On crossed product rings with twisted involutions, their module categories and L-theory

ON CR OSSED PR ODUCT RINGS WITH TWISTED INV OLUTIONS, THEIR MODULE CA TEGORIES AND L -THEOR Y AR THUR BAR TELS AND W OLFGANG L ¨ UCK Abstract. W e study the F arrell -Jones Conjectu re wi th coeﬃcien ts in an ad- ditive G -category with inv olution. This i s a v arian t of the L -theoretic F arr ell- Jones Con jecture whic h originally d eals with group rings wi th the standard inv olution. W e show that this for mulat ion of the conjectu re can b e applied to crossed pro duct rings R ∗ G equipp ed with twisted inv olutions and aut omati- cally implies the a priori more general ﬁbered version. Introduction The F arr el l-Jones Conje ctu r e for algebr aic L -the ory predicts for a gr oup G and a ring R with in volution r 7→ r that the so called assembly map asmb G,R n : H G n  E V C yc ( G ); L h−∞i R  → L h−∞i n ( RG ) (0.1) is bijective for all n ∈ Z . Here the targ et is the L -the ory of the g roup ring R G with the s tandard in v olution sending P g ∈ G r g · g to P g ∈ G r g · g − 1 . This is the group one wan ts to unders ta nd. It is a crucial ingredient in the s ur gery prog ram for the class iﬁcation of closed manifolds. The source is a muc h easier to handle term, namely , a G -homology theor y applied to the the classifying sp ac e E V C yc ( G ) of the family V C yc of virtual ly cyclic sub gr oups of G . There is also a K -theory version o f the F arrell- Jones Conjecture. The o riginal source for the (Fiber e d) F arrell- J ones Conjecture is the pap er by F ar rell-Jo nes [6, 1.6 on pa ge 257 a nd 1 .7 on pag e 262]. More information can b e found for instance in the survey a rticle [10]. In this pap er we study the F a rrell-J ones Co njecture with co eﬃcie nts in an ad- ditive G -ca teg ory with in volution. W e show that this more general for mulation of the conjecture allows to consider instead of the gro up ring RG the c rossed pr o duct ring with inv o lution R ∗ c,τ , w G (see Section 4), whic h is a g eneraliza tion o f the t wisted g roup ring , and to use mo re gener al involutions, for instance the one given by twisting the standar d inv olution with a gro up homomo rphism w 1 : G → {± 1 } . The data describing R ∗ c,τ G and more g eneral inv olutions are pr etty complicated. It turns out that it is conv enien t to put thes e in to a more general but easier to handle context, where the co eﬃcients are g iven by an additive G -categor ies A with inv o lution (see Deﬁnition 4.22). Deﬁnition 0.2 ( L -theore tic F a r rell-Jo nes Co njectur e ) . A gr oup G together with an a dditiv e G -c a tegory with inv olution A satisfy the L -the or etic F arr el l-Jones Con- je ctur e with c o eﬃcients in A if the ass e m bly map asmb G, A n : H G n  E V C yc ( G ); L h−∞i A  → H G n  pt; L h−∞i A  = L h−∞i n R G A  . (0.3) induced by the pr o jection E V C yc ( G ) → pt is bijectiv e for a ll n ∈ Z .. Date : O ctober, 2007. 2000 Mathematics Subject Classiﬁc ation. 18F25, 57R67. Key wor ds and phr ases. L -theoretic F arrell- Jones Conjecture of group rings with arbitrary coeﬃcients, additiv e categories with inv olution. 1 2 AR THUR BAR TELS AND WOLF GANG L ¨ UCK A g r oup G satisﬁes the L -the or etic F arr el l-Jones Conje ctur e if for any additive G - category with in volution A the L -the or etic F arr el l-Jones Conje ctur e with c o eﬃcients in A is true. Here R G A is a certa in homotopy colimit which yields an additive catego ry with inv o lution a nd we use the L -theor y asso cia ted to an additive ca tegory with inv olu- tion due to Ranicki (see [12], [1 3] a nd [14]). The G -homology theory H G n  − ; L h−∞i A  is brieﬂy r ecalled in Section 9. If R is a ring with inv olution, A is the additive ca t- egory with involution given b y ﬁnitely gene r ated free R -mo dules and we equip A with the trivial G -ac tion, then the as sembly map (0.3 ) ag rees with the one for R G in (0.1) (see Theo rem 0.4 below). This general setup is also a very useful fr amework when one is dealing with ca tegories app ear ing in controlled top olo g y , which is an impo rtant to o l for proving the F arr ell-Jones Conjecture for certa in gr oups. Next we sta te the main r esults o f this pap er. Theorem 0.4. Supp ose t hat G satisﬁes the L -the or etic F arr el l-Jones Conje ctu r e in the sense of Deﬁn ition 0.2. L et R b e ring with the data ( c, τ , w ) and R ∗ c,τ , w G b e the asso ciate d cr osse d pr o duct ring with involution as ex plaine d in Se ction 4. Then the assembly map asmb G,R c,τ,w n : H G n  E V C yc ( G ); L h−∞i R,c,τ ,w  → L h−∞i n ( R ∗ c,τ , w G ) (0.5) is bije ctive. Her e L h−∞i R,c,τ ,w is a fu n ctor fr om the orbit c ate gory Or ( G ) to t he c ate gory of sp e ctr a such that π n  L h−∞i R,c,τ ,w ( G/H )  for H ⊆ G agr e es with L h−∞i n ( R ∗ c | H ,τ | N ,w | H H ) . Another impor ta nt feature is that in this setting the (unﬁbere d) F arrell- J ones Conjecture do es a lready imply the ﬁb er ed version. Deﬁnition 0.6 (Fib ered L -theoretic F ar rell-Jones Conjecture) . A g roup G satisﬁes the ﬁb er e d L -the or etic F arr el l-Jones Conje ct u r e if for an y group homomorphism φ : K → G and a dditive G -categor y with inv olution A the assembly map asmb φ, A n : H K ∗  E φ ∗ V C yc ( G ); L h−∞i φ ∗ A  → L h−∞i n R K φ ∗ A  . is bijective for all n ∈ Z , where the family φ ∗ V C yc of subgroups o f K co nsists of subgroups L ⊆ K with φ ( L ) vir tually cy clic and φ ∗ A is the a dditive K -ca tegory with in volution obtained from A b y restric tion with φ . Obviously the ﬁb ered version for the group G of Deﬁnition 0.6 implies the v ersion for the gr oup G of Deﬁnition 0.2, take φ = id in Deﬁnition 0 .6. The con verse is also true. Theorem 0. 7. L et G b e a gr oup. Then G satisﬁes t he ﬁb er e d L -the or etic F arr el l- Jones Conje ctur e if and only if G satisﬁes t he L -the or etic F arr el l-Jones Conje ctu r e of Deﬁnition 0.2. A general statemen t of a Fib er e d Isomorphism Conje ctur e and the discussion of its inherita nce pro pe rties under subgroups and co limits of gr oups can be found in [1, Section 4] (se e also [6, App endix], [7, Theorem 7.1 ]). These very us eful inher- itance prop erties do no t hold for the unﬁb ered version of Deﬁnition 0 .2. The next three co rollar ies are immediate consequences of Theo rem 0 .7 a nd [1, T he o rem 3.3, Lemma 4 .4, Lemma 4.5 and L e mma 4.6 ]. Corollary 0.8. L et { G i | i ∈ I } b e a dir e cte d system (with not ne c essarily inje ctive) structur e maps and let G b e its c olimit colim i ∈ I G i . Supp ose that G i satisfy the F arr el l-Jones Conje ctu re of Deﬁnition 0.2 for every i ∈ I . Then G satisﬁes the F arr el l-Jones Conje ctur e of Deﬁnition 0.2. ON CR OSSED PRODUCT RINGS WITH TWISTED INVOLUTIONS, . . . 3 Corollary 0. 9. L et 1 → K → G p − → Q → 1 b e an ext ension of gr oups. Supp ose that t he gr oup Q and for any virtual ly cyclic sub gr oup V ⊆ Q the gr oup p − 1 ( V ) satisfy the F arr el l-Jones Conje ctur e of Deﬁnition 0.2. Then the gr oup G satisﬁes t he F arr el l-Jones Conje ctur e of Deﬁnition 0.2. Corollary 0. 10. If G satisﬁes t he F arr el l-Jones Conje cture of Deﬁnition 0.2, then any su b gr oup H ⊆ G satisﬁes the F arr el l-Jones Conje cture of Deﬁnit ion 0.2. Corollar y 0 .9 and Co r ollary 0.1 0 hav e als o b een proved in [8]. Remark 0.11. Supp os e that the F a rrell-Jo nes Conjecture of Deﬁnition 0.2 has bee n prov ed for the pro duct of t wo virtually cyclic subgroups. Then Cor ollary 0.9 and Cor ollary 0.10 imply that G × H satisfy the F a rrell-J ones Conjecture of Deﬁnition 0.2 if and only if b oth G and H satisfy the F a rrell-Jo nes Conjecture of Deﬁnition 0.2 It is sometimes useful to have strict structures on A , e.g., the in v olution is de s ired to b e stric t a nd there s hould b e a (s tr ictly asso cia tive) functoria l direct sum. The functorial direct s um is ac tually needed in some pro o fs in order to guar a ntee go o d functoriality pro p er ties of certain catego ries arising from cont rolled top ology . W e will show Theorem 0.12. The gr oup G satisﬁes the L -the or etic F arr el l-Jones Conje ctur e of Deﬁnition 0.2 if it satisﬁes the obvious version of it, wher e one only c onsiders addi- tive G -c ate gory with (strictly asso ciative) functorial dir e ct sum and str ict involution (se e Deﬁnition 10. 6). The F ar rell-Jo nes Conjecture with coeﬃcients (in K - and L -theory) has been int ro duced in [4]. O ur tr eatment here is mo r e g eneral in that we allow inv olutions that ar e no t necessa rily str ic t and also deal with twisted in volutions on the cr ossed pro duct r ing. All r esults mentioned here have obvious ana logues for K - theo ry whose pro o f is actually easier since one do es not hav e to deal with the inv o lutions. The work w as ﬁnancially suppo rted by the So nderforsch ungsb ereich 478 – Ge- ometrische Strukturen in der Mathematik – and the Max-Planck-F ors ch ung s preis of the seco nd autho r . The pap er is o rganized a s follows: 1. Additiv e categor ies w ith involution 2 Additiv e categor ies w ith weak ( G, v )-action 3 Making an a dditive ca tegories with w eak ( G, v )-a ction strict 4. Crossed pr o duct r ing s a nd inv olutions 5. Connected g roup oids a nd additive ca tegories 6. F rom cros s ed pro duct rings to additive catego ries 7. Connected g roup oids a nd additive ca tegories with in volutions 8. F rom cros s ed pro duct rings with inv olution to additive categ ories with inv olution 9. G -homology theor ies 10. Z -categor ie s a nd a dditive ca tegories with inv olutions 11. G -homology theor ies a nd restriction 12. Pro of o f the main theor ems References 1. Additive ca tegories with involution In this section we will review the no tio n of a n additive categor y with inv olution as it a pp e ars a nd is used in the liter ature. This will b e one of our main examples. Let A be an additive c ate gory , i.e., a small category A suc h that for tw o ob- jects A and B the morphism set mor A ( A, B ) has the structur e of a n ab elian 4 AR THUR BAR TELS AND WOLF GANG L ¨ UCK group a nd the direc t sum A ⊕ B of tw o ob jects A a nd B ex ists a nd the ob- vious compatibility conditions hold. A cov aria nt fu n ctor of additive c ate gories F : A 0 → A 1 is a cov ar iant functor such that for tw o o b jects A a nd B in A 0 the map mor A 0 ( A, B ) → mor A 1 ( F ( A ) , F ( B )) sending f to F ( f ) resp ects the ab elia n group structures and F ( A ⊕ B ) is a mo del for F ( A ) ⊕ F ( B ). The notion of a contra v aria nt functor of additive categ o ries is deﬁned ana lo gously . An involution ( I , E ) on an additive c ate gory A is contra v aria nt functor I : A → A (1.1) of additive categor ies to g ether with a na tur al eq uiv alence of such functors E : id A → I 2 := I ◦ I (1.2) such that for every ob ject A we hav e the equality of morphisms E ( I ( A )) = I ( E ( A ) − 1 ) : I ( A ) → I 3 ( A ) . (1.3) In the sequel w e o ften write I ( A ) = A ∗ and I ( f ) = f ∗ for a morphism f : A → B in A . If I 2 = id A and E ( A ) = id A for all ob jects A , then we call I = ( I , id ) a strict involution . Deﬁnition 1.4 (Additiv e category with inv olution) . An additive c ate gory with involution is a n additive categor y tog ether w ith a n inv olution ( I , E ). An additive c ate gory with strict involution is an additive categor y together with a s tr ict inv o lution I . The follo wing example is a key example and illustr ates wh y one cannot expect in concrete situa tion that the inv olution is strict. Example 1.5. Let R be a ring . Let R - FGP be the categor y of ﬁnitely generated pro jective R -mo dules. This becomes an additive category by the direct sum of R -mo dules and the element wis e additio n of R -homomor phisms. A ring with involution is a ring R together with a ma p R → R , r 7→ r s atisfying 1 = 1 , r + s = r + s and r · s = s · r for r, s ∈ R . Given a ring with involution R , deﬁne an in volution I on the additive categor y R - FGP as fo llows. Giv en a ﬁnitely generated pro jectiv e R -mo dule P , let I ( P ) = P ∗ be the ﬁnitely generated pro jectiv e hom R ( P, R ),where for r ∈ R and f ∈ hom R ( P, R ) the element r f ∈ hom R ( P, R ) is deﬁned by r f ( x ) = f ( x ) · r for x ∈ P . The desired natural transfo rmation E : id R - FGP → I 2 assigns to a ﬁnitely gener a ted pro jective R -mo dule P the R -isomor phis m P ∼ = − → ( P ∗ ) ∗ sending x ∈ P to hom R ( P, R ) → R , f 7→ f ( x ). A functor of additive c ate gorie s with involution ( F , T ) : A → B cons is ts of a cov a r iant functor F of the underlying a dditive ca teg ories together w ith a natural equiv alence T : F ◦ I A → I B ◦ F such tha t for every ob ject A in A the follo wing diagram commutes F ( A ) E B ( F ( A ))   F ( E A ( A )) / / F ( A ∗∗ ) T ( A ∗ )   F ( A ) ∗∗ T ( A ) ∗ / / F ( A ∗ ) ∗ (1.6) If T ( A ) = id A for all o b jects A , then we call F a strict functor o f additive categories with in volution. The c omp osite of functors of additive catego r ies with inv o lution ( F 1 , T 1 ) : A 1 → A 2 and ( F 2 , T 2 ) : A 2 → A 3 is deﬁned to b e ( F 2 ◦ F 1 , T 2 ◦ T 1 ), where F 2 ◦ F 1 is the ON CR OSSED PRODUCT RINGS WITH TWISTED INVOLUTIONS, . . . 5 comp osite of functor s of a dditive ca tegories and the natural equiv alence T 2 ◦ T 1 assigns to an ob ject A ∈ A 1 the isomorphism in A 3 F 2 ◦ F 1 ◦ I A 1 ( A ) F 2 ( T 1 ( A )) − − − − − − → F 2 ◦ I A 2 ◦ F 1 ( A ) T 2 ( F 1 ( A )) − − − − − − → I A 3 ◦ F 2 ◦ F 1 ( A ) . A natur al tr ansformation S : ( F 1 , T 1 ) → ( F 2 , T 2 ) of functors A 1 → A 2 of add itive c ate gories with involutions is a natural transformation S : F 1 → F 2 of functors of additiv e catego ries such that for every ob ject A in A the following diagram commutes F 1 ( I A 1 ( A )) T 1 ( A ) / / S ( I A 1 ( A ))   I A 2 ( F 1 ( A )) F 2 ( I A 1 ( A )) T 2 ( A ) / / I A 2 F 2 ( A )) I A 2 ( S ( A )) O O (1.7) 2. Additive ca tegories with weak ( G, v ) -action In the s equel G is a group and v : G → {± 1 } is a group homomorphism to the m ultiplicative group {± 1 } . In this section we w an t to introduce the notion of an a d- ditive category with weak ( G, v )-ac tio n such that the notio n of an additive category with inv o lution is the sp ecial case of an additive catego ry with weak ( Z / 2 , v )-a ction for v : Z / 2 → {± 1 } the unique g roup isomorphis m and we can also trea t G -actions up to na tural equiv alence. Notice that this will for ce us to dea l with cov ar iant and contra v aria nt functor s simultaneously . The homomorphism v will take ca re of that. W e call a functor +1-v a riant if it is cov ar iant and − 1-v a riant if it is contra v ariant. If F 1 : C 0 → C 1 is an ǫ 1 -v ar iant functor a nd F 2 : C 1 → C 2 is an ǫ 2 -v ar iant functor, then the comp osite F 2 ◦ F 1 : C 0 → C 2 is ǫ 1 ǫ 2 -v ar iant functor. If f : x 0 → x 1 is an isomor phism and ǫ ∈ {± 1 } , then deﬁne f ǫ : x 0 → x 1 to b e f if ǫ = 1 and f ǫ : x 1 → x 0 to be the inv erse of f if ǫ = − 1. If F : C 0 → C 1 is ǫ -v ar ia nt and f : x 0 ∼ = − → y 0 is an is omorphism in C 0 , then F ( f ) ǫ : F ( x 0 ) → F ( x 1 ) is an isomo rphism in C 1 . Deﬁnition 2.1 (Additive ca teg ory with weak ( G, v )-action) . Let G b e a gr oup together with a group homo morphism v : G → {± 1 } . An addi tive c ate gory with we ak ( G, v ) -action A is an a dditive ca teg ory together with the following data : • F or ev ery g ∈ G we have a v ( g )-v aria nt functor R g : A → A of additive categorie s; • F or every tw o elements g , h ∈ G there is a natural equiv alence of v ( g h )- v aria nt functors of additive categor ies L g,h : R gh → R h ◦ R g . W e requir e: (i) R e = id for e ∈ G the unit element; (ii) L g,e = L e,g = id for a ll g ∈ G ; (iii) The following diagr a m commutes for all g , h, k ∈ G and o b jects A in A R ghk ( A ) L gh,k ( A ) / / L g,hk ( A )   R k ( R gh ( A )) R k ( L g,h ( A )) v ( k )   R hk ( R g ( A )) L h,k ( R g ( A )) / / R k ( R h ( R g ( A ))) If for ev ery tw o elements g , h ∈ G we ha ve L g,h = id and in particular R gh = R h R g , we call A with these data an additive c ate gory with strict ( G, v ) - action or brieﬂy a additive ( G, v ) -c ate gory . If v is trivia l, we will omit it from the notation. 6 AR THUR BAR TELS AND WOLF GANG L ¨ UCK Let A and B be t w o additive categorie s with w eak ( G, v )-actio n and let ǫ ∈ {± 1 } . An ǫ -variant functor ( F , T ) : A → B of additive c ate gories with we ak ( G, v ) -action is a ǫ -v ar iant functor F : A → B of additive ca tegories together with a collec tion { T g | g ∈ G } of na tural transfor mations of ǫv ( g )-v aria nt functor s of a dditiv e categor ies T g : F ◦ R A g → R B g ◦ F . W e require that for all g , h ∈ G and all ob jects A in A the following diag ram commutes F ( R A hg ( A )) F ( L A h,g ( A )) ǫ / / T hg ( A )   F ( R A g ( R A h ( A ))) T g ( R h ( A )) / / R B g ( F ( R A h ( A ))) R B g ( T h ( A )) v ( g )   R B hg ( F ( A )) L B hg ( F ( A )) / / R B g ( R B h ( F ( A ))) (2.2) The comp osite ( F 2 , T 2 ) ◦ ( F 1 , T 1 ) : A 1 → A 3 of an ǫ 1 -v ar iant functor of a dditiv e categorie s with weak ( G, v )-action ( F 1 , T 1 ) : A 1 → A 2 and an ǫ 2 -v ar iant functor of additive categories with weak ( G, v )-a ction ( F 2 , T 2 ) : A 2 → A 3 is the ǫ 1 ǫ 2 -v ar iant functor of additive categories with weak ( G, v )-actio n whose underlying ǫ 1 ǫ 2 -v ar iant functor of additive categories is F 2 ◦ F 1 : A 1 → A 3 and whose require d natura l transformatio ns for g ∈ G are g iven for a n ob ject A in A 1 by F 2 ◦ F 1 ◦ R A 1 g ( A ) F 2 (( T 2 ) g ( A )) ǫ 2 − − − − − − − − − − → F 2 ◦ R A 2 g ◦ F 1 ( A ) ( T 2 ) g ( F 1 ( A )) − − − − − − − − → R A 3 g ◦ F 2 ◦ F 1 ( A ) . A natur al tr ansformation S : ( F 1 , T 1 ) → ( F 2 , T 2 ) of functors A 1 → A 2 of add itive c ate gories with we ak ( G, v ) -action is a na tural trans fo rmation S : F 1 → F 2 of func- tors of additive catego ries such that for all g ∈ G a nd ob jects A in A 1 the following diagram commutes F 1 ( R A 1 g ( A )) ( T 1 ) g ( A ) / / S ( R A 1 g ( A ))   R A 2 g ( F 1 ( A )) ( R A 2 g ( S ( A )) ) v ( g )   F 2 ( R A 1 g ( A )) ( T 2 ) g ( A ) / / R A 2 g ( F 2 ( A )) (2.3) An ǫ -variant functor F : A → B of additive c ate gories with s t rict ( G, v ) -action is an ǫ -v ar iant functor F : A → B of additive categor ies satisfying F ◦ R A 1 g = R A 2 g ◦ F for all g ∈ G . A natura l tr ansformatio n S : F 1 → F 2 of ǫ - variant functors A 1 → A 2 of additive c ate gories with strict ( G, v ) -action is a natura l tra nsformation S : F 1 → F 2 of additive catego ries satisfying S ( R A 1 g ( A )) = R A 2 g ( S ( A )) v ( g ) for a ll g ∈ G and ob jects A in S 1 . Example 2.4 (Additive categor ies with inv olution) . Given an a dditive category A , the structure of a n a dditive categor y with weak ( Z / 2 , v )-a c tion for v : Z / 2 → {± 1 } the unique gro up isomorphism is the same as an a dditive category with inv olution. Namely , let t ∈ Z / 2 b e the gener a tor. Given an in v olution ( I , E ) in the sense of Deﬁnition 1.4, deﬁne the structure o f an a dditive category with weak ( Z / 2 , v )-action in the sense o f Deﬁnition 2 .1 by putting R e = id , R t = I , L e,e = L t,e = L t,e = id a nd L t,t = E . Co ndition (iii) in Deﬁnition 2 .1 follows from condition (1.3 ). Given the structure of an additive categor y with weak ( Z / 2 , v )-a ction, deﬁne the inv olution ( E , I ) by E = R t and I = L t,t . The cor resp onding statement is true for functors of additive catego ries with weak ( Z / 2 , v )-a ction a nd natural transfor mations b etw een them, whe r e dia gram (1.6 ) corres po nds to diagram (2.2). Analogously w e ge t that the structure of a additive catego ry with strict ( Z / 2 , v )- action is the sa me as an a dditive ca tegory w ith s trict inv olution. ON CR OSSED PRODUCT RINGS WITH TWISTED INVOLUTIONS, . . . 7 3. Making an additive ca tegories with weak ( G, v ) -action strict Many interesting examples o ccur as additive categor ies with weak ( G, v )-a ction which ar e not necessarily strict. On the o ther hand a dditive categor ies with strict ( G, v )-action are easier to handle. W e explain how w e can turn a n additive categor y with weak ( G, v )-actio n A to an additive category with strict ( G, v )-a ction which we will deno te by S ( A ). Deﬁnition 3.1 ( S ( A )) . An ob ject in S ( A ) is a pair ( A, g ) consisting of a n ob- ject A ∈ A and an elemen t g ∈ G . A mo rphism ( A, g ) to ( B , h ) is a morphism φ : R g ( A ) → R h ( B ) in A . The comp o sition o f morphisms is g iven by the one in A . The ca teg ory S ( A ) inherits the structure of an a dditive category from A in the obvious way . Next we deﬁne the structure of an additive categor y with strict ( G, v )-a c tion on S ( A ). Deﬁne for g ∈ G a functor R S g : S ( A ) → S ( A ) of additive c a tegories as follows. Given an ob ject ( A, h ), deﬁne R S g ( A, h ) = ( A, hg ) . Given a morphism φ : ( A, h ) → ( B , k ) deﬁne R S g ( φ ) : R S g ( A, h ) = ( A, hg ) → R S g ( B , k ) = ( B , k g ) by the co mpo site of mo rphisms in A R hg ( A ) L h,g ( A ) − − − − − → R g ( R h ( A )) R g ( φ ) − − − − → R g ( R k ( B )) L k,g ( B ) − 1 − − − − − − − → R kg ( B ) . if v ( g ) = 1 and R S g ( φ ) : R S g ( B , k ) = ( B , k g ) → R S g ( A, h ) = ( A, hg ) by the co mpo site of mo rphisms in A R kg ( B ) L k,g ( B ) − − − − − → R g ( R k ( B )) R g ( φ ) − − − − → R g ( R h ( A )) L h,g ( A ) − 1 − − − − − − − → R hg ( A ) if v ( g ) = − 1 A dir ect computatio n shows that R S g is indeed a functor o f additive ca teg ories. W e conclude R S e = id S ( A ) from the conditions R e = id and L g,e = L e,g = id. W e hav e to c heck R S g 2 ◦ R S g 1 = R S g 1 g 2 . W e will do this for simplicity only in the case v ( g 1 ) = v ( g 2 ) = 1, the other cases are analog ous. Given a morphism φ : ( A, h ) → ( B , k ), the mor phis m R S g 1 g 2 ( φ ) is g iven by the comp os ite in A R hg 1 g 2 ( A ) L h,g 1 g 2 ( A ) − − − − − − − → R g 1 g 2 ( R h A )) R g 1 g 2 ( φ ) − − − − − − → R g 1 g 2 ( R k ( B )) L k,g 1 g 2 ( B ) − 1 − − − − − − − − − → R kg 1 g 2 ( B ) . The morphism R S g 2 ◦ R S g 1 ( φ ) is g iven by the comp os ite in A R hg 1 g 2 ( A ) L hg 1 ,g 2 ( A ) − − − − − − − → R g 2 ( R hg 1 ( A )) R g 2 ( L h,g 1 ( A )) − − − − − − − − − → R g 2 ( R g 1 ( R h ( A ))) R g 2 ( R g 1 ( φ ) ) − − − − − − − − → R g 2 ( R g 1 ( R k ( B ))) R g 2 ( L k,g 1 ( B ) − 1 ) − − − − − − − − − − − → R g 2 ( R kg 1 ( B )) L kg 1 ,g 2 ( B ) − 1 − − − − − − − − − → R kg 1 g 2 ( B ) . Next we c o mpute that these t w o morphisms agr ee. B ecause of condition (iii) in Deﬁnition 2.1 hav e R g 2 ( L h,g 1 ( A )) ◦ L hg 1 ,g 2 ( A ) = L g 1 ,g 2 ( R h ( A )) ◦ L h,g 1 g 2 ( A ); R g 2 ( L k,g 1 ( B )) ◦ L kg 1 ,g 2 ( B ) = L g 1 ,g 2 ( R k ( B )) ◦ L k,g 1 g 2 ( B ) . 8 AR THUR BAR TELS AND WOLF GANG L ¨ UCK Hence it suﬃces to show that the co mpo site R g 1 g 2 ( R h ( A )) L g 1 ,g 2 ( R h ( A )) − − − − − − − − − → R g 2 ( R g 1 ( R h ( A ))) R g 2 ( R g 1 ( φ ) ) − − − − − − − − → R g 2 ( R g 1 ( R k ( B ))) L g 1 ,g 2 ( R k ( B )) − 1 − − − − − − − − − − − → R g 1 g 2 ( R k ( B )) agrees with R g 1 g 2 ( R k ( B )) R g 1 g 2 ( φ ) − − − − − − → R g 1 g 2 ( R k ( B )) . This follows fr o m the fact that L g 1 ,g 2 : R g 1 g 2 → R g 2 ◦ R g 2 is a natura l equiv alence. Let ( F , T ) : A → B b e an ǫ -v a riant functor of additive categor ies with weak ( G, v )-action. It induces an ǫ -v ariant functor S ( F , T ) : S ( A ) → S ( B ) of a dditive categorie s w ith strict ( G, v )-a ction as follows. F o r simplicity w e will only treat the ca se ǫ = 1, the other case ǫ = − 1 is analogous. The functor S ( F, T ) se nds an ob ject ( A, h ) in S ( A ) to the ob ject ( F ( A ) , h ) in S ( B ). It sends a morphism φ : ( A, h ) → ( B , k ) in S ( A ) which is given by a morphism φ : R A h ( A ) → R A k ( B ) in A to the morphism S ( F , T )( φ ) : ( F ( A ) , h ) → ( F ( B ) , k ) in S ( B ) which is given b y the following co mpo site of mo r phisms in B R B h ( F ( A )) T h ( A ) − 1 − − − − − → F ( R A h ( A )) F ( φ ) − − − → F ( R A k ( B )) T k ( B ) − − − − → R A k ( F ( B )) . W e hav e to s how R S ( B ) g ◦ S ( F ) = S ( F ) ◦ R S ( A ) g for ev ery g ∈ G . W e only treat the cas e v ( g ) = 1. This is obvious on o b jects since b oth comp osites se nd an ob ject ( A, h ) to ( F ( A ) , hg ). Let φ : ( A, h ) → ( B , k ) be a mor phis m in S ( A ) which is given by a morphism φ : R A h ( A ) → R A k ( B ) in A . Then R S ( B ) g ◦ S ( F )( φ ) is the mor phism ( F ( A ) , hg ) → ( F ( B ) , k g ) in S ( B ) which is g iven by the co mp os ite in B R B hg ( F ( A )) L B h,g ( F ( A )) − − − − − − − → R B g ( R B h ( F ( A ))) R B g ( T h ( A ) − 1 ) − − − − − − − − − → R B g ( F ( R A h ( A ))) R B g ( F ( φ )) − − − − − − → R B g ( F ( R A k ( B ))) R B g ( T k ( B )) − − − − − − − → R B g ( R B k ( F ( B ))) L B k,g ( F ( B )) − 1 − − − − − − − − − → R B kg ( F ( B )) and S ( F ) ◦ R S ( A ) g ( φ ) is the morphism ( F ( A ) , hg ) → ( F ( B ) , k g ) in S ( B ) which is given by the co mpo site in B R B hg ( F ( A )) T hg ( A ) − 1 − − − − − − → F ( R A hg ( A )) F ( L A h,g ( A )) − − − − − − − → F ( R A g ( R A h ( A ))) F ( R A g ( φ )) − − − − − − → F ( R A g ( R A k ( B ))) F ( L A k,g ( B ) − 1 ) − − − − − − − − − → F ( R A kg ( B )) T kg ( A ) − − − − → R B kg ( F ( B )) . Since T g : F ◦ R A g → R B g ◦ F is a natural transformation, the follo wing diagr am commutes F ( R A g ( R A h ( A ))) F ( R A g ( φ )) / / T g ( R A h ( A ))   F ( R A g ( R A k ( B ))) T g ( R A k ( B ))   R B g ( F ( R A h ( A ))) R B g ( F ( φ )) / / R B g ( F ( R A k ( B ))) Hence it suﬃces to show that the co mpo site R B hg ( F ( A )) L B h,g ( F ( A )) − − − − − − − → R B g ( R B h ( F ( A ))) R B g ( T h ( A ) − 1 ) − − − − − − − − − → R B g ( F ( R A h ( A ))) T g ( R A h ( A )) − 1 − − − − − − − − − → F ( R A g ( R A h ( A ))) agrees with the comp osite R B hg ( F ( A )) T hg ( A ) − 1 − − − − − − → F ( R A hg ( A )) F ( L A h,g ( A )) − − − − − − − → F ( R A g ( R A h ( A ))) ON CR OSSED PRODUCT RINGS WITH TWISTED INVOLUTIONS, . . . 9 and that the co mpo site F ( R A g ( R A k ( B ))) T g ( R A k ( B )) − − − − − − − → R B g ( F ( R A k ( B ))) R B g ( T k ( B )) − − − − − − − → R B g ( R B k ( F ( B ))) L B k,g ( F ( B )) − 1 − − − − − − − − − → R B kg ( F ( B )) agrees with the comp osite F ( R A g ( R A k ( B ))) F ( L A k,g ( B ) − 1 ) − − − − − − − − − → F ( R A kg ( B )) T kg ( A ) − − − − → R B kg ( F ( B )) . This follo ws in both case s from the commutativit y o f the diagra m (2.2). This ﬁnishes the pr o of that S ( F , T ) is a functor of a dditiv e ca tegories with strict ( G, v )- action. Let S : ( F 1 , T 1 ) → ( F 2 , T 2 ) b e a natural transformation of ǫ -v ariant functors of additive catego ries with weak ( G, v )-actio n ( F 1 , T 1 ) : A 1 → A 2 and ( F 2 , T 2 ) : A 1 → A 2 . It induces a natural transfo rmation S ( S ) : S ( F 1 , T 1 ) → S ( F 2 , T 2 ) o f functors of additive categ ories with strict ( G, v )-a ction as follows. Given an ob ject ( A, g ) in S ( A ), we hav e to sp ecify a morphism in S ( A ) S ( S )( A ) : S ( F 1 , T 1 )( A, g ) = ( F 1 ( A ) , g ) → S ( F 2 , T 2 )( A, g ) = ( F 2 ( A ) , g ) , i.e., a morphism R A g ( F 1 ( A )) → R A g ( F 2 ( A )) in A . W e take R A g ( S ( A )) v ( g ) . W e leave it to the reader to check that this is indeed a natural tra nsformation of ǫ -v ariant functors of additive categor ies with strict ( G, v )-ac tio n using the commutativit y o f the diagram (2.3). Let ( G, v )- Add - Cat ǫ be the category of additive categorie s with w eak ( G, v )- action with ǫ -v a riant functors as mor phisms a nd let st rict -( G, v )- Add - Cat ǫ be the category of additive categories with strict ( G, v )-a ction with ǫ - v aria nt functors as morphisms. There is the for getful functor forget : strict -( G, v )- Add - Cat ǫ → ( G, v )- Add - Cat ǫ and the functor constructed a b ove S : ( G, v )- Add - Cat ǫ → strict -( G, v )- Add - Cat ǫ . Lemma 3.2. (i) We obtain an adjoi nt p air of functors ( S , forget) . (ii) We get for every additive c ate gory A with we ak ( G, v ) -action a functor of additive c ate gories with we ak ( G, v ) -action P A : A → for get( S ( A )) which is natu r al in A and whose u nderlying functor of additive c ate gories is an e quivalenc e of additive c ate gorie s. Pr o of. W e will only tr e a t the case, whe r e v is trivial a nd ǫ = 1, the o ther ca ses ar e analogo us. (i) W e ha ve to co ns truct for any additive c a tegory A w ith w eak G -action and any additive ca tegory B with s tr ict G -ac tio n to o ne ano ther inv erse ma ps α : func strict -( G,v )- Add - Cat ( S ( A ) , B ) → func ( G,v )- Add - Cat ( A , forget( B )) and β : func ( G,v )- Add - Cat ( A , forget( B )) → func strict -( G,v )- Add - Cat ( S ( A ) , B ) . F or a functor of additive ca tegories with str ic t G -action F : S ( A ) → B , the functor of additiv e ca tegories with weak G -ac tio n, α ( F ) : A → forg et( B ) is given by a functor α ( F ) : A → forget( B ) of additive categor ies and a collection of natural transformatio ns T ( F ) g : α ( F ) ◦ R A g → R B g ◦ α ( F ) sa tisfying certain compatibilit y conditions. W e ﬁr s t explain the functor α ( F ) : A → forg et( B ). It sends a morphism 10 AR THUR BAR TELS AND WOLF GANG L ¨ UCK f : A → B in A to the morphism in B whic h is g iven b y the v alue of F on the morphism ( A, e ) → ( B , e ) in S ( A ) deﬁned b y f . F o r g ∈ G the tr ansformatio n T ( F ) g ev aluated at an ob ject A in A is the mor phism α ( F )( R A g ( A )) = F ( R A g ( A ) , e ) → R B g ( α ( F )( A )) = R B g ( F ( A, e )) deﬁned as follows. It is given by the comp osite of the image under F of the morphism ( R A g ( A ) , e ) → R S ( A ) g ( A, e ) = ( A, g ) in S ( A ) whic h is deﬁned by the ident ity morphism id : R A g ( A ) → R A g ( A ) in A and the identit y F ( R S ( A ) g ( A, e )) = R B g ( F ( A, e )) which comes from the ass umption that F is a functor of str ic t a ddi- tive G -categ ories. One easily checks that α ( F ) satisﬁes condition (2 .2) since it is satisﬁed for F . Given a functor of a dditive categorie s with weak G -action ( F, T ) : A → forget( B ), the functor of additive categor ies with strict G -action β ( F, T ) : S ( A ) → B is deﬁned as follows. It s ends a n ob ject ( A, h ) to R B h ( F ( A )). A morphism φ : ( A, h ) → ( B , k ) in S ( A ) which is given by a morphism φ : R A h A → R A k B in A is sent to morphism in B given by the comp os ite R B h ( F ( A )) T h ( A ) − 1 − − − − − → F ( R A h ( A )) F ( φ ) − − − → F ( R A k ( B )) T k ( B ) − − − − → R B k ( F ( B )) . The following ca lc ula tion shows that β ( F, T ) is indeed a functor of additive cate- gories with strict G -ac tio n. Given an elemen t g ∈ G the morphism R S ( A ) g ( φ ) : ( A, hg ) → ( B , kg ) in S ( A ) is given by the morphism in A R A hg ( A ) L A hg ( A ) − − − − − → R A g ( R A h ( A )) R A g ( φ ) − − − − → R A g ( R A k ( B )) L A k,g ( B ) − 1 − − − − − − − → R A kg ( B ) . Hence β ( F, T ) ◦ R S ( A ) g ( φ ) is the mo rphism in B given by the comp os ite R B hg ( F ( A )) T hg ( A ) − 1 − − − − − − → F ( R A hg ( A )) F ( L A hg ( A )) − − − − − − − → F ( R A g ( R A h ( A ))) F ( R A g ( φ )) − − − − − − → F ( R A g ( R A k ( B ))) F ( L A k,g ( B ) − 1 ) − − − − − − − − − → F ( R A kg ( B ))) T kg ( B ) − − − − → R A kg ( F ( B ))) . The morphism R B g ( B ) ◦ β ( F, T )( φ ) in B is given by the c o mp o site R B g ( R B h ( F ( A ))) R B g ( T h ( A ) − 1 ) − − − − − − − − − → R B g ( F ( R A h ( A ))) R B g ( F ( φ )) − − − − − − → R B g ( F ( R A k ( B ))) R B g ( T k ( B )) − − − − − − − → R B g ( R B k ( F ( B ))) . Since B is a additive categ ory with strict G -a ction b y assumption, we have the equalities R B g ( R B h ( F ( A ))) = R B hg ( F ( A )) and R B g ( R B k ( B )) = R A kg ( F ( B ))). W e must show that under these identiﬁcations the t wo mor phisms in B above ag ree. Since T g is a natural tra nsformation F ◦ R A g → R B g ◦ F , the following diag ram commutes F ( R A g ( R A h ( A ))) F ( R A g ( φ )) / / T g ( R A h ( A ))   F ( R A g ( R A k ( B ))) T g ( R A k ( B ))   R B g ( F ( R A h ( A ))) R B g ( F ( φ )) / / R B g ( F ( R A k ( B ))) Hence it suﬃces to show that the co mpo sites R B g ( R B h ( F ( A ))) = R B hg ( F ( A )) T hg ( A ) − 1 − − − − − − → F ( R A hg ( A )) F ( L A hg ( A )) − − − − − − − → F ( R A g ( R A h ( A ))) ON CR OSSED PRODUCT RINGS WITH TWISTED INVOLUTIONS, . . . 11 and R B g ( R B h ( F ( A ))) R B g ( T h ( A ) − 1 ) − − − − − − − − − → R B g ( F ( R A h ( A ))) T g ( R A h ( A )) − 1 − − − − − − − − − → F ( R A g ( R A h ( A ))) agree and that the co mpo sites F ( R A g ( R A k ( B ))) F ( L A k,g ( B ) − 1 ) − − − − − − − − − → F ( R A kg ( B ))) T kg ( B ) − − − − → R A kg ( F ( B ))) = R B g ( R B k ( F ( B ))) and F ( R A g ( R A k ( B ))) T g ( R A k ( B )) − − − − − − − → R B g ( F ( R A k ( B ))) R B g ( T k ( B )) − − − − − − − → R B g ( R B k ( F ( B )) agree. This follows in b oth c ases from the co mm utativity of the diagra m (2 .2). This ﬁnishes the pro o f that β ( F ) is a functor of additive catego r ies with stric t G -actio n. W e leav e it to the reader to c heck that both comp osites β ◦ α and α ◦ β are the ident ity . (ii) The in A natura l functor of additive categ ories with weak ( G, v )-a ction P A : A → for get( S ( A )) is deﬁned to b e the a djoint of the iden tit y functor id : S ( A ) → S ( A ). Explicitly it sends a n ob ject A to the ob ject ( A, e ) and a mor phis m φ : A → B to the morphism ( A, e ) → ( B , e ) g iven by φ . Obviously P A induces a bijection mo r A ( A, B ) → mor S ( A ) ( P A ( A ) , P A ( B )) and for every ob ject ( A, g ) in S ( A ) there is an ob ject in the ima ge of P A which is isomorphic to ( A, g ), namely , P A ( R A g ( A )) = ( R A g ( A ) , e ). Hence the underlying functor R A is an equiv alence of additive categ ories.  4. Cr ossed product rings and involutions In this subsectio n we will introduce the concept of a cr ossed pro duct ring. Let R be a ring and let G b e a gr oup. Let e ∈ G b e the unit in G and denote by 1 the m ultiplicative unit in R . Suppo se that we are given ma ps of se ts c : G → aut( R ) , g 7→ c g ; (4.1) τ : G × G → R × . (4.2) W e requir e c τ ( g,g ′ ) ◦ c gg ′ = c g ◦ c g ′ ; (4.3) τ ( g , g ′ ) · τ ( g g ′ , g ′′ ) = c g ( τ ( g ′ , g ′′ )) · τ ( g , g ′ g ′′ ); (4.4) c e = id R ; (4.5) τ ( e, g ) = 1; (4.6) τ ( g , e ) = 1 , (4.7) for g , g ′ , g ′′ ∈ G , where c τ ( g,g ′ ) : R → R is conjuga tion with τ ( g , g ′ ), i.e., it sends r to τ ( g , g ′ ) rτ ( g , g ′ ) − 1 . Let R ∗ G = R ∗ c,τ G be the free R -mo dule with the set G a s basis. It b ecomes a ring with the following multiplication   X g ∈ G λ g g   · X h ∈ G µ h h ! = X g ∈ G     X g ′ ,g ′′ ∈ G, g ′ g ′′ = g λ g ′ c g ′ ( µ g ′′ ) τ ( g ′ , g ′′ )     g . This multiplication is uniq uely determined by the prop erties g · r = c g ( r ) · g and g · g ′ = τ ( g , g ′ ) · ( g g ′ ). The conditions (4.3) and (4.4) rela ting c and τ a re equiv alent to the co ndition that this multiplication is asso c iative. The other conditions (4.5), (4.6) and (4.7) are equiv alen t to the condition that the element 1 · e is a m ultiplicativ e unit in R ∗ G . W e ca ll R ∗ G = R ∗ c,τ G (4.8) 12 AR THUR BAR TELS AND WOLF GANG L ¨ UCK the cr osse d pr o duct of R a nd G with resp ect to c a nd τ . Example 4.9 . Let 1 → H i − → G p − → Q → 1 be an extension of groups. Let s : Q → G be a map sa tisfying p ◦ s = id a nd s ( e ) = e . W e do not re q uire s to b e a group ho- momorphism. Deﬁne c : Q → a ut( RH ) by c q ( P h ∈ H λ h h ) = P h ∈ H λ h s ( q ) hs ( q ) − 1 . Deﬁne τ : Q × Q → ( RH ) × by τ ( q, q ′ ) = s ( q ) s ( q ′ ) s ( q q ′ ) − 1 . Then we obtain a ring isomorphism RH ∗ Q → R G b y s ending P q ∈ Q λ q q to P q ∈ Q i ( λ q ) s ( q ), where i : RH → RG is the ring homomor phis m induced b y i : H → G . Notice that s is a group ho mo morphism if and o nly if τ is constant with v alue 1 ∈ R . Next we consider the additive categor y with inv olution R - FGP o f ﬁnitely gener - ated pr o jectiv e R -mo dules. F o r g ∈ G we obta in a functor res c g : R - FGP → R - FGP by r e striction with the ring automorphism c g : R → R . Deﬁne natural transforma - tion of functor s R - FGP → R - FGP L τ ( g,h ) : res c gh → res c h ◦ res c g by as signing to a ﬁnitely g enerated pro jectiv e R -mo dule the R - homomorphism res c gh P → res c h res c g P, p 7→ τ ( g , h ) p. This is indee d a R -linear map b ecause o f the following computatio n for r ∈ R and p ∈ P τ ( g , h ) c gh ( r ) = τ ( g , h ) c gh ( r ) τ ( g , h ) − 1 τ ( g , h ) = c τ ( g,h ) ◦ c gh ( r ) τ ( g , h ) = c g ◦ c h ( r ) τ ( g , h ) . Lemma 4. 10. We get fr om t he c ol le ctions { res c g | g ∈ G } and { L τ ( g,h ) | g , h ∈ G } the s t ructur e of an additive c ate gory with we ak G -action on R - FGP . Pr o of. Condition (4.4) implies that for every ﬁnitely g enerated pro jective R -mo dule the comp osites res c gg ′ g ′′ P L τ ( g,g ′ g ′′ ) − − − − − − → res c g ′ g ′′ res c g P L c g ( τ ( g ′ ,g ′′ )) − − − − − − − − → res c g ′′ res C g ′ res c g P and res c gg ′ g ′′ P L τ ( gg ′ ,g ′′ ) − − − − − − → res c g ′′ res c gg ′ P L τ ( g,g ′ ) − − − − − → res c g ′′ res C g ′ res c g P agree. This takes ca re o f condition (iii) in Deﬁnition 2.1. W e conclude (res c ( e ) = id , L τ ( g,e ) = id a nd L τ ( e,g ) = id for a ll g ∈ G fro m (4.5), (4.6) and (4.7).  Because of Lemma 4.10 we o btain tw o additive ca tegories with s trict G -action from the co nstructions of Sec tion 3 R - FGP c,τ := S ( R - FGP ); (4.11) ¿F rom now on as sume that R comes with an inv olution of rings r 7→ r . W e want to consider ex tensions of it to an inv olution o n R ∗ G . Supp os e that additiona lly we a r e g iven a map w : G → R . (4.12) W e requir e the following conditions for g , h ∈ G and r ∈ R w ( e ) = 1; (4.13) w ( g h ) = w ( h ) c h − 1 ( w ( g )) τ ( h − 1 , g − 1 ) c ( gh ) − 1  τ ( g , h )  − 1 ; (4.14) w ( g ) = w ( g ) c − 1 g  τ ( g , g − 1 ) τ ( g , g − 1 ) − 1  ; (4.15) c g ( r ) = c g   w ( g ) τ ( g − 1 , g )  − 1 r  w ( g ) τ ( g − 1 , g )   . (4.16) ON CR OSSED PRODUCT RINGS WITH TWISTED INVOLUTIONS, . . . 13 W e claim that there is precisely one in volution on R ∗ G with the prop erties that it extends the inv olution o n R and sends g to w ( g ) · g − 1 . The candidate for the inv o lution is X g ∈ G r g · g := X g ∈ G w ( g ) c g − 1 ( r g ) · g − 1 . (4.17) One easily concludes from the re q uirements and the axioms of an involution tha t this is the o nly p oss ible formula for such an inv olution. Namely , X g ∈ G r g · g = X g ∈ G r g · g = X g ∈ G g · r g = X g ∈ G w ( g ) · g − 1 · r g = X g ∈ G w ( g ) ·  g − 1 · r g · g  · g − 1 = X g ∈ G ( w ( g ) c g − 1 ( r g ) · g − 1 . Before we explain that this deﬁnition indeed satisﬁes the axioms for an inv olution, we show that the conditions a b o ut w abov e are necessary for this map to be an inv o lution on R ∗ G . So ass ume that w e hav e an inv olution o n R ∗ G that e x tends the inv o lution on R and sends g to w ( g ) · g − 1 for a given ma p w : G → R . Denote by 1 the multiplicativ e unit in bo th R and R ∗ G . F rom 1 · e = 1 = 1 = 1 · e = w ( e ) · e we co nclude (4.1 3) . The equality w ( g h ) c ( gh ) − 1  τ ( g , h )  · ( g h ) − 1 = τ ( g , h ) · g h = g · h = h · g = w ( h ) · h − 1 · w ( g ) · g − 1 = w ( h )  h − 1 · w ( g ) · h  · h − 1 · g − 1 = w ( h ) c h − 1 ( w ( g )) τ ( h − 1 , g − 1 ) · ( g h ) − 1 implies (4.14). If we take h = g − 1 in (4.14) and us e (4.1 3) , we g et 1 = w ( e ) = w ( g g − 1 ) = w ( g − 1 ) c g ( w ( g )) τ ( g , g − 1 ) τ ( g , g − 1 ) − 1 . (4.18) This implies tha t fo r a ll g ∈ G the element w ( g ) is a unit in R with inverse w ( g ) − 1 = c g − 1 ( w ( g − 1 )) τ ( g − 1 , g ) τ ( g − 1 , g ) − 1 . The equality g = g = w ( g ) · g − 1 = g − 1 · w ( g ) = w ( g − 1 ) · g · w ( g ) = w ( g − 1 ) ·  g · w ( g ) · g − 1  · g = w ( g − 1 ) c g  w ( g )  · g together with (4.18) implies w ( g − 1 ) c g  w ( g )  = 1 = w ( g − 1 ) c g ( w ( g )) τ ( g , g − 1 ) τ ( g , g − 1 ) − 1 . If we multiply this eq uation with w ( g − 1 ) − 1 and apply the inv erse c − 1 g of c g , w e derive condition (4.15). The eq uality r · w ( g ) · g − 1 = r · g = g · r = ( g · r · g − 1 ) · g = c g ( r ) · g = g · c g ( r ) = w ( g ) · g − 1 · c g ( r ) = w ( g ) ·  g − 1 · c g ( r ) · g  · g − 1 = w ( g ) · c g − 1  c g ( r )  · g − 1 implies that fo r all g ∈ G and r ∈ R we hav e r · w ( g ) = w ( g ) · c g − 1  c g ( r )  and hence c g ( r ) = c − 1 g − 1  w ( g ) − 1 r w ( g )  . ¿F rom the r elation (4.3) we conclude c τ ( g − 1 ,g ) = c g − 1 ◦ c g and hence c − 1 g − 1 = c g ◦ c − 1 τ ( g − 1 ,g ) . No w condition (4 .16) follows. 14 AR THUR BAR TELS AND WOLF GANG L ¨ UCK Finally we show that the conditions (4.13), (4.14), (4.15) and (4.16) on w do imply that we get an involution of r ing s o n R ∗ G by the formula (4.17). Ob viously this form ula is c o mpatible with the additive structure on R ∗ G and sends 1 to 1. In order to s how that it is an in v olution and compatible with the multiplicativ e structure we hav e to show g · h = h · g , r s = s · r , r · g = g · r , g · r = r · g , r = r and g = g for r , s ∈ R and g , h ∈ G . W e get r s = s · r and r = r from the fact that we start with a n inv olution on R . The other equations follow from the pro ofs ab ov e that (4.17) is the only p ossible candidate and that the conditions ab out w are necessa ry for the existence of the desired inv olution on R ∗ G , just re a d the v ario us equa tions and implications bac kwards. W e will denote the resulting r ing with in volution by R ∗ c,τ , w G. (4.19) Example 4.2 0. Supp ose that w e are in the situation of Ex ample 4.9. Supp o se that we are additionally given a group homo morphism w 1 : G → cent( R ) × to the ab elian group of inv ertible cen tral ele men ts in R satisfying w 1 ( g ) = w 1 ( g ) for all g ∈ G . The w 1 -twisted inv olution on RG is deﬁned by P g ∈ G r g · g = P g ∈ G r g w 1 ( g ) · g − 1 . It e x tends the w 1 | H -inv olution o n RH . W e obtain an inv olution on RH ∗ Q if we conjugate the w 1 -twisted inv olution with the isomorphism RH ∗ Q ∼ = − → R G which we hav e intro duced in E xample 4.9. This inv olution on RH ∗ Q s ends q ∈ Q to the element w 1 ( s ( q )) τ ( q − 1 , q ) − 1 · q − 1 bec ause of the following ca lculation in RG for q ∈ Q s ( q ) = w 1 ( s ( q )) · s ( q ) − 1 = w 1 ( s ( q )) · s ( q ) − 1 · s ( q − 1 ) − 1 · s ( q − 1 ) = w 1 ( s ( q )) ·  s ( q − 1 ) · s ( q )  − 1 · s ( q − 1 ) = w 1 ( s ( q )) ·  τ ( q − 1 , q ) s ( q − 1 q )  − 1 · s ( q − 1 ) = w 1 ( s ( q )) τ ( q − 1 , q ) − 1 · s ( q − 1 ) . Deﬁne w : Q → R H , q 7→ w 1 ( s ( q )) τ ( q − 1 , q ) − 1 . Then w satisﬁes the conditions (4.13), (4.1 4), (4.1 5) a nd (4.16) and the in volution on RH ∗ Q determined by w corresp onds under the iso mo rphism RH ∗ Q ∼ = − → RG to the w 1 -twisted involution o n RG . Let t g : res c g ◦ I R - FGP → I R - FGP ◦ res c g (4.21) be the natural tr a nsformation which assigns to a ﬁnitely genera ted pro jective R -mo dule P the R -isomor phism t g ( P ) : r e s c g P ∗ → (res c g P ) ∗ which sends the R - linear map f : P → R to the R -linear map t g ( P )( f ) : res c g P → R, p 7→ c − 1 g ( f ( p ))  w ( g ) τ ( g − 1 , g )  − 1 . W e ﬁr stly check that t g ( P )( f ) : r es c g P → R is R -linea r by the following computa- tion t g ( P )( f )( c g ( r ) p ) = c − 1 g ( f ( c g ( r ) p ))  w ( g ) τ ( g − 1 , g )  − 1 = c − 1 g ( c g ( r ) f ( p ))  w ( g ) τ ( g − 1 , g )  − 1 = c − 1 g ( c g ( r )) c − 1 g ( f ( p ))  w ( g ) τ ( g − 1 , g )  − 1 = rc − 1 g ( f ( p ))  w ( g ) τ ( g − 1 , g )  − 1 = rt g ( P )( f )( p ) . ON CR OSSED PRODUCT RINGS WITH TWISTED INVOLUTIONS, . . . 15 Finally w e c heck that t g ( P ) : res c g P ∗ → (res c g P ) ∗ is R -linea r b y the follo wing calculation for f ∈ P ∗ and p ∈ P t g ( P ) (( c g ( r ) f )) ( p ) = c − 1 g (( c g ( r ) f )( p ))  w ( g ) τ ( g − 1 , g )  − 1 = c − 1 g  f ( p ) c g ( r )   w ( g ) τ ( g − 1 , g )  − 1 = c − 1 g ( f ( p )) c − 1 g ( c g ( r ) )  w ( g ) τ ( g − 1 , g )  − 1 = c − 1 g ( f ( p )) c − 1 g  c g   w ( g ) τ ( g − 1 , g )  − 1 r  w ( g ) τ ( g − 1 , g )    w ( g ) τ ( g − 1 , g )  − 1 = c − 1 g ( f ( p )  w ( g ) τ ( g − 1 , g )  − 1 r  w ( g ) τ ( g − 1 , g )   w ( g ) τ ( g − 1 , g )  − 1 = c − 1 g ( f ( p )  w ( g ) τ ( g − 1 , g )  − 1 r = t g ( P )( f )( p ) r = ( rt g ( P )) ( f )( p ) . Deﬁnition 4.22. An additive G -c ate gory with involut ion A is an additive G - category , which is the same as an additive categ ory with strict G -action (see Deﬁ- nition 2.1), toge ther with an involution ( I , E ) of a dditive ca tegories (see (1.1) and (1.2)) with the following pro p erties: I : A → A is a contra v aria nt functor of ad- ditive G -categor ies, i.e., R g ◦ I = I ◦ R g for all g ∈ G , a nd E : id A → I ◦ I is a natural transformatio n o f functors of additive G -c a tegories , i.e., for every g ∈ G and every o b ject A in A the morphisms E ( R g ( A )) and R g ( E ( A )) from R g ( A ) to I 2 ◦ ( R g ( A ) = R g ◦ I 2 ( A ) agre e . Lemma 4.2 3. The additive c ate gory with strict G -action R - FGP c,τ of (4.11) in- herits the structu r e of an additive G -c ate gory with involution in the sense of Deﬁ- nition 4.22 . Pr o of. W e ﬁrstly show that I R - FGP : R - FGP → R - FGP together with the co llection o f the { t − 1 g : I R - FGP ◦ res c g → res c g ◦ I R - FGP | g ∈ G } (see (4.21)) is a contra v aria nt functor of additive categories with w eak G -action. W e have to verify that the diagram (2.2) commutes. This is equiv alent to show for every ﬁnitely genera ted pro jective R - mo dule P and g , h ∈ G that the following diagram commutes res c gh P ∗ t gh ( P ) / / L τ ( g,h ) ( P ∗ )   (res c gh P ) ∗ res c h res c g P ∗ res c h t g ( P ) / / res c h (res c g P ) ∗ t h (res g P ) / / (res c h res c g P ) ∗ L τ ( g,h ) ( P ) ∗ O O W e start with an element f : P → R in the left upp er corner. Its imag e under the up- per ho rizontal arr ow is p 7→ c − 1 gh ( f ( p ))  w ( g h ) τ (( g h ) − 1 , g h )  − 1 . Next we list succes- sively how its image lo oks lik e if we go in the an ticlo ckwise direction from the left up- per cor ner to the right upper co rner. W e ﬁrst get p 7→ f ( p ) τ ( g , h ). After the second map w e get p 7→ c − 1 g  f ( p ) τ ( g , h )   w ( g ) τ ( g − 1 , g )  − 1 . After applying the third map we obtain p 7→ c − 1 h  c − 1 g  f ( p ) τ ( g , h )   w ( g ) τ ( g − 1 , g )  − 1   w ( h ) τ ( h − 1 , h )  − 1 . Fi- nally we get p 7→ c − 1 h  c − 1 g  f ( τ ( g , h ) p ) τ ( g , h )   w ( g ) τ ( g − 1 , g )  − 1   w ( h ) τ ( h − 1 , h )  − 1 . Since f lies in P ∗ , we hav e f ( τ ( g , h ) p ) = τ ( g , h ) f ( p ). Hence it s uﬃces to show for 16 AR THUR BAR TELS AND WOLF GANG L ¨ UCK all r ∈ R c − 1 h  c − 1 g  τ ( g , h ) r τ ( g , h )   w ( g ) τ ( g − 1 , g )  − 1   w ( h ) τ ( h − 1 , h )  − 1 = c − 1 gh ( r )  w ( g h ) τ (( g h ) − 1 , g h )  − 1 . (Notice that now f has been eliminated.) B y applying c gh we see that this is equiv alent to showing c gh  c − 1 h  c − 1 g  τ ( g , h ) r τ ( g , h )  = rc gh   w ( g h ) τ (( g h ) − 1 , g h )  − 1  w ( h ) τ ( h − 1 , h )  c − 1 h  w ( g ) τ ( g − 1 , g )   . ¿F rom the rela tion (4.3) we co nc lude tha t c gh ◦ c h − 1 ◦ c g − 1 ( s ) = τ ( g , h ) − 1 sτ ( g , h ) holds fo r a ll s ∈ R . Hence it remains to s how τ ( g , h ) − 1  τ ( g , h ) r τ ( g , h )  τ ( g , h ) = rc gh   w ( g h ) τ (( g h ) − 1 , g h )  − 1 w ( h ) τ ( h − 1 , h ) c − 1 h  w ( g ) τ ( g − 1 , g )   . This r e duces to proving for g , h ∈ G τ ( g , h ) τ ( g , h ) = c gh  τ (( g h ) − 1 , g h ) − 1 w ( g h ) − 1 w ( h ) τ ( h − 1 , h ) c − 1 h  w ( g ) τ ( g − 1 , g )  . (Notice that now r has b een elimina ted.) By inserting condition (4.14) and the con- clusions c τ ( h − 1 ,h ) ◦ c − 1 h = c h − 1 and c τ (( gh ) − 1 ,gh ) ◦ c − 1 gh = c ( gh ) − 1 from conditions (4.3) and (4.5 ) we g et w ( g h ) − 1 w ( h ) τ ( h − 1 , h ) c − 1 h  w ( g ) τ ( g − 1 , g )  =  w ( h ) c h − 1 ( w ( g )) τ ( h − 1 , g − 1 ) c ( gh ) − 1  τ ( g , h )  − 1  − 1 w ( h ) τ ( h − 1 , h ) c − 1 h  w ( g ) τ ( g − 1 , g )  τ ( h − 1 , h ) − 1 τ ( h − 1 , h ) = c ( gh ) − 1  τ ( g , h )  τ ( h − 1 , g − 1 ) − 1 c h − 1 ( w ( g )) − 1 w ( h ) − 1 w ( h ) c h − 1  w ( g ) τ ( g − 1 , g )  τ ( h − 1 , h ) = c ( gh ) − 1  τ ( g , h )  τ ( h − 1 , g − 1 ) − 1 c h − 1 ( w ( g )) − 1 c h − 1 ( w ( g )) c h − 1  τ ( g − 1 , g )  τ ( h − 1 , h ) = c ( gh ) − 1  τ ( g , h )  τ ( h − 1 , g − 1 ) − 1 c h − 1  τ ( g − 1 , g )  τ ( h − 1 , h ) = τ (( g h ) − 1 , g h ) c − 1 gh  τ ( g , h )  τ (( g h ) − 1 , g h ) − 1 τ ( h − 1 , g − 1 ) − 1 c h − 1  τ ( g − 1 , g )  τ ( h − 1 , h ) . This implies c gh  τ (( g h ) − 1 , g h ) − 1 w ( g h ) − 1 w ( h ) τ ( h − 1 , h ) c − 1 h  w ( g ) τ ( g − 1 , g )  = c gh  τ (( g h ) − 1 , g h ) − 1 τ (( g h ) − 1 , g h ) c − 1 gh  τ ( g , h )  τ (( g h ) − 1 , g h ) − 1 τ ( h − 1 , g − 1 ) − 1 c h − 1  τ ( g − 1 , g )  τ ( h − 1 , h )  = τ ( g , h ) c gh  τ (( g h ) − 1 , g h ) − 1 τ ( h − 1 , g − 1 ) − 1 c h − 1  τ ( g − 1 , g )  τ ( h − 1 , h )  . Hence it remains to sho w τ ( g , h ) = c gh  τ (( g h ) − 1 , g h ) − 1 τ ( h − 1 , g − 1 ) − 1 c h − 1  τ ( g − 1 , g )  τ ( h − 1 , h )  . ON CR OSSED PRODUCT RINGS WITH TWISTED INVOLUTIONS, . . . 17 (Notice that we ha ve eliminated any expressio n inv o lving the involution.) ¿F rom condition (4 .3), (4.4) and (4 .5 ) we co nclude τ ( h − 1 , g − 1 ) τ (( g h ) − 1 , g ) = c h − 1 ( τ ( g − 1 , g )); τ ( g h ) − 1 , g ) τ ( h − 1 , h ) = c ( gh ) − 1 ( τ ( g , h )) τ (( g h ) − 1 , g h ); c − 1 gh = c τ (( gh ) − 1 ,gh ) − 1 ◦ c ( gh ) − 1 . Hence τ (( g h ) − 1 , g h ) − 1 τ ( h − 1 , g − 1 ) − 1 c h − 1  τ ( g − 1 , g )  τ ( h − 1 , h ) = τ (( g h ) − 1 , g h ) − 1 τ ( h − 1 , g − 1 ) − 1 τ ( h − 1 , g − 1 ) τ (( g h ) − 1 , g ) τ ( h − 1 , h ) = τ (( g h ) − 1 , g h ) − 1 τ (( g h ) − 1 , g ) τ ( h − 1 , h ) = τ (( g h ) − 1 , g h ) − 1 c ( gh ) − 1 ( τ ( g , h )) τ (( g h ) − 1 , g h ) = c − 1 gh ( τ ( g , h )) . This ﬁnishes the pro o f of the co mm utativity o f the diagr am (2.2). Next we s how that E R - FGP : id R - FGP → I R - FGP ◦ I R - FGP is a natural transforma - tion of contra v ariant functors of additive ca tegories with weak G -ac tio n. W e ha ve to show that the diagra m (2.3) comm utes. This is equiv alent to show for ev ery ﬁnitely generated pr o jective R -mo dule P the fo llowing diagram commutes res c g P E R - FGP (res c g P ) / / res c g E R - FGP ( P )   (res c g P ) ∗∗ t g ( P ) ∗   res c g ( P ∗∗ ) t g ( P ∗ ) / / (res c g P ∗ ) ∗ W e start with a n element p ∈ P in the left uppe r corner . It is s e nt under the left vertical a rrow to the element g iven by f 7→ f ( p ). The imag e of this element under the low er hor izontal is given by f 7→ c − 1 g ( f ( p ))  w ( g ) τ ( g − 1 , g )  − 1 . The imag e of p ∈ P under the upp e r horizontal arrow is f 7→ f ( p ). The image of this element under the right vertical ar row sends f to f ◦ t g ( P )( p ) = c − 1 g ( f ( p ))  w ( g ) τ ( g − 1 , g )  − 1 . ¿F rom the naturality of the co ns truction o f the additive catego ry with strict G - action R - FGP c,τ := S ( R - FGP ) (see Section 3) w e conclude that ( I R - FGP , { t g | g ∈ G } ) induces a functor of additive categ ories with strict G -a c tion I R - FGP c,τ : R - FGP c,τ → R - FGP c,τ and E R - FGP induces a natural trans fo rmation of functors o f additive categor ies with strict G -a ction E R - FGP c,τ : id R - FGP → I R - FGP c,τ ◦ I R - FGP c,τ . It r emains to prove that condition (1 .3) holds for ( I R - FGP c,τ , E R - FGP c,τ ). But this follows easily fro m the fact that co ndition (1.3) holds for ( I R - FGP , E R - FGP ).  The additiv e G -category with in v olution constructed in Lemma 4.23 will b e denoted in the se q uel by R - FGP c,τ , w . (4.24) 5. Connected groupoids and additive ca tegories Group oids are always to b e unders to o d to b e small. A gr oup oid is called c on- ne cte d if for tw o ob jects x and y there exis ts a morphism f : x → y . Let G be a connected group oid. Le t Add - Cat b e the categ ory of s ma ll additive catego ries. 18 AR THUR BAR TELS AND WOLF GANG L ¨ UCK Given a cont rav ariant functor F : G → Add - Cat , we deﬁne a new s ma ll additiv e category , which we call its homotopy c olimit (see for instance [1 5]) R G F (5.1) as fo llows. An ob ject is a pair ( x, A ) consisting of an ob ject x in G and an ob ject A in F ( x ). A morphism in R G F fro m ( x, A ) to ( y , B ) is a formal sum X f ∈ m or G ( x,y ) f · φ f where φ f : A → F ( f )( B ) is a morphism in F ( x ) and only ﬁnitely ma ny coeﬃ- cients φ f are diﬀerent fro m zer o. The comp osition of a morphism P f ∈ m or G ( x,y ) f · φ f : ( x, A ) → ( y , B ) and a morphism P g ∈ m or G ( y ,z ) g · φ g : ( y , B ) → ( z , C ) is giv en by the for m ula X h ∈ mor G ( x,z ) h · X f ∈ m or G ( x,y ) g ∈ m or G ( y ,z ) h = g ◦ f F ( f )( ψ g ) ◦ φ f ) ! . The decisive sp ecia l case is ( g · ψ ) ◦ ( f · φ ) = ( g ◦ f ) · ( F ( f )( ψ ) ◦ φ ) . The Z -mo dule str ucture on mor R G F ( x, y ) is given by   X f ∈ m or G ( x,y ) f · φ f   +   X f ∈ m or( G ) f · ψ f   = X f ∈ m or G ( x,y ) f · ( φ f + ψ f ) . A mo del fo r the sum of t w o ob jects ( x, A ) a nd ( x, B ) is ( x, A ⊕ B ) if A ⊕ B is a mo del for the sum o f A and B in F ( x ). Since G is b y as sumption connected, w e can choo se for any ob ject ( y , B ) in R G F and any ob ject x in G an isomorphism f : x → y and the ob jects ( x, F ( f )( B )) a nd ( y , B ) in R G F are isomorphic. Namely f · id F ( f )( B ) is an isomorphism ( x, F ( f )( B ) ∼ = − → ( y , B ) whose in v erse is f − 1 · id B · Hence the direct s um of tw o a r bitrary ob jects ( x, A ) and ( y , B ) exists in R G F . Notice that we need the connectedness of G only to show the existence of a direct sum. This will b eco me impo rtant later when w e deal with no n-connected group oids. This cons truction is functorial in F . Namely , if S : F 0 → F 1 is a natura l trans- formation of co ntra v ariant functors G → Add - Cat , then it induces a functor R G S : R G F 0 → R G F 1 (5.2) of additiv e catego ries as follows. It sends an ob ject ( x, A ) in R G F 0 to the ob ject ( x, S ( x )( A )) in R G F 1 . A morphism P f ∈ m or G ( x,y ) f · φ f : ( x, A ) → ( y , B ) is s ent to the morphism X f ∈ m or G ( x,y ) f · S ( x )( φ f ) : ( x, s ( x )( A )) → ( y , s ( y )( B )) . This makes sense since S ( x )( φ f ) is a morphism in F 1 ( x ) fr om S ( x )( A ) to S ( x )( F 0 ( f )( B )) = F 1 ( f )( S ( y )( B )). The de c isive sp ecia l cas e is that R G S sends ( f : x → y ) · φ to ( f : x → y ) · S ( x )( φ ). One eas ily checks that R G S is compatible with the structur e s ON CR OSSED PRODUCT RINGS WITH TWISTED INVOLUTIONS, . . . 19 of additive categor ies a nd we have  Z G S 2  ◦  Z G S 1  = Z G ( S 2 ◦ S 1 ); (5.3) Z G id F = id R G F . (5.4) The co ns truction is also functorial in G . Namely , let W : G 1 → G 2 be a cov a riant functor o f g roup oids. Then we obtain a cov aria n t functor W ∗ : R G 1 F ◦ W → R G 2 F (5.5) of additiv e categories as follows. An ob ject ( x 1 , A ) in R G 1 F ◦ W is sent to the ob ject ( W ( x 1 ) , A ) in R G 2 F . A morphism P f ∈ m or G 1 ( x 1 ,y 1 ) f · φ f : ( x 1 , A ) → ( y 1 , B ) in R G 1 F ◦ W is sen t to the mor phism X f ∈ m or G 2 ( W ( x 1 ) ,W ( y 1 )) f · X f 1 ∈ mor G 1 ( x 1 ,y 1 ) W ( f 1 )= f φ f 1 ! : ( W ( x 1 ) , A ) → ( W ( y 1 ) , B ) in R G 2 F . Here the dec isive sp ecia l case is tha t W ∗ sends the mo rphism f · φ to W ( f ) · φ . One eas ily checks that W ∗ is compa tible with the structures of a dditiv e categorie s and we hav e for co v aria nt functors W 1 : G 1 → G 2 , W 2 : G 2 → G 3 and a contra v aria nt functor F : G → Add - Cat ( W 2 ) ∗ ◦ ( W 1 ) ∗ = ( W 2 ◦ W 1 ) ∗ ; (5.6) (id G ) ∗ = id R G F . (5.7) These tw o constructions are compa tible. Namely , given a natural tra nsformation S 1 : F 1 → F 2 of con trav ar iant functors G → Add - Cat and a co v ar iant functor W : G 1 → G , we ge t  Z G S  ◦ W ∗ = W ∗ ◦  Z G 1 ( S ◦ W )  . (5.8) A functor F : C 0 → C 1 of categor ies is ca lled an e quivalenc e if there exis ts a functor F ′ : C 1 → C 0 with the prop erty tha t F ′ ◦ F is na turally equiv alen t to the ident ity functor id C 0 and F ◦ F ′ is na tur ally equiv alen t to the identit y functor id C 1 . A functor F is a natural equiv a lence if and only if it is ful l and fai thful , i.e., it induces a bijection on the isomorphism classes of ob jects and fo r an y tw o ob jects c, d in C 0 the induced map mor C 0 ( c, d ) → mor C 1 ( F ( c ) , F ( d )) is bijective. If C 0 and C 1 come with an a dditional structure such as of an additive category (with inv olution) and F is compatible with this structure, we re quire that F ′ and the t wo natural equiv alences F ′ ◦ F ≃ id C 0 and F ◦ F ′ ≃ id C 1 are compatible with these. In this case it s till true that F is an equiv alence of categories with this additional structure if and o nly if F is full and faithful. One ea sily chec k s Lemma 5.9. (i) L et W : G 1 → G b e an e quivalenc e of c onne cte d gr oup oids. L et F : G → Add - Cat b e a c ontr avaria nt functor. Then W ∗ : Z G 1 F ◦ W → Z G F is an e quivalenc e of additive c ate gorie s. (ii) L et G b e a c onne cte d gr oup oid. L et S : F 1 → F 2 b e a t r ansformation of c ontr avariant functors G → Add - Cat such that for every obje ct x in G 20 AR THUR BAR TELS AND WOLF GANG L ¨ UCK the functor S ( x ) : F 0 ( x ) → F 1 ( x ) is an e quivalenc e of additive c ate gories. Then Z G S : Z G F 1 → Z G F 2 is an e quivalenc e of additive c ate gorie s. 6. Fr om crossed pr oduct rings to additive ca tegories Example 6.1. Here is our ma in example of a contra v aria nt functor G → Add - Cat . Notice that a group G is the same as a group oid with one ob ject a nd hence a con- trav a riant functor from a group G to Add - Cat is the same as a n a dditive G -categor y what is the same as a n additive catego ry with strict G -a ction (see Deﬁnition 2.1). Let R b e a ring together with maps of sets c : G → aut( R ) , g 7→ c g ; τ : G × G → R × . satisfying (4.3), (4.4), (4 .5), (4.6 ) and (4.7 ). W e hav e intro duced the additive G -categor y R - FGP c,τ in (4.11). All the constructio n restric t to the subcateg ory R - FGF ⊆ R - FGP of ﬁnitely gener ated free R -mo dules and le ad to the additiv e G -categor y R - FGF c,τ := S ( R - FGP ); (6.2) Lemma 6. 3. Consider the data ( R, c, τ ) and t he ad ditive c ate gory R - FGF c,τ ap- p e aring in Ex ample 6.1. L et R G R - FGF c,τ b e the additive c ate gory deﬁne d in (5.1) . Sinc e G r e gar de d as a gr oup oid has pr e cisely one obje ct, we c an (and wil l) identify the set of obje cts in R G R - FGF c,τ with the set of obj e cts in R - FGF c,τ which c on- sists of p airs ( M , g ) for M a ﬁnitely gener ate d fr e e R -mo dule and g ∈ G . Denote by R G R - FGF c,τ  e the ful l sub c ate gory of R G R - FGF c,τ c onsisting of obje cts of the shap e ( M , e ) for e ∈ G the unit element. Denote by R ∗ G = R ∗ c,τ G t he cr osse d pr o duct ring (se e (4.8) ). Then (i) Ther e is an e quivalenc e of additive c ate gories α :  Z G R - FGF c,τ  e → R ∗ c,τ G - FGF ; (ii) The inclus ion  Z G R - FGF c,τ  e → Z G R - FGF c,τ is an e quivalenc e of additive c ate gorie s. Pr o of. (i) An ob ject ( M , e ) in  R G R - FGF c,τ  e is sent under α to the ﬁnitely gener- ated free R ∗ c,τ G -mo dule R ∗ c,τ G ⊗ R M . A mo rphism φ = P g ∈ G g ·  φ g : M → res c g ( N )  from ( M , e ) to ( N , e ) is se nt to the R ∗ c,τ G -homomorphism α ( φ ) : R ∗ c,τ G ⊗ R M → R ∗ c,τ G ⊗ R N , u ⊗ x 7→ X g ∈ G u · τ ( g − 1 , g ) − 1 · g − 1 ⊗ φ g ( x ) ON CR OSSED PRODUCT RINGS WITH TWISTED INVOLUTIONS, . . . 21 for u ∈ R ∗ c,τ G a nd x ∈ M . This is well-deﬁned, i.e., co mpatible with the tensor relation, by the following ca lculation fo r r ∈ R us ing (4.3) and (4.5). u · τ ( g − 1 , g ) − 1 · g − 1 ⊗ φ g ( rx ) = u · τ ( g − 1 , g ) − 1 · g − 1 ⊗ c g ( r ) φ g ( x ) = u · τ ( g − 1 , g ) − 1 · g − 1 · c g ( r ) ⊗ φ g ( x ) = u · τ ( g − 1 , g ) − 1 c g − 1 ( c g ( r )) · g − 1 ⊗ φ g ( x ) = u · τ ( g − 1 , g ) − 1 c g − 1 ( c g ( r )) τ ( g − 1 , g ) τ ( g − 1 , g ) − 1 · g − 1 ⊗ φ g ( x ) = u · c g − 1 g ( r ) τ ( g − 1 , g ) − 1 · g − 1 ⊗ φ g ( x ) = u · c e ( r ) τ ( g − 1 , g ) − 1 · g − 1 ⊗ φ g ( x ) = ( u · r ) τ ( g − 1 , g ) − 1 · g − 1 ⊗ φ g ( x ) . Next we show that α is a cov ariant functor. Ob viously α (id ( M ,e ) ) = id α ( M ,e ) . Co n- sider morphis ms φ = P g ∈ G g · φ g : ( M , e ) → ( N , e ) and ψ = P g ∈ G g · ψ g : ( N , e ) → ( P, e ) in R G R - FGF c,τ  e . A dir e ct computation shows for u ∈ R ∗ c,τ G a nd x ∈ M α ( ψ ) ( α ( φ )( u ⊗ x )) = α ( ψ ) X k ∈ G u · τ ( k − 1 , k ) − 1 · k − 1 ⊗ φ k ( x ) ! = X h ∈ G X k ∈ G u · τ ( k − 1 , k ) − 1 · k − 1 · τ ( h − 1 , h ) − 1 · h − 1 ⊗ ψ h ◦ φ k ( x ) = X h,k ∈ G u · τ ( k − 1 , k ) − 1 c k − 1 ( τ ( h − 1 , h ) − 1 ) · k − 1 · h − 1 ⊗ ψ h ◦ φ k ( x ) = X h,k ∈ G u · τ ( k − 1 , k ) − 1 c k − 1 ( τ ( h − 1 , h ) − 1 ) τ ( k − 1 , h − 1 ) · ( hk ) − 1 ⊗ ψ h ◦ φ k ( x ) and α ( ψ ◦ φ )( u ⊗ x ) = X g ∈ G u · τ ( g − 1 , g ) − 1 · g − 1 ⊗ ( ψ ◦ φ ) g ( x ) = X g ∈ G u · τ ( g − 1 , g ) − 1 · g − 1 ⊗     X hk ∈ G, hk = g r k ( ψ h ) ◦ φ k ( x )     = X g ∈ G u · τ ( g − 1 , g ) − 1 · g − 1 ⊗     X hk ∈ G, hk = g τ ( h, k ) − 1 ψ h ◦ φ k ( x )     = X g ∈ G X hk ∈ G, hk = g u · τ ( g − 1 , g ) − 1 · g − 1 · τ ( h, k ) − 1 ⊗ ψ h ◦ φ k ( x ) = X g ∈ G X hk ∈ G, hk = g u · τ ( g − 1 , g ) − 1 c g − 1 ( τ ( h, k ) − 1 ) · g − 1 ⊗ ψ h ◦ φ k ( x ) = X hk ∈ G u · τ (( hk ) − 1 , hk ) − 1 c ( hk ) − 1 ( τ ( h, k ) − 1 ) · ( hk ) − 1 ⊗ ψ h ◦ φ k ( x ) . Hence it remains to sho w for h, k ∈ G τ ( k − 1 , k ) − 1 c k − 1 ( τ ( h − 1 , h ) − 1 ) τ ( k − 1 , h − 1 ) = τ (( hk ) − 1 , hk ) − 1 c ( hk ) − 1 ( τ ( h, k ) − 1 ) , 22 AR THUR BAR TELS AND WOLF GANG L ¨ UCK or, equiv alently , τ ( k − 1 , h − 1 ) c ( hk ) − 1 ( τ ( h, k )) τ (( hk ) − 1 , hk ) = c k − 1 ( τ ( h − 1 , h )) τ ( k − 1 , k ) . Since (4.4) yields τ (( hk ) − 1 , h ) τ ( k − 1 , k ) = c ( hk ) − 1 ( τ ( h, k ) τ (( hk ) − 1 , hk ) , it suﬃces to show τ ( k − 1 , h − 1 ) τ (( hk ) − 1 , h ) = c k − 1 ( τ ( h − 1 , h )) . But this follo ws fr om (4.4) and (4.7). This ﬁnis hes the pro of that α is a cov ar iant functor. Obviously it is compatible with the str uctur es of an additive catego r y . One easily chec ks that α induces a bijectio n b etw e en the isomor phis m cla sses of ob jects. In order to show that α is a weak equiv alence, we hav e to show for tw o ob jects ( M , e ) a nd ( N , e ) that α induces a bijection mor ( R G R - FGF c,τ ) e (( M , e ) , ( N , e )) ∼ = − → hom R ∗ c,τ G ( R ∗ c,τ ⊗ R M , R ∗ c,τ ⊗ R N ) . Since α is co mpa tible with the structures of an additive category , it suﬃces to chec k this in the sp ecial case M = N = R , wher e it is obvious. (ii) An ob ject of the shap e ( M , g ) in R G R - FGF c,τ is isomorphic to the ob ject ( M , e ), namely a n isomo rphism ( M , g ) ∼ = − → ( M , e ) in R G R - FGF c,τ is given by g · id res c g ( M ) .  7. Connected gr oupoids and additive ca tegories with involutions Next w e w ant to enrich the constructions of Section 5 to additive categories with inv o lutions. Let Add - Cat inv be the ca tegory of additive categories with involution. Given a contrav ar iant functor ( F , T ) : G → Add - Cat inv , we wan t to deﬁne on the additive ca tegory R G F the str ucture o f an additive categor y with inv olution. Her e the pair ( F , T ) mea ns that we assig n to every o b ject x in G an a dditive category with involution F ( x ) a nd for every mor phism f : x → y in G we have a functor o f additive ca tegories with inv olution ( F ( f ) , T ( f )) : F ( y ) → F ( x ). Next w e construct for a functor G → Add - Cat inv an in volution of a dditiv e categorie s ( I R G F , E R G F ) (7.1) on the additive category R G F which w e have in tro duced in (5.1). On ob jects we put I R G F ( x, A ) := ( x, I G ( A )) = ( x, A ∗ ) . Let φ = P f ∈ m or G ( x,y ) f · φ f : ( x, A ) → ( y , B ) be a morphism in R G F . Deﬁne I R G F ( φ ) : B ∗ → A ∗ to b e the morphism φ ∗ = P f ∈ m or G ( y ,x ) f · ( φ ∗ ) f : ( y , B ∗ ) → ( x, A ∗ ) in R G F who se co mpo nent for f ∈ mor G ( y , x ) is given by the comp os ite ( φ ∗ ) f : B ∗ = F ( f )  F ( f − 1 )( B ∗ )  F ( f )( T ( f − 1 )( B )) − − − − − − − − − − − → F ( f )  F ( f − 1 )( B ) ∗  F ( f ) ( ( φ f − 1 ) ∗ ) − − − − − − − − − − → F ( f )( A ∗ ) . Next we show that I R G F is a con trav ariant functor. Ob viously I R G F sends the ident ity id A to id I R G F ( A ) . W e hav e to sho w I R G F ( ψ ◦ φ ) = I R G F ( φ ) ◦ I R G F ( ψ ) for morphisms φ = P h ∈ mor G ( x,y ) h · φ h : ( x, A ) → ( y , B ) and ψ : P k ∈ mor G ( y ,z ) k · ψ k : ( y , B → ( z , C ), or in short notation ( ψ ◦ φ ) ∗ = φ ∗ ◦ ψ ∗ . ON CR OSSED PRODUCT RINGS WITH TWISTED INVOLUTIONS, . . . 23 By deﬁnition ( φ ∗ ◦ ψ ∗ ) = P g ∈ m or G ( z ,x ) g · ( φ ∗ ◦ ψ ∗ ) g for ( φ ∗ ◦ ψ ∗ ) g := X k ∈ mor G ( z ,y ) , h ∈ mor G ( y ,x ) , hk = g F ( k )(( φ ∗ ) h ) ◦ ( ψ ∗ ) k . By deﬁnition ( ψ ∗ ) k : C ∗ = F ( k )( F ( k − 1 )( C ∗ )) F ( k )( T ( k − 1 )( C )) − − − − − − − − − − − → F ( k )( F ( k − 1 )( C ) ∗ ) F ( k )(( ψ k − 1 ) ∗ ) − − − − − − − − − → F ( k )( B ∗ ) and ( φ ∗ ) h : B ∗ = F ( h )( F ( h − 1 )( B ∗ )) F ( h )( T ( h − 1 )( B )) − − − − − − − − − − − → F ( h )( F ( h − 1 )( B ) ∗ ) F ( h )(( φ h − 1 ) ∗ ) − − − − − − − − − → F ( h )( A ∗ ) . Hence the c omp onent ( φ ∗ ◦ ψ ∗ ) g of ( φ ∗ ◦ ψ ∗ ) at g : z → x is given by the sum of morphisms from C ∗ to F ( g )( A ∗ ) X k ∈ mor G ( z ,y ) , h ∈ mor G ( y ,x ) , hk = g F ( k ) ( F ( h )(( φ h − 1 ) ∗ )) ◦ F ( k )  F ( h )( T ( h − 1 )( B ))  ◦ F ( k )(( ψ k − 1 ) ∗ ) ◦ F ( k )( T ( k − 1 )( C )) . The comp onent of ( ψ ◦ φ ) ∗ g of ( ψ ◦ φ ) ∗ at g : z → x is given by C ∗ = F ( g )  F ( g − 1 )( C ∗ )  F ( g )( T ( g − 1 )( C )) − − − − − − − − − − − → F ( g )  F ( g − 1 )( C ) ∗  F ( g ) ( (( ψ ◦ φ ) g − 1 ) ∗ ) − − − − − − − − − − − − → F ( g )( A ∗ ) . Since for g : z → x we hav e ( ψ ◦ φ ) g − 1 = X h ∈ mor G ( y ,z ) , k ∈ mor G ( x,y ) , hk = g − 1 F ( k )( ψ h ) ◦ φ k , the comp onent of ( ψ ◦ φ ) ∗ g of ( ψ ◦ φ ) ∗ at g : z → x is given by the sum of morphisms C ∗ to F ( g )( A ∗ ) X h ∈ mor G ( y ,z ) , k ∈ mor G ( x,y ) , hk = g − 1 F ( g ) (( φ k ) ∗ ) ◦ F ( g ) ( F ( k )( ψ h ) ∗ ) ◦ F ( g )( T ( g − 1 )( C )) . By changing the index ing b y replacing h with k − 1 and k by h − 1 , this trans forms to X k ∈ mor G ( z ,y ) , h ∈ mor G ( y ,x ) , hk = g F ( g ) ( ( φ h − 1 ) ∗ ) ◦ F ( g )  F ( h − 1 )( ψ k − 1 ) ∗  ◦ F ( g )( T ( g − 1 )( C )) . Hence we have to show for every k : z → y and h : y → x with hk = g that the t wo comp osites F ( k ) ( F ( h )(( φ h − 1 ) ∗ )) ◦ F ( k )  F ( h )( T ( h − 1 )( B ))  ◦ F ( k )(( ψ k − 1 ) ∗ ) ◦ F ( k )( T ( k − 1 )( C )) and ( F ( g )(( φ h − 1 ) ∗ ) ◦ F ( g )  F ( h − 1 )( ψ k − 1 ) ∗  ◦ F ( g )( T ( g − 1 )( C )) 24 AR THUR BAR TELS AND WOLF GANG L ¨ UCK agree. W e co mpute for the ﬁrst o ne F ( k ) ( F ( h )(( φ h − 1 ) ∗ )) ◦ F ( k )  F ( h )( T ( h − 1 )( B ))  ◦ F ( k )(( ψ k − 1 ) ∗ ) ◦ F ( k )( T ( k − 1 )( C )) = ( F ( g )(( φ h − 1 ) ∗ ) ◦ F ( g )( T ( h − 1 )( B )) ◦ F ( g )  F ( h − 1 )(( ψ k − 1 ) ∗ )  ◦ F ( g )  F ( h − 1 )( T ( k − 1 )( C ))  . Hence it remains to sho w that the comp osites F ( g − 1 )( C ∗ ) T ( g − 1 )( C ) − − − − − − − → F ( g − 1 )( C ) ∗ F ( h − 1 )( ψ k − 1 ) ∗ − − − − − − − − − − → F ( h − 1 )( B ) ∗ and F ( g − 1 )( C ∗ ) = F ( h − 1 )  F ( k − 1 )( C ∗ )  F ( h − 1 )( T ( k − 1 )( C )) − − − − − − − − − − − − − → F ( h − 1 )( F ( k − 1 )( C ) ∗ ) F ( h − 1 )(( ψ k − 1 ) ∗ ) − − − − − − − − − − − → F ( h − 1 )( B ∗ ) T ( h − 1 )( B ) − − − − − − − → F ( h − 1 )( B ) ∗ agree. The seco nd one agr ees with the comp os ite F ( g − 1 )( C ∗ ) = F ( h − 1 )  F ( k − 1 )( C ∗ )  F ( h − 1 )( T ( k − 1 )( C )) − − − − − − − − − − − − − → F ( h − 1 )( F ( k − 1 )( C ) ∗ ) T ( h − 1 )( F ( k − 1 )( C )) − − − − − − − − − − − − − → F ( h − 1 )( F ( k − 1 )( C )) ∗ F ( h − 1 )( ψ k − 1 ) ∗ − − − − − − − − − − → F ( h − 1 )( B ) ∗ since T ( h − 1 ) is a natura l tra nsformation F ( h − 1 ) ◦ I F ( y ) → I F ( x ) ◦ F ( h − 1 ). Since ( F ( h − 1 ) , T ( h − 1 )) ◦ ( F ( k − 1 ) , T ( k − 1 )) = ( F ( k − 1 h − 1 ) , T ( k − 1 h − 1 )) = ( F ( g − 1 ) , T ( g − 1 )) the map T ( g − 1 )( C ) can b e written as the comp osite T ( g − 1 )( C ) : F ( g − 1 )( C ∗ ) = F ( h − 1 )  F ( k − 1 )( C ∗ )  F ( h − 1 ) ( T ( k − 1 )( C ) ) − − − − − − − − − − − − − → F ( h − 1 )  F ( k − 1 )( C ) ∗  T ( h − 1 ) ( F ( k − 1 )( C ) ) − − − − − − − − − − − − − → F ( h − 1 )  F ( k − 1 )( C )  ∗ = F ( g )( C ) ∗ . This ﬁnishes the pro o f that I R G F is a contra v aria nt functor. The natur a l equiv alence E R G F : id R G F → I R G F ◦ I R G F assigns to an ob ject ( x, A ) in R G F the iso morphism id x ·  E G ( A ) : A ∼ = − → A ∗∗  : ( x, A ) → ( x, A ∗∗ ) . W e hav e to c heck that E R G F is a na tur al equiv alence. Co nsider a mor phism φ = P f ∈ m or G ( x,y ) f · ( φ f : ( x, A ) → ( y , B )) in R G F . Then  I R G F ◦ I R G F  ( φ ) has as the comp onent for f : x → y the comp osite A ∗∗ = F ( f )  F ( f − 1 )( A ∗∗ )  F ( f ) ( ( T ( f − 1 )( A ∗ ) ) − − − − − − − − − − − − − → F ( f )  F ( f − 1 )( A ∗ ) ∗  F ( f ) ( F ( f − 1 )(( φ f ) ∗ ) ∗ ) − − − − − − − − − − − − − − − → F ( f )  F ( f − 1 )( F ( f )( B ) ∗ ) ∗  F ( f ) ( F ( f − 1 )( T ( f )( B )) ∗ ) − − − − − − − − − − − − − − − − → F ( f )  F ( f − 1 )( F ( f )( B ∗ )) ∗  = F ( f )( B ∗∗ ) . ON CR OSSED PRODUCT RINGS WITH TWISTED INVOLUTIONS, . . . 25 Hence  I R G F ◦ I R G F  ( φ ) ◦ E R G F ( x, A ) ha s as co mpo nent for f : x → y the co m- po site A E A ( A ) − − − − → A ∗∗ = F ( f )  F ( f − 1 )( A ∗∗ )  F ( f ) ( ( T ( f − 1 )( A ∗ ) ) − − − − − − − − − − − − − → F ( f )  F ( f − 1 )( A ∗ ) ∗  F ( f ) ( F ( f − 1 )( φ ∗ f ) ∗ ) − − − − − − − − − − − − − → F ( f )  F ( f − 1 )( F ( f )( B ) ∗ ) ∗  F ( f ) ( F ( f − 1 )( T ( f )( B )) ∗ ) − − − − − − − − − − − − − − − − → F ( f )  F ( f − 1 )( F ( f )( B ∗ )) ∗  = F ( f )( B ∗∗ ) . The comp onent of E R G F ( y , B ) ◦ φ a t f : x → y is the co mpo site A φ f − − → F ( f )( B ) F ( f )( E A ( B )) − − − − − − − − → F ( f )( B ∗∗ ) . It rema ins to show that these tw o morphisms A → F ( f )( B ∗∗ ) ag ree. The following t wo diagrams commute s ince E A and T ( f − 1 ) are natural transfor mations A E A ( A ) / / φ f   A ∗∗ = F ( f )  F ( f − 1 )( A ∗∗ )  φ ∗∗ f = F ( f ) ( F ( f − 1 )( φ ∗∗ f )) )   F ( f )( B ) E A ( F ( f )( B )) / / F ( f )( B ) ∗∗ = F ( f )  F ( f − 1 )( F ( f )( B ) ∗∗ )  and F ( f )  F ( f − 1 )( A ∗∗ )  F ( f )( T ( f − 1 )( A ∗ )) / / F ( f ) ( F ( f − 1 )( φ ∗∗ f )) )   F ( f )  F ( f − 1 )( A ∗ ) ∗  F ( f ) ( F ( f − 1 )( φ ∗ f ) ∗ )   F ( f )  F ( f − 1 )( F ( f )( B ) ∗∗ )  F ( f ) ( T ( f − 1 )( F ( f )( B ) ∗ ) ) / / F ( f )  F ( f − 1 )( F ( f )( B ) ∗ ) ∗  . Hence we have to show that F ( f )( B ) F ( f )( E A ( B )) − − − − − − − − → F ( f )( B ∗∗ ) agrees with the comp osite F ( f )( B ) E A ( F ( f )( B )) − − − − − − − − → F ( f )( B ) ∗∗ = F ( f )  F ( f − 1 )( F ( f )( B ) ∗∗ )  F ( f ) ( T ( f − 1 )( F ( f )( B ) ∗ ) ) − − − − − − − − − − − − − − − − → F ( f )  F ( f − 1 )( F ( f )( B ) ∗ ) ∗  F ( f ) ( F ( f − 1 )( T ( f )( B )) ∗ ) − − − − − − − − − − − − − − − − → F ( f )  F ( f − 1 )( F ( f )( B ∗ )) ∗  = F ( f )( B ∗∗ ) . (Notice that φ is not in volv ed anymore.) The following diagra m commutes by the axioms (see (1.6)) F ( f )( B ) E A ( F ( f )( B )) / / F ( f )( E A ( B ))   F ( f )( B ) ∗∗ T ( f )( B ) ∗   F ( f )( B ∗∗ ) T ( f )( B ∗ ) / / F ( f )( B ∗ ) ∗ 26 AR THUR BAR TELS AND WOLF GANG L ¨ UCK Hence it remains to show the commutativit y of the following diag r am (which do es not inv olv e φ and E A anymore). F ( f )( B ) ∗∗ = F ( f )  F ( f − 1 )( F ( f )( B ) ∗ )  ∗ F ( f ) ( T ( f − 1 )( F ( f )( B ) ∗ ) ) / / T ( f )( B ) ∗   F ( f )  F ( f − 1 )( F ( f )( B ) ∗ ) ∗  F ( f ) ( F ( f − 1 )( T ( f )( B )) ∗ )   F ( f )( B ∗ ) ∗ F ( f )( B ∗∗ ) = F ( f )  F ( f − 1 )( F ( f )( B ∗ )) ∗  T ( f )( B ∗ ) o o Since ( F ( f ) , T ( f )) ◦ ( F ( f − 1 ) , T ( f − 1 )) = id, we have T ( f )  F ( f − 1 )( F ( f )( B ) ∗ )  ◦ F ( f )  T ( f − 1 )( F ( f )( B ) ∗ )  = id . Hence it suﬃces to prove the commutativit y o f the fo llowing diagram F ( f )( B ) ∗∗ = F ( f )  F ( f − 1 )( F ( f )( B ) ∗ )  ∗ T ( f )( B ) ∗   F ( f )  F ( f − 1 )( F ( f )( B ) ∗ ) ∗  F ( f ) ( F ( f − 1 )( T ( f )( B )) ∗ )   T ( f ) ( F ( f − 1 )( F ( f )( B ) ∗ ) ) o o F ( f )( B ∗ ) ∗ F ( f )( B ∗∗ ) = F ( f )  F ( f − 1 )( F ( f )( B ∗ )) ∗  . T ( f )( B ∗ ) o o This follows b eca use this dia gram is obtained by applying the natural trans fo rma- tion T ( f ) to the morphism F ( f − 1 )( F ( f )( B ∗ )) F ( f − 1 )( T ( f )( B )) − − − − − − − − − − − → F ( f − 1 )( F ( f )( B ) ∗ ) . The conditio n (1.3) is sa tis ﬁe d for ( I R G F , E R G F ) since it holds for ( I A , E A ). W e will denote the r esulting additiv e c a tegory R A F with in volution ( I R G F , E R G F ) by R G ( F, T ) . (7.2) Let ( F 0 , T 0 ) a nd ( F 1 , T 1 ) b e tw o contra v aria nt functors G → Add - Cat inv . Let ( S, U ) : ( F 0 , T 0 ) → ( F 1 , T 1 ) b e a na tural transfo r mation of suc h functors . This means tha t we for each ob ject x in G we hav e an equiv alence ( S ( x ) , U ( x )) : F 0 ( x ) → F 1 ( y ) of additiv e catego r ies with inv olution such that for all f : x → y in G the following diag ram of functor s of a dditive ca tegories with in volution commutes F 0 ( y ) ( S ( y ) ,U ( y )) / / ( F 0 ( f ) ,T 0 ( f ))   F 1 ( y ) ( F 1 ( f ) ,T 1 ( f ))   F 0 ( x ) ( S ( x ) ,U ( x )) / / F 1 ( x ) (7.3) Then b o th R G ( F 0 , T 0 ) and R G ( F 1 , T 1 ) are additive categor ie s with inv olutions. The functor of additive categories R G S : R G F 0 → R G F 1 deﬁned in (5.2) extends to a functor o f a dditive ca tegories with inv o lution R G ( S, U ) : R G ( F 0 , T 0 ) → R G ( F 1 , T 1 ) (7.4) as follows. W e have to sp ecify a natura l equiv a le nce b U :  Z G S  ◦ I R G ( F 0 ,T 0 ) → I R G ( F 1 ,T 1 ) ◦ Z G S. F or an ob ject ( x, A ) in R G F 0 the isomorphism b U ( x, A ) :  Z G S  ◦ I R G ( F 0 ,T 0 ) ( x, A ) → I R G ( F 1 ,T 1 ) ◦ Z G S ( x, A ) ON CR OSSED PRODUCT RINGS WITH TWISTED INVOLUTIONS, . . . 27 is g iven by the isomo rphism id x · U ( x )( A ) : ( x, S ( x )( A ∗ )) → ( x, S ( x )( A ) ∗ ) in R G F 1 . Next we chec k that b U is a natural equiv alence . Let P f ∈ m or G ( x,y ) f · φ f : ( x, A ) → ( y , B ) b e a morphism in R G F 0 , where b y deﬁnition φ f : A → F ( f )( B ) is a morphism in the a dditive category F 0 ( x ). W e hav e to s how the co mmu tativity of the following diagr a m in the additive categor y R G F 1  R G S  ◦ I R G ( F 0 ,T 0 ) ( y , B ) ( R G S ) ◦ I R G ( F 0 ,T 0 ) ( φ ) / / b U ( y ,B )    R G S  ◦ I R G ( F 0 ,T 0 ) ( x, A ) b U ( x, A )   I R G ( F 1 ,T 1 ) ◦ R G S ( y , B ) I R G ( F 1 ,T 1 ) ◦ R G S ( φ ) / / I R G ( F 1 ,T 1 ) ◦ R G S ( x, A ) The morphism I R G ( F 0 ,T 0 ) ( φ ) in R G F 0 is given by φ ∗ = X f ∈ m or G ( x,y ) f · ( φ ∗ ) f : ( y , B ∗ ) → ( x, A ∗ ) , where the comp onent ( φ ∗ ) f is the comp osite ( φ ∗ ) f : B ∗ = F 0 ( f )  F 0 ( f − 1 )( B ∗ )  F 0 ( f ) ( T 0 ( f − 1 )( B ) ) − − − − − − − − − − − − → F 0 ( f )  F 0 ( f − 1 )( B ) ∗  F 0 ( f )(( φ f − 1 ) ∗ ) − − − − − − − − − − → F 0 ( f ) ( A ∗ ) . The morphism  R G S  ◦ I R G ( F 0 ,T 0 ) ( φ ) : ( y , B ∗ ) → ( x, A ∗ ) in R G F 1 is given by P f ∈ m or G ( x,y ) f · ψ f : ( y , B ∗ ) → ( x, A ∗ ), where ψ f is the co mpo site ψ f : S ( y )( B ∗ ) = S ( y )  F 0 ( f )  F 0 ( f − 1 )( B ∗ )  S ( y ) ( F 0 ( f ) ( T 0 ( f − 1 )( B ) )) − − − − − − − − − − − − − − − − − → S ( y )  F 0 ( f )  F 0 ( f − 1 )( B ) ∗  S ( y )( F 0 ( f )(( φ f − 1 ) ∗ )) − − − − − − − − − − − − − − → S ( y ) ( F 0 ( f ) ( A ∗ )) = F 1 ( f ) ( S ( x ) ( A ∗ )) . Hence the morphism b U ( x, A ) ◦  R G S  ◦ I R G ( F 0 ,T 0 ) ( φ ) : ( y , B ∗ ) → ( x, A ∗ ) in R G F 1 is given by P f ∈ m or G ( x,y ) f · µ f : ( y , B ∗ ) → ( x, A ∗ ), where µ f is the comp osite in F 1 ( y ) µ f : S ( y )( B ∗ ) = S ( y )  F 0 ( f )  F 0 ( f − 1 )( B ∗ )  S ( y ) ( F 0 ( f ) ( T 0 ( f − 1 )( B ) )) − − − − − − − − − − − − − − − − − → S ( y )  F 0 ( f )  F 0 ( f − 1 )( B ) ∗  S ( y )( F 0 ( f )(( φ f − 1 ) ∗ )) − − − − − − − − − − − − − − → S ( y ) ( F 0 ( f ) ( A ∗ )) = F 1 ( f ) ( S ( x ) ( A ∗ )) F 1 ( f )( U ( x )( A )) − − − − − − − − − − → F 1 ( f ) ( S ( x )( A ) ∗ ) . The morphism R G S ( φ ) : ( x, S ( x )( A )) → ( y , S ( y )( B )) in R G F 1 is g iven by X f ∈ m or G ( x,y ) f · ( S ( x )( φ f ) : S ( x )( A ) → S ( x )( F 0 ( f )( B ) = F 1 ( f )( S ( y )( B )) . 28 AR THUR BAR TELS AND WOLF GANG L ¨ UCK The morphism I R G ( F 1 ,T 1 ) ◦ R G S ( φ ) : ( y , S ( y )( B ) ∗ ) → ( x, S ( x )( A ) ∗ ) in R G F 1 is given by P f ∈ m or G ( y ,x ) f · ν f , where ν f is the co mpo site in F 1 ( y ). ν f : S ( y )( B ) ∗ = F 1 ( f )  F 1 ( f − 1 ) ( S ( y )( B ) ∗ )  F 1 ( f ) ( T 1 ( f − 1 )( S ( y )( B )) ) − − − − − − − − − − − − − − − − → F 1 ( f )  F 1 ( f − 1 ) ( S ( y )( B )) ∗  = F 1 ( f )  S ( x )  F 0 ( f − 1 )( B )  ∗  F 1 ( f ) ( ( S ( x )( φ f − 1 )) ∗ ) − − − − − − − − − − − − − − → F 1 ( f ) ( S ( x )( A ) ∗ ) . The mor phism I R G ( F 1 ,T 1 ) ◦ R G S ( φ ) ◦ b U ( y , B ) : ( y , S ( y )( B ∗ )) → ( x, S ( x )( A ) ∗ ) in R G F 1 is g iven by P f ∈ m or G ( y ,x ) f · ω f , where ω f is the comp osite in F 1 ( y ). ω f : S ( y )( B ∗ ) U ( y )( B ) − − − − − → S ( y )( B ) ∗ = F 1 ( f )  F 1 ( f − 1 ) ( S ( y )( B ) ∗ )  F 1 ( f ) ( T 1 ( f − 1 )( S ( y )( B )) ) − − − − − − − − − − − − − − − − → F 1 ( f )  F 1 ( f − 1 ) ( S ( y )( B )) ∗  = F 1 ( f )  S ( x )  F 0 ( f − 1 )( B )  ∗  F 1 ( f ) ( ( S ( x )( φ f − 1 )) ∗ ) − − − − − − − − − − − − − − → F 1 ( f ) ( S ( x )( A ) ∗ ) . Hence w e have to show for all f : y → x in mor G ( y , x ) that the tw o c omp osites in F 1 ( y ) S ( y )( B ∗ ) = S ( y )  F 0 ( f )  F 0 ( f − 1 )( B ∗ )  S ( y ) ( F 0 ( f ) ( T 0 ( f − 1 )( B ) )) − − − − − − − − − − − − − − − − − → S ( y )  F 0 ( f )  F 0 ( f − 1 )( B ) ∗  S ( y )( F 0 ( f )(( φ f − 1 ) ∗ )) − − − − − − − − − − − − − − → S ( y ) ( F 0 ( f ) ( A ∗ )) = F 1 ( f ) ( S ( x ) ( A ∗ )) F 1 ( f )( U ( x )( A )) − − − − − − − − − − → F 1 ( f ) ( S ( x )( A ) ∗ ) and S ( y )( B ∗ ) U ( y )( B ) − − − − − → S ( y )( B ) ∗ = F 1 ( f )  F 1 ( f − 1 ) ( S ( y )( B ) ∗ )  F 1 ( f ) ( T 1 ( f − 1 )( S ( y )( B )) ) − − − − − − − − − − − − − − − − → F 1 ( f )  F 1 ( f − 1 ) ( S ( y )( B )) ∗  = F 1 ( f )  S ( x )  F 0 ( f − 1 )( B )  ∗  F 1 ( f ) ( ( S ( x )( φ f − 1 )) ∗ ) − − − − − − − − − − − − − − → F 1 ( f ) ( S ( x )( A ) ∗ ) . agree. Since S is a na tural transformatio n from F 0 → F 1 , the ﬁrst compo site ca n be rewritten as the comp os ite S ( y )( B ∗ ) = F 1 ( f )  S ( x )  F 0 ( f − 1 )( B ∗ )  F 1 ( f ) ( S ( x ) ( T 0 ( f − 1 )( B ) )) − − − − − − − − − − − − − − − − − → F 1 ( f )  S ( x )  F 0 ( f − 1 )( B ) ∗  F 1 ( f )( S ( x )(( φ f − 1 ) ∗ )) − − − − − − − − − − − − − − → F 1 ( f ) ( S ( x ) ( A ∗ )) F 1 ( f )( U ( x )( A )) − − − − − − − − − − → F 1 ( f ) ( S ( x )( A ) ∗ ) . ON CR OSSED PRODUCT RINGS WITH TWISTED INVOLUTIONS, . . . 29 Since U ( x ) is a natura l transfor ma tion from S ( x ) ◦ I F 0 ( x ) to I F 1 ( x ) ◦ S ( x ), this agre es with the comp osite S ( y )( B ∗ ) = F 1 ( f )  S ( x )  F 0 ( f − 1 )( B ∗ )  F 1 ( f ) ( S ( x ) ( T 0 ( f − 1 )( B ) )) − − − − − − − − − − − − − − − − − → F 1 ( f )  S ( x )  F 0 ( f − 1 )( B ) ∗  F 1 ( f ) ( U ( x ) ( F 0 ( f − 1 )( B ) )) − − − − − − − − − − − − − − − − − → F 1 ( f )  S ( x )  F 0 ( f − 1 )( B )  ∗  F 1 ( f )(( S ( x )( φ f − 1 )) ∗ ) − − − − − − − − − − − − − − → F 1 ( f ) ( S ( x )( A ) ∗ ) . Hence it suﬃces to show that the following t wo comp os ites ag ree S ( y )( B ∗ ) = F 1 ( f )  S ( x )  F 0 ( f − 1 )( B ∗ )  F 1 ( f ) ( S ( x ) ( T 0 ( f − 1 )( B ) )) − − − − − − − − − − − − − − − − − → F 1 ( f )  S ( x )  F 0 ( f − 1 )( B ) ∗  F 1 ( f ) ( U ( x ) ( F 0 ( f − 1 )( B ) )) − − − − − − − − − − − − − − − − − → F 1 ( f )  S ( x )  F 0 ( f − 1 )( B )  ∗  and S ( y )( B ∗ ) U ( y )( B ) − − − − − → S ( y )( B ) ∗ = F 1 ( f )  F 1 ( f − 1 ) ( S ( y )( B ) ∗ )  F 1 ( f ) ( T 1 ( f − 1 )( S ( y )( B )) ) − − − − − − − − − − − − − − − − → F 1 ( f )  F 1 ( f − 1 ) ( S ( y )( B )) ∗  = F 1 ( f )  S ( x )  F 0 ( f − 1 )( B )  ∗  (Notice that φ f − 1 has b een elimina ted.) This will follow by applying F 1 ( f ) to the following diag ram, provided we can s how that it do es commute. S ( x )  F 0 ( f − 1 )( B ∗ )  = F 1 ( f − 1 ) ( S ( y )( B ∗ )) S ( x ) ( T 0 ( f − 1 )( B ) ) / / F 1 ( f − 1 )( U ( y )( B ))   S ( x )  F 0 ( f − 1 )( B ) ∗  U ( x ) ( F 0 ( f − 1 )( B ) )   F 1 ( f − 1 ) ( S ( y )( B ) ∗ ) T 1 ( f − 1 )( S ( y )( B )) / / S ( x )  F 0 ( f − 1 )( B )  ∗ But the latter diagram co mmu tes bec ause we require the following equality of func- tors of a dditiv e ca tegories with inv o lution for f − 1 : x → y (se e (7.3)) ( F 0 ( f − 1 ) , T 0 ( f − 1 )) ◦ ( S ( x ) , U ( x )) = ( S ( y ) , U ( y )) ◦ ( F 1 ( f − 1 ) , T 1 ( f − 1 )) . This ﬁnishes the pro of that b U is a natural equiv alence. One easily chec ks that condition (1 .6) is satisﬁed by b U since it holds for U ( x ) for a ll ob jects x in G . This ﬁnishes the constr uction of the functor of additive catego ries with in volution ( S, U ) (see (7.4 )). One ea sily chec k s  Z G ( S 2 , U 2 )  ◦  Z G ( S 1 , U 1 )  = Z G ( S 2 , U 2 ) ◦ ( S 1 , U 1 ) (7.5) Z G id F = id R G F . (7.6) Given a functor of gr oup oids W : G 1 → G and a functor ( F, T ) : G → Add - Cat inv , the comp osition with W a yields a functor ( F ◦ W , T ◦ W ). Hence b oth R G 1 ( F, T ) ◦ W a nd R G ( F, T ) are additive categories with in v olutions. O ne ea sily c hecks that I R G F ◦ W ∗ = W ∗ ◦ I R G 1 F ◦ W holds fo r the functor W ∗ deﬁned in (5.5). Hence ( W ∗ , id) : R G 1 ( F, T ) ◦ W → R G ( F, T ) . (7.7) 30 AR THUR BAR TELS AND WOLF GANG L ¨ UCK is a functor of additive categ o ries with inv olution. One easily chec ks (( W 2 ) ∗ , id) ◦ (( W 1 ) ∗ , id) = (( W 2 ◦ W 1 ) ∗ , id); (7.8) (id G ) ∗ = id R G F . (7.9) These tw o c o nstructions are c ompatible. Namely , we get  Z G ( S, U )  ◦ ( W ∗ , id) = ( W ∗ , id) ◦  Z G 1 ( S ◦ W, U ◦ W )  . (7.10) One ea sily chec k s Lemma 7.11. (i) L et W : G 1 → G b e an e quivalenc e of c onne cte d gr oup oids. L et ( F, T ) : G → Add - Cat inv b e a c ontr avaria nt functor. Then W ∗ : Z G 1 ( F, T ) ◦ W → Z G ( F, T ) is an e quivalenc e of additive c ate gorie s with involution. (ii) L et G b e a c onne ct e d gro up oid. L et S : ( F 1 , T 1 ) → ( F 2 , T 2 ) b e a tra ns- formation of c ontr avariant functors G → Add - Cat inv such that for every obje ct x in G the functor S ( x ) : F 0 ( x ) → F 1 ( x ) is an e quivalenc e of additive c ate gories. Then Z G S : Z G ( F 1 , T 1 ) → Z G ( F 2 , T 2 ) is an e quivalenc e of additive c ate gorie s with involution. 8. Fr om crossed pr oduct rings with inv olution to additive ca tegories with involution Next we w an t to extend E xample 6.1 and Lemma 6.3 to rings and additive categorie s with in v olutions. Let R b e a ring and let G b e a gro up. Supp os e that we a r e g iven ma ps of sets c : G → aut( R ) , g 7→ c g ; τ : G × G → R × ; w : G → R , satisfying conditions (4.3), (4.4), (4.5), (4.6), (4.7 ), (4.13), (4.14), (4.15), and (4.1 6). W e hav e co nstructed in Section 4 a n in volution on the crossed pro duct R ∗ G = R ∗ c,τ G . W e hav e denoted this ring with involution b y R ∗ G = R ∗ c,τ , w G (see (4.19)). The additiv e category R ∗ G - FGF inher its the structure of an additive catego r y with inv o lution (see Exa mple 1.5). W e hav e in tro duced notion of a n additive G -categ ory with in volution in Deﬁ- nition 4 .22 a nd co nstructed an explicit example R - FGP c,τ , w in (4.24). All these constructions restrict to the sub categor y R - FGF ⊆ R - FGP of ﬁnitely g e ne r ated free R -mo dules. Th us we obtain the additive G -c ategory with inv o lution R - FGP c,τ , w (8.1) Lemma 8. 2. Consider t he data ( R , c, τ , w ) and the additive G -c ate gory with involu- tion R - FGF c,τ , w of (8.1) . L et R G R - FGF c,τ , w b e the additive c ate gory with involution deﬁne d in (7.1) . Sinc e G r e gar de d as a gr oup oid has pr e cisely one obje ct , we c an (and wil l) identify t he set of obje cts in R G R - FGF c,τ , w with t he set of obje cts in R - FGF c,τ , w which c onsists of p airs ( M , g ) for M a ﬁnitely gener ate d fr e e R -mo du le and g ∈ G . Denote by R G R - FGF c,τ , w  e the ful l s ub c ate gory of R G F GF R c,τ , w c onsisting of obje cts of the shap e ( M , e ) for e ∈ G t he unit element. Denote by R ∗ G = R ∗ c,τ , w G the ring with involution given by the cr osse d pr o duct ring (se e 4.19). Then ON CR OSSED PRODUCT RINGS WITH TWISTED INVOLUTIONS, . . . 31 (i) Ther e is an e quivalenc e of additive c ate gories with involution ( α, β ) :  Z G R - FGF c,τ , w  e → R ∗ c,τ , w G - FGF ; (ii) The inclus ion  Z G R - FGF c,τ , w  e → Z G R - FGF c,τ , w is an e quivalenc e of additive c ate gorie s with involution. Pr o of. (i) W e ha ve a lready co nstructed a n equiv alence o f ca tegories α :  Z G R - FGF c,τ  e → R ∗ c,τ G - FGF ; in Lemma 6 .3 (i). W e wan t to s how that α is compa tible with the involution, i.e ., there is a functor of catego ries with inv olutions ( α, β ) :  Z G R - FGF c,τ , w  e → R ∗ c,τ , w G - FGF . The natura l equiv alenc e β : α ◦ I ( R G R - FGF c,τ,w ) e → I R ∗ c,τ,w G - FGF ◦ α a ssigns to a n ob ject ( M , e ) in R G R - FGF c,τ , w  e the R ∗ c,τ G -isomorphism β ( M , e ) : R ∗ c,τ , w G ⊗ R M ∗ ∼ = − → ( R ∗ c,τ , w G ⊗ R M ) ∗ given by β ( M , e )( u ⊗ f )( v ⊗ m ) = v f ( m ) u for f ∈ M ∗ , u ∈ R ∗ c,τ G and m ∈ M . Obviously β is compatible with the structures of additive catego ries. Next we chec k that β is a natural trans fo rmation. W e hav e to show for a mor- phism φ : ( M , e ) → ( N , e ) in R G R - FGF c,τ , w  e that the following diag ram comm utes R ∗ G ⊗ R N ∗ β ( N ,e )   α ( φ ∗ ) / / R ∗ G ⊗ R M ∗ β ( M ,e )   ( R ∗ G ⊗ R N ) ∗ α ( φ ) ∗ / / ( R ∗ G ⊗ R M ) ∗ Recall tha t a mor phism φ = X g ∈ G g · φ g : ( M , e ) → ( N , e ) in R G R - FGF c,τ , w  e is g iven b y a collection of mor phisms φ g : ( M , e ) → R g ( N , e ) = ( N , g ) in R - FGF c,τ , w for g ∈ G , wher e φ g is a R -homo morphism M → res c g N . W e wan t to unravel what the dual mor phis m φ ∗ = X g ∈ G g · ( φ ∗ ) g : ( N , e ) ∗ = ( N ∗ , e ) → ( M , e ) ∗ = ( M ∗ , e ) in R G R - FGF c,τ , w  e is. It is given by a c ollection of morphisms { ( φ ∗ ) g : ( N ∗ , e ) → R g ( M ∗ , e ) = ( M ∗ , g ) | g ∈ G } in R - FGF c,τ , w , w he r e ( φ ∗ ) g is a R -ho momorphism N ∗ → res c g M ∗ . In R - FGF c,τ , w the morphism ( φ ∗ ) g is g iven by the comp osite ( N ∗ , e ) = ( N ∗ , g − 1 ) · g = R g ( N ∗ , g − 1 ) R g (( φ g − 1 ) ∗ ) − − − − − − − − → R g ( M ∗ , e ) = ( M ∗ , g ) . The morphism ( φ g − 1 ) ∗ is g iven by the comp osite res c g − 1 N ∗ t g − 1 ( N ) − − − − − →  res c g − 1 N  ∗ ( φ g − 1 ) ∗ − − − − − → M ∗ . Explicitly this is the map N ∗ → M ∗ , f ( x ) 7→ c − 1 g − 1  f ◦ φ g − 1 ( x )   w ( g − 1 ) τ ( g , g − 1 )  − 1 . 32 AR THUR BAR TELS AND WOLF GANG L ¨ UCK The morphism R g (( φ g − 1 ) ∗ ) is the co mpo site N ∗ = res c g − 1 g N ∗ L τ ( g − 1 ,g ) − − − − − − → res g res c g − 1 N ∗ res c g ( φ g − 1 ) ∗ − − − − − − − − → res c g M ∗ . Hence the R -linea r map ( φ ∗ ) g : N ∗ → res c g M ∗ sends f ∈ N ∗ to the e le ment in M ∗ given by x 7→ c − 1 g − 1  f ◦ φ g − 1 ( x ) τ ( g − 1 , g )   w ( g − 1 ) τ ( g , g − 1 )  − 1 . This implies tha t the R ∗ G -ho momorphism α ( φ ∗ ) : R ∗ G ⊗ R N ∗ → R ∗ G ⊗ R M ∗ sends u ⊗ f for u ∈ R ∗ G and f ∈ N ∗ to the R -linear map M → R given by X g ∈ G u · τ ( g − 1 , g ) − 1 · g − 1 ⊗  c − 1 g − 1 ◦ f ◦ φ g − 1  c − 1 g − 1  τ ( g − 1 , g )   w ( g − 1 ) τ ( g , g − 1 )  − 1 . W e conclude that the comp osite β ( M , e ) ◦ α ( φ ∗ ) sends u ⊗ f for u ∈ R ∗ G and f ∈ N ∗ to the R -linear map R ∗ G ⊗ R M → R ∗ G which maps v ⊗ x fo r v ∈ R ∗ G and x ∈ M to the e lement in R ∗ G X g ∈ G v ·  c − 1 g − 1 ◦ f ◦ φ g − 1  ( x ) c − 1 g − 1  τ ( g − 1 , g )   w ( g − 1 ) τ ( g , g − 1 )  − 1 · u · τ ( g − 1 , g ) − 1 · g − 1 . W e compute that the comp os ite α ( φ ) ∗ ◦ β ( N , e ) sends u ⊗ f for u ∈ R ∗ G and f ∈ N ∗ to the R -linear map R ∗ G ⊗ R M → R ∗ G which maps v ⊗ x fo r v ∈ R ∗ G and x ∈ M to the e lement in R ∗ G β ( N , e )( u ⊗ f ) ( α ( φ )( v ⊗ x )) = β ( N , e )( u ⊗ f )   X g ∈ G v · τ ( g − 1 , g ) − 1 · g − 1 ⊗ φ g ( x )   = X g ∈ G β ( N , e )( u ⊗ f )  v · τ ( g − 1 , g ) − 1 · g − 1 ⊗ φ g ( x )  = X g ∈ G v · τ ( g − 1 , g ) − 1 · g − 1 · f ( φ g ( x )) · u. Hence it suﬃces to show for ea ch g ∈ G , u, v ∈ R ∗ G a nd x ∈ M v  c − 1 g − 1 ◦ f ◦ φ g − 1  ( x ) c − 1 g − 1  τ ( g − 1 , g )   w ( g − 1 ) τ ( g , g − 1 )  − 1 · u · τ ( g − 1 , g ) − 1 · g − 1 = v · τ ( g , g − 1 ) − 1 · g · f  φ g − 1 ( x )  · u. Since u · τ ( g − 1 , g ) − 1 · g − 1 = w ( g − 1 ) c g ( τ ( g − 1 , g ) − 1 ) · g · u ; g · f  φ g − 1 ( x )  · u = c g  f  φ g − 1 ( x )  · g · u, it remains to s how for a ll g ∈ G a nd x ∈ M  c − 1 g − 1 ◦ f ◦ φ g − 1  ( x ) c − 1 g − 1  τ ( g − 1 , g )   w ( g − 1 ) τ ( g , g − 1 )  − 1 · w ( g − 1 ) c g ( τ ( g − 1 , g ) − 1 ) = τ ( g , g − 1 ) − 1 · c g  f  φ g − 1 ( x )  . ON CR OSSED PRODUCT RINGS WITH TWISTED INVOLUTIONS, . . . 33 If we put r = f ◦ φ g − 1 ( x ), this b eco mes equiv alent to sho wing for all g ∈ G and r ∈ R c − 1 g − 1 ( r ) c − 1 g − 1  τ ( g − 1 , g )   w ( g − 1 ) τ ( g , g − 1 )  − 1 · w ( g − 1 ) c g ( τ ( g − 1 , g ) − 1 ) = τ ( g , g − 1 ) − 1 · c g ( r ) . This is e quiv alent to showing τ ( g , g − 1 ) c − 1 g − 1  r τ ( g − 1 , g )  τ ( g , g − 1 ) − 1 = c g ( rτ ( g − 1 , g )) . F rom (4.3) and (4.5) we conclude for x ∈ R τ ( g , g − 1 ) c − 1 g − 1 ( x ) τ ( g , g − 1 ) − 1 = c g ( x ) , and the cla im follows, i.e., β is a na tural equiv alence. It r emains to chec k that the following diagr am (see (1.6)) commutes for every ob ject ( M , e ) in R - FGF c,τ [ G ] e . R ∗ G ⊗ R M E R ∗ c,τ,w G - FGF ( R ∗ G ⊗ R M ) / / α ( E R - FGF c,τ,w ( M ,e ))   ( R ∗ G ⊗ R M ) ∗∗ ( β ( M , e )) ∗   R ∗ G ⊗ R M ∗∗ β (( M , e ) ∗ ) / / ( R ∗ G ⊗ R M ∗ ) ∗ W e consider a n element u ⊗ x in the left upp er corner for u ∈ R ∗ G and x ∈ M . It is sent by the upper horizontal a rrow to the elemen t in ( R ∗ G ⊗ R M ) ∗∗ which maps h ∈ ( R ∗ G ⊗ R M ) ∗ to h ( u ⊗ x ). This element is mapped b y the right vertical arrow to the element in ( R ∗ G ⊗ R M ∗ ) ∗ which sends v ⊗ f for v ∈ R ∗ G and f ∈ M ∗ to β ( M , e )( v ⊗ f )( u ⊗ x ) = uf ( x ) v = v f ( x ) u. The left vertical ar row sends u ⊗ x to u ⊗ I R - FGF ( x ), where I R - FGF ( x ) sends f ∈ M ∗ to f ( x ). This element is mapp ed by the low e r hor izontal arrow to the ele men t in ( R ∗ G ⊗ R M ∗ ) ∗ which sends v ⊗ f for v ∈ R ∗ G and f ∈ M ∗ to v I R - FGF ( x )( f ) u = vf ( x ) u. This ﬁnishes the pro o f that ( α, β ) :  Z G R - FGF c,τ , w  e → R ∗ c,τ , w G - FGF is a n equiv alence of a dditiv e catego r y w ith involutions. (ii) This has alr eady b een proved in Lemma 6.3 (ii).  9. G -homology theories In this sectio n we co nstruct G -ho mology theo ries and dis c uss induction. Deﬁnition 9. 1 (T ransp or t gro upo id) . Let G be a gro up and let ξ b e a G -set. Deﬁne the tr ansp ort gr oup oid G G ( ξ ) to b e the following gro upo id. The set of ob jects is ξ itself. F or x 1 , x 2 ∈ ξ the s e t of mor phisms fro m x 1 to x 2 consists of those elements g in G fo r which g x 1 = x 2 holds. Compositio n of morphisms comes from the g r oup m ultiplication in G . A G - map α : ξ → η of G -sets induces a cov a riant functor G G ( α ) : G G ( ξ ) → G G ( η ) by sending a n ob ject x ∈ ξ to the o b ject α ( x ) ∈ η . A morphism g : x 1 → x 2 is sent to the mor phism g : α ( x 1 ) → α ( x 2 ). 34 AR THUR BAR TELS AND WOLF GANG L ¨ UCK Fix a functor E : Add - Cat inv → Spectra which sends weak equiv alences of additive ca tegories with in volutions to weak ho- motopy equiv alence s of sp ectra. Let G be a g roup. Let Gr oupoids ↓ G b e the c a tegory of connected gr oup oids ov er G considered as a group oid w ith one ob ject, i.e., ob jects a cov ar iant functor s F : G → G with a connected group oid as so ur ce a nd G a s target a nd a mor phism from F 0 : G 0 → G to F 1 : G 1 → G is a cov aria nt functor W : G 0 → G 1 satisfying F 1 ◦ W = F 0 . F or a G -s et S let pr G : G G ( S ) → G ( G/G ) = G be the functor induced by the pro jection S → G/G . The transp or t category yields a functor G G : Or G → Groupoids ↓ G by sending G/H to pr G : G G ( G/H ) → G G ( G/G ) = G . Let A b e an additive G -c a tegory with inv olution in the sense of Deﬁnition 4.2 2. W e obtain a functor E A : Or G → Spectra , G/H 7→ E  R G ( G/H ) A ◦ pr G  . (9.2) Asso ciated to it ther e is a G -homo logy theory in the sense of [9, Section 1] H G ∗ ( − ; E A ) (9.3) such that H G ∗ ( G/H ; E A ) ∼ = π n ( E A ( G/H )) holds for every n ∈ Z and every sub- group H ⊆ G . Namely , deﬁne for a G - C W -c omplex X H G n ( X ; E A ) = π n  map G ( G/ ? , X ) + ∧ Or ( G ) E A ( G/ ?)  . F or more details ab o ut sp ectra and spaces ov er a categor y and asso ciated homo logy theories we r efer to [5 ]. (Notice that there ∧ Or ( G ) is deno ted by ⊗ Or ( G ) .) Lemma 9. 4. L et f : A → B b e a we ak e quivalenc e of additive G - c ate gories with involution. Then the induc e d map H G n ( X ; E f ) : H G n ( X ; E A ) ∼ = − → H G n ( X ; E A ) is a bije ction for al l n ∈ Z . Pr o of. This follows fro m Lemma 7 .1 1 and [5, Lemma 4.6].  Let φ : K → G be a gro up ho mo morphism. Giv en a K - C W -complex X , let G × φ X be the G - C W -complex obta ined fr om X by induction with φ . If H G ∗ ( − ) is a G -homo logy theory , then H G ( φ ∗ ( − )) is a K -ho mology theor y . The next result is essentially the sa me as the pro of of the existence of an induction str uc tur e in [1 , Lemma 6 .1]. Lemma 9.5. L et φ : K → G b e a gr oup homomorp hism. L et A b e an additive G - c ate gory with involution in the sense of D eﬁ nition 4.22. L et res φ A b e the additive K -c ate gory with involution obtaine d fr om A by r estriction with φ . Then ther e is a tr ansformation of K -homolo gy the ories σ ∗ : H K ∗ ( − ; E res φ A ) → H G ∗ ( φ ∗ ( − ); E A ) If X is a K - C W -c omplex on which ker( φ ) acts trivial ly, then σ n : H K n ( X ; E res φ A ) ∼ = − → H G n ( φ ∗ X ; E A ) is bije ctive for al l n ∈ Z . ON CR OSSED PRODUCT RINGS WITH TWISTED INVOLUTIONS, . . . 35 Pr o of. W e have to co ns truct for every K - C W -c omplex X a natura l transformation (9.6) map K ( K/ ? , X ) + ∧ Or ( K ) E Z G K ( K/ ?) res φ A ◦ pr K ! → map G ( G/ ? , φ ∗ X ) + ∧ Or ( G ) E Z G G ( G/ ?) A ◦ pr G ! . The group homomorphism φ induces for e very tra nsitive K -set ξ a functor, natura l in ξ , G φ ( ξ ) : G K ( ξ ) → G G ( φ ∗ ξ ) which sends an ob ject x ∈ ξ to the ob ject ( e , x ) in G × φ ξ and s ends a morphism given by k ∈ K to the morphism giv en by φ ( k ). W e o btain for every transitive K -set ξ a functor of additive categ ories with inv olutions, natur al in ξ (see (7.7)) G φ ( ξ ) ∗ : Z G K ( ξ ) res φ A ◦ pr K = Z G K ( ξ ) A ◦ pr G ◦G φ ( ξ ) → Z G G ( φ ∗ ξ ) A ◦ pr G . Thu s we obta in a ma p of sp ectr a map K ( K/ ? , X ) + ∧ Or ( K ) E Z G K ( K/ ?) res φ A ◦ pr K ! → map K ( K/ ? , X ) + ∧ Or ( K ) E Z G G ( φ ∗ ( K/ ?)) A ◦ pr G ! . ¿F rom the adjunction of induction and restr iction with the functor Or ( φ ) : Or ( K ) → Or ( G ) , K /H 7→ φ ∗ K/H , and the ca nonical map o f c o ntra v ariant Or ( G )-spaces Or ( φ ) ∗ (map K ( K/ ? , X )) → map G ( G/ ? , φ ∗ X ) , which is a n iso morphism for a K - C W -complexes X , we o bta in maps o f sp ectra map K ( K/ ? , X ) + ∧ Or ( K ) E Z G G ( φ ∗ ( K/ ?)) A ◦ pr G ! ∼ = map K ( K/ ? , X ) + ∧ Or ( K ) Or ( φ ) ∗ E Z G G ( G/ ?) A ◦ pr G !! ∼ = Or ( φ ) ∗ (map K ( K/ ? , X )) + ∧ Or ( G ) E Z G G ( G/ ?) A ◦ pr G ! ∼ = map G ( G/ ? , φ ∗ X ) + ∧ Or ( G ) E Z G G ( G/ ?) A ◦ pr G ! . Now the desir ed map of sp ectra (9.6) is the comp osite of the tw o maps ab ove. The pro of that τ n ( X ) is bijectiv e if k er( φ ) acts freely on X is the s ame as the one o f [1, Lemma 1.5 ].  10. Z -ca tegories and additive ca tegories with inv olutions F or tec hnical reason it will be useful that A comes with a (strictly asso cia - tive) functorial direct sum. It will b e used in the deﬁnition of the catego r y ind φ A in (11.5) and in functorial constr uctio ns abo ut categories arising in controlled to po l- ogy . (See for instance [2, Sectio n 2.2], [3, Section 3].) 36 AR THUR BAR TELS AND WOLF GANG L ¨ UCK Deﬁnition 10.1 ( Z -categ ory (with inv olution)) . A Z -c ate gory A is an additiv e category exc e pt that we drop the condition that ﬁnite dir ect sums do exists. More precisely , a Z -ca teg ory A is a sma ll categor y such that for tw o ob jects A a nd B the morphism set mor A ( A, B ) has the structure of an abelia n g roup and compo sition yields bilinea r ma ps mor A ( A, B ) × mor A ( B , C ) → mor A ( B , C ). The notion of a Z -c ate gory with involution A is deﬁned analogous ly . Na mely , we require the existence of the pa ir ( I A , E A ) with the same axio ms as in Section 1 except that we forget everything ab out ﬁnite direct sums. Of course an a dditiv e category (with in volution) is a Z -categor y (with inv olu- tion), just forget the existence of the direct sum of tw o o b jects. Given a Z -categor y A , we ca n enlarge it to an additive c ategory A ⊕ with a func- torial direct sums as fo llows. The ob jects in A ⊕ are n -tuples A = ( A 1 , A 2 , . . . , A n ) consisting of ob jects A i in A for i = 1 , 2 , . . . , n and n = 0 , 1 , 2 , . . . , where we think of the empt y set as 0-tuple whic h w e denote b y 0. the Z -mo dule of morphis ms fro m A = ( A 1 , . . . , A m ) to B = ( B 1 , . . . , B n ) is given by mor A ⊕ ( A , B ) := M 1 ≤ i ≤ m, 1 ≤ j ≤ n mor A ( A i , B j ) . Given a mo rphism f : A → B , we denote by f i,j : A i → B j the comp onent whic h belo ngs to i ∈ { 1 , . . . , m } and j ∈ { 1 , . . . , n } . If A or B is the empty tuple, then mor A ⊕ ( A, B ) is deﬁned to b e the trivial Z -mo dule. The comp osition of f : A → B and g : B → C for ob jects A = ( A 1 , . . . , A m ), B = ( B 1 , . . . , B n ) and C = ( C 1 , . . . , C p ) is deﬁned by ( g ◦ f ) i,k := n X j =1 g j,k ◦ f i,j . The sum on A ⊕ is deﬁned on ob jects by sticking the tuples tog e ther, i.e., for A = ( A 1 , . . . , A m ) and B = ( B 1 , . . . , B n ) deﬁne A ⊕ B := ( A 1 , . . . , A m , B 1 , . . . , B n ) . The deﬁnition of the sum of t wo morphisms is now obvious. The zero ob ject is given by the empt y tuple 0. The construction is strictly asso ciative. These data deﬁne the structure o f a n additive category w ith functoria l direct sum on A ⊕ . Notice that this is more than a n additive categor y since for an a dditiv e ca tegory the existence of the dire c t s um of tw o o b jects is req uired but not a functorial mo del. In the sequel functor ial direct sum is alw ays to b e under sto o d to b e strictly asso ciative, i.e., we hav e for thr ee o b jects A 1 , A 2 and A 3 the equality ( A 1 ⊕ A 2 ) ⊕ A 3 = A 1 ⊕ ( A 2 ⊕ A 3 ) and we will and can o mit the brack ets from now o n in the notion. W e hav e constructed a functor from the ca tegory of Z -ca tegories to the category of a dditive ca tegories with functorial direct sum ⊕ : Z - Cat → Add - Ca t ⊕ , A 7→ A ⊕ . Let forget : Add - Cat ⊕ → Z - Cat be the forg etful functor. Lemma 10.2. (i) We obtain an adjoi nt p air of functors ( ⊕ , forget) . (ii) We get for every Z -c ate gory A a functor of Z - c ate gories Q A : A → for get( A ⊕ ) which is natur al in A . If A is alr e ady an additive c ate gory, Q A is an e quivalenc e of additive c ate gories. ON CR OSSED PRODUCT RINGS WITH TWISTED INVOLUTIONS, . . . 37 Pr o of. (i) W e hav e to constr uc t for every Z - c a tegory A and ev ery additive categor y B with functor ial dir ect sum to o ne a no ther inv erse ma ps α : func Add - Cat ⊕ ( A ⊕ , B ) → func Z - Cat ( A , forget( B )) and β : func Z - Cat ( A , forget( B )) → func Add - Cat ⊕ ( A ⊕ , B ) . Given F : A ⊕ → B , deﬁne α ( F ) : A → B to b e the comp osite of F with the obvious inclusion Q A : A → A ⊕ which sends A to ( A ). Given F : A → forg et( B ), deﬁne β ( F ) : A ⊕ → B b y sending ( A 1 , A 2 , . . . , A n ) to F ( A 1 ) ⊕ F ( A 2 ) ⊕ · · · ⊕ F ( A n ). (ii) W e have deﬁned Q A already above. It is the adjoint of the iden tit y on A ⊕ . Obviously Q A induces a bijection mor A ( A, B ) → mor A ⊕ ( Q A ( A ) , Q A ( B )) for every ob jects A, B ∈ A . Supp o se that A is a n additiv e category . Then every o b ject ( A 1 , A 2 , . . . , A n ) in A ⊕ is isomorphic to an o b ject in the image o f P A , namely to P A ( A 1 ⊕ A 2 ⊕ · · · A n ) = ( A 1 ⊕ A 2 ⊕ · · · ⊕ A n ). Hence Q A is an equiv alence of additive ca tegories .  Deﬁnition 10.3 (Additiv e category with functorial direct sum and in volution) . An additive c ate gory with fun ctorial sum and involution is an additive category with (strictly ass o ciative) functoria l sum ⊕ a nd inv olution ( I , E ) which are str ictly compatible with one ano ther, i.e., if A 1 and A 2 are tw o ob jects in A , then I ( A 1 ⊕ A 2 ) = I ( A 1 ) ⊕ I ( A 2 ) and E ( A 1 ⊕ A 2 ) = E ( A 1 ) ⊕ E ( A 2 ) hold. One easily chec ks that if the Z -category A comes with an inv olution ( I A , E A ), the a dditive categor y A ⊕ constructed ab ov e inherits the str ucture o f a n a dditive category with functorial direct s um and in volution in the sense o f Deﬁnition 10.3. Namely , deﬁne I A ⊕ (( A 1 , A 2 , . . . , A n )) = ( I A ( A 1 ) , I A ( A 1 ) , . . . , I A ( A 1 )) ; E A ⊕ (( A 1 , A 2 , . . . , A n )) = E A ( A 1 ) ⊕ E A ( A 2 ) ⊕ · · · ⊕ E A ( A 1 ) . W e obtain a functor from the ca teg ory of Z -categories with in volution to the category of a dditive ca tegories with functorial direct sum and inv olution ⊕ : Z - Cat inv → Add - Cat inv ⊕ , A 7→ A ⊕ . Let forget : Add - Cat inv ⊕ → Z - Cat inv be the forgetful functor. One eas ily extends the pro of of Lemma 10 .2 to the case with in volution. Lemma 10.4. (i) We obtain an adjoi nt p air of functors ( ⊕ , forget) . (ii) We get for every Z -c ate gory with involution A a functor of Z -c ate gories with involution Q A : A → for get( A ⊕ ) which is natur al in A . If A is alr e ady an additive c ate gory with involution, t hen Q A is an e quivalenc e of additive c ate gorie s with involution. Deﬁnition 10.5. A Z - G -c ate gory with involution A is the same as an additiv e G -categor y in the se nse of Deﬁnition 4.22 exc e pt tha t one for gets ab out the dir ect sum. Deﬁnition 10.6 (Additive G -category with functoria l sum and (strict) inv olution) . An add itive G -c ate gory with functorial su m and involution is an additiv e G -ca tegory with (strictly ass o ciative) functoria l sum ⊕ a nd inv olution ( I , E ) which are str ictly compatible w ith o ne another, i.e ., w e hav e: 38 AR THUR BAR TELS AND WOLF GANG L ¨ UCK (i) If A 1 and A 2 are tw o o b jects in A , then I ( A 1 ⊕ A 2 ) = I ( A 1 ) ⊕ I ( A 2 ) and E ( A 1 ⊕ A 2 ) = E ( A 1 ) ⊕ E ( A 2 ) hold; (ii) If A 1 and A 2 are tw o ob jects in A and g ∈ G , then R g ( A 1 ) ⊕ R g ( A 2 ) = R g ( A 1 ⊕ A 2 ) holds; (iii) If A is an ob ject in A , then I ( R g ( A )) = R g ( I ( A )) and E ( R g ( A )) = R g ( E ( A )) ho ld. If the inv olution is str ict in the se ns e o f Section 1, i.e., E = id a nd I ◦ I = id, we call A a n additive G -c ate gory with functorial sum and strict involution . Deﬁne a Z - G -categor y with (strict) inv olution analogo usly , just forge t the direc t sum. W e obtain a functor fr o m the categor y of Z - G -categ ories with in volution to the category of a dditive ca tegories with functorial direct sum and inv olution ⊕ : Z - G - Cat inv → Add - G - Cat inv ⊕ , A 7→ A ⊕ . Let forget : Add - G - Cat inv ⊕ → Z - Cat inv be the forgetful functor. One eas ily extends the pro of of Lemma 10 .2 to the case with G -action and inv olution. Lemma 10.7. (i) We obtain an adjoi nt p air of functors ( ⊕ , forget) . (ii) We get for every Z - G -c ate gory with involution A a fu n ctor of Z -c ate gories with involution Q A : A → for get( A ⊕ ) which is natur al in A . If A is alr e ady an additive G -c ate gory with involution, then Q A is an e quivalenc e of additive G -c ate gories with involution; (iii) The c orr esp onding deﬁnitions and re sults c arry over t o the c ase of strict involutions. Remark 10. 8. Given an additiv e G -ca tegory A and a G -s et T , we hav e constructed the additive G -categ o ry  R G ( T ) A ◦ pr G  ⊕ . Let A ∗ G T b e the a dditive G -ca tegory deﬁned in [4, Deﬁnition 2 .1]. W e o btain a functor o f Z -categor ie s ρ ( T ) : Z G ( T ) A ◦ pr G → A ∗ G T by sending an ob ject ( x, A ) to the ob ject { B t | t ∈ T } for which B x = A if x = t and B x = 0 if x 6 = t . It induces a functor of additive categ ories with functorial direct s um ρ ( T ) ⊕ : Z G ( T ) A ◦ pr G ! ⊕ → ( A ∗ G T ) ⊕ . Recall tha t we have the functor of Z -ca tegories Q A∗ G T : A ∗ G T → ( A ∗ G T ) ⊕ . One ea sily checks that b oth ρ ( T ) ⊕ and Q A∗ G T are equiv alences of additive cate- gories and na tur al in T . If A is an additive G -ca tegory with str ict in volution, then we obtain o n the source and the ta rget of ρ ( T ) ⊕ and o f Q A∗ G T strict involutions such that b oth ρ ( T ) ⊕ and Q A∗ G T are equiv ale nces o f additive ca tegories with stric t involution. This implies that the G -ho mology theories co nstructed for K - and L -theory here and in [4, Deﬁnition 2 .1] are natur ally isomo rphic and lea d to isomo r phic assembly maps. ON CR OSSED PRODUCT RINGS WITH TWISTED INVOLUTIONS, . . . 39 11. G -homology theories and restriction Fix a functor E : Add - Cat inv → Spectra (11.1) which sends weak equiv alences of additive ca tegories with in volutions to weak ho- motopy eq uiv alences of sp ectra. W e call it c omp atible with dir e ct sums if for any family of a dditiv e catego ries with in v olutions {A i | i ∈ I } the map induced by the canonical inclus ions A i → L i ∈ I A i for i ∈ I _ i ∈ I E ( A i ) → E M i ∈ I A i ! is a w eak homotopy equiv alence of sp ectra. Example 11.2. The most imp ortant examples for E will b e for us the functor which sends an a dditive category A to its non-connective alg ebraic K - theory sp ec- trum K A in the sense of Pedersen-W eib el [11], a nd the functor whic h s ends an additive categ ory with in v olution A to its a lg ebraic L −∞ -sp ectrum L −∞ A in the sense o f Ranicki (s e e [12 ], [13] and [1 4]). Bo th functors send weak equiv a le nces to weak homoto py equiv a lences and are compatible with direct s ums. The latter fol- lows from the fact that they are compatible with ﬁnite direct sums and compatible with directed c o limits. This is proven for r ings in [1, Lemma 5.2], the pro of car ries ov er to a dditive ca teg ories with involution. Given a G - C W -co mplex X and a gro up homomorphism φ : K → G , le t φ ∗ X be the K - C W -complex obtained from X b y restr iction with φ . Giv en a K -homology theory H K ∗ , we obtain a G -homology theory b y s e nding a G - C W -complex X to H K ∗ ( φ ∗ X ). Recall that we hav e assig ned to an additive G -catego r y A with inv o- lution a G -homology theory H G ∗ ( − ; E A ) in (9 .3 ). The ma in result of this section is Theorem 1 1.3. Supp ose that the functor E of (11.1) is c omp atible with dir e ct sums. L et φ : K → G b e a gr oup homomorphism. L et A b e a Z - K -c ate gory with involution in t he sense of Deﬁnition 10.5. L et ind φ A b e the G - Z -c ate gory with involution deﬁne d in (11.5) . Then ther e is a natur al e quivalenc e of G -homolo gy the ories τ ∗ : H K  φ ∗ ( − ); E A ⊕  ∼ = − → H G  − ; E (ind φ A ) ⊕  . Its pro of needs some prepa r ation. Given a contra v ariant functor F : G → Add - Cat inv from a group oid in to the category Add - Cat inv of additive categor ie s with inv olution, we have deﬁned an additive ca tegory with in volution R G F in (7.2), provided that G is connected. W e wan t to drop the as s umption that G is connected. The co nnectedness of G was o nly used in the cons tr uction of the direct sum of t w o ob jects in R G F . Hence e verything go es thr ough if we re ﬁne us to the construction of Z -categ ories with inv olution. Namely , if we dr op the connectivity assumption on G , a ll cons tructions and a ll the functoriality prop er ties expla ined in Sectio n 7 rema in tr ue if we work within the category Z - Cat inv instead of Add - Cat inv . Let G and K b e g roups. Consider a (left) K -s e t ξ and a K - G -biset η . Then G a cts from the right on the tra ns po rt gr oup oid G K ( η ). Namely , for an elemen t g ∈ G the map R g : η → η , x 7→ xg is K -equiv ariant and induces a functor G K ( R g ) : G K ( η ) → G K ( η ). Consider a K - Z -category with inv olution A . Le t pr K : G K ( η ) → G K ( K/K ) = K be the functor induced by the pro jection η → K /K . Then A ◦ pr K is a contra v ar iant 40 AR THUR BAR TELS AND WOLF GANG L ¨ UCK functor G K ( η ) → Z - Cat inv . W e o btain a Z -categor y with inv olution R G K ( η ) A ◦ pr K (compare (7.2)). Given g ∈ G , the functor G G ( R g ) : G K ( η ) → G K ( η ) induces a functor o f Z - c ategories with inv olution (co mpare (7.7)) G G ( R g ) : Z G K ( η ) A ◦ pr K = Z G K ( η ) A ◦ pr K ◦G K ( R g ) → Z G K ( η ) A ◦ pr K , which strictly comm utes with the inv olution. Thus R G K ( η ) A ◦ pr K bec omes a Z - G -categor y with in volution in the sense of Deﬁnition 1 0.5. W e conclude that  R G K ( η ) A ◦ pr K  ◦ pr G is a contra v aria nt functor G G ( ξ ) → Z - Cat inv . W e obtain a Z -categor y with inv olution (co mpare (7.2)) Z G G ( ξ ) Z G K ( η ) A ◦ pr K ! ◦ pr G . Consider η × ξ as a left G × K set by ( g , k ) · ( y , x ) = ( k y g − 1 , g x ). Then A ◦ pr G × K is a contrav ar iant functor G G × K ( η × ξ ) → Z - Cat inv . W e o btain a Z -categ ory with inv o lution (compa re (7.2)) Z G G × K ( η × ξ ) A ◦ pr G × K . Lemma 11.4. Ther e is an isomorp hism of Z - c ate gories with involution ω : Z G G ( ξ ) Z G K ( η ) A ◦ pr K ! ◦ pr G ∼ = − → Z G G × K ( η × ξ ) A ◦ pr G × K which is natur al in b oth ξ and η . Pr o of. An ob ject in R G G ( ξ )  R G K ( η ) A ◦ pr K  ◦ pr G is given by ( x, ( y , A )), where x ∈ ξ is an ob ject in G G ( ξ ) and ( y , A ) is an o b ject in R G K ( η ) A ◦ pr K which is given by an ob ject y ∈ η in G K ( η ) and an ob ject A in A . The ob ject ( x, ( y , A )) is sent under ω to the ob ject (( y , x ) , A ) given by the ob ject ( y , x ) in G G × K ( η × ξ ) and the ob ject A ∈ A . A morphism φ in R G G ( ξ )  R G K ( η ) A ◦ pr K  ◦ pr G from ( x 1 , ( y 1 , A 1 )) to ( x 1 , ( y 2 , A 2 )) is g iven by g · ψ for a mo rphism g : x 1 → x 2 in G G ( ξ ) and a mo rphism ψ : ( y 1 , A 1 ) → G K ( R g ) ∗ ( y 2 , A 2 ). The morphism ψ itself is given by k · ν for a mor phism k : y 1 → y 2 g in G K ( η ) and a morphism ν : A → r k ( A ) in A . Deﬁne the image of φ under ω to be the morphism in R G G × K ( η × ξ ) A ◦ pr given by the mor phism ( g − 1 , k ) : ( y 1 , x 1 ) → ( y 2 , x 2 ) in G G × K ( η × ξ ) and the morphism φ : A → r k ( A ). This makes sense since r k ( A ) is the ima ge of A under the functor A ◦ pr( g − 1 , k ). One easily chec ks that ω is an isomo rphism o f Z -categor ie s with inv olution and natural with resp ect to ξ and η .  Let φ : K → G b e a gro up homomorphis m and ξ b e a G -set. Let φ ∗ ξ b e the K - set obtained from the G -se t ξ by re s triction with φ . Consider a K - Z -category with inv o lution A in the sense of Deﬁnition 10.5. Let φ ∗ G be the K - G -biset for which m ultiplication with ( k , g ) ∈ K × G sends x ∈ G to φ ( k ) xg − 1 . W e have explained ab ov e how R G K ( φ ∗ G ) A can be consider ed as a G - Z -category with inv olution. W e will denote it by ind φ A := Z G K ( φ ∗ G ) A . (11.5) ON CR OSSED PRODUCT RINGS WITH TWISTED INVOLUTIONS, . . . 41 Lemma 11.6. F or every G -set ξ ther e is a n atur al e quivalenc e of Z -c at e gories with involutions τ : Z G G ( ξ ) ind φ A ≃ − → Z G K ( φ ∗ ξ ) A ◦ pr K . It is natur al in ξ . Pr o of. Because of Lemma 11.4 it suﬃces to constr uct a natural equiv alence τ : Z G G × K ( φ ∗ G × ξ ) A ◦ pr ≃ − → Z G K ( φ ∗ ξ ) A ◦ pr K . Consider the functor W : G G × K ( φ ∗ G × ξ ) → G K ( φ ∗ ξ ) sending an ob ject ( x, y ) ∈ G × ξ in G G × K ( φ ∗ G × ξ ) to the ob ject xy ∈ ξ in G K ( φ ∗ ξ ) and a mor phism ( g , k ) : ( x 1 , y 1 ) → ( x 2 , y 2 ) to the morphism k : xy 1 → xy 2 . Now deﬁne τ to b e W ∗ : Z G G × K ( φ ∗ G × ξ ) A ◦ pr = Z G G × K ( φ ∗ G × ξ ) A ◦ pr K ◦ W ≃ − → Z G K ( φ ∗ ξ ) A (see (7.7)). Since W is a weak equiv ale nc e of g roup oids, τ is a weak equiv alence of additive catego ries with inv olution b y L e mma 5.9 (i). O ne easily checks that this construction is natural in ξ .  Now we can give the pro o f of Theorem 11.3. Pr o of. In the sequel we write E ⊕ : Z - Cat inv → Spectra for the c omp osite of the functor E o f (11.1) and the functor Z - Cat inv → Add - Cat inv sending A to A ⊕ . Given a G - C W -complex X , we have to deﬁne a weak equiv a lence o f s pe c tr a map K ( K/ ? , φ ∗ X ) + ∧ Or ( K ) E ⊕ Z G K ( K/ ?) A ◦ pr K ! → map G ( G/ ? , X ) + ∧ Or ( G ) E ⊕ Z G G ( G/ ?) ind φ A ◦ pr G ! . The left hand side can b e rewr itten as map K ( K/ ? , φ ∗ X ) + ∧ Or ( K ) E ⊕ Z G K ( K/ ?) A ◦ pr K ! = map G ( φ ∗ ( K/ ?) , X ) + ∧ Or ( K ) E ⊕ Z G K ( K/ ?) A ◦ pr K ! = map G ( G/ ? , X ) + ∧ Or ( G ) map K ( φ ∗ ( K/ ??) , G/ ?) + ∧ Or ( K ) E ⊕ Z G K ( K/ ??) A ◦ pr K ! = map G ( G/ ? , X ) + ∧ Or ( G ) map K ( K/ ?? , φ ∗ G/ ?) + ∧ Or ( K ) E ⊕ Z G K ( K/ ??) A ◦ pr K ! . Because o f Le mma 11 .6 the r ight hand side ca n b e identiﬁed with map G ( G/ ? , X ) + ∧ Or ( G ) E ⊕ Z G G ( G/ ?) ind φ A ◦ pr G ! = map G ( G/ ? , X ) + ∧ Or ( G ) E ⊕ Z G K ( φ ∗ G/ ?) A ◦ pr K ! . 42 AR THUR BAR TELS AND WOLF GANG L ¨ UCK Hence we need to c onstruct for ev ery K - s et ξ a weak homotopy equiv alence, natural in ξ ρ ( ξ ) : map K ( K/ ?? , ξ ) + ∧ Or ( K ) E ⊕ Z G K ( K/ ??) A ◦ pr K ! → E ⊕ Z G K ( ξ ) A ◦ pr K ! . The ma p ρ ( ξ ) sends a n ele men t in the source given b y ( φ, z ) for a K -ma p φ : K / ?? → ξ and z ∈ E ⊕  R G K ( K/ ??) A ◦ pr K  to E ⊕  G K ( φ ) ∗  ( z ), where G K ( φ ) ∗ : Z G K ( K/ ??) A ◦ pr K = Z G K ( K/ ??) A ◦ pr K ◦G K ( φ ) → Z G K ( ξ ) A ◦ pr K has been deﬁned in (7.7). Obviously it is natura l in ξ a nd is an is o morphism if ξ is a K -o r bit. F or a family of K - s ets { ξ i | i ∈ I } there is a natural iso mo rphism of sp ectra _ i ∈ I map K ( K/ ?? , ξ i ) + ∧ Or ( K ) E ⊕ Z G K ( K/ ??) A ◦ pr K !! ∼ = − → map K K/ ?? , a i ∈ I ξ i ! + ∧ Or ( K ) E ⊕ Z G K ( K/ ??) A ◦ pr K ! . W e hav e a i ∈ I G K ( ξ i ) ∼ = G K a i ∈ I ξ i ! ; a i ∈ I Z G K ( ξ i ) A ◦ pr k ∼ = Z ‘ i ∈ I G K ( ξ i ) A ◦ pr k ; M i ∈ I Z G K ( ξ i ) A ◦ pr k ! ⊕ ∼ = a i ∈ I Z G K ( ξ i ) A ◦ pr k ! ⊕ . By a ssumption E is co mpatible with direct sums. Hence we obtain a weak equiv a - lence _ i ∈ I E ⊕ Z G K ( ξ i ) A ◦ pr K ! ≃ − → E ⊕ Z G K ( ‘ i ∈ I ξ i ) A ◦ pr K ! . W e conclude that ρ  ` i ∈ I ξ i  is a w eak homoto py eq uiv alence if and o nly if W i ∈ I ρ ( ξ i ) is a weak homotopy equiv a le nc e . Since a K -set is the disjoint union of its K -orbits and a wedge of w eak homotopy equiv alences of spec tr a is ag a in a weak homotopy equiv alence, ρ ( ξ ) is a weak homotopy equiv a lence for every K -set ξ . T his ﬁnishes the pro of of Theo r em 1 1 .3.  12. Proof of the main theorems In this section we can ﬁnally give the pro ofs of Theorem 0.4, Theor em 0.7 and Theorem 0.12. Pr o of of The or em 0.4. This follows fro m Lemma 7.11 and Lemma 8.2.  Pr o of of The or em 0.7. Let φ : K → G b e a group ho momorphism and let B b e a additive K -ca tegory with inv o lution. W e have to show that the following a ssembly map is bijective asmb K, B n : H K ∗ ( E φ ∗ V C y c ( K ); L B ) → H K n (pt; L B ) = L n  R K B  . ON CR OSSED PRODUCT RINGS WITH TWISTED INVOLUTIONS, . . . 43 Since φ ∗ E V C yc ( G ) is a mo del for E φ ∗ V C yc ( K ), this follows from the co mm utative diagram H K n ( E φ ∗ V C yc ( K ); L B ) H K n (id; L Q B ) ∼ =   H K n (pr; L B ) / / H K n (pt; L B ) = L n R K B  H K n (id; L Q B ) ∼ =   H K n ( E φ ∗ V C y c ( K ); L B ⊕ ) τ φ n ( E V C yc ( G )) ∼ =   H K n (pr; L B ⊕ ) / / H K n (pt; L B ⊕ ) = L n R K B ⊕  τ φ n (pt) ∼ =   H G n  E V C yc ( G ); L (ind φ B ) ⊕  H G n  pr; L (ind φ B ) ⊕  / / H G n  pt; L (ind φ B ) ⊕  = L n  R G (ind φ B ) ⊕  where pr denotes the pro jection on to the one-p oint-space pt and Q B : B → B ⊕ is the natural equiv alence coming from Lemma 10.7 and the vertical ar rows are isomorphisms be c ause of Lemma 9 .4 a nd Theo r em 1 1 .3.  Pr o of of The or em 0.1 2. Given an a dditive categ ory A with in volution A , w e can consider it as an a dditive categ ory with ( Z / 2 , v )-o p er ation a s explained in Exam- ple 2 .4. If we apply Lemma 3.2, we obtain an additive ca teg ory with strict involution S Z / 2 ( A ) together with a weak equiv a lence o f additive catego ries with inv olutions P A : A → S Z / 2 ( A ) If A is an additive G -c a tegory with inv olution in the s ense o f Deﬁnition 4.2 2, then P A is an equiv alence of additive G -ca teg ories with involution. If we apply Lemma 10.7, w e obtain an additive G -categor y with functoria l di- rect sum and str ic t inv olution S Z / 2 ( A ) ⊕ in the sense o f Deﬁnition 10 .6 a nd an equiv alence of additive G -ca teg ories with strict inv olution Q S Z / 2 ( A ) : S Z / 2 ( A ) → S Z / 2 ( A ) ⊕ The comp osite f := Q S Z / 2 ( A ) ◦ P A : A → S Z / 2 ( A ) ⊕ is a weak equiv alence of additive G -category with inv olution. Now the cla im follows from the following commutativ e diagra m H G n ( E V C yc ( G ); L A ) H K n (id; L f ) ∼ =   H K n (pr; L A ) / / H G n (pt; L A ) = L n R G A  H K n (id; L f ) ∼ =   H G n  E V C yc ( G ); L S Z / 2 ( A ) ⊕  H K n  pr; L S Z / 2 ( A ) ⊕  / / H G n  pt; L A  = L n R G S Z / 2 ( A ) ⊕  whose vertical a rrows ar e iso morphisms by Lemma 9 .4.  References [1] A. Bartels, S. Ech terhoﬀ, and W. L ¨ uck. Inheritance of i somorphism conjectures under colim- its. Preprin treihe SFB 478 — Geometrische St rukturen in der Mathematik, Heft 452, M ¨ unster, arXiv:math.KT/0702460, 2007. [2] A. Bartels, T. F arrell, L. Jones, and H. Reich. O n the isomorphism conjecture in al gebraic K -theory . T op olo gy , 43(1):157–213, 2004. [3] A. Bartels, W. L ¨ uck , and H. Reich. The K -theoretic Farrell-Jones Conjecture for hyperbol i c groups. Preprintreihe SFB 478 — Geometrisc he Strukturen in der Mathematik, Heft 434, M ¨ unster, arX iv:math.GT/0609685, 2007. 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Math. , 543:193–234 , 2002. [10] W. L ¨ uc k and H. Reich. The Baum-Connes and the Farrell - Jones conjectures i n K - and L - theory . In Handb o ok of K -the ory. V ol. 1, 2 , pages 703–842. Spri nger, Berl in, 2005. [11] E. K. Pede rsen and C. A. W eib el. A non-connectiv e delooping of algebraic K -theory . In Algebr aic and Ge ometric T op olo gy; pr o c. co nf. Rutgers Uni. , New Brunswick 1983 , volume 1126 of L ectur e notes in mathematics , pages 166–181. Springer, 1985. [12] A. A. Ranicki. Additive L -theory . K -The ory , 3(2):163–195, 1989. [13] A. A. Ranicki. Algebr aic L -theo ry and top olo gica l manifolds . Cambridge Univ ersity Pr ess, Camb ridge, 1992. [14] A. A. Ranicki. Lo wer K - and L -the ory . Cambridge Universit y Press, Cambridge, 1992. [15] R. W. Thomason. Homotop y colimits in the category of small categories. Math. Pr o c. Cam- bridge Philos. So c. , 85(1):91–10 9, 1979. Imperial College London, Huxley Building, London SW7 2AZ, UK E-mail addr ess : a.bartels@imp erial.ac. uk URL : http:/ /ma.ic.ac .uk/~abartels/ Westf ¨ alische Wilhelms-Universit ¨ at M ¨ unster, Mathema tisches Institut, Einsteinstr. 62 , D-48149 M ¨ unster, Germ any E-mail addr ess : lueck@math.un i-muenste r.de URL : http:/ /www.math .uni-muenster.de/u/lueck

On crossed product rings with twisted involutions, their module categories and L-theory

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