive rings coincide with the H C -excisive ones, a nd that they ar e the H -unital rings, who s e deﬁnition is r ecalled in Subsection A.3 of the App endix. Rings with lo cal units, a nd mor e g enerally s - unital rings are H - unital by [31, Co r. 4.5 ]. By Pro po s ition A.6.4, A ⋊ G is H -unital for every H -unital G -ring A . It is c le ar fro m the deﬁnition of H -unitality that H - unital r ings are closed under ﬁltering colimits. Thus it suﬃce s to v erify Standing Assumption iii) for ﬁnite X , and this is [3 1, Cor o llary 9.8]. Assumption iv) follows from Example 6.2. Finally assumption v) is prov ed in P rop osition A.4.6.  7. Assembl y f or Ho chs ch il d and cyclic homology Let G b e a gr oup, S a G -set a nd R a unital G -ring. W e have a direct sum decomp osition (7.1) C ( R ⋊ G G ( S )) = M ( g ) ∈ c on( G ) C ( g ) ( R ⋊ G G ( S )) here con( G ) is the set of co njug a cy classes and C ( g ) ( R ⋊ G G ( S )) n is gener ated by those elementary tensors x 0 ⋊ g 0 ⊗ · · · ⊗ x n ⋊ g n with g 0 · · · g n ∈ ( g ). If g ∈ G , we write R g for R consider e d as a bimo dule over itself with the usua l left multiplication and the right m ultiplication given by x · r = xg ( r ). In Pr op o sition 7.5, we shall need the a bsolute Ho chschild homology of R with co eﬃcients in R g . In gener al if M is any R -bimo dule, w e w r ite H H ( R, M ) for the Ho chsc hild co mplex with co eﬃcients in M ([29, § 9 .1.1]). Prop osition 7 .5 b elow computes the G -equiv ar iant ho mo logy of a G -simplicial set X with co eﬃcients in H H ( R ) for an arbitrary unital G -r ing R . The case when G acts trivially on R was obtained by L ¨ uc k a nd Reic h in [18]. The case when X is a point may b e reg arded as a transpo rt g roup oid version of Lo r enz’ co mputation of H H ( R ⋊ G ) [17]; H C ( R ⋊ G ) was computed by F e ˘ ıgin and Tsyg an in [7 ]. Our pro of uses ideas fro m each of the three cited a rticles. Lemma 7. 2. Le t G b e a gr oup, S a G -set, g ∈ G , and Z g ⊂ G the c entr ali zer of g . Write E Z g := E ( Z g , { 1 } ) . Then t her e is a natu r al we ak e quivalenc e of simplicial ab elian gr oups (7.3) Z [ E Z g ] ⊗ Z [ Z g ] ( Z [ S g ] ⊗ H H ( R, R g )) ∼ − → H H ( g ) ( R ⋊ G G ( S )) T aking homotopy gr oups one obtains t he re lative T or gr oups [29, 8.7.5] : π ∗ H H ( g ) ( R ⋊ G G ( S )) = T or [( R ⊗ R op ) ⋊ Z g ] / Z ∗ ( R, R g ) Pr o of. Note tha t  Z [ E Z g ] ⊗ Z [ Z g ] ( Z [ S g ] ⊗ H H ( R, R g ))  n = Z [ Z g ] ⊗ n ⊗ Z [ S g ] ⊗ R ⊗ n +1 Deﬁne a map α : Z [ E Z g ] ⊗ Z [ Z g ] ( Z [ S g ] ⊗ H H ( R , R g )) → H H ( g ) ( R ⋊ G G ( S )) α ( z 1 ⊗ · · · ⊗ z n ⊗ s ⊗ x 0 ⊗ · · · ⊗ x n ) = x 0 ⋊ ( z 1 · · · z n ) − 1 g ⊗ ( z 1 · · · z n )( x 1 ) ⋊ z 1 ⊗ ( z 2 · · · z n )( x 2 ) ⋊ z 2 ⊗ · · · ⊗ z n ( x n ) ⋊ z n ∈ hom R ⋊ G G ( S ) ( z 1 · · · z n s, s ) ⊗ · · · ⊗ ho m R ⋊ G G ( S ) ( s, z n s ) One chec ks that α is a simplicial homomorphism. W rite U = ( R ⊗ R op ) ⋊ Z g . T o prove that α is a weak equiv alence, and also that its domain and co do ma in 24 GUILLERMO COR TI ˜ NAS AND EUGENIA E LLIS bo th compute T o r U / Z ∗ ( R, R g ), it suﬃces to ﬁnd simplicial r esolutions P ∼ − → R g and Q ∼ − → R g by r elatively pr o jectiv e U -mo dules and a simplicial mo dule homomo r- phism ˆ α : P → Q covering the iden tit y of R g and suc h that R ⊗ U ˆ α = α . W e need some nota tion. W rite E ( Z g , M ) for the simplicial Z [ Z g ]-mo dule r esolution o f a left Z g -mo dule M asso ciated to the cotriple N 7→ Z [ Z g ] ⊗ N [29, 8.6 .11]. Let C bar ( R, R g ) b e the bar r e solution (A.2); Z g acts diagona lly on Z [ S g ] ⊗ C bar ( R, R g ). W rite P = E ( Z g , Z [ S g ] ⊗ C bar ( R, R g )) for the diagonal of the bis implicial mo dule ([ p ] , [ q ]) 7→ E p ( Z g , Z [ S g ] ⊗ C bar ( R, R g ) q ). By co nstruction, P ∼ − → R g is a s im- plicial U -mo dule resolution, and every U -mo dule P n is extended from Z , whence relatively pro jectiv e. Next, given k ∈ G , consider the simplicial submodule V ( k ) ⊂ C bar ( R ⋊ G G ( S )) gener ated by the elementary tensors x 0 ⋊ h 0 ⊗ · · · ⊗ x n +1 ⋊ h n +1 ∈ hom R ⋊ G G ( S ) ( h 1 · · · h n +1 s, k s ) ⊗ · · · ⊗ hom R ⋊ G G ( S ) ( s, h n +1 s ) with s ∈ S and h 0 · · · h n +1 = k ( n ≥ 0). Put Q = V ( g ); note Q is stable under m ultiplication b y ele ments o f the form a ⋊ z ⊗ b ⋊ z − 1 ∈ ( R ⋊ G ) ⊗ ( R ⋊ G ) op with z ∈ Z g . W e hav e a r ing homomorphism ι : U → ( R ⋊ G ) ⊗ ( R ⋊ G ) op a ⊗ b ⋊ z 7→ a ⋊ z ⊗ b z ⋊ z − 1 Thu s Q is a simplicial U - module. W e have an iso morphism of graded U - modules θ : M ¯ h ∈ G/ Z g M k ∈ G U ⊗ V ( k ) → Q θ (( a ⊗ b ⋊ z ) ⊗ v )) = ι ( a ⊗ b ⋊ z ) · (1 ⋊ h ⊗ v ⊗ 1 ⋊ ( hk ) − 1 g ) In particular ea c h U -module Q n is extended from Z . Next observe that the aug- men tation of C bar ( R ⋊ G ) r estricts to a n augmentation (7.4) Q → R ⋊ g ∼ = R g and that the c a nonical co n tracting chain homotopy x 7→ 1 ⊗ x induces a co n tracting homotopy for (7.4 ). Th us (7.4) is a simplicial resolution by relatively pr o jective U - mo dules. Consider the map ˆ α : P → Q ˆ α ( z 0 ⊗ · · · ⊗ z n ⊗ s ⊗ x 0 ⊗ · · · ⊗ x n +1 ) = ( z 0 · · · z n )( x 0 ) ⋊ z 0 ⊗ ( z 1 · · · z n )( x 1 ) ⋊ z 1 ⊗ · · · ⊗ z n ( x n ) ⋊ z n ⊗ x n +1 ⋊ ( z 0 · · · z n ) − 1 g ∈ hom R ⋊ G G ( S ) ( z − 1 0 s, s ) ⊗ · · · ⊗ ho m R ⋊ G G ( S ) ( s, ( z 0 · · · z n ) − 1 s ) One c hecks that ˆ α is a simplicial U -mo dule homomorphis m cov ering the identit y of R g and that R ⊗ ˆ α = α , concluding the pro of.  Prop osition 7.5. (Comp ar e [18, 9.16 ] ) L et G b e a gr oup, X ∈ S G . F or e ach ξ ∈ con( G ) cho ose a r epr esentative g ξ . Then ther e is an isomorphism M ξ ∈ con( G ) H ∗ ( Z g ξ , Z [ X g ξ ] ⊗ H H ( R , R g ξ )) ∼ = − → H G ∗ ( X, H H ( R )) ISOMORPHISM CONJECTURES WITH PROPER COEFFICIE NTS 25 natur al in X and R , which dep ends on t he choic e of r epr esentatives { g ξ : ξ ∈ con( G ) } . Her e H ( Z g , − ) is hyp erho molo gy of c omple xes of Z g -mo dules, and the tensor pr o duct is e quipp e d with the diagonal actio n. Pr o of. By (7.1) we have H G ∗ ( X, H H ( R )) = M ξ ∈ con( G ) H G ∗ ( X, H H ( ξ ) ( R )) By Lemma 7.2 and the deﬁnition of equiv a r iant ho mology , if g ∈ ξ , then H G ( X, H H ξ ( R )) = Z Or G H H ξ ( R ⋊ G G ( G/H )) ⊗ Z [map( G/H, X )] ∼ ← − Z Or G  Z [ E Z g ] ⊗ Z [ Z g ] [ Z [map( G/ h g i , G/H )] ⊗ H H ( R, R g )]  ⊗ Z [map( G/H, X )] = Z [ E Z g ] ⊗ Z [ Z g ] ( Z [ X g ] ⊗ H H ( R , R g )) = H ( Z g , Z [ X g ] ⊗ H H ( R, R g ))  Prop osition 7. 6. L et G b e a gr oup, F a fa mily of sub gr oups of G and R a unit al G -ring. Assume that F c ontains al l cyclic sub gr oups of G . Then H G ( − , H H ( R )) pr eserves ( G, F ) -we ak e quivalenc es. In p articular, t he assembly map H G ∗ ( E ( G, F ) , H H ( R )) → H H ∗ ( R ⋊ G ) is an isomorphism. The analo gue statements for cyclic homolo gy also hold. Pr o of. The ﬁrst statement abo ut Ho ch schild homology follows from 7.5, and the fact that if K is a group, then H ( K, − ) preserves quasi-isomor phisms. The s econd follows fr o m the ﬁrst a nd the fac t that E ( G, F ) → ∗ is a n equiv alence. Next, given a cyclic mo dule M , co nsider the sub complex HC n ( M ) = ker( S n : H C ( M ) → H C ( M )[ − 2 n ]) Note that 0 = H C 0 ( M ) ⊂ H H ( M ) = H C 1 ( M ) ⊂ HC 2 ( M ) ⊂ · · · ⊂ [ n HC n ( M ) = H C ( M ) is an exhaustive ﬁltration. Hence, b ecause H G ( X, − ) preserves ﬁltering colimits ( X ∈ S G ), to prov e the statement of the lemma for cyclic homolo gy , it is suﬃcient to show that fo r each n , H G ( − , HC n ( R )) pres erves ( G, F )-equiv a lences of G -simplicial sets. Observe that if M is a cyclic mo dule, then we ha ve an exact sequence 0 → HC n ( M ) → HC n +1 ( M ) → H H ( M )[ − 2 n ] → 0 Using the sequence ab ov e and what we hav e already proved, one shows b y induction that H G ( − , HC n ( R )) pres e rves ( G, F )-equiv alences. This ﬁnishes the pr o of.  26 GUILLERMO COR TI ˜ NAS AND EUGENIA E LLIS 8. The Chern ch a racter and infinitesimal K -theor y 8.1. No nconnectiv e Che rn c haracter. Let C b e a Z -line a r categ ory . By results of Randy McCar th y [2 2, § 3.3 and § 4 .4 ] w e hav e a Chern character (8.1.1) K Q (Idem C ⊕ ) → | τ ≥ 0 H N (Idem C ⊕ ) | going from the K -theory simplicial set to the simplicial set obtained via the Do ld- Kan corr esp o ndence from the go o d truncation of the negative cyclic complex with- out negative terms. In this section we use this to obtain a ma p K ( C ) → | H N ( C ) | going from the nonconnective K -theory spectrum of Section 4 to the sp ectrum obtained from the negative cyclic complex via Dold-Kan corresp ondence. W e shall need the following result of McCar th y . Prop osition 8.1. 2. [22, Thm. 2 .3.4] L et D b e a Z -line ar c ate gory and C ⊂ D a ful l sub c ate gory. Assum e that for every obje ct d ∈ D ther e exists an n = n ( d ) , a ﬁnite se quenc e c 1 , . . . , c n of obje cts of C , and morphisms φ i : c i → d and ψ i : d → c i such that P i φ i ψ i = 1 d . Then the inclusion funct or C → D induc es a quasi-isomorphism C ( C ) → C ( D ) . Lemma 8.1 .3. L et C b e an additive c ate gory, and let • b e t he only obje ct of Γ( Z ) . Consider the functor µ : Γ Z ⊗ C → Γ( C ) µ ( • , c ) = ( c, c, . . . ) , µ ( f ⊗ α ) ij = f ij α Then i) The functor µ is ful ly faithful. ii) L et F ( − ) b e as in (4.1.1) . F or every obje ct x ∈ Γ( C ) t her e exist morphisms φ c : µ ( • , c ) → x and ψ c : x → µ ( • , c ) , c ∈ F ( x ) such t hat P c ∈ F ( x ) φ c ψ c = 1 x . iii) The functor µ induc es a ful ly faithful fun ctor ¯ µ : Σ ⊗ C → Σ( C ) . Pr o of. Part i) is proved in [3, Lemma 4.7 .1] for the ca se when C has only o ne o b ject; the sa me argument applies in general. T o prov e ii), let x ∈ Γ( C ) be an ob ject. If c ∈ F ( x ), write I ( c ) = { n ∈ N : x n = c } , and let χ I ( c ) be the characteristic function. Put φ c : µ ( • , c ) → x, ψ c : x → µ ( • , c ) , ( φ c ) i,j = ( ψ c ) i,j = δ i,j χ I ( c ) ( j )1 c One chec ks that X c ∈ F ( x ) φ c ψ c = 1 x This prov es ii). Next, consider the exact sequence 0 → M ∞ Z → Γ Z π → Σ Z → 0 As is explained in [3, pp 92], it follo ws from r esults of N¨ ob eling [23] that the se q uence ab ov e is split a s a sequence of ab elian groups. Hence if c, d ∈ C , then ker( π ⊗ 1 : ho m Γ Z ⊗C (( • , c ) , ( • , d )) → ho m Σ Z ⊗C (( • , c ) , ( • , d ))) = M ∞ Z ⊗ hom C ( c, d ) Next observe that if α ∈ ho m C ( c, d ) and f ∈ M ∞ Z , then µ ( f ⊗ α ) is a ﬁnite morphism. Hence µ passes to the quo tien t, inducing a functor ¯ µ : Σ Z ⊗ C → Σ( C ). ISOMORPHISM CONJECTURES WITH PROPER COEFFICIE NTS 27 If c, d ∈ ob C and w e put x = µ ( • , c ), y = µ ( • , d ) then we hav e a map o f exact sequences 0 → M ∞ Z ⊗ hom C ( c, d )   / / Γ Z ⊗ hom C ( c, d ) / /   Σ Z ⊗ hom C ( c, d ) → 0   0 → hom Fin( C ) ( x, y ) / / hom Γ( C ) ( x, y ) / / hom Σ( C ) ( x, y ) → 0 Here Fin( C ) ⊂ Γ( C ) is the sub category of ﬁnite morphisms. The second v ertical map is an iso morphism by part i). In pa rticular the ﬁrst map is injectiv e; further more, one chec ks that it is onto. It follows that the third vertical ma p is an isomorphism; this prov es iii).  Prop osition 8.1.4. L et C b e a Z -line ar c ate gory. Then: i) C ( C ) → C ( C ⊕ ) is a quasi-isomorphism. ii) If C is additiv e, then C ( C ) → C (Idem C ) is a qu asi-isomorph ism. iii) The m aps C (Γ( Z ) ⊗ C ) → C (Γ( C )) , C (Γ( C )) → 0 and C (Σ( Z ) ⊗ C ) → C (Σ( C )) ar e qu asi-isomorp hisms. iv) The se quenc e Idem C ⊕ → Γ C ⊕ → Σ C ⊕ induc es a distinguishe d triangle of Ho chschild , cyclic, ne gative cyclic and p erio dic cyclic c omplexes. Pr o of. The ﬁr st t w o assertions are straightforw ard applications of Prop osition 8.1.2. That C (Γ( Z ) ⊗ C ) → C (Γ( C )) a nd C (Σ( Z ) ⊗ C ) → C (Σ( C )) ar e quasi-is omorphisms follows fro m Pr opo sition 8.1.2 and Lemma 8 .1.3. In particula r we hav e qua si- isomorphisms C (Γ( C )) C ( A (Γ Z ⊗ C ) / A (ob(Γ Z ⊗ C )) ∼ o o C ( A (Γ Z ⊗ C )) ∼ o o C (Γ A ( C )) ∼ / / H H (Γ A ( C )) But b ecause A ( C ) is H -unital, H H (Γ A ( C )) is acyclic by [31, Thm. 1 0.1]. T o pr ov e iv), consider the c o mm utative diagr am (8.1.5) C   / / Γ C / /   Σ C   Idem C ⊕ / / Γ C ⊕ / / Σ C ⊕ By i) and ii), the ﬁrst vertical map induces quasi-iso morphisms of cyclic mo dules. If R is a unital ring ﬂat as a Z -mo dule, then the quasi- isomorphism C ( C ) → C ( C ⊕ ) of i) induces a quasi- isomorphism C ( R ⊗ C ) = C ( R ) ⊗ C ( C ) → C ( R ⊗ C ⊕ ). In pa rticular this applies when R = Γ Z , Σ Z . Hence the sec ond a nd third v ertical maps in (8 .1.5) are quasi-isomor phis ms as well, by iii). By Lemma 6.1 the cyclic mo dules of the top row are quasi-isomorphic to the cyclic modules of their asso ciated rings; th us iv) reduces to the fact, pr oved in [31, § 10 ], that the sequenc e C ( A ( C )) / / C (Γ A ( C )) / / C (Σ A ( C )) induces distinguished tria ngles for H H , H C , H N and H P .  28 GUILLERMO COR TI ˜ NAS AND EUGENIA E LLIS Let C ( n ) be a s in (4.1 .3). Observe that by Pr opo sition 8 .1.4, we hav e an equiv a - lence | τ ≥ 0 H N ( C ( n ) ) | ∼ − → | τ ≥ 0 H N ( C )[+ n ] | . Comp osing with the map K Q ( C ( n ) ) → | τ ≥ 0 H N ( C ( n ) ) | we obtain a sequence K Q ( C ( n ) ) → τ ≥ 0 H N ( C )[+ n ] which induces a map of nonco nnec tiv e s p ectra (8.1.6) ch : K ( C ) → | H N ( C ) | R emark 8.1.7 . If C has only one ob ject, then the Chern character (8.1.6) agrees with the usual one. This follows from (4.1.6) and the ring ana lo gue of Prop ositio n 8 .1.4, part iv), proved in [31, § 10]. F urthermore, for any Z -linear categ ory C , the character (8.1.6) agr ees with that o f A ( C ). Indeed, K ( A ( C )) ∼ − → K ( C ) by Pro po sition 4.2.8, and the pro of of Pr opo sition 8.1.4 makes clear that the homolo gy sequences of iv) are equiv alent to the cor resp onding sequences for A ( C ). 8.2. K nil and the rel ativ e Chern c haracter. Let E : Z − Cat → Spt be a functor and C ∈ Z − Ca t. Consider the homotopy ﬁb er E nil ( C ) = hoﬁb er( E ( C ) → E ( C ⊗ Z ∆ • )) W rite ch ∆ : K H ( C ) = K ( C ⊗ Z ∆ • ) → H N ( C ⊗ Z ∆ • ) for the r e sult of applying the map K → H N dimensionwise. W e have a map of sp ectra ch nil : K nil ( C ) → H N nil ( C ) which ﬁts into a map of homotopy ﬁbr ations K nil ( C ) ch nil   / / K ( C ) ch   / / K H ( C ) ch ∆   | H N nil ( C ) | / / | H N ( C ) | / / | H N ( C ⊗ Z ∆ • ) | Lemma 8.2 .1. L et C b e a Q -line ar c ate gory. Then ther e is a homotopy c ommuta- tive diagr am with vertic al we ak e quivalenc es | H N nil ( C ) | ι   / / | H N ( C ) | / / | H N ( C ⊗ Z ∆ • ) | ≀   Ω − 1 | H C ( C ) | / / | H N ( C ) | / / H P ( C ) Pr o of. By Example 6.2, this is a statement ab out the Q -a lgebra A ( C ). The latter is prov ed in [8, Theorem 4 .1].  By [30, Prop. 1.6 ], if A is a Q -alg ebra the groups K nil ∗ ( A ) are Q - vectorspa c e s. Hence for every ring A we ha ve a map (8.2.2) q : K nil ( A ) ⊗ Q → K nil ( A ⊗ Q ) which is a n equiv alence if A is a Q -alge br a. W e write ν = ιch nil ( − ⊗ Q ) q : K nil ( C ) ⊗ Q → Ω − 1 | H C ( C ⊗ Q ) | ∼ ← − Ω − 1 | H C ( C ) | ⊗ Q (8.2.3) K ninf ( C ) = hoﬁb er( ν ) W e remark that ν is a v aria n t of the rela tiv e character intro duced by W eib el in [28]. ISOMORPHISM CONJECTURES WITH PROPER COEFFICIE NTS 29 Prop osition 8.2.4. K ninf : Z − Cat → Spt s atisﬁ es the standing assumptions. In addition, it is ex cisive and K ninf ∗ c ommut es with ﬁltering c olimits. Pr o of. It is prov ed in [1] that (8.2.5) K inf , Q := hoﬁb er( ch Q : K ( − ) ⊗ Q → H N ( − ⊗ Q )) is excisive; it follows tha t K ninf is excisive to o. Next obser v e that K ninf satisﬁes iii) and v) of the standing assumptions 3.3.2 for unital rings, a nd iv) for all C ∈ Z − Cat, since both K nil and H C do. B ecause K ninf is excisive, this implies that it satisﬁes all standing assumptions , by Remark 3.3.5. Finally K ninf ∗ commutes with ﬁlter ing colimits b ecause b oth K nil ∗ and H C ∗ do.  9. Rings o f po l ynomial functions on a simplicial set 9.1. Fi niteness. An ob ject K in a category A is smal l if hom A ( K, − ) preserves colimits. If A = S , then X is small if and o nly if it has only a ﬁnite num b er of nondegenera te simplices, or, equiv alently , if there ex ists a ﬁnite s e t of nonneg a tiv e int egers n 1 , . . . , n r and a surjection r a i =1 ∆ n i ։ X Small simplicial sets are called ﬁnite . Similar ly , a G -simplicial s et is small if there are n 1 , . . . , n r ≥ 0 a nd a G -equiv a riant surjection r a i =1 ∆ n i × G ։ X Let F b e a family of subgroups o f G . A ﬁnite ( G, F )-complex is G -simplicial set obtained by attaching ﬁnitely many cells of the form ∆ n × G/H with H ∈ F . A G -ﬁnite simplicial set is a ﬁnite ( G, A ll )-complex. The concept of G -ﬁniteness is the simplicial set v ersion of the concept of G -compactness. Indeed one chec ks that a G -simplicial set X is G -ﬁnite if and o nly if X/G is ﬁnite as a simplicia l set. 9.2. Lo call y ﬁnite simpl i cial sets. If X is a simplicial set and σ ∈ X is a simplex, we write < σ > ⊂ X for the simplicial subset generated by σ . W e hav e < σ > n = { α ∗ ( σ ) : α ∈ hom([ n ] , [dim σ ]) } The star of σ is the following set of simplices of X : St( σ ) = St X ( σ ) = { τ ∈ X : < τ > ∩ < σ > 6 = ∅ } The close d star is the simplicial subset St( σ ) = < St( σ ) > generated by St( σ ). If M is a set of simplices of X we put St X ( M ) = ∪ σ ∈ M St X ( σ ), St X ( M ) = < St X ( M ) > . W e a lso deﬁne the link o f M as Link( M ) = St X ( M ) \ St X ( M ). Lemma 9. 2.1. L et X b e a simplic ial set; write N X for the set of nonde gener ate simplic es. The fol lowing ar e e quivalent. i) ( ∀ σ ∈ X ) { τ ∈ N X : < τ > ⊃ < σ > } is a ﬁn ite set. ii) F or every σ ∈ X , St X ( σ ) is a ﬁnite simplicial set. Pr o of. If σ ∈ X , then < σ > has ﬁnitely many nondegenera te simplices, and thus the se t { < τ > ∩ < σ > : τ ∈ X } is ﬁnite. Hence if i) holds, there are ﬁnitely many τ ∈ N X s uc h that < τ > ∩ < σ > 6 = ∅ ; in other words, N X ∩ St X ( σ ) is a ﬁnite 30 GUILLERMO COR TI ˜ NAS AND EUGENIA E LLIS set, and therefor e St X ( σ ) is a ﬁnite simplicial set. Th us i) ⇒ ii). Next note that < τ > ⊃ < σ > implies τ ∈ St X ( σ ), whence ii) ⇒ i).  W e say that X is lo c al ly ﬁnite if it satisﬁe s the equiv alen t conditions of the lemma ab ov e. 9.3. R i ngs of p olynomi al functions on a simpl icial set. If X is a simplicial set and A is a ring, we put A X = hom S ( X, A ∆ • ) The simplicial ring A ∆ • = A ⊗ Z ∆ • is deﬁned as in (5.1). Note X 7→ A X , f 7→ f ∗ gives a functor S op → Ring s. By its very deﬁnition, the functor A − sends colimits to limits; if I is a small ca tegory and X : I → S is a functor, then A colim i X i = lim i A X i Example 9.3.1 . Any simplicial set X is the union of the sub ob jects generated b y each of its nondegenera te simplices ; in symbols X = colim σ ∈ N X < σ > Thu s we obta in (9.3.2) A X = lim σ ∈ N X A <σ> = { φ ∈ Y σ ∈ N X A <σ> : φ ( σ ) | <σ> ∩ <τ > = φ ( τ ) | <σ> ∩ <τ > , σ, τ ∈ N X } If φ ∈ A X , then its su pp ort is supp( φ ) = < { σ ∈ X : φ ( σ ) 6 = 0 } > Note that if φ, ψ ∈ A X and f : X → Y is a simplicial ma p, then (9.3.3) supp( φ · ψ ) ⊂ s upp( φ ) ∩ supp( ψ ) supp( f ∗ ( φ )) ⊂ f − 1 (supp( φ )) W e say that φ is ﬁnitely supp orte d if supp( φ ) is a ﬁnite simplicia l set. Note φ is ﬁnitely supported if and only if there is only a ﬁnite num ber of nondegenerate simplices σ s uc h that φ ( σ ) 6 = 0. P ut A ( X ) = { f ∈ A X : supp( f ) is ﬁnite. } If X is ﬁnite, then clear ly A X = A ( X ) . In general, A ( X ) ⊂ A X is a tw o-sided ideal, by (9.3.3). W e r e mark that if f : X → Y is an arbitra ry map of simplicial sets, then the ass o c ia ted r ing homomorphism f ∗ : A Y → A X do es not necessa rily send A ( Y ) int o A ( X ) . How ever, if f happ ens to b e pr op er , i.e. if f − 1 ( K ) is ﬁnite for every ﬁnite K ⊂ Y , then f ∗ ( A ( Y ) ) ⊂ A ( X ) , by (9 .3.3). Hence A ( − ) is a functor on the category o f simplicial sets and proper maps. Next w e consider the b ehaviour of this functor with resp ect to c o limits. First of a ll, if { X i } is a family o f simplicial sets, then we have (9.3.4) A ( ` X i ) = M i A ( X i ) Here L indicates the direct sum of ab elian g roups, equipp ed with coor dinatewise m ultiplication. Seco nd, A ( − ) maps co equalizers of prop er maps to equalizer s; if { f j : X → Y } is a family o f prop e r maps, then (9.3.5) A (co eq j { f j : X → Y } ) = eq j { f ∗ j : A ( Y ) → A ( X ) } ISOMORPHISM CONJECTURES WITH PROPER COEFFICIE NTS 31 Next reca ll that if I is a s mall catego ry and X : I → S is a functor, then the colimit of X can b e c omputed as a co equa lizer: colim i X i = co eq( a α ∈ Ar( I ) X s ( α ) ∂ 0 ⇒ ∂ 1 a i ∈ Ob( I ) X i ) Here Ob( I ) and Ar ( I ) are r espe c tiv ely the sets o f ob jects and o f arrows o f I , and if α ∈ Ar( I ) then s ( α ) ∈ Ob( I ) is its s ource; we also write r ( α ) for the ra nge of α . The maps ∂ 0 and ∂ 1 are deﬁned a s follows. The re striction of ∂ i to the co p y of X s ( α ) indexed b y α is the inclusion X s ( α ) ⊂ ` j X j if i = 0 and the comp osite o f X ( α ) followed by the inc lus ion X r ( α ) ⊂ ` j X j if i = 1. The conditions tha t ∂ 0 and ∂ 1 be prop er are equiv alent to the following ∂ 0 ) Ea ch ob ject of I is the sour c e of ﬁnitely many arrows. ∂ 1 ) Ea ch o b ject of I is the range of ﬁnitely many arrows, and X s ends each map of I to a prop er map. Example 9.3.6 . F or example the functor σ 7→ < σ > fro m the set of nondeg enerate simplices of X , ordered by σ ≤ τ if < σ > ⊂ < τ > , alwa ys satisﬁes ∂ 1 ; condition ∂ 0 is precisely conditio n i) of Lemma 9.2.1. Hence ∂ 0 is satisﬁed if and only if X is lo cally ﬁnite, and in that case we hav e A ( X ) = eq( M σ ∈ N X A <σ> ∂ ∗ 0 ⇒ ∂ ∗ 1 M <τ > ⊂ <σ>, σ,τ ∈ N X A <τ > ) Lemma 9 .3.7. If X is a lo c al ly ﬁnite simplici al set, t hen Z ( X ) is a fr e e ab elian gr oup. Pr o of. By [3, 3.1 .3] the lemma is true when X is ﬁnite. Hence if X is any simplicial set, and σ ∈ X is a simplex, then Z <σ> is free. If X loca lly ﬁnite, then by Example 9.3.6, Z ( X ) is a subg roup of a free gr o up, and therefore it is fr e e.  9.4. Extendi ng p olynomial functions. Theorem 9.4.1. L et X b e a simplicia l set, Y ⊂ X a simplicial subset and A a ring. Le t φ ∈ A Y and K = s upp φ . Then ther e exists ψ ∈ A X with supp ψ ⊂ St X K such t hat ψ | Link X ( K ) = 0 and ψ | Y = φ . Pr o of. W e ha ve K ⊂ St Y K ⊂ St Y K , whence φ | Link Y ( K ) = 0. Note St X K ∩ Y = St Y K ; thus φ v anishes on Link X ( K ) ∩ Y . Hence we may extend φ to a ma p φ ′ : Y ′ = Y ∪ Link X ( K ) → A ∆ • by φ ′ | Link X ( K ) = 0. Put Y ” = Y ∪ St X K . B ecause Y ′ ⊂ Y ” is a coﬁbration and A ∆ • ։ 0 is a tr iv ial ﬁbr ation, w e may further extend φ ′ to a map φ ” : Y ” → A ∆ • . By c o nstruction, { σ ∈ X : φ ” ( σ ) 6 = 0 } ⊂ St X K , and φ ” v a nishes on Link X K . Hence we may ﬁnally extend φ ” to a map ψ : X → A ∆ • , by letting ψ ( σ ) = 0 if σ / ∈ St X K . T his concludes the pr o o f.  Corollary 9.4.2 . If X is lo c al ly ﬁnite and Y ⊂ X is a simplicial su bset, then the r est riction map A ( X ) → A ( Y ) is surje ctive. Pr o of. It follows from Theorem 9.4.1, us ing 9.2.1.  Prop osition 9.4. 3. (Comp ar e [4, L e mma 2 .5] ) L et A b e a nonzer o ring. The fol lowing ar e e quivalent fo r a simp licial set X . i) F or every simplex σ ∈ X ther e exists φ ∈ A ( X ) such that φ ( σ ) 6 = 0 . 32 GUILLERMO COR TI ˜ NAS AND EUGENIA E LLIS ii) X is lo c al ly ﬁnite. Pr o of. O bserve that if σ , τ ∈ X are simplices with < τ > ⊃ < σ > and φ ∈ A X satisﬁes φ ( σ ) 6 = 0, then φ ( τ ) 6 = 0. If X is not lo cally ﬁnite, then by Lemma 9.2 .1, there exists a simplex σ ∈ X which is co n tained in inﬁnitely man y nondeg enerate simplices. By the previous o bs erv ation, φ ( σ ) = 0 for ev ery φ ∈ A ( X ) . W e have prov ed that i) ⇒ ii). Assume conversely that X is lo cally ﬁnite, a nd le t σ b e a simplex of X . W e want to show that there exists φ ∈ A ( X ) such tha t φ ( σ ) 6 = 0 . W e may assume that σ is nondegener ate. Let Y = < σ > ⊂ X be the sub-simplicial set generated by σ ; b y C o rollary 9 .4.2, it suﬃces to show that A Y 6 = 0. Now Y is an n -dimensional quotient of ∆ n , whence S n = ∆ n /∂ ∆ n is a quotient of Y . So w e may further r educe to showing A S n is nonzero. Now A S n = Z n A ∆ • = n \ i =0 ker( d i : A ∆ n → A ∆ n − 1 ) But if 0 6 = a ∈ A , then a t 0 . . . t n is a nonzero element of Z n A ∆ • .  9.5. Excis ion properties . Prop osition 9.5.1. If X is a lo c al ly ﬁnite simplicial set, then Z ( X ) is s -unital. Pr o of. Le t φ 1 , . . . , φ n ∈ Z ( X ) , and let K = S i supp( φ i ). By Theorem 9.4.1 ther e is µ ∈ Z ( X ) such that µ | K = 1 is the constant map. Thus (9.5.2) φ i = φ i µ ( ∀ i ) .  Prop osition 9. 5.3. If A is K -excisive and X is lo c al ly ﬁnite, then Z ( X ) ⊗ A is K -ex cisive. Pr o of. F ollows fro m Lemma 9.3.7 and Prop ositions 9.5.1 and A.5.3.  R emark 9 .5.4 . If A is a ring and X a locally ﬁnite simplicial set, then there is a natural map Z ( X ) ⊗ A → A ( X ) It was proved in [3, 3 .1 .3] that this ma p is an isomor phism if X is ﬁnite. 10. Proper G -rings 10.1. Prop er rings ov er a G -simplicial set. Fix a group G and consider rings equipp e d with an action of G by ring automor phisms. W e write G − Rings for the category of suc h rings a nd equiv ariant ring ho momorphisms. If C ∈ G − Rings is commutativ e but not necessar ily unita l and A ∈ G − Rings, then by a c omp atible ( G, C ) - algebr a structur e on A w e understand a C -bimo dule structure on A such that the following iden tities hold for a, b ∈ A , c ∈ C , and g ∈ G : c · a = a · c c · ( ab ) = ( c · a ) b = a ( c · b ) g ( c · a ) = g ( c ) · g ( a ) (10.1.1) ISOMORPHISM CONJECTURES WITH PROPER COEFFICIE NTS 33 If X is a G -simplicia l set and A ∈ G − Rings, then we say that A is pr op er ov er X if it car ries a compatible ( G, Z ( X ) ) algebr a s tructure such that (10.1.2) Z ( X ) · A = A If F is a family of subgroups of G , w e sa y that A is ( G, F ) -pr op er if it is prop er ov er some ( G, F ) complex X . Example 10 .1.3 . Fix a group G , a family o f subgroups F and a ( G, F )-co mplex X . B y Pro pos ition 9.5 .1, w e hav e Z ( X ) · Z ( X ) = Z ( X ) ; thus Z ( X ) is prop er ov er X . Hence if A is a G -r ing with a compatible ( G, Z ( X ) )-action, then Z ( X ) · A is prop er ov er X . If A is proper o ver X , and B is any ring, then A ⊗ B is prop er over X . In particular, Z ( X ) ⊗ B is prop er. If T ∈ T op is the geometric realiz a tion of X , and F is either R or C , then the F -algebra P = C comp ( T ) o f compa c tly supp orted contin uo us functions T → F is prop er ov er X . T o chec k tha t Z ( X ) · P = P , o bserve that if f ∈ P then its supp ort meets ﬁnitely many maximal simplices; write K ⊂ X for their union. By Coro llary 9 .4.2, there exists φ ∈ Z ( X ) which is c onstantly equa l to 1 on K ; th us f = φ · f ∈ Z ( X ) · P . Let X be a lo cally ﬁnite simplicial set, and Y ⊂ X a subob ject. P ut I ( Y ) = { φ : supp φ ⊂ Y } ⊳ Z ( X ) Note that if ψ ∈ Z ( Y ) and ˆ ψ ∈ Z ( X ) restricts to ψ , then the pr o duct ψ · φ := ˆ ψ φ depe nds only o n ψ . This deﬁnes a compatible action o f Z ( Y ) on I ( Y ) whic h makes the latter ring prop er over Y . Mor e gener a lly , if A ∈ Rings ha s a compatible ( G, Z ( X ) )-structure, we put (10.1.4) A ( Y ) = I ( Y ) · A ⊳ A Observe that A ( Y ) is an idea l of A , pro per over Y . In particula r if X is a ( G, F )- complex, then A ( Y ) is ( G, F )-prop er for all Y ⊂ X . Lemma 10.1.5 . L et A b e a G -ring. Assume that A is ( G, F ) -pr op er. Then A has an exhaustive ﬁltra tion { A ( K ) } by ide als such that e ach A ( K ) pr op er o ver a ﬁnite ( G, F ) -c omplex K . Pr o of. By hypothesis , there exists a ( G, F )-complex X suc h that A is prop er over X . Consider the ﬁltra tio n { A ( K ) } where A ( K ) is deﬁned in (10.1 .4) and K r uns among the G -ﬁnite simplicial subsets of X . By the dis cussion abov e, A ( K ) ⊂ A is an ideal, prop er ov er K . It is clear that { I ( K ) } and { A ( K ) } are ﬁlter ing systems and that ∪ K I ( K ) = Z ( X ) . W e claim fur ther more that A = ∪ K A ( K ). B y deﬁnition of Z ( X ) -algebra , A = Z ( X ) · A . Hence if a ∈ A , then there exist φ 1 , . . . , φ n ∈ Z ( X ) and a 1 , . . . , a n ∈ A such that a = P i φ i a i . Hence a ∈ A ( K ) for K = ∪ i G · supp( φ i ).  Lemma 10. 1.6. (cf. [9, pp. 51] ) L et A ∈ G − Rings b e pr op er over a lo c al ly ﬁn ite G -simplicial set X , and let f : X → Y b e an e quivariant map with Y lo c al ly ﬁnite. Then the map f ∗ : Z Y → Z X induc es a c omp atible ( G, Z ( Y ) ) -algebr a stru ctur e on A which makes it pr op er ove r Y . Pr o of. W e b egin b y showing that the compatible ( G, Z ( X ) )-algebra structure on A extends to a co mpatible ( G, Z X )-mo dule structure. By the lemma ab ov e, if a ∈ A 34 GUILLERMO COR TI ˜ NAS AND EUGENIA E LLIS then there e xists a ﬁnite s implicial subset K ⊂ X such that a ∈ A ( K ) = I ( K ) · A . By Theorem 9.4 .1 ther e exists µ K ∈ Z X , with supp( µ K ) ⊂ St( K ) such that (10.1.7) µ K a = a ∀ a ∈ A ( K ) . Because X is lo cally ﬁnite, St( K ) is ﬁnite and µ K ∈ Z ( X ) . Thus we ha ve a map A ( K ) → I ( St( K ))) ⊗ A ( K ), a 7→ µ K ⊗ a . Now I (St( K )) is a n idea l in Z X by (9.3.3); using the mu ltiplication o f Z X we obtain a map (10.1.8) Z X ⊗ A ( K ) → A ( St( K )) , φ ⊗ a 7→ ( φ · µ K ) a. If L ⊃ K , and we choos e an element µ L as ab ov e, then fo r a ∈ A ( K ) and φ ∈ Z X we hav e: ( φ · µ L ) · a = ( φ · µ L ) · ( µ K · a ) = ( φ · µ K ) a This shows that (1 0.1.8) is indep endent o f the choice o f the element µ K of (1 0.1.7), and that we hav e a well-deﬁned a ction Z X ⊗ A → A . Co mpatibilit y with the G - action follows from the fact that g · µ K is the iden tit y on g · K . The remaining compatibility conditions ar e immediate. Now A b ecomes an Z ( Y ) -mo dule through f ∗ . If K ⊂ X is a ﬁnite simplicia l subset, then L = f ( K ) ⊂ Y is ﬁnite, and since Y is lo cally ﬁnite, there is a µ L ∈ Z ( Y ) which is the identit y on L , and thus f ∗ ( µ L ) is the iden tit y on K . It follows that the action of Z ( Y ) on A satisﬁes (10.1.2). The remaining ( G, Z ( Y ) )-compatibility conditions of (10 .1) are straightforward.  10.2. Induction. Let G b e a gro up, H ⊂ G a s ubgroup and A an H -ring. Consider BigInd G H ( A ) = { f : G → A : f ( g h ) = h − 1 f ( g ) } Note tha t BigInd G H ( A ) is a G -ring with op erations deﬁned p oint wise , and where G acts by left mu ltiplication. If f ∈ Big Ind G H ( A ) and x = sH ∈ G/H , then the v alue of f at any g ∈ x determines f on the whole x ; in particular , supp( f ) ∩ sH 6 = ∅ ⇒ sH ⊂ supp( f ) ( sH ∈ G/H ) Hence supp( f ) = a sH ∩ supp( f ) 6 = ∅ sH Consider the pro jectio n π : G → G/H . Put Ind G H ( A ) = { f ∈ BigInd G H ( A ) : # π (supp( f )) < ∞} One chec ks that Ind G H ( A ) ⊂ Big Ind G H ( A ) is a subring; we sha ll presently in tro duce some of its typical elements. If s ∈ G , we write χ s : G → Z for the characteristic function. If a ∈ A and s ∈ G , then ξ H ( s, a ) = X h ∈ H h − 1 ( a ) χ sh ∈ Ind G H ( A ) Let r : G/H → G be a p ointed section and R = r ( G/H ). Every elemen t φ ∈ BigInd G H ( A ) can b e wr itten as a for ma l sum (10.2.1) φ = X s ∈R ξ H ( s, φ ( s )) Note that φ ∈ Ind G H ( A ) if a nd only if the sum ab ov e is ﬁnite. In particular Ind G H ( A ) = X s ∈ G,a ∈ A Z ξ H ( s, a ) ⊂ B igInd G H ( A ) ISOMORPHISM CONJECTURES WITH PROPER COEFFICIE NTS 35 Next obser v e that, fo r each ﬁxe d s ∈ G , the ma p ξ H ( s, − ) : A → Big Ind G H ( A ) is a ring homo mo rphism. Moreov er, we hav e the following relatio ns g ξ H ( s, a ) = ξ H ( g s, a ) (10.2.2) ξ H ( sh, a ) = ξ H ( s, ha ) (10.2.3) ξ H ( s, a ) ξ H ( t, b ) =  0 if sH 6 = tH ξ H ( s, ab ) if s = t (10.2.4) It follows that ( s, a ) 7→ ξ H ( s, a ) gives a G -e quiv aria nt map G × H A → Ind G H ( A ) . Here G × H A = G × A/ ∼ , wher e ( g 1 , a 1 ) ∼ ( g 2 , a 2 ) ⇐ ⇒ h = g − 1 1 g 2 ∈ H and a 1 = ha 2 . Extending by linearity we obtain an iso mo rphism of left G -mo dules Z [ G ] ⊗ Z [ H ] A → Ind G H ( A ) Thu s we may think of Ind G H ( A ) as the G -mo dule induced from the H -mo dule A equipp e d with a ring structure compatible with that o f A . In fact (10 .2.4) implies that if r : G/H → G is a section as a bove, then (10.2.5) Z ( G/H ) ⊗ A → Ind G H ( A ) , χ x ⊗ a 7→ ξ H ( r ( x ) , a ) is a (nonequiv a riant) ring is omorphism. Lemma 10.2.6. L et X b e an H -simplicial set; put Ind G H ( X ) = G × H X Ther e is a natur al, G -e quivaria nt isomo rphism Z (Ind G H ( X ) ) ∼ = Ind G H ( Z ( X ) ) . Pr o of. Le t π : G × X → Ind G H ( X ) b e the pro jection. W e hav e a G -ring iso momor- phism θ : BigInd G H ( Z X ) → Z Ind G H ( X ) , θ ( f )( π ( g , x )) = f ( g )( x ) F or s ∈ G and φ ∈ Z X , θ ( ξ H ( s, φ )) π ( g , x ) =  φ ( s − 1 g x ) if g ∈ sH 0 else. In particular, for θ ( ξ H ( s, φ )) no t to v a nish on π ( g , x ), we m ust hav e g = sh and x ∈ h − 1 { φ 6 = 0 } for so me h ∈ H . Hence s upp( θ ( ξ H ( s, φ ))) ⊂ π ( { s } × supp( φ )) which is a ﬁnite simplicial set if φ ∈ Z ( X ) . T he r efore θ maps Ind G H ( Z ( X ) ) inside Z (Ind G H ( X ) ) . It r emains to show that θ − 1 ( Z (Ind G H ( X ) ) ) ⊂ Ind G H ( Z ( X ) ). Let { g i } ⊂ G be a full s e t of represent atives o f G/H . Every element o f G × H X can b e written uniquely as π ( g i , x ) for some i and some x ∈ X . Hence as a simplicial se t, Ind G H ( X ) is the dis jo in t union o f the Y i = π ( { g i } × X ). In par ticular if φ ∈ Z (Ind G H ( X ) ) , then its supp ort meets ﬁnitely ma ny of the Y i , a nd supp( φ ) ∩ Y i is a ﬁnite simplicial set. Thu s there is a ﬁnite nu mber of i such that ψ = θ − 1 ( φ ) is nonzero on g i H , a nd its restriction to each of these subsets ta k es v alues in Z ( X ) . By (10.2.1), this implies that ψ ∈ Ind G H ( Z ( X ) ), as we had to prove.  36 GUILLERMO COR TI ˜ NAS AND EUGENIA E LLIS If C, A ∈ H − Rings with C commutativ e a nd w e ha ve a compatible ( H, C )- algebra structure on A , then Ind G H ( A ) car ries a compatible ( G, Ind G H ( C ))-algebra structure, given b y ξ H ( s, c ) · ξ H ( t, a ) =  ξ H ( s, c · a ) s = t 0 sH 6 = tH If moreover C · A = A , then Ind G H ( C ) · Ind G H ( A ) = Ind G H ( A ). W e r ecord a particula r case of this in the following Lemma 10.2.7. If A ∈ H − Rings is pr op er ov er an H -simplicial set X , t hen the G -ring Ind G H ( A ) is pr op er over Ind G H ( X ) . Pr o of. It follows from Lemma 10.2 .6 a nd the discussion ab ove.  10.3. Compressio n. Let A ∈ G − Rings, and H ⊂ G a subgroup. Assume that A is prop er over G/H . Let χ H ∈ Z ( G/H ) be the characteristic function of H . The c ompr ession of A over H is the subr ing Comp G H ( A ) = χ H · A Note the action of G on A restricts to an action of H on Comp G H ( A ), which makes it into a n ob ject of H − Rings. Prop osition 10.3.1. (Comp ar e [9, L e mma 12.3, and par a graph a fter 12.4 ] ) i) If B ∈ H − Rings , then Ind G H ( B ) is pr op er over G/H , and B → Comp G H Ind G H B , b 7→ ξ H (1 , b ) is an H -e qu ivariant isomorphism. ii) If A ∈ G − Rings is pr op er over G/H , then Ind G H Comp G H ( A ) → A, ξ H ( s, χ H a ) 7→ χ sH s ( a ) is a G -e quivaria nt isomorphism. Pr o of. Any B ∈ H − Rings is prope r over the 1 -po in t space ∗ . Hence Ind G H ( B ) is prop er over Ind G H ( ∗ ) = G/H , by L e mma 10 .2.7. The pro of that the maps of i) and ii) a re isomo rphisms is stra ightf orward; to show e q uiv ariance , o ne uses (10.2 .2) and (10.2.3).  10.4. A discrete v arian t of Green’s im primitivity theorem. Let G b e a group, H ⊂ G a subgroup and A a n H - ring.. Obs e rve that, b y deﬁnition, the G -ring Ind G H ( A ) is a G -subr ing of the r ing map( G, A ∆ • ) = map( G, A ) = A G (note that this is not the same as the subr ing of G -inv ariants of A ). Since A ( G ) ⊳ A G is a G -ideal, we may regar d A ( G ) as a le ft Ind G H ( A )-module via left m ultiplication in A G , and moreover, this action is co mpatible with that of G , in the sense that the t wo together deﬁne a left Ind G H ( A ) ⋊ G -mo dule structure on A ( G ) . W e may also regar d A ( G ) as a right module ov er A ⋊ H , via [ φ · ( a ⋊ h )]( g ) = h − 1 ( φ ( g h − 1 ) a ) One chec ks that these tw o actions satisfy ( f ⋊ g ) · [ φ · ( a ⋊ h )] = [( f ⋊ g ) · φ ] · ( a ⋊ h ) ISOMORPHISM CONJECTURES WITH PROPER COEFFICIE NTS 37 Hence they make A ( G ) int o an (Ind G H ( A ) ⋊ G, A ⋊ H )-bimo dule. In particular left m ultiplication by elemen ts of Ind G H ( A ) ⋊ G induces a ring homomorphism (10.4.1) Ind G H ( A ) ⋊ G → E nd A ⋊ H ( A ( G ) ) Observe that the decomp osition G = ` x ∈ G/H x induces (10.4.2) A ( G ) = M x ∈ G/H A ( x ) and that A ( x ) · ( A ⋊ H ) ⊂ A ( x ) . Hence (10.4.2) is a direct sum of right A ⋊ H - mo dules. Thus we may think of an elemen t T ∈ End A ⋊ H ( A ( G ) ) as a matrix T = [ T x,y ] x,y ∈ G/H , where T x,y : A ( y ) → A ( x ) is a ho momorphism of right A ⋊ H -mo dules, and is such that fo r each v ∈ A ( y ) , T x,y ( v ) = 0 for all but a ﬁnite num ber o f x . Moreov er A ⋊ H → A ( gH ) , a ⋊ h 7→ χ g · ( a ⋊ h ) = χ gh h − 1 ( a ) is an isomor phis m of right A ⋊ H - modules . Fix a full s et of repres e ntatives R of G/H , with 1 ∈ R , write M R ∈ Z − Rings for the ring of R × R -matrices with ﬁnitely ma ny nonzer o c o eﬃcients in Z , and put M R ( A ⋊ H ) = M R ⊗ ( A ⋊ H ). W e hav e a ring homomorphism M R ( A ⋊ H ) → E nd A ⋊ H ( A ( G ) ) M 7→ ( X y ∈R χ y · α y 7→ X x ∈R χ x X y ∈R m x,y α y ) F urthermo re, we hav e a map G → R , which sends each s ∈ G to the repres en tative ˆ s ∈ R o f sH . Using this map we obtain an isomo r phism M G/H ∼ = M R which s ends the matrix unit E sH,tH to E ˆ s, ˆ t . By c o mpo s ition, w e obtain a ring ho momorphism (10.4.3) M G/H ( A ⋊ H ) → E nd A ⋊ H ( A ( G ) ) ∼ = End A ⋊ H (( A ⋊ H ) ( G/H ) ) R emark 10.4.4 . If A happ ens to b e unital, then bo th (10.4.1) and (10.4.3) are injectiv e. Theorem 10. 4.5. L et G b e a gr oup, H ⊂ G a sub gr oup, and A ∈ H − Rings . Then ther e is an isomorphism Ind G H ( A ) ⋊ G ∼ = M G/H ( A ⋊ H ) su ch that the fol lowing diagr ams c ommute Ind G H ( A ) ⋊ G (10.4.1) / / ∼ = ( ( P P P P P P P P P P P P End A ⋊ H ( A ( G ) ) M G/H ( A ⋊ H ) (10.4.3) 6 6 m m m m m m m m m m m m m Ind G H ( A ) ⋊ G ∼ = / / M G/H ( A ⋊ H ) A ⋊ H ξ H (1 , − ) ⋊ id f f N N N N N N N N N N N e H,H ⊗− 7 7 p p p p p p p p p p p Pr o of. W e use the notatio n introduced in the paragr aph preceding the theo r em. If s ∈ G , put φ ( s ) = ˆ s − 1 s ∈ H . Note that φ ( sh ) = φ ( s ) h ( s ∈ G, h ∈ H ). One 38 GUILLERMO COR TI ˜ NAS AND EUGENIA E LLIS chec k s that the following map is a well-deﬁned, bijective ring ho momorphism with the requir e d pro per ties α : Ind G H ( A ) ⋊ G → M G/H ( A ⋊ H ) , α ( ξ H ( s, a ) ⋊ g ) = e sH,g − 1 sH ⊗ φ ( s ) ( a ) ⋊ φ ( s ) φ ( g − 1 s ) − 1  R emark 10.4.6 . The isomor phis m of the theorem ab ov e is natural in A , but not in the pair ( G, H ), as it depends on a choice of a full set of representativ es R of G/H , o r what is the same, o f a choice o f p ointed s ection G/H → G of the canonical pro jection. 10.5. Restriction. Let B be a G -ring , H ⊂ G a subgro up. W rite Res H G B for the H - ring obtained by r estriction to H o f the action o f G on B . Lemma 10.5 .1. If B is a G -ring, t hen Ind G H Res H G B → Z ( G/H ) ⊗ B , ξ H ( s, b ) 7→ χ sH ⊗ s ( b ) is a G -ring isomorphism. Pr o of. Str a ightf orward.  Now supp ose K ⊂ G is a nother subgroup. Let x ∈ H \ G/K . P ut (10.5.2) Res H G Ind G K ( A )[ x ] = { f ∈ Ind G K ( A ) : supp( f ) ⊂ x } ∈ H − Rin gs W e have (10.5.3) Res H G Ind G K ( A ) = M x ∈ H \ G/K Res H G Ind G K ( A )[ x ] W rite x = H θ K for some θ ∈ G . Co nsider the subgroup H ⊃ H θ = H ∩ θK θ − 1 W e shall see presently that the H -ring (10 .5 .2) is prop er ov er H /H θ . Consider the subgroup K ⊃ K θ − 1 = θ − 1 H θ ∩ K Conjugation by θ − 1 deﬁnes an iso morphism c θ − 1 : H θ → K θ − 1 , c θ − 1 ( h ) = θ − 1 hθ Hence we may view Res K θ − 1 K A a s an H θ -ring via c θ − 1 ; we write c ∗ θ − 1 (Res K θ − 1 K A ) for the resulting H θ -ring. Lemma 10.5.4. The map α : Res H G Ind G K ( A )[ H θ K ] → Ind H H θ ( c ∗ θ − 1 (Res K θ − 1 K ( A )) α ( f )( h ) = f ( hθ ) is an isomorphism of H -rings. Pr o of. O ne checks that if m ∈ H θ , then α ( f )( hm ) = m − 1 α ( f )( h ). It is clea r that α is H -equiv ariant. A calculation shows that α ( ξ K ( hθ, a )) = ξ H θ ( h, a ). It follows that α is a n isomorphism.  ISOMORPHISM CONJECTURES WITH PROPER COEFFICIE NTS 39 11. Induction and equiv ariant homology Lemma 11.1. L et G b e a gr oup, K ⊂ G a sub gr oup, A a K -ring, and E : Z − Cat → Spt a functor satisfying t he standing assu mptions. Then A is E -excisive if and only if I nd G K ( A ) is E -excisive. Pr o of. The ma p (10.2.5) gives a nonequiv a riant isomorphism Ind G K ( A ) ∼ = Z ( G/K ) ⊗ A = M x ∈ G/K A The equiv alence o f the lemma follows from Standing Ass umption v).  Let G , K and A be as in Lemma 11.1, and let X b e a G -s implicial set. If A is unital, then for each subgro up S ⊂ K we hav e a functor A ⋊ G K ( K/S ) → Ind G K ( A ) ⋊ G G ( G/S ) k S 7→ k S, a ⋊ k 7→ ξ K (1 , a ) ⋊ k If A is any E -e xcisive ring, the map ab ov e is deﬁned for the unitaliza tion ˜ A ; a pplying E , ta king ﬁb ers re la tiv e to the augmentation ˜ A → Z , and using the s ta nding assumptions, we g et a map E ( A ⋊ G K ( K/S )) → E (Ind G K ( A ) ⋊ G G ( G/S )). The maps X S + ∧ E ( A ⋊ G K ( K/S )) → X S + ∧ E (Ind G K ( A ) ⋊ G G ( G/S )) → H G ( X, E (Ind G K A )) assemble to (11.2) Ind : H K ( X, E ( A )) → H G ( X, E (Ind G K ( A ))) Prop osition 1 1 .3. ( Comp ar e [9, P rop osition 12.9] ) L et A b e an E -excisive G -ring. Then the map (11.2) is an e quivalenc e. Pr o of. As a functor o f G -simplicial sets, equiv ariant homology satisﬁes excisio n and comm utes with ﬁltering colimits (see [6]). B e c ause of this, and because X is obtained by gluing to gether cells of the form Ind G H (∆ n ), H ∈ A l l , it suﬃces to prov e the prop osition for X = Ind G H ( T ) wher e H acts tr ivially on T . Let R be a full set of r epresentativ es of K \ G/H . W e hav e Ind G H ( T ) = T × G/H = a θ ∈R T × K θ H ∼ = a θ ∈R T × K/K θ Here as in Subse c tion 10.5, K θ = c θ ( H ) ∩ K . Thus H K (Ind G H ( T ) , E ( A )) = T + ∧ _ θ ∈R E ( A ⋊ G K ( K/K θ )) On the other ha nd, H G (Ind G H ( T ) , E (Ind G K ( A )) = T + ∧ E (Ind G K ( A ) ⋊ G G ( G/H )) W e have to show that _ θ ∈R E ( A ⋊ G K ( K/K θ )) → E (Ind G K ( A ) ⋊ G G ( G/H )) 40 GUILLERMO COR TI ˜ NAS AND EUGENIA E LLIS is an equiv alence. By standing assumptions iv) and v) we may replace the map ab ov e by that induced by the corres ponding ring ho momorphism (11.4) M θ ∈R A ( A ⋊ G K ( K/K θ )) → A (Ind G K ( A ) ⋊ G G ( G/H )) Here A ( A ⋊ G K ( K/K θ )) → A (Ind G K ( A ) ⋊ G G ( G/H )) is induced b y ξ K (1 , − ) : A → Ind G K ( A ) and b y the inclusions K ⊂ G and K /K θ → G/H , k K θ 7→ k θ H . One chec k s that the following diagram commutes A (Ind G K ( A ) ⋊ G G ( G/H )) 3.2.6 ∼ * * U U U U U U U U U U U U U U U U A ( A ⋊ G K ( K/K θ )) ξ K (1 , − ) ⋊ inc 4 4 j j j j j j j j j j j j j j j j M G/H (Ind G K ( A ) ⋊ H ) A ⋊ K θ O O ≀ 1 ⋊ c θ − 1   ξ K ( θ − 1 , − ) ⋊ c θ − 1 / / Ind G K ( A )[ H θ − 1 K ] ⋊ H e θH,θH O O c ∗ θ ( A ) ⋊ H θ − 1 e H θ − 1 ,H θ − 1 / / M H/H θ − 1 ( c ∗ θ ( A ) ⋊ H θ − 1 ) 10.4.5 ∼ / / Ind H H θ − 1 ( c ∗ θ ( A )) ⋊ H ∼ 10.5.4 O O Because the low er rectangle co mm utes, E ( A ⋊ K θ → Ind G K ( A )[ H θ − 1 K ] ⋊ H ) is an equiv alence, by matrix stability . Again by matr ix stability and by Lemma 3 .2.6, applying E to the top left vertical arrow is an equiv a lence. Hence to pr ov e that E applied to (11 .4) is an equiv alence, it suﬃces to s how that E applied to (11.5) Ind G K ( A ) ⋊ H = L θ ∈R Ind G K ( A )[ H θ K ] ⋊ H P θ e θH,θH / / M G/H (Ind G K ( A ) ⋊ H ) is one. But another application of matr ix stabilit y (using [2, Prop. 2 .2.6]) sho ws that E applied to (11 .5) gives the same map in HoSpt as E applied to the inclusion e H,H : Ind G K ( A ) ⋊ H → M G/H (Ind G K ( A ) ⋊ H ) . This concludes the pr oo f.  Theorem 11.6 . L et E : Z − Cat → Spt b e a functor satisfying the standing assumptions 3.3.2. Also let G b e a gr oup, F a family of su b gr oups of G and B an E - excisive ring, p r op er o ver a 0 -dimensional ( G, F ) -c omplex X . Then H G ( − , E ( B )) maps ( G, F ) -e quivalenc es to e quivalenc es. In p articular, t he assembly map H G ( E ( G, F ) , E ( B )) → E ( B ⋊ G ) is an e quivalenc e. Pr o of. W e hav e X = ` i G/K i for some K i ∈ F , and Z ( X ) = L i Z ( G/K i ) . The r ing B i = Z ( G/K i ) · B is pro per ov er G/K i , and is excisiv e by Standing assumption v ). Again by Standing assumption v), it suﬃces to prov e the assertion of the theorem individually for ea c h B i ; in other words, we ma y assume X = G/ K for some K ∈ F . Hence for A = Co mp K G B we hav e B = Ind G K A , b y Prop osition 1 0.3.1. More ov er , ISOMORPHISM CONJECTURES WITH PROPER COEFFICIE NTS 41 by Lemma 11 .1, A is E -excisive. Let Y → Z b e a ( G, F )-equiv alence. W e have a commutativ e diagr am H G ( Y , E ( B )) / / H G ( Z, E ( B )) H K ( Y , E ( A )) Ind O O / / H K ( Z, E ( A )) Ind O O The b ottom horizontal arrow is an equiv alence because K ∈ F . The t w o vertical arrows a r e equiv alences by Prop osition 11.3. It follows that the top horizo ntal arrow is an equiv a lence to o.  12. Assembl y as a connecting map Throughout this sec tio n, we consider a ﬁxed functor E : Z − Cat → Spt, a nd –except when otherwise stated– we a s sume that, in addition to the standing a s- sumptions, it satisﬁes the following: Se ctional A s s umptions 12.1 . vi) E ∗ commutes with ﬁlter ing colimits. vii) If A is E - e x cisive and L has loc al units and is ﬂat a s a Z -module, then L ⊗ A is E - excisive. 12.1. Preliminaries. Mapping c ones. Let f : A → B b e a ring homomorphism; the mapping c one o f f is deﬁned as the pullback Γ f   / / Γ B   Σ A Σ f / / Σ B Lemma 12. 1.1. Le t E : Z − Cat → Spt b e a functor satisfying b oth the standing and the se ctional assumptions, and f : A → B a homomorphism of E -ex cisive rings. Then i) E (Γ B ) is we akly c ont r actible. ii) E (Σ B ) ∼ − → Σ E ( B ) . iii) The fol lowing is a di stinguishe d triangle in HoSpt E ( B ) → E (Γ f ) → Σ E ( A ) Σ E ( f ) − → Σ E ( B ) Pr o of. By Lemma 8.1.3, Γ B = Γ Z ⊗ B , whence it is E -exc is ive, b y sectional as- sumption 1 2.1 vii). Part i) follows fro m matrix stability a nd the fact that Γ Z is a ring with inﬁnite sums (see e.g. [2, Prop. 2.3.1]). Parts ii) and iii) follow from i) and excision.  Matrix rings and gr oup actions. Lemma 12. 1.2. L et G b e a gr oup, A a G -ring and X a G -s et . Write M X for the ring M X e quipp e d with t he G - action g ( e x,y ) = e gx, gy 42 GUILLERMO COR TI ˜ NAS AND EUGENIA E LLIS The m ap ( M X A ) ⋊ G → M X ( A ⋊ G ) , ( e x,y ⊗ a ) ⋊ g 7→ e x,g − 1 y ⊗ ( a ⋊ g ) is a G -e quivaria nt isomorphism of rings. 12.2. Dirac extensions. Le t G be a group, F a family of subgroups, E : Z − Cat → Spt a functor satisfying the standing assumptions, and A an E -excisive ring. A Dir ac ex tension for ( G, F , A, E ) consists o f an extension of E -excis iv e G -rings (12.2.1) 0 → B → Q → P → 0 together with a zig-zag A = Z 0 f 0 / / Z 1 Z 2 f 2 o o f 3 / / . . . Z n = B such that a) E ( f i ⋊ H ) is an equiv a lence for every subgroup H ⊂ G . b) E ∗ ( Q ⋊ H ) = 0 for every H ∈ F . c) H G ( − , E ( P )) sends ( G, F )-e q uiv alences to equiv a lences. R emark 12.2.2 . Condition a) tog ether with sta nding assumptions iii) a nd iv) and Lemma 3.2.6 imply that the zig - zag f = { f i } induces an equiv alence H G ( X, E ( A )) ∼ − → H G ( X, E ( B )) for every G -space X . Similar ly , it follows from condition b) that H G ∗ ( Y , E ( Q )) = 0 for every ( G, F )-co mplex Y . Prop osition 12.2 .3. L et E : Z − Cat → Spt b e a funct or satisfying the standing assumptions, G a gr oup, F a family of sub gr oups of G , and A a G -ring. L et (12 .2 .1) b e a Dir ac extension for ( G, F , A, E ) . Then t her e ar e an ex act se quenc e E ∗ +1 ( A ⋊ G ) → E ∗ +1 ( Q ⋊ G ) → E ∗ +1 ( P ⋊ G ) ∂ → E ∗ ( A ⋊ G ) an isomorphism H G ∗ ( E ( G, F ) , E ( A )) ∼ = E ∗ +1 ( P ⋊ G ) , and a c ommutative diagr am H G ∗ ( E ( G, F ) , E ( A )) ∼ = ) ) R R R R R R R R R R R R R R Assembly / / E ∗ ( A ⋊ G ) E ∗ +1 ( P ⋊ G ) ∂ 7 7 o o o o o o o o o o o o Pr o of. By Prop osition 3.3.9 and Remark 12.2.2 we hav e a distinguis hed triangle (12.2.4) H G ( X, E ( A )) / / H G ( X, E ( Q )) / / H G ( X, E ( P )) ∂ X / / Σ H G ( X, E ( A )) for every G -simplicial s e t X . The propo sition follows b y compar ison of the long exact sequence o f homotop y associa ted to the triangles fo r X = E ( G, F ), a nd X = ∗ , using that, ag ain by Remark 12.2 .2, w e hav e H G ∗ ( E ( G, F ) , E ( Q )) = 0.  R emark 1 2.2.5 . If X ∈ S G and cX ∼ ։ X is a ( G, F )-coﬁbrant replacement, then the same ar gumen t a s tha t of the proo f of Prop osition 12 .2.3 shows that the map H G ( cX , E ( A )) → H G ( X, E ( A )) is an equiv a le nce if and only if the b oundary ma p ∂ X in the seq uence (12.2.4) is an equiv alence. ISOMORPHISM CONJECTURES WITH PROPER COEFFICIE NTS 43 12.3. A canonical Dirac extens ion. Let G b e a gro up and F a family of sub- groups. Consider the discrete G -simplicial se ts X = X F = a H ∈F G/H , Y = G/G a X The gro up G a cts on Y and thus on the ring M Y of Y × Y -matrices with ﬁnitely many nonzer o integral co eﬃcients. The p oint y 0 corres p onding to the unique o rbit of G/G is ﬁxed by G , whence the map ι : Z → M Y , λ → λE y 0 ,y 0 is G -equiv ar iant. In particula r we hav e a directed system o f G -rings { id ⊗ ι : ( M ∞ M Y ) ⊗ n → ( M ∞ M Y ) ⊗ n +1 } n . Put F 0 = colim n ( M ∞ M Y ) ⊗ n Since X is discrete, the r ing of ﬁnitely supp orted functions brea ks up into a sum Z ( X ) = M x ∈ X k χ x Multiplication b y an ele ment of M Y gives an Z -linear endomo rphism of Z ( Y ) . This deﬁnes an equiv ariant monomorphis m M Y → End Z ( Z ( Y ) ) whose image consists o f those linear transformations T such tha t the matrix of T with resp ect to the basis { χ y : y ∈ Y } has ﬁnitely many nonzero en tries. Note tha t m ultiplication by χ x in Z ( X ) ⊂ Z ( Y ) is in this image. Thus we have an equiv a r iant injectiv e ring homomo rphism ρ : Z ( X ) → M Y F or e a ch n ≥ 1, consider the G -ring F n = n O i =1 Γ ρ ! ⊗ F 0 The inclusion M ∞ M Y → Γ ρ induces an inclusio n F n ⊂ F n +1 for ea c h n ≥ 0. Put F ∞ = [ n ≥ 0 F n If A ∈ Ring s, we a lso write F n A = F n ⊗ A ( n ≥ 0). W e have Lemma 12.3.1. i) F n ⊂ F ∞ is an ide al ( n < ∞ ) . ii) F or e ach n ≥ 0 , F n and F n +1 / F n ∼ = Σ Z ( X ) ⊗ F n have lo c al units, ar e ( G, F ) -pr op er rings and ar e ﬂat as ab elian gr oups. iii) If H ∈ F , χ H ∈ Z ( G/H ) ⊂ Z ( X ) is the char acteristic fun ction, and A is a G -ring, we have a c ommutative diagr am ( Z ( G/H ) ⊗ F n A ) ⋊ H ⊂ ( Z ( X ) ⊗ F n A ) ⋊ H ( ρ ⊗ 1) ⋊ id / / ( M Y F n A ) ⋊ H ≀ (12.1.2)   F n A ⋊ H χ H ⊗ 1 O O e H,H ⊗− / / M Y ( F n A ⋊ H ) 44 GUILLERMO COR TI ˜ NAS AND EUGENIA E LLIS Pr o of. Part i) is clear . Because M Y is proper over Y , F n is proper over Y for all n , by 10.1 .3. Similarly , (12.3.2) F n +1 / F n = Σ Z ( X ) ⊗ F n is pro p er. That F n is ﬂat is clear for n = 0; the general case follows b y inductio n, using (12 .3.2). The r ing F 0 has lo cal units b ecause M Y and M ∞ do. T o prov e that F n has lo cal units for n ≥ 1, it suﬃces to show that Γ ρ do es. W e may and do identify Γ ρ with the inv erse ima ge o f Σ( ρ ( Z ( X ) )) under the pro jection π : Γ M Y → Σ M Y ; th us Γ ρ = Γ ρ ( Z ( X ) ) + M ∞ M Y ⊂ Γ M Y One checks that if φ 1 , . . . , φ r ∈ Γ ρ , then there are ﬁnite subsets F 1 ⊂ X and F 2 ⊂ N such that fo r y 0 = G/G ∈ Y , the element e = 1 ⊗ X x ∈ F 1 e x,x + X p ∈ F 2 e p,p ⊗ e y 0 ,y 0 ∈ Γ ρ satisﬁes e 2 = e and e φ i = φ i e = φ i for all i = 1 , . . . , r . This prov es part ii); part iii) is straig h tforward.  Theorem 12.3. 3. (Comp ar e [5, Theo rem 5.18] ) L et E : Z − Ca t → Spt b e a functor satisfying b oth the standing and the se ctional assumptions. L et G a gr oup, F a family of sub gr oups, and A an E -excisive G -ring. Then F 0 A → F ∞ A → F ∞ A/ F 0 A is a Dir ac ext ension for ( G, F , E , A ) . Pr o of. The three rings in the e x tension of the theorem ar e E -ex cisive, by Lemma 12.3.1 ii) and sectiona l a ssumption 1 2.1 vii). T he map E ( A ⋊ H ) → E ( F 0 A ⋊ H ) is an equiv a le nce for all subgr oups H ⊂ G by L e mma 12.1.2, standing a ssumptions ii) and iii) a nd sectional assumption v i). Next we prove that if c X → X is a coﬁbrant repla cemen t, then H G ( cX , E ( F ∞ A/ F 0 A )) → H G ( X, E ( F ∞ A/ F 0 A )) is an equiv alence. By e x cision and sec tio nal a ssumption vi), it suﬃces to show that (12.3.4) H G ( cX , E ( F n A/ F 0 A )) → H G ( X, E ( F n A/ F 0 A )) ( n ≥ 1) is an equiv a lence. Consider the extensio n 0 → F n A/ F 0 A → F n +1 A/ F 0 A → F n +1 A/ F n A → 0 By Prop ositio n 3.3 .9, cX → X gives a map of homoto py ﬁbration sequences H G ( cX , E ( F n A/ F 0 A )) / /   H G ( X, E ( F n A/ F 0 A ))   H G ( cX , E ( F n +1 A/ F 0 A )) / /   H G ( X, E ( F n +1 A/ F 0 A ))   H G ( cX , E ( F n +1 A/ F n A )) / / H G ( X, E ( F n +1 A/ F n A )) By Lemma 12 .3.1 a nd Theorem 11 .6, the b ottom hor izontal map is an equiv alence. Hence (12.3.4) is an eq uiv alence for each n , by induction. It remains to show that E ∗ ( F ∞ A ⋊ H ) = 0 for each H ∈ F . Beca use E ∗ preserves ﬁltering co limits by assumption, we may further res trict our selves to proving that the ma p j n : ISOMORPHISM CONJECTURES WITH PROPER COEFFICIE NTS 45 E ∗ ( F n A ⋊ H ) → E ∗ ( F n +1 A ⋊ H ) induced by inclusion is zero for a ll n . By Lemma 12.1.1 w e have a long exac t se q uence ( q ∈ Z ) E q ( F n A ⋊ H ) j n / / E q ( F n +1 A ⋊ H ) / / E q − 1 ( Z ( X ) ⊗ F n A ⋊ H ) ∂   E q − 1 ( F n A ⋊ H ) where ∂ = E q − 1 ( ρ ⊗ 1 ⋊ 1). By Lemma 12.3 .1, part iii), ∂ is a split sur jection. It follows that j n = 0; this co ncludes the pro of.  Example 12.3 .5 . The h yp othesis of Theorem 12.3.3 are satisﬁed, fo r exa mple, by the functorial sp ectra K , K ninf and K H . 13. Isomorphism conjectures with proper coefficients 13.1. The excisiv e case. Theorem 13. 1.1. L et E : Z − Cat → Spt b e a fun ctor. Assume that E satisﬁes t he standing assumptions 3.3.2, t hat it is excisive and t hat E ∗ c ommut es with ﬁ lt ering c olimits. L et A b e a ( G, F ) -pr op er G -ring. Then the functor H G ( − , E ( A )) sends ( G, F ) -e quivalenc es to e qu ivalenc es. In p articular the assembly map H G ( E ( G, F ) , E ( A )) → E ( A ⋊ G ) is an e quivalenc e. Pr o of. By deﬁnition of prop erness, there is a lo cally ﬁnite ( G, F )-complex X such that A is pr ope r over X . W e co nsider ﬁrs t the case when X is ﬁnite dimensional. If dim X = 0, the theore m follows from Theo r em 1 1.6. Let n > 0 and a ssume the theorem true in dimensio ns < n . If dim X = n , and Y ⊂ X is the n − 1-skeleton, we hav e a pushout diag ram ` i Ind G H i (∆ n ) / / X ` i Ind G H i ( ∂ ∆ n ) / / O O Y O O Here H i ∈ F a nd the hor izontal arrows ar e prop er, since X is assumed lo cally ﬁnite. Hence we o btain a pullback diag ram (13.1.2) L i Z (∆ n ) ⊗ Z ( G/H i )   Z ( X ) o o   L i Z ( ∂ ∆ n ) ⊗ Z ( G/H i ) Z ( Y ) o o Let I = ker( Z ( X ) → Z ( Y ) ) b e the kernel of the res triction map; b ecause the dia g ram ab ov e is cartesian, I ∼ = L i ker( Z (∆ n ) ⊗ Z ( G/H i ) → L i Z ( ∂ ∆ n ) ⊗ Z ( G/H i ) ). The quotient A/I · A is prop er ov er Y , and I · A is prop er ov er ` i Ind G H i (∆ n ), whence also ov er the zero-dimensio na l ` i G/H i , by Lemma 10.1 .6. Thus the theor em is true for b oth A/I · A and I · A ; be cause E is excisive by hypothesis, this implies that the theorem is a lso true for A . This prov es the theorem for X ﬁnite dimensio nal. 46 GUILLERMO COR TI ˜ NAS AND EUGENIA E LLIS The g eneral case follows fro m this using Lemma 1 0.1.5 and the hypo thesis that E ∗ commutes with ﬁlter ing colimits.  Example 13.1.3 . Both K H and K ninf satisfy the hypothes is of Theor em 13.1.1. R emark 13.1.4 . The proo f of Theor e m 13.1.1 makes clear that if the hypo thes is that E ∗ commutes with ﬁltering colimits is dropp ed, then the theo rem remains true for A prop er over a ﬁnite dimens io nal ( G, F )-co mplex. On the other hand, the hypothesis that E b e excisive is key , since the standing assumptions alo ne do not guara ntee that the excision arguments of the pro of go through, not ev en for A = Z . The argument uses that the common k ernel of the v ertical maps of (13.1.2) be E -excis iv e; by standing assumption 3.3.2 v) this is equiv alent to saying that I n = k er( Z ∆ n → Z ∂ ∆ n ) is E -excisive. How ever I n is not K -excisive, b ecause T or ˜ I n 1 ( Z , I n ) = I n /I 2 n 6 = 0 (see Subs e c tion A.1). 13.2. The K -theory i somorphism conjec ture with prop er co e ﬃcien ts. Theorem 13.2. 1. L et G b e a gr oup, F a family of sub gr oups of G , and A a G -ring. Assume t hat F c ontains al l the cyclic su b gr oups, and that A is pr op er over a lo c al ly ﬁnite ( G, F ) -c omplex. Also assum e that A ⊗ Q is K -excisive. Then H G ( − , K ( A )) sends ( G, F ) - e quivalenc es to r ational e quivalenc es. If m or e over A is a Q -algebr a, then H G ( − , K ( A )) sends ( G, F ) -e quivalenc es to inte gr al e quivalenc es. In p articular the assembly map H G ∗ ( E ( G, F ) , K ( A )) → K ∗ ( A ⋊ G ) is a r ational isomorphi sm if A is a ( G, F ) - pr op er ring, and an inte gr al isomorphism if in add ition A is a Q -algebr a. Pr o of. By Theorem 13.1 .1, H G ( − , K H ( A )) maps ( G, F )-equiv alences to eq uiv a- lences. Hence using the ﬁbration K nil → K → K H we s e e tha t it suﬃces to show that the statement o f the theor em is true with K nil substituted for K . B ecause the map (8.2.2) is an eq uiv alence for Q -alg ebras, it suﬃces to prov e that if A is a ( G, F )-prop er r ing, then H G ( − , K nil ( A )) sends ( G, F )-equiv alences to rationa l equiv alences. Cons ider the ﬁbration K ninf → K nil ⊗ Q → Ω − 1 | H C ( − ⊗ Q ) | Because F con tains a ll c y clic subgro ups and A ⊗ Q is H -unital, H G ( − , H C ( A ⊗ Q )) sends ( G, F )-eq uiv alences to equiv a lences, by Prop osition 7.6 and Coro llary 3.3.1 1. Similarly , H G ( − , K ninf A )) sends ( G, F )-e q uiv alences to equiv alences, by Theorem 13.1.1 and Pr op osition 8.2.4. It follows tha t the sa me is true of H G ( − , K nil ( A ) ⊗ Q ). This completes the pro o f.  Example 13.2.2 . If X is a ( G, F )-complex lo ca lly ﬁnite as a simplicial set and B is K -ex cisive, then Z ( X ) ⊗ B is ( G, F )-prop er by E xample 10 .1.3 and is K -excisive by Prop osition 9 .5.3. If T is the g eometric r ealization of X and F = R , C , then the ring C comp ( T ) o f F -v alued compactly supp orted contin uous functions is prop er ov er X , again by Example 10.1.3, a nd therefore it ( G, F )-prope r . In fa c t the arg umen t giv en in 10.1 .3 to show that Z ( X ) · C comp ( T ) = C comp ( T ) shows that C comp ( T ) is s -unital and therefore K -excisive, by Exa mple A.3.5. Hence C comp ( T ) ⊗ B is K -excisive if B is , by P rop osition A.5.3. ISOMORPHISM CONJECTURES WITH PROPER COEFFICIE NTS 47 A. Appendix: K -excisive and H -unit al rings A.1. The groups T or ˜ A ∗ ( − , A ) . Let M = Z , Z /n Z , Q . Theorems of Suslin [25] (for M = Z , Z /n Z ) and Suslin-W o dzicki [26] (for M = Q ) establish that a ring A is excisive for K -theory with co eﬃcients in M if and only if T or ˜ A ∗ ( M , A ) = 0 Example A.1.1 . A ring A is said to hav e the (right) triple fac torization pr op erty if for every ﬁnite family a 1 , . . . , a n ∈ A there exis t b 1 , . . . , b n , c, d ∈ A such that a i = b i cd and { a ∈ A : ad = 0 } = { a ∈ A : acd = 0 } It was pr oved in [26, Theorem C] that rings having the triple factorizatio n pr o per t y are K -ex cisive. In par ticular, rings with lo cal units are K -excisive. Let M b e a n ab elian g roup; rega rd M as an ˜ A -mo dule through the augmentation ˜ A → Z . W e sha ll introduce a functoria l ab e lian group ¯ Q ( A, M ) whic h computes T or ˜ A ∗ ( M , A ). Consider the functor ⊥ : ˜ A − mod → ˜ A − mod , ⊥ N = M x ∈ N ˜ A. The functor ⊥ is the free ˜ A -mo dule cotriple [29, 8.6.6 ]. Let Q ( A ) → A b e the canonical simplicial resolutio n by fre e ˜ A -mo dules asso ciated to ⊥ [29, 8.7.2]; by deﬁnition, its n -th term is Q n ( A ) = ⊥ n +1 A . Put ¯ Q ( A, M ) = M ⊗ ˜ A Q ( A ) . W e have π ∗ ( ¯ Q ( A, M )) = T or ˜ A ∗ ( M , A ) W e a bbreviate ¯ Q ( A ) = ¯ Q ( A, Z ). Note that ¯ Q ( A, M ) = M ⊗ ¯ Q ( A ) W e have ¯ Q 0 ( A ) = Z [ A ] , ¯ Q n +1 ( A ) = Z [ Q n ( A )] . Lemma A.1.2. L et F ∼ ։ A b e a s implicial r esolution in Rings and M an ab elian gr oup. L et diag ¯ Q ( F ) b e the diagonal of t he bisimplicial a b elian gr oup ¯ Q ( F ) . Then T or ˜ A ∗ ( M , A ) = π ∗ ( M ⊗ diag ¯ Q ( F )) Pr o of. Be cause F → A is a s implicial reso lutio n in Rings, ¯ Q 0 ( F ) = Z [ F ] → Z [ A ] = ¯ Q 0 ( A ) is a free s implicial reso lution in A b of the free ab elian gr oup Z [ A ]. Observe that if G → N is a free resolution o f a free a belia n group N , then ˜ A ⊗ G → ˜ A ⊗ N is a free simplicia l ˜ A -mo dule re solution, and Z [ ˜ A ⊗ G ] → Z [ ˜ A ⊗ N ] is a free simplicial Z -mo dule r e solution. Thus for each n , ¯ Q n ( F ) → ¯ Q n ( A ) is a free reso lution of the free ab elian group ¯ Q n ( A ), and thus it remains a resolution after tensoring by M . It follows that M ⊗ diag ¯ Q ( F ) computes T or ˜ A ∗ ( M , A ).  Prop osition A. 1.3. L et F ∼ ։ A b e a s implici al r esolution and M an ab elian gr oup. Then ther e is a ﬁrst quadr ant sp e ctr al se quenc e E 2 p,q = π q (T or ˜ F p ( M , F )) ⇒ T or ˜ A p + q ( M , A ) 48 GUILLERMO COR TI ˜ NAS AND EUGENIA E LLIS Pr o of. This is just the sp ectral sequence of the bisimplicial ab elian group ([ p ] , [ q ]) 7→ ¯ Q p ( F q , M ).  Corollary A.1.4. Le t F ∼ ։ A b e fr e e simplicial r esolution in Rings . Then π ∗ ( M ⊗ ( F /F 2 )) = T or ˜ A ∗ ( M , A ) Pr o of. In view of the previo us prop osition, and o f the fact that T or ˜ B 0 ( M , B ) = M ⊗ B /B 2 for ev ery ring B , it suﬃces to show that if V is a free abelian gro up, and T V the tenso r algebra , then T o r ˜ T V n ( M , T V ) = 0 for n ≥ 1. But this is clea r, since T V is free as a ˜ T V -module; indeed, the mult iplication map ˜ T V ⊗ V → T V is an isomorphis m.  A.2. Bar compl e x. Let A be a ring. Consider the complex P ( A ) given by P n ( A ) = ˜ A ⊗ A ⊗ n +1 ( n ≥ 0), with b oundar y map b ”( a − 1 ⊗ a 0 ⊗ a 1 ⊗ · · · ⊗ a n ) = n − 1 X i = − 1 ( − 1) i a − 1 ⊗ · · · ⊗ a i a i +1 ⊗ · · · ⊗ a n The m ultiplicatio n map µ : P 0 ( A ) = ˜ A ⊗ A → A gives a surjective quasi-is omorphism µ : P ( A ) ։ A [29, 8.6.12]. A canonical Z -linear s e ction o f µ is j = 1 ⊗ − : A → ˜ A ⊗ A . Le t ǫ : ˜ A → A , ǫ ( a, n ) = a . A Z -linear homoto p y j µ → 1 is deﬁned by s : P n ( A ) → P n +1 ( A ) , s ( a − 1 ⊗ · · · ⊗ a n ) = 1 ⊗ ǫ ( a − 1 ) ⊗ a 0 ⊗ · · · ⊗ a n Thu s P ( A ) is a re s olution of A by ˜ A -mo dules, and moreover these ˜ A -mo dules a re scalar extensions of Z -mo dules. Put C bar ( A ) = Z ⊗ ˜ A P ( A ) , b ′ = Z ⊗ ˜ A b ” If A is ﬂa t as Z -mo dule, then C bar ( A ) computes T or ˜ A ∗ ( Z , A ) and C bar ( A, M ) = M ⊗ C bar ( A ) computes T or ˜ A ∗ ( M , A ). In general, the ho mology o f C bar ( A ) can b e int erpreted as the T or gr oups relative to the extension Z → ˜ A . F or an arbitrar y ring A , one ca n use the natural ho mo top y s to give a natural map Q ( A ) → P ( A ) The induced ma p M ⊗ ¯ Q ( A ) → M ⊗ C bar ( A ) is a quasi-ho momorphism if A is ﬂat as a Z -mo dule. In particular, we have the following. Lemma A.2.1 . L et F ∼ ։ A b e a simplici al r esolution by ﬂat rings, and M an ab elian gr oup. Then T or ˜ A ∗ ( M , A ) = H ∗ (T ot( M ⊗ C bar ( F ))) A.3. H -unital rings. A r ing A is ca lled H -unital if for every ab elian gr oup V , the complex C bar ( A ) ⊗ V is acy c lic. R emark A.3.1 . Note that for A ﬂat as a Z -mo dule, H -unita lit y is eq uiv alent to the acyclicity o f C bar ( A ), that is, to the v a nishing of the gro ups T or ˜ A ∗ ( Z , A ). Thus for a ﬂat ring H -unitality equa ls K -ex cisiveness. ISOMORPHISM CONJECTURES WITH PROPER COEFFICIE NTS 49 Pur e exact se quenc es. Let (A.3.2) 0 → A → B → C → 0 be an exa c t se quence of rings . W e say that (A.3.2) is pure if for e very ab elian gr oup V , the seque nc e o f ab elian gro ups 0 → A ⊗ V → B ⊗ V → C ⊗ V → 0 is exac t. Pure injective and pure surjective maps, and pur e acyclic complex es a re deﬁned in the ob vious way . If X ( − ) is a functor ial chain co mplex, then we say that A is pur e X -excisive if fo r every pure exact seq uence (A.3.2), X ( A ) → X ( B ) → X ( C ) is a distinguished triangle. The following theo rem was pr oved by M. W o dzicki in [31]. Theorem A.3.3. ( W o dzicki) The fol lowing c onditions ar e e qu ivalent for a ring A . i) A is H -unital. ii) A is pur e C bar -excisive. iii) A is pur e H H -excisive. iv) A is pur e H C -excisive. Example A.3.4 . An y linearly s plit sequence (A.3.2) is pure. In particular , any sequence (A.3.2 ) with A a Q -algebra is pure, since a n y Q -vectorspac e is injective a s an ab elian group. Thus for a Q -alg ebra A , W o dzicki’s theo r em remains v a lid if we omit the word “pure” ev erywhere. F urthermor e, b y the Suslin-W o dzicki theorem cited a bove, for A a Q -algebra the co nditions of Theorem A.3.3 ar e a ls o equiv alen t to A b eing K Q -excisive. In fact it is well-known that for a Q -algebra A , b eing K Q -excisive is equiv alent to b eing K -excisive; as expla ined in [1, Lemma 4.1] this well-kno wn fact follows from the main result of [27]. See [26, Lemma 1.9] for a diﬀerent pro of. Example A.3.5 . Each s -unital r ing is H -unital, by [31, Cor. 4.5 ]. Thus any s -unital ring which is ﬂat as a Z -mo dule is K -excisive, by Remark A.3.1. A.4. Co limits. The bar co mplex ma nifestly commutes with ﬁltering colimits, and th us H -unital rings are closed under them. The ne x t prop osition establishes the analogue of this pr ope r t y fo r K -excisive r ings. Prop osition A. 4.1. L et { A i } b e a ﬁltering system of rings, and let M b e an ab elian gr oup. W rite A = colim A i . Then T or ˜ A ∗ ( M , A ) = colim i T or ˜ A i ∗ ( M , A i ) Pr o of. W rite ⊥ : Ring s → Rings, ⊥ B = T ( Z [ B ]) for the cotriple ass oc iated with the forgetful functor Rings → S ets and its adjoint. W rite F ( A ) ∼ ։ A for the cotriple reso lution F ( A ) n = ⊥ n +1 A ([29, § 8/6]). W e have F ( A ) = co lim i F ( A i ). Thu s T o t( M ⊗ C bar F ( A )) = colim i M ⊗ C bar F ( A i ). Hence we ar e do ne by Lemma A.2.1.  Corollary A.4.2. K -excisive rings ar e close d under ﬁ ltering c olimits. 50 GUILLERMO COR TI ˜ NAS AND EUGENIA E LLIS Let M 0 and M 1 be chain complexes of ab elian gr oups, and let f ∈ [1] n . Put T f ( M 0 , M 1 ) = M f (1) ⊗ · · · ⊗ M f ( n ) Let M 0 ⋆ M 1 = M n ≥ 0 M f ∈ m ap([ n ] , [1]) T f ( M 0 , M 1 ) Lemma A.4.3. L et A and B b e rings. Then C bar ( A ⊕ B ) = ( C bar ( A )[ − 1] ⋆ C bar ( B )[ − 1])[+ 1] Pr o of. If D is a ring then C bar ( D ) = T ( D [ − 1])[+ 1 ] as graded ab elian groups. Hence for ` the copro duct of ring s, w e hav e C bar ( A ⊕ B ) = T ( A [ − 1 ] ⊕ B [ − 1])[+ 1] =( T ( A [ − 1]) a T ( B [ − 1 ]))[+1] =( C bar ( A )[ − 1] ⋆ C bar ( B )[ − 1])[+ 1] It is is stra ig h tforward to chec k that the identiﬁcations ab ov e are compatible with bo undary maps.  Prop osition A.4.4 . L et { A i } b e a family of rings and A = L i A i . Then A is K -ex cisive if and only if e ach A i is, and in that c ase L i K ( A i ) → K ( A ) is an e quivalenc e. Pr o of. Le t B and C be ring s, and let F → B and G → C b e free simplicial resolutions in Rings. Then F ⊕ G → B ⊕ C is a ﬂa t simplicial r esolution. Fix q ≥ 0, and put C 0 = C bar ( F q ), C 1 = C bar ( G q ). Let p ≥ 1 , and f ∈ [1] p . Then by the K ¨ unneth formula H n ( T f ( C 0 [ − 1] , C 1 [ − 1])[+1]) = T f ( H ∗ ( C 0 ) , H ∗ ( C 1 )) n +1 =  T f ( F q /F 2 q , G q /G 2 q ) p = n + 1 0 p 6 = n + 1 Hence the sec ond page of the sp ectral sequence for the double complex of Lemma A.2.1 is E 2 p,q = M f ∈ [ 1] p +1 π q ( T f ( F /F 2 , G/ G 2 )) If B and C a r e K -excisive, we hav e E 2 = 0, by the Eilenber g-Zilb er theo r em and the K¨ unneth formula, and th us B ⊕ C is ag a in K -excisive. It follows from this and from Prop osition A.4.1 that if { A i } is a family of K -excisive ring s as in the prop osition, then A is K - excisive. If B and C ar e arbitrar y , then E 2 0 ,q = T o r ˜ B q ( Z , B ) ⊕ T or ˜ C q ( Z , C ) E 2 p, 0 = M f ∈ [ 1] p +1 T f ( B /B 2 , C /C 2 ) Hence if B ⊕ C is excisive, E 2 ∗ , 0 = 0. It follows that E 2 0 , 1 = 0, a nd there fo re π 1 ( T f ( F /F 2 , G/ G 2 )) involv es direct summands of tensor pr o ducts of the form E 2 p, 0 ⊗ E 2 0 , 1 and its symmetric, and both of these a re zero. Thus E 2 ∗ , 1 = 0. A recursive arg umen t shows that E 2 = 0, whence bo th B and C are K -excisive. If now A and { A i } are as in the prop o sition, A is excisive, and j ∈ I , then setting ISOMORPHISM CONJECTURES WITH PROPER COEFFICIE NTS 51 B = A j and C = L i 6 = j A i ab ov e, we obtain that A j is K -excisive. The las t asser- tion o f the propo sition is w ell-known if eac h A i is unital. Mor e genera lly , a ssume all A i are K - excisive, and co nsider the exact s e q uence (A.4.5) 0 → A → M i ˜ A i → M i Z → 0 W e have a commutativ e diagram with homoto p y ﬁbration rows L i K ( A i ) / /   L i K ( ˜ A i )   / / L i K ( Z )   K ( A ) / / K ( L i ˜ A i ) / / K ( L i Z ) Because the middle and right vertical a rrows ar e eq uiv alences, it follows that the left one is an equiv ale nce too .  Prop osition A.4.6 . L et { A i } b e a family of rings and A = L i A i . Then A is H - unital if and only if e ach A i is, and in that c ase L i H H ( A i ) → H H ( A ) and L i H C ( A i ) → H C ( A ) ar e quasi-isomorphisms. Pr o of. The la st assertion is prov ed by the same argument as its K -theo r etic co un- terpart. By Theorem A.3.3 and Lemma A.4.3, if B and C are rings and B is H - unital, then C bar ( B ⊕ C ) ⊗ V → C bar ( C ) ⊗ V is a q uasi-isomor phism for every ab elian group V . Thus if also C is H - unital, then so is B ⊕ C . Using this a nd the fact that H -unitalit y is preserved under ﬁltering co limits, it follo ws that if { A i } is a family of H -unital rings, then A = L i A i is H - unital. Supp ose c o n versely that A is H -unital, and consider the pure extension (A.4.5). A similar argument as that of the pro o f of Pr op osition A.4.4 s hows that L i H H ( A i ) → H H ( A ) is a quasi-isomo rphism. Next ﬁx an index j and let 0 → A j → B → C → 0 be a pure extens ion. Then 0 → A → M i 6 = j A i ⊕ ˜ B → M i 6 = j A i ⊕ ˜ C → 0 is a pure extension. Applying H H yields a distinguished triangle quasi-isomo rphic to M i H H ( A i ) → M i 6 = j H H ( A i ) ⊕ H H ( B ) ⊕ H H ( Z ) → M i 6 = j H H ( A i ) ⊕ H H ( C ) ⊕ H H ( Z ) Removing summands, we obtain a tr iangle H H ( A j ) → H H ( B ) → H H ( C ) W e hav e shown that A j satisﬁes excision for pur e extensions in Ho chsc hild homo l- ogy; by Theor em A.3.3, this implies tha t A j is H -unital.  52 GUILLERMO COR TI ˜ NAS AND EUGENIA E LLIS A.5. T ens or pro ducts. It w as prov ed by Suslin a nd W o dzicki [26, Theorem 7.10] that the tenso r pro duct of H -unital r ing s is H -unital. Here w e establis h a weak analogue of this pr ope r t y fo r K -excisive r ings. Let A b e a ring. Put L − 1 A = A, L n +1 A = ker ( A ⊗ L n ( A ) µ → L n ( A )) ( n ≥ − 1) Here µ is the mult iplication map. Lemma A.5. 1. L et A b e a K -excisive ring, and V an ab elian gr oup. Assume b oth A and V ar e ﬂat over Z . Then L n − 1 A is ﬂat as an ab elian gr oup and T or ^ A ⊗ T V n ( Z , A ⊗ T V ) = L n − 1 A ⊗ V ⊗ n +1 ( n ≥ 0) . Pr o of. If M is a le ft A -mo dule such tha t (A.5.2) A · M = M , and L ( M ) = k er( A ⊗ M → M ) is the kernel of the m ultiplication map, then we hav e a shor t exact sequence 0 → L ( M ) ⊗ T ≥ n +1 V → ^ A ⊗ T V ⊗ M ⊗ V ⊗ n → M ⊗ T ≥ n V → 0 By deﬁnition, L n A = L n +1 A . B y [26, Theo rem 7.8 and Lemma 7.6], M = L n A satisﬁes (A.5.2) for a ll n , and moreover, it is a ﬂat ab elian gr o up, b y induction. Thu s for n ≥ 1, the sequence 0 → L n − 1 ( M ) ⊗ T ≥ n +1 V → ^ A ⊗ T V ⊗ L n − 2 M ⊗ V ⊗ n → L n − 2 M ⊗ T ≥ n V → 0 is exact. Hence T or ^ A ⊗ T V i ( Z , A ⊗ T V ) =T or ^ A ⊗ T V i ( Z , L − 1 A ⊗ T ≥ 1 V ) =T or ^ A ⊗ T V 0 ( Z , L i − 1 A ⊗ T ≥ i +1 V ) = L i − 1 A ⊗ V ⊗ i +1  Prop osition A. 5.3. L et A and B b e K -excisive rings, at le ast one of them ﬂat as a Z -mo dule. Then A ⊗ B is K -ex cisive. Pr o of. Ass ume A is ﬂat. Let F ∼ ։ B be a simplicial reso lution by free rings. Then A ⊗ F ∼ ։ A ⊗ B is a reso lution by ﬂat rings . By Lemma A.5.1, the seco nd page of the sp ectral sequence o f Prop osition A.1.3 is E 2 p,q = π q ( L p − 1 A ⊗ ( F /F 2 ) ⊗ p +1 ) = L p − 1 A ⊗ π q (( F /F 2 ) ⊗ p +1 ) which equals zero by Corollary A.1.4 and the K¨ unneth form ula, since B is K -excis iv e by assumption, a nd L p − 1 A is ﬂa t b y Lemma A.5.1.  ISOMORPHISM CONJECTURES WITH PROPER COEFFICIE NTS 53 A.6. Cros sed pro ducts. Let G b e a g roup and π : Z [ G ] → Z the augmentation g 7→ 1. Put J G = ker π Lemma A.6.1. L et V b e a Z [ G ] -mo dule, fr e e as an ab elian gr oup. Then T or ^ T V ⋊ G n ( Z , T V ⋊ G ) = V ⊗ n +1 ⊗ J G ⊗ n ⊗ Z [ G ] n ≥ 0 Pr o of. Note tha t the subset V ⊗ n ⊕ T V ≥ n +1 ⋊ G ⊂ T V ⋊ G is a left ide a l, and that the map (A.6.2) ^ T V ⋊ G ⊗ V ⊗ n → V ⊗ n ⊕ T V ≥ n +1 ⋊ G 1 ⊗ y 7→ y x ⋊ g ⊗ y 7→ xg ( y ) ⋊ g is a ^ T V ⋊ G -module isomorphis m. Let M b e a Z [ G ]-mo dule. Co nsider the map V ⊗ n ⊗ M ⊕ ( T V ≥ n +1 ⋊ G ) ⊗ M → T V ≥ n V ⊗ M , ( x, ( y ⋊ g ) ⊗ m ) 7→ x + y ⊗ g m T enso r ing the isomorphis m (A.6.2) with M and comp osing, we obtain a Z -split surjective ho mo morphism of ^ T V ⋊ G -modules ^ T V ⋊ G ⊗ V ⊗ n ⊗ M ։ T V ≥ n ⊗ M This map ﬁts in a n exact sequenc e 0 → T ≥ n +1 V ⊗ J G ⊗ M → ^ T V ⋊ G ⊗ V ⊗ n ⊗ M → T ≥ n V ⊗ M → 0 If M is ﬂat as an a belia n gro up, then the middle term in the exact sequence above is a ﬂat ^ T V ⋊ G -module. Applying this successively , starting with M = Z [ G ], w e obtain T or ^ T V ⋊ G n ( Z , T V ⋊ G ) = T or ^ T V ⋊ G 0 ( Z , T V ≥ n +1 ⊗ J G ⊗ n ⊗ Z [ G ]) = V ⊗ n +1 ⊗ J G ⊗ n ⊗ Z [ G ]  Prop osition A.6.3 . L et G b e a gr oup and A ∈ G − Rings . Assume A is K -excisive. Then A ⋊ G is K -excisive. Pr o of. Note that the forgetful functor from G − Rings to sets has a left a djoin t; namely X 7→ T ( Z [ G × X ]). Hence A admits a free re s olution F ∼ ։ A such that each F n is a G -ring; for example we may take the cotriple resolutio n asso ciated to the adjoint pair just described. Since F is a simplicial G -ring, we ca n take its cro s sed pro duct with G , to obtain a Z -ﬂat resolution F ⋊ G ∼ ։ A ⋊ G . Now pro ceed as in the pro of of Pr opo sition A.5 .3, using Lemma A.6.1.  Prop osition A.6.4 . L et G b e a gr oup and A ∈ G − Rings . Assum e A is H -unital. Then A ⋊ G is H -unital. Pr o of. The bar resolution E ( G, M ) ([2 9, § 6.5]) is functorial on the G -mo dule M . Applying it dimensionwise to C bar ( A ), we obtain a simplicial chain complex E ( G, C bar ( A )). W e may view the latter a s a double chain complex with A ⊗ q +1 ⊗ Z [ G p +1 ] in the ( p, q ) spot. Removing the ﬁrs t r ow a nd the ﬁrs t column yie lds a double complex whose total chain complex we shall ca ll M [ − 1 ]. Note M is a c hain 54 GUILLERMO COR TI ˜ NAS AND EUGENIA E LLIS complex of A ⋊ G -modules and homomorphisms. W e ha v e M 0 ∼ = ( A ⋊ G ) ⊗ 2 , a nd the m ultiplication map ( A ⋊ G ) ⊗ 2 → A ⋊ G induces a surjection onto the kernel L of the augmentation A ⋊ G → A , a ⋊ g → a . Note that the h y pothes is that A is H - unital implies that the augmented co mplex (A.6.5) · · · → M 1 → M 0 → L is pure acyclic. Since each M n is extended, (A.6.5) is a pur e pseudo -free res o lution in the terminolo gy of [26, 7.7 ]. On the other hand, b ecause A is H -unital, the m ultiplication map µ : A ⊗ 2 → A is pure sur jectiv e; th us µ ◦ ( id ⊗ g ) is pure sur jectiv e for ea c h g ∈ G . It follows from this that the mult iplication map ( A ⋊ G ) ⊗ 2 → A ⋊ G is pure sur jectiv e. W e have shown that A ⋊ G s atisﬁes conditio n d) of [26, Theor em 7.8], which by lo c. cit. implies that A ⋊ G is H - unital.  References [1] G. Corti ˜ nas. The obstruction to excision i n K -theory and i n cyclic homology . Invent. Math. 164:143–173 , 2006. [2] G. Corti ˜ nas. A lgebraic v. top ological K-theory: a f riendly match. T opics in algebr aic and top olo g ic al K -the ory , 103–165 Lecture Notes in Mathematics 2008, Spr i nger, Berlin, 2011. [3] G. C or ti˜ nas and A. Thom. Biv ariant algebraic K -theory. Journal f ¨ ur die R eine und Ange- wandte Mathematik (Cr el le’s Journal) , 610:71–123, 2007. [4] J. Cun tz. Noncomm utative simplicial complexes and the Baum-Connes conjec ture. Ge om. F unct. 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Matem ´ atica-IMAS, F CEyN-UBA, Ciud ad Universit aria P ab 1, 1428 Buenos Aires, Argentina E-mail addr ess : euge nia@cmat. edu.uy CMA T, F acul t ad de Ciencias-UDELAR, Igu ´ a 4225 , 11 400 Mon tevideo, Urugua y

Isomorphism conjectures with proper coefficients

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