Clifford modules and invariants of quadratic forms

CLIFF ORD MODULES AND INV ARIANT S OF QUADRA TIC F ORMS M. KAROUBI 0. Intr oduction F or an y in teger k > 0 , the Bott class ρ k in top ological complex K -theory is w ell kno wn [7], [12, pg. 259]. If V is a c omplex v ec tor bundle on a compact sp ace X , ρ k ( V ) is defined as the image o f 1 by the comp osition K ( X ) ϕ − → K ( V ) ψ k − → K ( V ) ϕ − 1 − → K ( X ) , where ϕ is Thom’s isomorphism in complex K -theory and ψ k is the Adams op eratio n. This c haracteristic class is nat ura l and satisfies the follo wing prop erties whic h insure its uniq ueness (by the splitting prin- ciple): 1) ρ k ( V ⊕ W ) = ρ k ( V ) .ρ k ( W ) 2) ρ k ( L ) = 1 ⊕ L ⊕ ... ⊕ L k − 1 if L is a line bundle. The Bott class ma y be e xtended to the full K -theory group if w e in v ert the n um b er k in the gro up K ( X ). It induces a morphism fr o m K ( X ) t o the m ultiplicativ e group K ( X ) [1 /k ] × . The Bott class is some- times called ”cannibalistic”, since b o t h its origin and des tinatio n are K -groups. As p ointed out b y Serre [17], the definition of the Bott class and its ”square ro ot”, in tro duced in Lemma 3.5, may b e generalized to λ -rings, for instance in the the ory of group represen tations or in equiv ariant top ological K -theory . The purp o se of this pap er is to give a hermitian analog o f the Bott class. W e shall define it on hermitian K - theory , with target algebraic K -theory . F or instance, let X = Sp ec( R ) , where R is a commu tat ive ring with k ! inv ertible a nd let V b e an algebraic v ector bundle on X pro vided with a nondegenerate quadratic form 1 . W e shall asso ciate to V a ”hermitian Bott class”, designated b y ρ k ( V ) , whic h tak es its v alues in the same t yp e of multiplicativ e group K ( X ) [1 /k ] × , where K ( X ) is algebraic K -theory . W e write ρ k instead of ρ k in order to distinguish t he new class from the old one, although they are closely related (cf. Theorem 3.4).W e also note that the ”cannibalistic” ch ara cter of the new class ρ k is a voide d Date : 8 May 2010. 1 and a ls o a spinor ial structure: see b elow. 1 2 CLIFF OR D MOD ULES AND IN V ARIANTS OF QUADRA TIC FORMS since the source and the target are differen t g roups. W e refer to [11 ] for some basic notions in hermitian K -theory , except that w e follow more standard nota tions, writing this theory K Q ( X ) , instead of L ( X ) as in [11]. In order to define the new class ρ k ( V ), we need a sligh t enric hmen t of hermitian K -theory , using ”spinorial mo dules” and not only qua- dratic ones. More precisely , a spinorial mo dule is giv en b y a couple ( V , E ) , where V is a quadratic mo dule and E is a finitely gene rated pro jectiv e mo dule, suc h that the Clifford algebra C ( V ) is isomorphic to End( E ) . The asso ciated G r o thendiec k gro up K Spin( X ) is related to the hermitian K -group K Q ( X ) by an exact sequence 0 − → Pic( X ) θ − → K Spin ( X ) ϕ − → K Q ( X ) γ − → BW ( X ) , where BW ( X ) denotes the Brauer-W all group of X . As a set, BW( X ) is isomorphic to the sum of three ´ etale coho mo lo gy groups [18] [8 , The- orem 3.6]. There is a t wisted group rule on this direct sum, (compar e with [9]). In particular, for the sp ectrum of fields, the morphism γ is induced b y the ra nk, the discriminan t and the Hasse-Witt in v arian t [18]. F rom t his p oin t of view, the class ρ k w e shall define on K Spin( X ) ma y b e considered as a secondary in v arian t. The hy p erb olic functor K ( X ) − → K Q ( X ) admits a natural factor- ization H : K ( X ) − → K Spin( X ) − → K Q ( X ) . The class ρ k is mo r e precisely a ho momorphism ρ k : K Spin( X ) − → K ( X ) [1 /k ] × , suc h that w e hav e a f actorization with the classical Bott class ρ k : K ( X ) ρ k − → K ( X ) [1 /k ] × H ց ր ρ k K Spin( X ) An imp o rtan t example is when the bundle of Clifford algebras C ( V ) has a trivial class in BW( X ). In that case, C ( V ) is the bundle o f endomorphisms of a Z / 2-gra ded v ector bundle E (see the App endix) and w e can interpret ρ k as defined on a suitable sub quotien t of K Q ( X ) , thanks to the exact sequenc e ab o v e. If k is o dd, using a result of Serre [17], we can ”correct” t he class ρ k in to another class ρ k whic h is defined on the ”spinorial Witt gro up” W Spin( X ) = Cok er [ K ( X ) − → K Spin( X )] and whic h take s its v alues in the 2- torsion of the m ultiplicativ e gro up K ( X ) [1 /k ] × / (Pic( X )) ( k − 1) / 2 . With the same metho d, for n > 0 , we define Bott classes in ” higher spinorial K -theory”: ρ k : K Spin n ( X ) − → K n ( X ) [1 /k ] CLIFF OR D MOD ULES AND IN V ARIANTS OF QUADRA TIC FORMS 3 There is a canonical homomorphism K Spin n ( X ) − → K Q n ( X ) whic h is injective if n ≥ 2 and bijectiv e if n > 2 . F or a ll n ≥ 0, the follo wing diagr a m comm utes K n ( X ) ρ k − → K n ( X ) [1 /k ] H ց ր ρ k K Spin n ( X ) . In Section 4 , w e mak e the link with T op o logy , sho wing that ρ k is es- sen tially Bott’s class defined for spinorial bundles ( whereas ρ k is related to complex v ector bundles as w e ha ve seen b efor e). Sections 5 a nd 6 are dev oted t o characteristic classes for Azuma y a algebras, esp ecially generalizations of Adams op era t io ns. Finally , in Section 7, w e show ho w t o av oid spinorial structures b y defining ρ k on the full hermitian K - g roup K Q ( X ). The target of ρ k is no w an a lgebraic vers ion of ” t wisted K -theory” [14]. W e reco v er the previous hermitian Bott class in a presence of a spinorial structure. T erminology . It will b e implicit in this pap er that tensor pro ducts of Z / 2-g r aded mo dules or algebras are gra ded tensor pro ducts. Akno wledgmen ts . As w e shall see man y times through the pa p er, our metho ds a r e g reatly inspired by the pap ers of Bott [7], A tiy ah [1], A tiy ah, Bott and Shapiro [2], and Bass [6]. W e are indebted to Serre for the Lemma 3.5, concerning the ”square ro ot” of the classical Bott class. If k is o dd, w e use this Lemma in or der to define the characteristic class ρ k men tioned ab ov e for the Witt group. In Section 7 , a more refined square ro ot is used. Finally , w e are indebted to Deligne, Kn us and Tignol for useful remarks ab out op era t io ns o n Azuma y a algebras whic h are defined briefly in Sections 5 and 6. Here is a summary of the pap er by Sections: 1. Clifford algebras and the spinorial group. Orientation of a qua- dratic mo dule 2. Op erat io ns o n Clifford mo dules 3. Bott classes in hermitian K - theory 4. Relation with T op o logy 5. Oriente d Azuma y a algebras 6. Adams op eratio ns revisited 7. Twisted hermitian Bot t classes App endix. A remark ab out the Brauer-W all group. 1. Cliff ord algebras and the sp inorial group. Orient a tion of a quadra tic module In this Section, w e closely follow a pa p er of Bass [6]. The essen tial prerequisites are recalled here for the reader’s con v enience and in order to fix the nota tions. 4 CLIFF OR D MOD ULES AND IN V ARIANTS OF QUADRA TIC FORMS Let R b e a comm utativ e ring and let V b e a finitely generated pro jec- tiv e R -mo dule pro vided with a nondegenerate quadratic form q . W e de- note by C ( V , q ) , or simply C ( V ) , the asso ciated Clifford algebra whic h is naturally Z / 2-graded. The canonical map fr o m V to C ( V ) is an injection and w e shall implicitly identify V with its imag e. The Clifford group Γ( V ) is the subgroup of C ( V ) × , whose elemen ts u a re homogeneous and satisfy the condition uV u − 1 ⊂ V . W e define a homomor phism from Γ( V ) to the orthogonal group φ : Γ( V ) − → O( V ) b y the form ula φ ( u )( v ) = ( − 1) deg ( u ) u.v .u − 1 . The gro up w e are interes ted in is the 0- degree part of Γ( V ) , i.e. Γ 0 ( V ) = Γ( V ) ∩ C 0 ( V ) . W e then hav e an exact seq uence pro v ed in [6, pg . 172]: 1 → R ∗ → Γ 0 ( V ) → SO( V ) . The gro up SO( V ) in this sequence is defined as the ke rnel of the ”de- terminan t map” det : O ( V ) → Z / 2( R ) , where Z / 2( R ) is the set of lo cally constan t functions fr o m Sp ec( R ) to Z / 2. This set ma y b e iden tified with the Bo olean ring of idempotents in the ring R, a ccording to [6, pg. 159]. The addition of idemp o ten ts is defined as follows ( e, e ′ ) 7− → e + e ′ − ee ′ . The determinant map is then a group homomorphism. If Sp ec( R ) is connected and if 2 is in v ertible in R, w e reco v er the usual notion of determinan t whic h tak es its v alues in the m ultiplicativ e group ± 1 . W e define an antiautomorphism of order 2 (called an in v olution through t his pap er): a 7− → a of the Clifford algebra b y ex tension of t he iden tit y on V (w e c hange here the not a tion o f Bass who writes this in v olution a 7− → t a ). If a ∈ Γ( V ), its ”spinorial norm” N ( a ) is giv en by the formula N ( a ) = a a. It is easy to see that N ( a ) ∈ R × ⊂ C ( V ) × . The spinorial group Spin( V ) is then the subgroup of Γ 0 ( V ) whose elemen ts are of spinorial no r m 1 . W e hav e an exact sequence 1 → µ 2 ( R ) → Spin( V ) → SO ( V ) → D isc( R ) . CLIFF OR D MOD ULES AND IN V ARIANTS OF QUADRA TIC FORMS 5 Here µ 2 ( R ) is the group of 2- ro ots of the unity in R. It is reduced to ± 1 if R is an integral domain and if 2 is inv ertible in R . On the other hand, Disc( R ) is an extension 1 → R ∗ /R ∗ 2 → Disc( R ) → Pic 2 ( R ) → 1 , where Pic 2 ( R ) is the 2-torsion of the Picard group [6, pg. 176]. The homomorphism SN : SO( V ) → Disc( R ) , whic h is the g eneralization of the spinorial no rm if R is a field, is quite subtle and is a lso detailed in [6]. The map SN stabilizes and defines a homomorphism (where SO( R ) = col im m SO( H ( R m )) χ : SO( R ) → D isc( R ) . The fo llowing theorem is prov ed in [6, pg. 194 ]. Theorem 1.1. T he determinant map an d the spin orial norm define a homomorphism e χ : O( R ) → Z / 2 ( R ) ⊕ Disc( R ) which is surje ctive. It induc es a split epimorphism K Q 1 ( R ) → Z / 2( R ) ⊕ Disc( R ) . The fo llowing corolla r y is immediate. Corollary 1.2. We have a c entr a l extension 1 → µ 2 ( R ) → Spin( R ) → SO 0 ( R ) → 1 , wher e SO 0 ( R ) is the kernel of the epimorphis m e χ define d ab ove. Let us now assume that 2 is in ve rtible in R and that the quadratic form q is defined b y a symmetric bilinear form f , i.e. q ( x ) = f ( x, x ) . The symmetric bilinear form a sso ciated to q is then ( x, y ) 7→ 2 f ( x, y ) . Let us also assume that V is a n R -mo dule of constan t rank whic h is ev en, sa y n = 2 m. In this case, the n th exterior p o w er λ n ( V ) is an R -mo dule of rank 1 whic h ma y be provided with t he quadratic fo rm asso ciated to q . W e say that V is orien table (in the quadratic sense) if λ n ( V ) is isomorphic to R with the standard quadratic form θ : x 7− → x 2 (up to a scaling factor whic h is a square). W e say that V is orien ted if w e fix an isometry b etw een λ n ( V ) and ( R, θ ) . If V is free with a g iv en basis, this is equiv alen t to say ing that the symmetric matrix asso ciated to f is of determinan t 1 . Remark 1.3. One may use the orien tation on V to define on C 0 ( V ) a symmetric bilinear form Φ 0 : C 0 ( V ) × C 0 ( V ) → C 0 ( V ) σ → λ n ( V ) ∼ = R. 6 CLIFF OR D MOD ULES AND IN V ARIANTS OF QUADRA TIC FORMS The last map σ is defined b y the canonical filtra tion of the Cliffor d al- gebra, the asso ciated g raded a lgebra b eing the exterior alg ebra. In the same w a y , we define an an tisymmetric f o rm by taking the comp osition Φ 1 : C 1 ( V ) × C 1 ( V ) → C 0 ( V ) σ → λ n ( V ) ∼ = R. The f ollo wing theorem is not really needed for our purp oses but is w orth recording. Theorem 1.4. The pr evious biline ar forms Φ 0 and Φ 1 ar e non de - gener ate, i.e. induc e isomorphisms b e twe en C ( V ) and its dual as an R -mo dule. Pr o of. W e can c hec k this Theorem b y lo calizing at an y maximal ideal ( m ) (see for instance [3, pg. 49 ]) . In this case, there exists an orthog- onal basis ( e 1 , ..., e n ) of V ( m ) . Since V is oriented, w e may c ho ose this basis such that the product q ( e 1 ) ...q ( e n ) is equal to 1 . It is also w ell kno wn that the v arious pro ducts e I = e i 1 ... e i r form a basis of the free R ( m ) -mo dule C ( V ( m ) ). Here the m ultiindex I = ( i 1 , ..., i r ) is chose n suc h tha t i 1 < i 2 < ... < i r . By a direct computation w e hav e Φ( e I , e J ) = ± 1 if I ∪ J = { 1 , ..., n } and 0 otherwise, for Φ = Φ 0 or Φ 1 . The refore, these bilinear forms are non degenerate. Moreo v er, they are hy p erb olic at eac h lo calization.  Remark 1.5. Since V is oriented, the group SO( V ) acts naturally on C ( V ) and w e get tw o na tural represen tations of this group in the orthogonal and symplectic gr oups asso ciated to the previous bilinear forms Φ 0 and Φ 1 . Let us no w consider the submo dule N of C ( V ) whose elemen ts u satisfy the identit y u .v = − v .u for an y elemen t v in V ⊂ C ( V ) . The canonical surjection V − → λ n ( V ) induces a ho momorphism τ : N − → λ n ( V ) . Prop osition 1.6. T he homomorphis m τ is an isom o rp hism b etwe en N and λ n ( V ) . Mor e over, N is include d in C 0 ( V ) . Pr o of. W e a g ain lo calize with resp ect to a ll maximal ideals ( m ) of R and consider a n orthogona l basis { e i } of V ( m ) as ab ov e. Then we see that the pro duct e 1 ...e n generates N and we get the required isomor- phism b et we en N ( m ) and λ n ( V ) ( m ) .  Remark 1.7. If we assume tha t V is o rien ted and o f ev en rank, t he previous prop osition provide s us with a canonical elemen t u in C 0 ( V ) whic h an ticomm utes with all elemen ts v in V , suc h that u 2 = 1. More- o v er, u. u = 1 and therefore u b elongs to the spinorial gro up Spin( V ) . CLIFF OR D MOD ULES AND IN V ARIANTS OF QUADRA TIC FORMS 7 An imp ortant example is the cas e when the Clifford algebra C ( V ) has a trivial class in the Brauer- W all g roup of R, denoted by BW( R ) 2 . In other w ords, C ( V ) is isomorphic to the algebra End ( E ) of a graded v ector space E = E 0 ⊕ E 1 where E 0 and E 1 are not reduced to 0 (see the App endix) . The only p ossible c hoices for u are then one o f the tw o follo wing matrices  1 0 0 − 1  or  − 1 0 0 1  W e alwa ys c ho ose E s uch that u is of the first type and, b y a top o- logical analogy , w e shall sa y that V is ”spinorial” . F or instance, let R b e the ring of real con tinuous functions on a compact space X and let V b e a real v ector bundle pro vided with a p ositiv e definite quadratic form. The triviality of the Clifford bundle C ( V ) in BW( X ) is then equiv alent t o the followin g prop erties: the rank o f V is a m ultiple of 8 and the t wo first Stiefel-Whitney classes w 1 ( V ) and w 2 ( V ) are trivial (see [9]). Remark 1.8. Strictly sp eaking, in the top o lo gical situatio n, the classi- cal spinorialit y prop erty does not imply that the rank of V is a m ultiple of 8 . W e put this extra condition in order to ensure the trivialization of C ( V ) in the Brauer-W all group of R . 2. Opera tions on Cliff ord modules As it is we ll k nown, at least for fields, the standard non trivial in- v arian ts of quadratic forms ( V , q ) a re t he discriminan t and the Hasse- Witt in v arian t. They are enco ded in the class of the Clifford algebra C ( V ) = C ( V , q ) in the Brauer-W all g r o up o f R, which we call BW( R ), as in the previous Sec tion. F or an y comm utativ e ring R , this group BW( R ) has b een computed b y W all and Caenep eel [18][8]. As a set, it is the sum of the first three ´ etale cohomology groups o f X = Sp ec( R ) but with a twis ted group rule (compare with [9]). W e view this class of C ( V ) in BW( R ) as a ”primary” in v arian t. In order to define ”sec- ondary” inv ariants , we ma y pro ceed as usual b y assuming first that this class is trivial. Therefore, w e hav e an isomorphism C ( V ) ∼ = End( E ) , where E is a Z / 2-graded R -mo dule whic h is pro jectiv e and finitely generated. W e alw a ys choose E suc h that the a sso ciated elemen t u defined in the previous section is the matrix  1 0 0 − 1  . Ho w ev er, E is not uniquely defined by these conditions. If End( E ) ∼ = End( E ′ ) , 2 W e shall also us e the notation BW( X ) if X = Spec( R ) , as we wrote b efor e. 8 CLIFF OR D MOD ULES AND IN V ARIANTS OF QUADRA TIC FORMS w e hav e E ′ ∼ = E ⊗ L, where L is a mo dule of rank 1 , concen trated in degree 0 according to our c hoice of u (this is a simple consequenc e o f Morita equiv alence). As in the in tro duction, w e ma y fo r malize the previous considerations b etter thanks to the following definition. A ”spinorial mo dule” is a couple ( V , E ) , where E is a finitely generated pro jectiv e mo dule and V = ( V , q ) is a quadratic orien ted mo dule, suc h that C ( V ) is isomorphic to End( E ) with the c hoice of u ab o v e. W e define t he ”sum” ( V , E ) + ( V ′ , E ′ ) a s ( V ⊕ V ′ , E ⊗ E ′ ) a nd the gr oup K Spin( R ) by the usual Grothendiec k construction. Prop osition 2.1. We ha v e an exact se quen c e 0 − → Pic ( R ) θ − → K Spin( R ) ϕ − → K Q ( R ) γ − → BW( R ) , wher e the homomorphisms γ , ϕ and θ ar e d e fine d b elow. Pr o of. The map γ w as defined previously: it a sso ciates to the qua- dratic mo dule ( V , q ) the class of the Clifford algebra C ( V ) = C ( V , q ) in B W ( R ) . W e note that γ is no t necessarily surjectiv e, ev en on the 2-torsion part: see [9, pg. 11] for counterexample s. The map ϕ sen ds a couple ( V , E ) to the class o f the quadratic mo dule V . Finally , θ a sso- ciates to a mo dule L of rank one the difference 3 ( H ( R ) , Λ ( R ) ⊗ L ) − ( H ( R ) , Λ ( R )) , where H is the hy p erb olic functor and Λ the exterior algebra functor, view ed a s a mo dule functor. This map θ is a homo- morphism since the image of L ⊗ L ′ ma y b e written as follows ( H ( R ) , Λ ( R ) ⊗ L ⊗ L ′ ) − ( H ( R ) , Λ ( R ) ⊗ L ) +( H ( R ) , Λ ( R ) ⊗ L ) − ( H ( R ) , Λ( R )) . This image is also ( H ( R ) , Λ ( R ) ⊗ L ′ ) − ( H ( R ) , Λ( R )) + ( H ( R ) , Λ ( R ) ⊗ L ) − ( H ( R ) , Λ( R )) whic h is θ ( L ) + θ ( L ′ ) . In order to complete the pro o f , it remains to sho w that the induced map σ : Pic( R ) − → Ker( ϕ ) is a n isomorphism. 1) The map σ is surjectiv e. Any elemen t of K er( ϕ ) may b e written ( V , E ) − ( V , E ′ ) . Therefore, w e ha v e E ∼ = E ′ ⊗ L, where L is of rank 1 . If w e add to this eleme nt ( H ( R ) , Λ( R ) ⊗ L − 1 ) − ( H ( R ) , Λ( R )) , whic h b elongs to Im( ϕ ) , w e find 0 . 2) The map σ is injective . W e define a map backw ards σ ′ : Ker( ϕ ) − → Pic( R ) , 3 Note that we ca n re place R by R n in this formula. CLIFF OR D MOD ULES AND IN V ARIANTS OF QUADRA TIC FORMS 9 b y sending the differe nce ( V , E ) − ( V , E ′ ) to t he uniq ue L suc h that E ∼ = E ′ ⊗ L. It is clear that σ ′ · σ = I d, which pro ve s the injectivit y of σ .  Before go ing any further, w e need a con v enien t definition, due to A tiy ah, Bott and Shapiro [2], of the graded Grothendiec k group Gr K ( A ) of a Z / 2 -graded algebra A. It is defined as the cok ernel of the restriction map K ( A b ⊗ C 0 , 2 ) − → K ( A b ⊗ C 0 , 1 ) , where C 0 ,r is in general the Clifford alg ebra of R r with the standard quadratic form P r i =1 ( x i ) 2 . W e note that if A is concen trated in degree 0 , w e recov er the usual definition of the G rothendiec k group K ( A ) , under the assumption that 2 is in v ertible in A, whic h w e a ssume f rom no w on. This f ollo ws from the f act that a Z / 2-g raded structure on a mo dule M is equiv alen t to an inv olution on M . Another imp orta n t example is A = C ( V , q ) , where V is orien ted and of ev en rank. In order to compute the g raded Grothendiec k gro up o f A , w e use the elemen t u in tro duced in 1.7 to define a natural isomorphism A b ⊗ C 0 ,r − → A ⊗ C 0 ,r . It is induced b y the map ( v , t ) 7→ v ⊗ 1 + u ⊗ t, where v ∈ V ⊂ C ( V , q ) and t ∈ R r ⊂ C 0 ,r . If E is a Z / 2-graded R -mo dule, the same argumen t ma y b e applied to A = End( E ) . The graded Grothendiec k group again coincides with the usual one. Since w e consider only these examples in our pap er, w e simply write K ( A ) instead o f Gr K ( A ) from no w on. The graded alg ebras w e are in terested in are the Clifford algebras Λ k = C ( V , k q ) , where k > 0 is an in v ertible integer in R . The intere st of this family of a lg ebras is the fo llowing. Let M b e a Z / 2-graded mo dule ov er Λ 1 . Then its k th -p ow er M b ⊗ k is a graded mo dule o v er t he crossed pro duct algebra S k ⋉ C ( V ) b ⊗ k ) ∼ = S k ⋉ C ( V k ) , where S k is the symmetric gro up on k letters. One has to remark that the action of the symmetric group S k on M b ⊗ k tak es in to account the gra ding a s in [1, pg. 176]: the transp osition ( i, j ) acts on a decomp osable ho mo g eneous tensor m 1 ⊗ ... ⊗ m i ⊗ ... ⊗ m j ⊗ ... ⊗ m k as the p erm utation o f m i and m j , up to the sign ( − 1) deg( m i ) deg ( m j ) . Let us no w consider the diago nal V − → V k . It is an isometry if w e pro vide V with the q uadratic form k q . The refore, w e hav e a w ell defined map Λ k − → C ( V k ) 10 CLIFF OR D MOD ULES AND IN V ARIANTS OF QUADRA TIC FORMS whic h is equiv ar ia n t with resp ect to the action of the symmetric gr o up S k . It fo llo ws that the corresp ondence M 7− → M b ⊗ k induces ( b y restriction of the scalars) a ”p o w er map” P : K (Λ 1 ) − → K S k (Λ k ) , where K S k denotes equiv arian t K -theory , the group S k acting trivially on Λ k . Let us give more details a b out this definition. First, we not ice that V k splits as the direct sum of ( V , k q ) a nd its orthog onal mo dule W . This implies that C ( V ) b ⊗ k ∼ = C ( V k ) ∼ = C ( W ) b ⊗ Λ k is a Λ k -mo dule whic h is finitely generated and pro jectiv e. Therefore, the ” restriction of scalars” functor from the category of finitely generated pro jectiv e mo dules ov er C ( V k ) to the a na logous category of mo dules ov er Λ k is w ell define d. Secondly , w e ha v e to show that the map P , whic h is a priori defined in terms of mo dules, can b e extended to a map b et w een graded Grothendiec k gr oups. This ma y be sho wn by using a trick due to Atiy ah whic h is detailed in [1, pg. 175] 4 . Finally , we notice that P is a set map, not a group homomorphism. In o rder to define K -theory op eratio ns in this setting, we may pro- ceed in at least t w o w a ys. F irst, follow ing Grot hendiec k, w e consider a Z / 2-graded module M and its k th -exterior pow er in the gra ded sense . The sp ecific map K S k (Λ k ) − → K (Λ k ) whic h defines the k th -exterior p ow e r is the following: we tak e the quotien t of M b ⊗ k b y the relations iden tifying to 0 all the elemen ts of type m − ε ( σ ) m σ . In t his for mula, m is an elemen t of M b ⊗ k , m σ its imag e under the a ction of the elemen t σ in the symmetric group, with signature ε ( σ ). The comp osition K (Λ 1 ) − → K S k (Λ k ) − → K (Λ k ) defines t he a nalog of G rothendiec k’s λ - op erations: λ k : K (Λ 1 ) − → K (Λ k ) , as detailed in [12, pg. 2 52] f or instance. Remark 2.2. If M is a graded mo dule concen trated in degree 0 (resp. 1) λ k ( M ) is the usual exterior p ow er (resp. symmetric p ow er) with an extra Λ k -mo dule structure. The diagonal map from V in to V × V enables us to define a ”cup- pro duct”: it is induced b y the t ensor pr o duct of mo dules with Clifford actions: K (Λ k ) × K (Λ l ) − → K (Λ k + l ) 4 More precisely , At iyah is consider ing co mplexes in his arg umen t but the same idea may b e applied to Z / 2- graded mo dules . CLIFF OR D MOD ULES AND IN V ARIANTS OF QUADRA TIC FORMS 11 The follo wing theorem is a consequence of the classical prop erty of the usual exterior (graded) pow ers, extended to t his sligh tly more general situation. Theorem 2.3. L et M and N b e two Λ 1 -mo dules. Then one has natur al isomorphisms of Λ r -mo dules λ r ( M ⊕ N ) ∼ = X k + l = r λ k ( M ) ⊗ λ l ( N ) . Pr o of. It is more con v enien t to consider the direct sum of all the λ k ( M ), whic h w e view as the Z -graded exterior a lgebra Λ( M ) . Since the natural algebra isomor phism Λ( M ) ⊗ Λ( N ) − → Λ( M ⊕ N ) is compat ible with the Clifford structures, t he theorem is prov ed.  F rom these λ -op erations, it is classical to asso ciate ”Adams op era- tions” Ψ k . F or any elemen t x of K (Λ 1 ) , w e define Ψ k ( x ) ∈ K (Λ k ) by the form ula Ψ k ( x ) = Q k ( λ 1 ( x ) , ..., λ k ( x )) , where Q k is the Newton p olynomial (cf. [12 , pg. 253] for instance). The fo llowing theorem is a formal consequence of the previous one. Theorem 2.4. L et x and y b e two elemen ts of K (Λ 1 ) . Then one has the identity Ψ k ( x + y ) = Ψ k ( x ) + Ψ k ( y ) in the gr oup K (Λ k ) . Pr o of. F ollowin g Adams [12, pg. 257], w e note that the series Ψ − t ( x ) = ∞ P k =1 ( − 1) k t k Ψ k ( x ) is t he lo garithm differential of λ t ( x ) m ultiplied by − t, i.e. − t λ ′ t ( x ) λ t ( x ) . This can b e che ck ed b y a formal ”splitting principle” as in [12] for instance. The additivit y of the Adams op eratio n follows from the fa ct that the logarithm differential of a pro duct is the sum of the loga r ithm differen tials of eac h factor.  Another imp ortan t and less obvious prop erty of the Adams op era - tions is the follo wing. Theorem 2.5. L et us assume that k ! is invertible in R and let x and y b e two elements of K (Λ 1 ) . Then o n e has the fo l lowing identity i n the gr oup K (Λ 2 k ) . Ψ k ( x · y ) = Ψ k ( x ) · Ψ k ( y ) . 12 CLIFF OR D MOD ULES AND IN V ARIANTS OF QUADRA TIC FORMS Pr o of. In order to pro v e this t heorem, w e use the second description of the op erations λ k and Ψ k due to A tiy ah [1, § 2 ], whic h w e transp ose in our situation. In order to define op erations in K - theory , Atiy ah considers the following comp osition (where R ( S k ) denotes t he in tegral represen tation group ring of the symmetric group S k ): K (Λ 1 ) P − → K S k (Λ k ) ∼ = − → K (Λ k ) ⊗ R ( S k ) χ − → K (Λ k ) . In this seque nce, P is the k th -p ow er map intro duced b efore. The second map is defined b y using our hypothesis tha t k ! is in ve rtible in R. More precisely , a n y S k -mo dule is semi-simple and is therefore the direct sum of its isotop y summands: if π runs through all the (integral) irreducible represen tations o f the symmetric group S k , the natura l map ⊕ Hom( π , T ) ⊗ π − → T is an isomorphism (note that π is of degree 0). Therefore, by linearity , the equiv ar ia n t K -theory K S k ( C ( V , k q )) = K S k (Λ k ) ma y b e written as K (Λ k ) ⊗ R ( S k ) . Fina lly , the map χ is defined once a homomorphism χ k : R ( S k ) − → Z is giv en. F or instance, the Grothendiec k op eration λ k ( M ) is obt a ined through the sp ecific homomorphism χ k equal to 0 for a ll the irreducible represen tations of S k , exce pt the sign represen tation ε , where χ k ( ε ) = 1 . Moreo v er, w e can define the pro duct of tw o o p erations asso ciated to χ k and χ l using the ring structure on the direct sum ⊕ Hom( R ( S r ) , Z ) , as detailed in [1, pg. 16 9]. This structure is induced by the pairing Hom( R ( S k ) , Z ) × Hom( R ( S l ) , Z ) − → Hom( R ( S k × S l ) , Z ) − → Hom ( R ( S k + l ) , Z ) . In particular, as prov ed formally b y A tiy ah [1, pg. 179], the Adams op eration Ψ k is induced to the homomo r phism Ψ : R ( S k ) − → Z asso ciating to a class of represen tations ρ the tr a ce of ρ ( c k ) , where c k is the cycle (1 , 2 , ..., k ) . With this interpretation, the m ultiplicativit y of the Adams o p eration is obv ious.  Remark 2.6. W e conjecture that the previous theorem is true without the hypothesis that k ! is in v ertible in R . If w e assume that 2 k is in ve rt- ible in R, and that R contains the k th -ro ot s of the unity , w e prop ose another closely related op eration Ψ k in Section 6. W e conjecture that Ψ k = Ψ k . This is at least true if k ! is in v ertible in R . CLIFF OR D MOD ULES AND IN V ARIANTS OF QUADRA TIC FORMS 13 3. Bott clas s es in hermitian K -theor y Let us assume that k ! is inv ertible in R . W e cons ider a spinorial mo dule ( V , E ) , as in the previous Section. The fo llo wing maps are detailed b elo w: θ k : K ( R ) α − → ∼ = K ( C ( V , q )) P → K S k ( C ( V , k q )) − → ∼ = K ( C ( V , k q )) ⊗ R ( S k ) Ψ ′ − → K ( C ( V , k q )) ( α q ) − 1 − → ∼ = K ( R ) . The morphism α is the Morita isomorphism b et w een K ( R ) and K ( C ( V , q )) ∼ = K (End( E )) a nd P is the k th -p ow er map defined in the previous Sec- tion. The morphism Ψ ′ is induc ed b y Ψ : R ( S k ) − → Z also defined there. Finally , for the definition of α q , w e remark that the isomorphism b et w een C ( V , q ) and End( E ) implies the existence of an R -mo dule map f : V − → End( E 0 ⊕ E 1 ) suc h that f ( v ) =  0 σ ( v ) τ ( v ) 0  with σ ( v ) τ ( v ) = τ ( v ) σ ( v ) = q ( v ) . 1 . W e now define a ” k -twiste d map” f k : V − → End( E 0 ⊕ E 1 ) b y the form ula f k ( v ) =  0 k σ ( v ) τ ( v ) 0  . Since ( f k ( v )) 2 = k q ( v ), f k induces a homomo r phim b et we en C ( V , k q ) and End( E 0 ⊕ E 1 ) whic h is clearly an isomorphism, as w e can see b y lo calizing at all maximal ideals. The map α q is then induce d b y the same type of Morita isomorphism w e used to define α . . Theorem 3.1. L et ( V , E ) b e a sp inorial mo dule and le t M b e an R - mo dule. Then the i m age of M by the pr evious c omp osition θ k is define d by the fol low ing formula θ κ ( M ) = ρ k ( V , E ) . Ψ k ( M ) . Ther efor e, θ k is de termi n e d by θ κ (1) = ρ k ( V , E ) , which we shal l simply write ρ k ( V , q ) or ρ k ( V ) i f the quadr atic form q a n d the mo dule E ar e implicit. We c al l ρ k ( V ) the ”he rmitian B ott class” of V . Mor e over, we have the multiplic ativity form ula ρ k ( V ⊕ W ) = ρ k ( V ) · ρ k ( W ) in the Gr othendie ck gr oup K ( R ) . 14 CLIFF OR D MOD ULES AND IN V ARIANTS OF QUADRA TIC FORMS Pr o of. The first formula follows from the multiplicativit y of the Adams op eration prov ed in Theorem 2.5 . The second one follo ws from the same m ultiplicativit y prop ert y a nd the w ell-known isomorphism C ( V ⊕ W ) ∼ = C ( V ) ⊗ C ( W ) (graded tensor pro duct as alw ays , according to our con v en tions).  Theorem 3.2. L et ( V , − q ) b e the mo dule V pr ovide d with the opp osite quadr atic form. The n we ha ve the identity ρ k ( V , q ) = ρ k ( V , − q ) . Pr o of. According to our h ypothesis, the Clifford algebra C ( V ) is ori- en ted, since it is isomorphic t o End( E ) . Therefore, w e can use the el- emen t u defined in 1.7 to sho w that C ( V , q ) is isomorphic t o C ( V , − q ) (more generally , C ( V , q ) is isomorphic to C ( V , k q ) if k is inv e rtible). More explicitly , we ke ep the same E a s t he mo dule of spinors, so that C ( V , q ) ∼ = C ( V , − q ) ∼ = End( E ) . W e no w write the comm utativ e dia- gram K ( R ) → K ( C ( V , q )) Ψ k − → K ( C ( V , k q ) ) → K ( R ) ↓ I d ↓ ∼ = ↓ ∼ = ↓ I d K ( R ) → K ( C ( V , − q )) Ψ k − → K ( C ( V , − k q ) ) → K ( R ) .  Remark 3.3. The isomorphism b et w een the Clifford algebras C ( V , q ) and C ( V , − q ) is defined b y using the elemen t u of degree 0 and of square 1 in C ( V , q ) which an ticomm utes with all the elemen ts of V . It is easy to see that the k - tensor pro duct u k = u ⊗ ... ⊗ u satisfies the same prop erties for the Clifford algebra C ( V k , q ⊕ ... ⊕ q ) . Therefore, w e hav e an analo g ous comm utativ e diag ram with the p ow er map P instead o f the Adams op eration Ψ k : K ( C ( V , q )) P − → K ( C ( V , k q ) ) ⊗ R ( S k ) ↓ ∼ = ↓ ∼ = K ( C ( V , − q )) P − → K ( C ( V , − k q ) ) ⊗ R ( S k ) Theorem 3.4. L et ( V , q ) b e the hyp erb olic mo dule H ( P ) and E = Λ( P ) b e the asso ciate d mo dule of spinors. Then ρ k ( V , E ) is the classic al B ott class ρ k ( P ) of the R -mo dule P . Pr o of. According to [6, pg. 166], the Clifford algebra C ( V ) is isomor- phic to End(Λ P ) as a Z / 2-gr a ded algebra, whic h give s a meaning to our definition. The clas s ρ k ( V , q ) may b e iden tified with the ”formal quotien t” Ψ k (Λ P ) / Λ P whic h satisfies the algebraic splitting princi- ple. Therefore, in o rder to pro v e the theorem, it is enough to consider the case when P = L is of ra nk one. W e ha v e then Λ L = 1 − L, Ψ k (Λ L ) = 1 − L k and therefore, Ψ k (Λ L ) / Λ L = 1 + L + ... + L k − 1 .  CLIFF OR D MOD ULES AND IN V ARIANTS OF QUADRA TIC FORMS 15 In order to extend the definition o f the hermitian Bott class to ”spino- rial K -theory”, w e remark that an y elemen t x of K Spin( R ) ma y b e written a s x = V − H ( R m ) , where V is a quadratic mo dule ( t he mo dule of spinors E b eing implicit). Moreo v er, V − H ( R m ) = V ′ − H ( R m ′ ) iff we hav e an isomorphism V ⊕ H ( R m ′ ) ⊕ H ( R s ) ∼ = V ′ ⊕ H ( R m ) ⊕ H ( R s ) for some s. Therefore, if w e in v ert k in the G rothendiec k group K ( R ) , the follo wing definition ρ k ( x ) = ρ k ( V − H ( R m )) = ρ k ( V ) /k m do es not dep end of the c hoice o f V and m. The previous definitions a re not completely satisfactory if we are in terested in c haracteristic classes for the ”spinorial Witt gr o up” of R, denoted b y W Spin( R ) and defined as the cok ernel of the h yperb olic map K ( R ) − → K Spin( R ) . One w a y to deal with this problem is to consider the underlying mo d- ule V 0 of ( V , E ) . According to our hypothesis, V 0 is a mo dule of ev en rank, orien ted and isomorphic to its dual. The followin g lemm a is a particular case of a theorem due to Serre [17]. F or completeness’ sake , w e summarize Serre’s fo r mula in this sp ecial case. Lemma 3.5. L et us assume that k is o dd. With the pr evi o us hyp othe- sis, the classic al B ott c la ss ρ k ( V 0 ) is c anonic al ly a squar e in K ( R ) . Pr o of. Let Ω k b e the ring of integers in the k -cyclotomic extension of Q and let z be a primitiv e k th -ro ot of the unit y . In the computations b elo w, we alwa ys embed an ab elian group G in G ⊗ Z Ω k . Let us no w write G V 0 ( t ) = 1 + tλ 1 ( V 0 ) + ... + t n λ n ( V 0 ) . F rom the a lg ebraic splitting principle, it follows that ρ k ( V 0 ) = k − 1 Y r =1 G V 0 ( − z r ) . The iden tit y λ j ( V 0 ) = λ n − j ( V 0 ) implies that G V 0 ( t ) = t n G V 0 (1 /t ) . W e then deduce from [17] that ρ k ( V 0 ) has a square ro ot 5 whic h w e may c ho ose to b e p ρ k ( V 0 ) = ( − 1) n ( k − 1) / 4 ( k − 1) / 2 Y r =1 G V 0 ( − z r ) · z − nr / 2 . 5 W e hav e inserted a normaliza tion sign ( − 1 ) n ( k − 1) / 4 befo re Serre’s formula [17] for a re ason expla ined in the computation b elow. 16 CLIFF OR D MOD ULES AND IN V ARIANTS OF QUADRA TIC FORMS This square ro ot is in v arian t under the action of the G a lois group ( Z /k ) ∗ of the cyclotomic extension which is generated by the trans- formations z 7− → z j , where j ∈ ( Z /k ) ∗ . Therefore, it b elongs to K ( R ) , as a subgroup o f K ( R ) ⊗ Z Ω k .  The previous lemma enables us t o ”correct” t he hermitian Bott class in the follow ing wa y . W e put ρ k ( V ) = ρ k ( V )( p ρ k ( V 0 )) − 1 . If V is a h yp erb olic mo dule H ( W ) = W ⊕ W ∗ , w e hav e ρ k ( V ) = ρ k ( W ) . On the other hand, w e ha v e λ t ( W ⊕ W ∗ ) = J ( t ) · t n/ 2 · J (1 /t ) · σ, where J ( t ) = λ t ( W ) = 1 + tλ 1 ( W ) + ... + t n/ 2 λ n/ 2 ( W ) and σ = λ n/ 2 ( W ∗ ) . Therefore, p ρ k (( W ⊕ W ∗ ) = ( − 1) n ( k − 1) / 4 ( k − 1) / 2 Y r =1 σ · J ( − z r ) · ( − z r ) n/ 2 .J ( − 1 /z r ) · ( z − nr / 2 ) = σ ( k − 1) / 2 · k − 1 Q r =1 J ( − z r ) = σ ( k − 1) / 2 · ρ k ( W ) = σ ( k − 1) / 2 · ρ k ( H ( W )) . F rom this computation, it follo ws t ha t ρ k ( V ) is a [ k − 1) / 2] th -p ow er of a n elemen t of the Picard group of R if V is hyperb olic. Moreo v er, ρ k ( V ) 2 = ( ρ k ( V )) 2 ( ρ k ( V 0 )) − 1 = ρ k ( V , q )) ρ k ( V , − q ))( ρ k ( V 0 )) − 1 = ρ k ( H ( V 0 ))( ρ k ( V 0 )) − 1 = ρ k ( V 0 )( ρ k ( V 0 )) − 1 = 1 . Summarizing this discussion, we ha v e prov ed the following theorem: Theorem 3.6. L et k > 0 b e an o dd numb er . Then the c orr e ct e d her- mitian Bott c l a ss ρ k ( V ) = ρ k ( V )( p ρ k ( V 0 )) − 1 induc es a homomorphism also c al le d ρ k : ρ k : W Spin( R ) − → ( K ( R ) [1 / k ]) × / Pic( R ) ( k − 1) / 2 , wher e the right hand sid e is vie we d as a multiplic ative gr oup. Mor e over, the image of ρ k lies in the 2 -torsion of this gr oup. Remark 3.7. The case k ev en do es not fit with this strategy . Ho w ev er, w e shall see in the next Section that ρ 2 is no t trivial in general on the Witt group. Remark 3.8. W e hav e chosen the Adams op eration to define the her- mitian Bott class. W e could a s w ell consider an y op eratio n induced b y a homomo r phism R ( S k ) − → Z . The o nly reason fo r o ur c hoice is the v ery pleasan t properties of the Adams op erations with resp ect to direct sums and tensor pro ducts of Clifford mo dules. CLIFF OR D MOD ULES AND IN V ARIANTS OF QUADRA TIC FORMS 17 W e w ould lik e t o extend the previous considerations to higher her- mitian K -theory . More precisely , the o rthogonal g roup w e are con- sidering t o define this K -t heory is the gro up O m,m ( R ) whic h is the group of isometries of H ( R m ) , to gether with its direct limit O( R ) = ColimO m,m ( R ) . F or n ≥ 1 , the higher hermitian K -groups are defined in a w ay parallel to higher K -g r o ups, using Quillen’s + construction, b y the form ula K Q n ( R ) = π n ( B O( R ) + ) . Ho w ev er, f rom classical group considerations, as w e already hav e seen, the spinorial group b eha v es b etter than the orthogonal g roup for our purp oses. Therefore, w e shall replace the group O m,m ( R ) b y the as- so ciated spinorial g r oup Spin m , m ( R ) defined in Section 1, which direct limit is denoted b y Spin( R ). W e ha ve the follow ing tw o exact sequenc es (where the first one splits): 1 − → SO 0 ( R ) − → O( R ) − → Z 2 ( R ) ⊕ Disc( R ) − → 1 , 1 − → µ 2 ( R ) − → Spin( R ) − → SO 0 ( R ) − → 1 . Using classical to ols of Quillen’s + construction [10], one can sho w that the maps SO 0 ( R ) − → O( R ) and Spin( R ) − → SO 0 ( R ) induce isomor- phisms π n ( B SO 0 ( R ) + ) ∼ = π n ( B O( R ) + ) for n > 1 . π n ( B Spin( R ) + ) ∼ = π n ( B SO 0 ( R ) + ) for n > 2 . Moreo v er, t he maps π 1 ( B SO 0 ( R ) + ) − → π 1 ( B O( R ) + ) and π 2 ( B Spin( R ) + ) − → π 2 ( B SO 0 ( R ) + ) are injectiv e. Our extension of the Bo tt class to higher hermitian K -gr o ups will b e a map also called ρ k : ρ k : π n ( B Spin( R ) + ) − → π n ( B GL( R ) + ) [1 / k ] = K n ( R ) [1 /k ] In order to define suc h a map, w e w ork geometrically , using the description of the v a rious K -t heories in t erms of flat bundles as detailed in the app endix 1 to [13]. An y elemen t of π n ( B Spin( R ) + ) = K Spin n ( R ) for instance is repres en ted b y a formal difference x = V − T , where V and T are flat spinorial bundles of the same rank, sa y 2 m, o v er a homology sphere X = e S n of dimension n. W e ma y a lso assume that the fib ers of V and T are hyperb olic e.g. H ( R m ) and that T is ” virtually trivial”, whic h means that T is the pull-back o f a flat bundle o v er a n acyclic space. More precise ly , we should first consider a flat principal bundle Q o f structural gro up Spin m , m ( R ) suc h that V = Q × Spin m , m ( R ) H ( R m ) . 18 CLIFF OR D MOD ULES AND IN V ARIANTS OF QUADRA TIC FORMS On the other hand, Spin m , m ( R ) acts on C ( H ( R m )) = End(Λ R m ) b y inner automorphisms. Therefore, the bundle of Clifford algebras C ( V ) asso ciated to V is the bundle of endomorphisms of the flat bundle Q × Spin ′ m , m ( R ) Λ R m . W e now apply our general recip e of Section 2 on each fib er of V and T . In other w ords, for the bundle V for instance, w e consider the following comp osition K ( X ) α − → K C ( V ) ( X ) Ψ k − → K C ( V ( k )) ( X ) α − 1 k − → K ( X ) . In this sequ ence, w e write K ( Y ) for the group of homotopy classes of maps fro m Y to the clas sifying space o f a lgebraic K -theory whic h is homotopically equiv alent to K 0 ( R ) × B GL ( R ) + . Its elemen ts are repren ted b y flat bundles ov er spaces X ho mologically equiv alent to Y . The notatio n K C ( V ) ( X ) means the (g raded) K -theory of flat bundles pro vided with a graded C ( V )- mo dule structure. The image of 1 b y the comp osition α − 1 k . Ψ k .α defines an elemen t of K ( X ) , which we call ρ k ( V ) . On the other hand, since T is virtually triv- ial, w e hav e ρ k ( T ) = k m . W e then define ρ k ( x ) in t he group K n ( R ) [1 /k ] b y the form ula ρ k ( x ) = ρ k ( V ) /k m Theorem 3.9. F or n ≥ 1 , the c orr es p ondanc e x 7→ ρ k ( x ) induc es a gr oup homom orphism ρ k : K Spin n ( R ) − → K n ( R ) [1 /k ] c al le d the n -hermitian Bott class. Pr o of. The map x 7→ ρ k ( x ) is w ell-defined b y general homoto p y con- siderations. In order to c hec k that w e g et a g roup homomorphism, we write t he direct sum of the K n ( R ) [1 /k ] as the m ultiplicativ e group 1 + K ∗ > 0 ( R ) [1 /k ] where the v arious pro ducts b et w een the K n -groups are reduced to 0 . If we now tak e t w o elemen ts x and y in K Spin n ( R ) , we write ρ k ( x ) as 1 + u and ρ k ( y ) as 1 + v . Then ρ k ( x + y ) = ρ k ( x ) · ρ k ( y ) = (1 + u ) · (1 + v ) = 1 + u + v since u · v = 0 .  4. Re la tion with Topology Let V b e a real vector bundle on a compact space X pro vided with a p ositiv e definite quadratic form. W e a ssume that V is spinorial of rank 8 n, so that the bundle of Clifford algebras ma y b e written as End( E ) , where E is the Z / 2- graded v ector bundle of ”spinors” (see f or instance CLIFF OR D MOD ULES AND IN V ARIANTS OF QUADRA TIC FORMS 19 the app endix and [9]). F ollowin g Bot t [7], w e define the class ρ k top ( V ) as the image of 1 by the comp osition of the homomorphisms K R ( X ) ϕ − → K R ( V ) ψ k − → K R ( V ) ϕ − 1 − → K R ( X ) , where K R denotes real K -theory and ϕ is Thom’s isomorphism for this theory . One purp ose in this section is to show that this top ological class ρ k top ( V ) coincides with our hermitian Bott class ρ k ( V ) in the group K R ( X ) ∼ = K ( R ) , where R is the ring of real contin uous functions on X . The pro of of this statemen t requires a careful definition of the group K R ( V ) , since V is not a compact space. A p ossibilit y is to define this group as follo ws (see [12 , § 2] for instance). One considers couples ( G, D ) , where G is a Z / 2-gr aded real vec tor bundle on V , pro vided with a metric, and D an endomorphism of G with the followin g pr o p erties: 1) D is a n isomorphism outside a compact subset of V and w e iden tify ( G, D ) to 0 if this compact set is empt y 2) D is self-adjoint and of degree 1 . One can sho w that the Grothendiec k group asso ciated to this semi- group o f couples is t he reduced K -theory of the one-p oin t compactifi- cation of V . With this description, Thom’s isomorphism K R ( X ) ϕ − → K R ( V ) is easy to describ e. It asso ciates to a v ector bundle F o n X the couple τ = ( G, D ) = ( F ⊗ E , 1 ⊗ ρ ( v )) . In this formula, C ( V ) ∼ = End( E ) and ρ ( v ) denotes the C lifford m ul- tiplication b y ρ ( v ) ov er a p oin t v ∈ V ⊂ C ( V ) . W e no te that ρ ( v ) is an isomorphism outside the 0- section o f V . Therefore, Thom’s isomor- phism ma y b e inte rpreted as Morita’s equiv alence. In fact, ϕ is the follo wing comp osition (where K ( C ( V )) is the K - theory of the r ing of sections of t he a lgebra bundle C ( V )) K R ( X ) − → K ( C ( V )) t − → K R ( V ) , according to [12] for instance. Remark 4.1. This class ρ k top ( V ), whic h requires a metric and a spino- rial structure on V , is differen t in general fro m the alg ebraic class ρ k ( V ), defined for λ -rings . On the other hand, the quotient b etw een ρ k top ( V ) and ( ρ k ( V )) 2 is a 2-torsion class whic h is not trivial in general, as it is sho wn in an example at the end of this Section. If w e apply the Adams op eration to the previous couple τ = ( G, D ), one finds (Ψ k ( F ) · Ψ k ( E ) , Ψ k (1 ⊗ ρ ( v ))) with obv ious definitions. Strictly sp eaking, Ψ k ( E ) should b e though t of as a virtual mo dule ov er C ( V k ) and then w e use the diagonal V − → V k in order to view Ψ k ( E ) as a virtual mo dule ov er C ( V ). W e use here the functorial definition of 20 CLIFF OR D MOD ULES AND IN V ARIANTS OF QUADRA TIC FORMS the Adams op eration detailed in the pro of of Theorem 2 . 5 . Moreov er, since k has a square ro ot a s a p ositiv e real n um b er, w e alw a ys ha v e C ( V , k q ) ∼ = C ( V , q ) . T o sum up, w e ha v e prov ed the following theorem: Theorem 4.2. L et R b e the ring of r e al c ontinuous functions on a c omp act sp ac e X and let V b e a r e al spinorial bund le of r ank 8 n on X . Then the top olo gic al Bott class ρ k top ( V ) in K ( R ) c oincides with the hermitian Bott cla s s ρ k ( V ) of V , viewe d as a fini tely gener ate d pr oje ctive mo dule pr ovide d with a p ositive defi nite quadr atic f o rm an d a spinorial structur e. F or completeness’ sak e, let us mak e some explic it computations of this hermitian Bott class when X is a sphere of dimension 8 m. Let V b e a real or iented v ector bundle of rank 4 t o n S 8 m , generating the reduced real K - group e K R ( S 8 m ) and let W = V ⊕ V b e its complexification whic h generates e K C ( S 8 m ) , where e K C is redu ced complex K -theory . Let us denote by W R the underlying real v ector bundle w ith the asso ciat ed spinorial structure. According to [12, Prop o sition 7.2 7], w e ha ve the form ula c ( ρ k top ( W R )) = ρ k C ( W ) = ρ k ( W ) , where c : K R ( S 8 m ) ∼ = − → K C ( S 8 m ) denotes the complexification. There- fore, w e ar e reduced to computing the class ρ k for complex v ector bun- dles on even dimensional spheres X. If X = S 2 , K C ( S 2 ) is free of rank 2, generated by 1 and the Hopf line bundle L. The classical Bott class is then computed from the formula ρ k ( L ) = 1 + L + ... + L k − 1 . Since ( L − 1) 2 = 0 , another w ay to write this sum is to consider T a ylor’s expansion of the p olynomial 1 + X + ... + X k − 1 at X = 1 . W e get the formula ρ k ( L ) = k + [1 + 2 + ... + ( k − 1)] ( L − 1) = k + k ( k − 1)( L − 1) / 2 . If x 2 denotes the class L − 1 , w e also can write ρ k ( x 2 ) = 1 + [1 + 2 + ... + ( k − 1)] /k · x 2 . W e compute in the same w a y the Bott class on e K C ( S 4 ) ∼ = Z , g enerated b y the pro duct x 4 = ( L 1 − 1) · ( L 2 − 1) where L 1 and L 2 are tw o copies of the Hopf line bundle on S 2 . Since w e again ha v e ( L i − 1) 2 = 0 , it is sufficien t to compute the first terms of T aylor’s expansion o f the p olynomial f ( X , Y ) = 1 + X Y + X 2 Y 2 + ... + X k − 1 Y k − 1 CLIFF OR D MOD ULES AND IN V ARIANTS OF QUADRA TIC FORMS 21 at the p oin t (1 , 1) . W e get the second deriv ative ( δ 2 f /δ xδ y ) /k 2 at the p oin t (1 , 1 ) m ultiplied b y x 4 . In other words, we hav e ρ k ( x 4 ) = 1 +  1 + 2 2 + ... + ( k − 1) 2 )  /k 2 · x 4 . More generally , o n e K C ( S 2 r ) ∼ = Z , g enerated b y x 2 r = ( L 1 − 1) · · · ( L r − 1) , w e find the formula ρ k ( x 2 r ) = 1 + [1 + 2 r + ... + ( k − 1) r ] /k r · x 2 r . Since c ( ρ k ( x 8 m )) = ρ k top ( y 8 m ) , where y 8 m (resp x 8 m ) generates e K R ( S 8 m ) (resp. e K C ( S 8 m )), w e deduce from the last formu la the followin g pro p o- sition. Prop osition 4.3. L et V b e a r e al ve ctor bund le of r ank 8 t gene r ating the r e al r e duc e d K -the ory of the spher e S 8 m and let y 8 m = V − 8 t. We then have the formula ρ k top ( y 8 m ) = 1 +  1 + 2 4 m + ... + ( k − 1) 4 m  · y 8 m . Remark 4.4. If w e assume that k is o dd, the sum 1 + 2 r + ... + ( k − 1) r has the same pa r ity as 1 + 2 + ... + ( k − 1) or equiv alen tly ( k − 1) / 2 which is a lso o dd for an infinite n um b er of o dd k ′ s. A more delicate example is the case of the sphere X = S 8 m +2 with m > 0 . It is well kno wn tha t the realification map Z ∼ = e K C ( S 8 m +2 ) − → e K R ( S 8 m +2 ) ∼ = Z / 2 is surjectiv e. Let V b e a complex vector bundle ov er S 8 m +2 whic h generates e K C ( S 8 m +2 ). W e consider the fo llo wing diagram K C ( V ) Ψ k − → K C ( V ) ↑ φ C ↓ φ − 1 C K C ( S 8 m +2 ) K C ( S 8 m +2 ) , where φ C is Thom’s isomorphism in complex K - theory . By definition, w e hav e ρ k ( V ) = φ − 1 C (Ψ k ( φ C (1))) . Since m > 0 , V is also a spinorial bundle and we therefore ha v e a comm utativ e diagram up to isomorphism K C ( V ) r − → K R ( V ) ↑ φ C ↑ φ R K C ( S 8 m +2 ) r − → K R ( S 8 m +2 ) , where φ R is Thom’s isomorphism in real K -theory and r is the realifi- cation. Since the Adams op eration Ψ k comm utes with r , w e hav e the iden tit y ρ k top ( y 8 m +2 ) = 1 +  1 + 2 4 m +1 + ... + ( k − 1) 4 m +1  · y 8 m +2 = 1 + y 8 m +2 22 CLIFF OR D MOD ULES AND IN V ARIANTS OF QUADRA TIC FORMS if k a nd ( k − 1) / 2 are o dd. Let V 0 b e the underlying real vec tor bundle of V . The last iden tity implies tha t ρ k top ( V 0 ) = (1 + y 8 m +2 ) · k 4 t if V 0 is of rank 8 t. Therefore ρ k top ( V 0 ) cannot b e a square, eve n modulo the Picard g roup (whic h is trivial in this case). This implies that the corrected hermitian Bot t class defined in 3 .6: ρ k : W Spin( R ) − → K ( R ) × / (Pic( R )) ( k − 1) / 2 = K ( R ) × is not t rivial either (w e recall that R is the ring of real con tin uous functions o n the sphere S 8 m +2 ) . Remark 4.5. W e should add a f ew w ords if k is ev en. If k = 2 f o r instance and if X is the sphere S 8 n , w e find that ρ 2 ( y 8 n ) = 1 / 2 4 n · y 8 n W e get the same result for bundles with negativ e definite quadratic forms. Since the hy p erb olic map K ( C R ( S 8 n )) ∼ = Z − → K Q ( C R ( S 8 n )) ∼ = Z ⊕ Z is the diagonal, w e see that the class ρ 2 of an h yp erb o lic mo dule b elongs to 1 / 2 4 n − 1 Z . Therefore, at least for this example, the class ρ 2 also detects non trivial Witt classes. 5. Oriented Azuma y a algebras Another purpose of this pap er is the extension of our definitions to Azuma y a algebras [4][6], b ey ond the example of Clifford algebras. W e first consider the non graded case. Definition 5.1. Let A b e an Azumay a alg ebra. W e sa y that A is ”orien ted” if the p erm utation of the t w o copies of A in A ⊗ 2 is giv en b y an inner a utomorphism asso ciated to an elemen t τ ∈ ( A ⊗ 2 ) × of order 2 . As a matter of fact, as it w as p ointed out to us b y Kn us andTignol, an y Azuma y a alg ebra is oriented 6 . This is a theorem quoted by Kn us and Oj a nguren[15, Prop osition 4.1, p.. 112.] and attributed to O. Goldman. W e shall illustrate it by a few ty pical example s. The first easy but fundamen tal example is A = End( P ) , where P is a faithful finitely generated pro jectiv e mo dule. W e identit y B = A ⊗ A with End( P ⊗ P ) and B × with Aut( P ⊗ P ) . The elemen t τ required is simply the p erm utation of the tw o copies of P , view ed as an elemen t of ( A ⊗ A ) × = Aut( P ⊗ P ) , as it can b e sho wn by a direct computatio n. Let no w D be a division algebra o v er a field F . W e claim that D is also oriented. In or der to show this, w e conside r the tensor pro duct 6 How ever, the situation is differen t in the Z/2-graded case as we shall show below. CLIFF OR D MOD ULES AND IN V ARIANTS OF QUADRA TIC FORMS 23 A = D ⊗ F F 1 , where F 1 is a finite Galois extension o f F , suc h that A is F -isomorphic to a matrix algebra M n ( F 1 ) = End( F n 1 ) and is therefore orien ted according to our first example. Let G b e t he Galois gro up of F 1 o v er F , so that D is the fixed a lgebra of G acting on A. If w e comp ose this action b y the usual action of the Galo is group on M n ( F 1 ) , we get automorphisms of M n ( F 1 ) as a F 1 -algebra whic h are inner by Sk olem- No ether’s theorem. If g ∈ G , w e let α g b e an elemen t of Aut ( F n 1 ) so that the action ρ ( g ) of g on A is give n b y the comp osition of the inner automorphism associated to a g with the usual Galo is action on M n ( F 1 ) . Let no w τ ′ b e the p ermutation of the t w o copies of A in the tensor pro duct A ⊗ F 1 A. It is induced by the inner aut o morphism asso ciated to a sp ecific elemen t τ in Aut( F n 1 ⊗ F 1 F n 1 ) of order 2 whic h comm utes with ρ ( g ) ⊗ ρ ( g ) Therefore, τ is in v arian t by the action of G and b elongs to ( D ⊗ F D ) × , considered as a subgroup of ( A ⊗ F 1 A ) × . F rom a differen t p oin t of view, let us consider the algebra R of com- plex con tin uous functions on a connected compact space X . According to a w ell-kno wn dictionnary of Serre a nd Swan, one ma y consider an Azuma y a alg ebra A o v er R as a bundle e A of algebras o ve r X with fib er End( P ) , where P = C n . The structural group of t his bundle is the pro- jectiv e linear gr o up Aut( P ) / C × . In the same w ay , t he structural gro up of A ⊗ 2 is Aut( P ⊗ P ) / C × . Therefore, the inner automorphism of A ⊗ 2 , p erm uting the tw o copies of A, is induced b y the p ermutation of the t w o copies o f P . This is well defined globally since this permutation comm utes with the tra nsition functions o f e A . Let A b e any Azuma y a algebra. W e w ould lik e to lift the action σ k of the symme tric gro up S k on A ⊗ k to ( A ⊗ k ) × , suc h that we hav e a comm utativ e diagram ( A ⊗ k ) × e σ k ր ↓ γ S k σ k − → Aut( A ⊗ k ) , where e σ k is a g r o up homomorphism and γ induces inner a ut o mor- phisms. This program is achie ve d in the ”Bo ok of In v olutions” [16, Prop osition 10.1, pg. 115.], using again the ”Goldman elemen t” quoted ab ov e. F or completeness’ sak e, w e shall sk etc h a pro of b elo w, since we shall need it in the graded case to o. In order to define e σ k , we use the classical description of the symmetric group in terms of generators τ i = ( i, i + 1) , i = 1 , ..., k − 1 , with t he relations ( τ i ) 2 = 1 , τ i τ i +1 τ i = τ i +1 τ i τ i +1 and τ i τ j = τ j τ i if [ i − j | > 1 . Since A is oriented, w e may view the τ i in ( A ⊗ k ) × as the tensor pro duct of τ b y the appropriat e n um b er of 1 = I d A . W e easily che ck the previous relations, except the t ypical one τ 1 τ 2 τ 1 = τ 2 τ 1 τ 2 . 24 CLIFF OR D MOD ULES AND IN V AR IANTS OF QUADRA TIC FORMS (one may replace the couple (1 , 2 ) by ( i, i + 1 )) . How ev er, w e already ha v e τ 1 τ 2 τ 1 = λτ 2 τ 1 τ 2 , where λ ∈ R × . The identit y ( τ 1 τ 2 τ 1 ) 2 = ( τ 2 τ 1 τ 2 ) 2 = 1 also implies that ( λ ) 2 = 1 . The solution to our lifting problem is then to k eep the τ i for i o dd and replace the τ i for i ev en by λτ i , in order to get the required relations among the τ ′ s. The previous considerations ma y b e translated in the framew ork of Z / 2-graded Azuma ya a lgebras [6, pg. 160]. In this case, we m ust require the elemen t τ in t he definition to b e of degree 0 . Unfortunately , in general, a Clifford algebra is not oriente d in the gra ded sense. As a counterex ample, we may c ho ose A = C 0 , 1 . Then the p ermutation of the t w o copies of A in A ⊗ A = C 0 , 2 is giv en b y the inner automorphism asso ciated to e 1 + e 2 whic h is of degree 1 and not of degree 0 , as require d in our definition. Ho w ev er, if V is a mo dule whic h is orien ted and of ev en r a nk, the asso ciated Clifford algebra C ( V ) is oriented as we shall sho w b elow. T o start with, let V and V ′ b e t w o quadratic mo dules suc h that V is of ev en ra nk and oriented. The argument used in Section 1 shows the ex istence of an eleme nt v in C 0 ( V ) ⊗ C 0 ( V ′ ) ⊂ C ( V ) b ⊗ C ( V ′ ) ∼ = C ( V ⊕ V ′ ) whic h antic ommutes with the elemen ts of V and comm utes with the elemen ts of V ′ : one puts v = u ⊗ 1 with the notations of Section 1 (see Remark 1.7). Moreo v er, ( v ) 2 = 1 a nd v ∈ Spin( V ⊕ V ′ ) . Let us choose V ′ = V and put T = V ⊕ V . Since 2 is inv ertible in R, T is isomorphic t o the ortho g onal sum T 1 ⊕ T 2 , where T 1 = { v , − v } a nd T 2 = { v , v } . Thanks to this isomorphism, the p ermutation of the t w o summands o f V ⊕ V is translated into the inv olution ( t 1 , t 2 ) 7→ ( − t 1 , t 2 ) on T 1 ⊕ T 2 . Therefore, the previous arg umen t sho ws the existence of a canonical eleme nt u 12 ∈ Spin( V ⊕ V ) of square 1 suc h that the transformation x 7→ u − 1 12 .x.u 12 p erm utes the tw o summands of V ⊕ V . It follow s immediately that the Clifford algebra C ( V ) is oriented ( in the gra ded sense) if V is oriented and of ev en rank. F rom the previous general considerations, we deduce a natural rep- resen tation o f the symmetric group S k in the group Spin( V k ) whic h lifts t he canonical r epresen tation of S k in SO( V k ) . T o sum up, w e ha v e pro v ed the following Theorem: Theorem 5.1. L et V b e a quadr atic mo dule of even r ank w h ich is oriente d and let σ k : S k − → SO( V k ) b e the standar d r epr esentation. Then ther e is a c anonic al lifting e σ k : S k − → Spin ( V k ) , CLIFF OR D MOD ULES AND IN V AR IANTS OF QUADRA TIC FORMS 25 such that the fol lowing d i a gr am c ommutes Spin( V k ) e σ k, ր ↓ π S k σ k − → SO( V k ) . In other wor ds, the Cliffor d algebr a C ( V ) is a Z / 2 - g r ade d oriente d Azumaya algebr a. Remarks 5.2. One c an also make an expli c it c omputation in the Clif- for d algebr a C ( V k ) w i th the obvio us elements τ i = u i,i +1 . ; one che cks they satisfy t he r e quir e d r elations for the gener ators of the symmetric gr oup S k . Mor e over, these liftings for various k ′ s ar e of c ourse c om- p atible with e ach other. If R is an inte gr al doma i n, we n o te that e σ k is unique, onc e e σ 2 is give n . 6. Adams ope ra tions revisite d In this Section we assume t ha t V is a quadratic R -mo dule which is orien ted a nd of ev en ra nk, so that the Clifford algebra is a Z/2 -graded orien ted Azumay a algebra. If k ! is in v ertible in R we ha v e defined Adams op eratio ns in a func- torial w a y: Ψ k : K ( C ( V )) − → K ( C ( V ( k ))) . The purpose of this Section is to define similars op eration Ψ k under another ty p e of hy p o thesis: 2 k is in v ertible in R and R con tains the ring o f in tegers in the k -cyclotomic extension of Q whic h is Ω k = Z ( ω ) = Z [ x ] / (Φ k ( x )) . Here Φ k ( x ) is the cyclotomic p olynomial and ω is the class of x . W e conjecture t ha t Ψ k = Ψ k (whic h is defined via Newton p olynomials from the λ -op eratio ns) but we are not able to prov e it, except when k ! is inv e rtible in R. W e also w ant to extend these op erations Ψ k and Ψ k to or iented Azuma y a algebras (not o nly Clifford algebras) whic h w ere defined in the previous Section. The idea to define Ψ k is a remark b y A tiyah [1, F ormula 2 . 7] (used already in Section 3) that Adams op era t ions ma y b e defined using the cyclic g roup Z /k instead of the symmetric group S k (if k ! is in v ertible in R ) . More precisely , t he Adams op eratio n Ψ k is induced by the ho- momorphism R ( S k ) − → Z whic h associates to a repre sen tation σ its c haracter on the cycle (1 , 2 ..., k ) . Therefore, if w e put F = E ⊗ k , we see that Ψ k ( E ) = k − 1 P j =0 F ω j · ω j , 26 CLIFF OR D MOD ULES AND IN V AR IANTS OF QUADRA TIC FORMS where ω is a primitive k th -ro ot of unit y and where F ω j is the eigenm o dule corresp onding to the eigen v alue ω j . The previous sum belong s in fact to the subgroup K ( C ( V ( k )) of K ( C ( V ( k )) ⊗ Z Ω k . If we only assume tha t k is inv ertible in R, w e can consider the previous sum a s a new operat io n. More precisely , w e define Ψ k ( E ) = k − 1 P j =0 F ω j · ω j . In this new setting, this sum b elongs to the group K ( C ( V ( k ) ) ) ⊗ Z Ω k and not necess arily to the subgroup K ( C ( V ( k ) ) ). This definition mak es sense since w e ha v e assume d k in v ertible in R, so that F splits as the direct sum o f the eigenmo dules asso ciated to the eigen v alues ω j , where 0 ≤ j ≤ k − 1 . Since w e work in the Z / 2- graded case, w e also ha v e to assume that 2 is inv ertible in R . If k is prime, b ecause of the underlying action of the symmetric group on F , the eigenmodules F ω j are isomorphic to eac h other when 1 ≤ j ≤ k − 1 , so that this definition of Ψ k ( E ) reduces to F 0 − F ω . W e ma y be more precise and c hoose as a mo del of the symme tric group S k the group of p erm utations of the set Z /k . One generator T of the cyclic group Z /k is the p ermutation x 7→ x + 1 . If α is a generator of t he m ultiplicativ e cyc lic group ( Z /k ) × , the p erm utation x 7− → α s x, where s runs from 1 to k − 2 , enables us to iden tify all t he eigenmo dules F ω j , j = 2 , .., k − 1 with F ω . W e t herefore g et the following theorem: Theorem 6.1. L et E b e a gr ade d C ( V ) -mo dule and let us assume that 2 k is invertible in R and that the k th -r o ots of unity b elong to R . We define Ψ k ( E ) in the gr oup K ( C ( V ( k ))) ⊗ Z Ω k by the fol lowing formula Ψ k ( E ) = k − 1 P j =0 F ω j · ω j . If E 0 and E 1 ar e two such mo dules, we have Ψ k ( E 0 ⊗ E 1 ) = Ψ k ( E 0 ) · Ψ k ( E 1 ) in the Gr o then die ck g r oups K ( C ( V (2 k ))) ⊗ Z Ω k . Mor e over, if k is prime, Ψ k ( E ) b elongs to K ( C ( V ( k ))) ⊂ K ( C ( V ( k ))) ⊗ Z Ω k and we have the fol low i n g formula in K ( C ( V ( k ) : Ψ k ( E 0 ⊕ E 1 ) = Ψ k ( E 0 ) + Ψ k ( E 1 ) . Final ly, the op er ation Ψ k c oincides with the usual A dams op er ation Ψ k if k ! is invertible in R . Pr o of. When k is prime, w e hav e the isomorphism ( E 0 ⊕ E 1 ) ⊗ k ∼ = ( E 0 ) ⊗ k ⊕ ( E 1 ) ⊗ k ⊕ Γ , where Γ is a mo dule of type ( H ) k with an action of S k p erm uting the factors of ( H ) k . F rom elemen tary algebra, w e see that Γ is not CLIFF OR D MOD ULES AND IN V AR IANTS OF QUADRA TIC FORMS 27 con tributing to the computation of Ψ k ( E 0 ⊕ E 1 ) , hence the second form ula. F or the first for mula, w e compute Ψ k ( E 0 ⊗ E 1 ) b y lo oking for ma lly at the eigenmo dules of T ⊗ T acting on ( E 0 ) ⊗ k ⊗ ( E 1 ) ⊗ k , considered as a mo dule o ve r C ( V ( k )) ⊗ C ( V ( k )) . They are of course a sso ciated to the eigen v alues ω i ⊗ ω j = ω i + j . Using the remark ab ov e, we can write Ψ k ( E 0 ⊗ E 1 ) = k − 1 P r =0  ( E 0 ⊗ E 1 ) ⊗ k  r · ω r = k − 1 P r =0 P i + j = r  ( E 0 ) ⊗ k  i · ω i ·  ( E 1 ) ⊗ k  j · ω j = Ψ k ( E 0 ) · Ψ k ( E 1 ) . Finally , for k ! in v ertible in R, the fact that Ψ k = Ψ k is just the remark made b y Atiy ah [1] quoted ab o v e.  Let now A b e any Z / 2-graded Azumay a algebra which is orien ted. W e would lik e to define op erations o n the K - theory of A of the follo wing t yp e K ( A ) − → K ( A ⊗ k ) . F or this, w e a g ain follow the sche me defined by A tiyah [1, F orm ula 2 . 7 ] (if k ! is inv e rtible in R ) . The o nly p oin t whic h requires some care is the definition of the ”p ow er map” K ( A ) − → K S k ( A ⊗ k ) = K ( A ⊗ k ) ⊗ R ( S k ) . A priori, the target of this map is the K - group of the cross-pro duct algebra S k ⋉ A ⊗ k . How ev er, as we ha ve seen in the previous Section, the represen tation of S k in Aut( A ⊗ k ) lifts as a homomorphism f r o m S k to ( A ⊗ k ) × . Therefore, this cross pro duct algebra is the t ensor pro duct of t he g r oup algebra Z [ S k ] with A ⊗ k . Therefore, any ho mo mo r phism λ : R ( S k ) − → Z giv es rise to an op eratio n λ ∗ : K ( A ) − → K ( A ⊗ k ) , as w e sho w ed in Section 3. How ev er, o ne has to b e careful that this op eration dep ends on the orientation ch osen on A, i.e. on the lifting of the represen tation σ k : S k − → Aut( A ⊗ k ) to a represen tation e σ k : S k − → ( A ⊗ k ) × , in suc h a w ay that the diagra m ( A ⊗ k ) × e σ k ր ↓ S k σ k − → Aut( A ⊗ k ) comm utes. If R is an in tegral domain, this lifting is defined up to the sign represen tation. Ho w ev er, in this case, w e get a canonical c hoice of e σ k as follows . Let F b e the quotien t ring of R and F its algebraic 28 CLIFF OR D MOD ULES AND IN V AR IANTS OF QUADRA TIC FORMS closure. If we extend the scalar to F , A b ecomes a matrix alg ebra End( E ) ov er F , in whic h case ( A ⊗ k ) × is iden tified with Aut( E k ) . W e then c ho ose the sign of the lif t ing e σ k in such a wa y that it corresp onds to the canonical lifting S k − → Aut( E k ) by extension of the scalars. Let us b e more explicit and define the k th -exterior p ow e r λ k ( M ) of M as an A ⊗ k -mo dule in our setting. W e take the quotien t of M ⊗ k b y the usual relations (where the m i are homogeneous elemen ts): m s (1) ⊗ m s (2) ⊗ ... ⊗ m s ( k ) = ε ( s ) e σ k ( s ) deg ( m s ) m 1 ⊗ m 2 ⊗ ... ⊗ m k . Here ε ( s ) is the signature of the permutation s , e σ k the lifting defined ab ov e and deg( m s ) the signature of t he represen tation s restricted to elemen ts of o dd degree. W e note that λ k ( M ) is a graded mo dule o v er A ⊗ k . Example. Let A = End( E ) with the trivial grading and let e σ k b e the cano nical lift ing . Then, b y Morita equiv alence, all left A -mo dules M ma y b e written as E ⊗ N , where N is an R - mo dule. It is t hen easy to see that λ k ( M ) ∼ = E ⊗ k ⊗ λ k ( N ) , where λ k ( N ) is the usual k th - exterior p ow er ov er the commutativ e ring R, E ⊗ k b eing view ed a s a mo dule o v er A ⊗ k ∼ = End( E ⊗ k ) . W e note that if w e c hange t he sign of the orien tatio n, w e get the symmetric p ow er E ⊗ k ⊗ S k ( N ) instead of the exterior p o we r. It is conv enien t to consider the full exterior algebra Λ( M ) of M whic h is the direct sum of all the λ k ( M ) . As usual, Λ( M ) is t he solution of a univ ersal pro blem. If g : M − → C is an R -mo dule map where C is an R -algebra and if g ( m s (1) ) g ( m s (2) ) , ...g ( m s ( k ) ) = ε ( s ) e σ k ( s ) deg ( m s ) s ( m 1 ) s ( m 2 ) ...s ( m k ) , there is an algebra map Λ( M ) − → C whic h mak es t he ob vious dia gram comm utativ e. If M is a finitely generated pro jectiv e A -mo dule, λ k ( M ) as a finitely generated pro jectiv e A ⊗ k -mo dule: this is a consequence of the follo wing theorem. Theorem 6.2. L et A b e an oriente d Z / 2 -gr ade d Azumaya algebr a and let M and N b e two finitely ge ner ate d pr oje ctive A -mo dules T h en the exterior algeb r a o f M ⊕ N is c anonic al ly isomorphic to Λ( M ) ⊗ R Λ( N ) . Mor e over, in e ach de gr e e k , we get an i s omorphism of A ⊗ k -mo dules. Pr o of. The canonical map from M ⊕ N to Λ ( M ) ⊗ R Λ( N ) induces the usual isomorphism Λ( M ⊕ N ) − → Λ( M ) ⊗ R Λ( N ) . In eac h degree k , this map induces an isomorphism b et w een λ k ( M ⊕ N ) and the sum of the λ i ( M ) ⊗ R λ k − i ( N ) , view ed a s A ⊗ k -mo dules.  F ollo wing Grothendiec k and Atiy ah again, we define λ - o p erations o n K -groups: λ k : K ( A ) − → K ( A ⊗ k ) CLIFF OR D MOD ULES AND IN V AR IANTS OF QUADRA TIC FORMS 29 satisfying the usual iden tity λ r ( M ⊕ N ) = P k + l = r λ k ( M ) · λ l ( N ) as A ⊗ ( k + l ) -mo dules. W e can also define the Adams o p erations by the usual formalism. W e may view op erat io ns in this type of K -t heory a s comp ositions K ( A ) P − → K ( A ⊗ k ) ⊗ Z R ( S k ) θ − → K ( A ⊗ k ) . Here P is the p ow er map defined thro ugh the lifting e σ k ab ov e. The sec- ond map θ is induced b y an homomorphism R ( S k ) − → Z . In particular, the Adams o p eration Ψ k is g iv en b y the homomor phism R ( S k ) − → Z whic h asso ciates to a represen tation ρ its trace of the cycle (1 , 2 , ..., k ) . Remark 6.3. A careful analysis of these considerations sho ws tha t w e don’t need k ! to b e inv ertible in order to define the λ -op erations in the non graded case. Ho we ve r, w e need 2 to b e in v ertible in the graded case and, moreo v er, k ! in v ertible in order to define t he Adams op erations with go o d formal prop erties. Another appro a c h to the Adams op erations, as w e show ed a t the b eginning of this Section, only assumes that 2 k is in v ertible in R and that R con tains the k th -ro ot s of unit y . If E is a finitely generated pro jectiv e A -mo dule, the tensor pow er E ⊗ k is an S k ⋉ A ⊗ k -mo dule . W e can ”un t wist” the t w o actions of S k and A ⊗ k , thanks to the orien tatio n of A and w e end up with an A ⊗ k -mo dule F , with an indep endant action of S k . W e put formally Ψ k ( E ) = k − 1 P j =0 F j · ω j where F j is t he eigenmo dule asso ciated t o the eigenv alue ω j . The pre- vious sum lies in K ( A ⊗ k ) ⊗ Z Ω k and ev en in the subgroup K ( A ⊗ k ) if k is prime. This second definition is very pleasan t, since the formal pro p- erties of the Adams op erations can b e c hec k ed easily with this form ula (at least f o r k prime). W e conjecture that Ψ k = Ψ k in this case to o. 7. Twiste d he rmitian Bott classes W e are going to define mor e subtle op erations, asso ciated not only to the K -theory of A but also to the K -theory of A ⊗ B , where A = C ( V ) and B = C ( W ) a r e tw o Cliffor d algebras. W e no longer assume that V and W are of ev en rank or orien ted. How ev er, w e assume k o dd, 2 k in ve rtible in R and that the k th -ro ot s of unity b elong to R . W e also replace the symmetric group S k b y the cyclic group Z /k in our previous argumen ts. The reason for this c hange is the fo llo wing 30 CLIFF OR D MOD ULES AND IN V AR IANTS OF QUADRA TIC FORMS remark. The nat ura l represen tation σ k : Z /k − → O ( V k ) has its image in the subgroup SO 0 ( V k ) defined in Section 1 and lif t s uniquely to a represen tation of Z /k in Spin( V k ) , so that the following diagram comm utes: Spin( V k ) ր ↓ Z /k − → SO 0 ( V k ) . This lifting do es not exist in general for the symmetric group S k , except if V is ev en dimensional and orien ted, as we hav e seen in Section 5. Let no w M b e a finitely g enerated pro jectiv e mo dule o v er A ⊗ B = C ( V ) ⊗ C ( W ) = C ( V ⊕ W ) . W e can comp ose the p o w er map K ( A ⊗ B ) − → K ( Z /k ⋉ ( A ⊗ B ) ⊗ k ) ∼ = K ( Z /k ⋉ C ( V k ⊕ W k )) with the ”half-diagonal” K ( Z /k ⋉ C ( V k ⊕ W k )) − → K ( Z /k ⋉ C ( V ( k ) ⊕ W k )) ∼ = K ( C ( V ( k )) ⊗ ( Z /k ⋉ C ( W k ))) , as w e did in Section 2 for W = 0 . F rom the conside ratio ns in Section 6 , w e can ”unt wist” the action of Z /k on the Z / 2- graded Azuma y a algebra C ( W k ) , so t hat Z /k ⋉ C ( W k ) is isomorphic to the usual group algebra Z [ Z /k ] ⊗ C ( W k ) . Using the metho ds o f Section 2 and of the previous Section , w e get a more precise p ow er map: K ( C ( V ) ⊗ C ( W )) − → K ( C ( V ( k )) ⊗ C ( W k )) ⊗ R ( Z /k ) . Therefore, according to A tiyah ag a in [1], any homomorphism λ : R ( Z /k ) − → Ω k giv es rise to a ”twis ted operat ion” λ ∗ : K ( C ( V ) ⊗ C ( W )) − → K ( C ( V ( k )) ⊗ C ( W k )) ⊗ Ω k . W e apply this f ormalism to W = V ( − 1) , in whic h case C ( W ) is the (graded) opp osite algebra of C ( V ) . Therefore, K ( C ( V ) ⊗ C ( W )) ∼ = K ( R ) b y Morita equiv alence. If w e c ho ose for λ the map a b ov e, w e define the ”tw isted hermitian Bott class” as the image of 1 by the comp osition K ( R ) ∼ = K ( C ( V ) ⊗ C ( W )) − → K ( C ( V ( k )) ⊗ C ( W k )) ⊗ R ( Z /k ) − → K ( C ( V ( k )) ⊗ C ( W k ) ⊗ Ω k . W e hav e pro v ed the following theorem: Theorem 7.1. L et V b e an arbitr ary quadr atic m o dule. The ”twiste d hermitian Bott class” ρ k ( V ) b elongs to the fol lowing gr oup ρ k ( V ) ∈ K ( C ( V ( k )) ⊗ C ( W k )) ⊗ Z Ω k . It satisfies the multiplic ative pr op erty ρ k ( V 1 ⊕ V 2 ) = ρ k ( V 1 ) · ρ k ( V 2 ) , CLIFF OR D MOD ULES AND IN V AR IANTS OF QUADRA TIC FORMS 31 taking into ac c ount the identific ation of algebr as: C (( V 1 ⊕ V 2 )( k )) ⊗ C ( W k 1 ⊕ W k 2 ) ∼ =  C ( V 1 ( k )) ⊗ C ( W k 1 ))  ⊗  C ( V 2 ( k )) ⊗ C ( W k 2 )  . . Remark 7.2. Let ( V , E ) b e a spinorial mo dule and let us identify the four Z/2-gr aded algebras C ( V ) , End( E ) , C ( V ( k )) and C ( W ). Then, b y Morita equiv alence, w e see that the twisted hermitian Bott class coincides (non canonically) with the unt wiste d one. Remark 7.3. It is easy to show that V 4 is an orien table quadratic mo dule whic h implies b y 1.7 that the Clifford algebra C ( V 4 ) is isomor- phic to its opp o site. Let now k b e an o dd square whic h implies that k ≡ 1 mo d 8 . Since C ( V ( k ) ) ∼ = C ( V ) and C ( W k ) a r e Morita eq uiv a- len t to C ( W ), the targ et group of the t wisted hermitian Bott class is isomorphic to K ( C ( V ) ⊗ C ( W )) ⊗ Z Ω k ∼ = K ( R ) ⊗ Z Ω k . This sho ws that w e hav e a comm utativ e diagram up to isomorphism K Spin( R ) − → K Q ( R ) ↓ ↓ K ( R ) [1 /k ] × − → [ K ( R ) ⊗ Z Ω k ] [1 /k ] × , where the ve rtical ma ps are defined b y hermitian Bott classes, twisted and un tw isted. Finally , as w e did in Section 3, w e can ”correct” the t wisted hermitian Bott class by using the result of Serre ab out the square ro ot of t he classical Bo t t class [17 ]. More precisely , if V is a self-dual mo dule of dimension n , there is a n explicit class σ k ( V ) in K ( R ) ⊗ Z Ω k whic h only dep ends on the exterior p ow ers o f V , suc h that σ k ( V ) 2 = δ ( k − 1) / 2 ρ k ( V ) , with δ = ( − 1) n λ n ( V ) . The corrected t wisted hermitian Bott class is then defined b y the for m ula ρ k ( V ) = ρ k ( V )( σ k ( V )) − 1 , taking into accoun t the f a ct that K ( C ( V ( k ) ) ⊗ C ( W k )) is a mo dule o v er the ring K ( R ) . W e hav e ρ k ( V ) ∈ ± (Pic( R )) ( k − 1) / 2 if V is hyperb o lic 7 , as w e show ed in Section 3. Therefore, the previous formu la for ρ k defines a morphism also called ρ k , b etw een the classical Witt group W ( R ) and t wisted K - theory mo dulo ± (Pic( R )) ( k − 1) / 2 (as a multiplicativ e group), more precisely ρ k : W ( R ) − →  K ( C ( V ( k )) ⊗ C ( W k )) ⊗ Z Ω k  [1 /k ] / × ± (Pic( R )) ( k − 1) / 2 . 7 The sig n ambiguit y is unav oidable, since V is of ar bitrary dimension. 32 CLIFF OR D MOD ULES AND IN V AR IANTS OF QUADRA TIC FORMS Remark 7.4. If k ≡ 1 mo d 4 , w e can m ultiply σ k ( V ) b y the sign ( − 1) n ( k − 1)4 , as we did in Section 3. If we apply this sign c hange, the new corrected tw isted hermitian Bott class tak es it s v alues in the group  K ( C ( V ( k )) ⊗ C ( W k )) ⊗ Z Ω k  [1 /k ] / × / (Pic( R )) ( k − 1) / 2 , without an y sign ambiguit y . 8. Ap pendix. A remark ab out the Brauer-W all group The purp ose o f this a pp endix is to pro ve the follow ing theorem whic h is also found in [4, Prop o sition 5 . 3 and Corollary 5 . 4] for the non graded case. It is added to this pap er for completeness’ sak e with a Z / 2-graded v arian t. Theorem 8.1. L et R b e a c omm utative ring. L et A b e an R -algebr a which is pr oje ctive, finitely ge ner ate d and fa ithful as an R -mo dule. L et P and Q b e faithful pr oje ctive finitely ge n er ate d R -mo dules such that A ⊗ End( P ) ∼ = End( Q ) . Then A is isomorphic to s o me End ( E ) , wher e E is als o faithful, pr oje c- tive and finitely gen e r ate d. The same statement is true for Z / 2 -gr ade d algebr as and mo dules if 2 is invertible in R . In order to prov e the theorem, w e need the following classical lemma: Lemma 8.2. L et P b e a faithful fini tely gener ate d pr oje ct ive R -m o dule. Then ther e exists an R -mo dule Q such that F = P ⊗ Q is fr e e. Mor e- over, if P is Z / 2 -gr ade d and if 2 is i n vertible in R , we may cho ose Q such that F = R 2 m = R m ⊕ R m , with the obvious gr ad ing. Pr o of. (compare w ith [9, pg. 14] and [5, Corollary 16 . 2 ]). Since an y mo dule P of this type is lo cally the image of a pro jection o p erator p of rank r > 0 , w e can lo ok a t the ”unive rsal example”. This univ ersal ring R is generated b y v ariables p j i where 1 ≤ i ≤ n and 1 ≤ j ≤ n suc h that the matrix p = ( p j i ) is idem p o ten t of tra ce r. According to [5, p. 39], since R is of finite stable r a nge, the elemen t y = [ P ] − [ r ] is nilp otent in the G rothendiec k g roup K ( R ) , say y N = 0 fo r some N . Let us no w consider the elemen t x = r N − 1 − r N − 2 y + ... + ( − 1) N − 1 y N − 1 . W e hav e the identit y ( r + y ) M x = M ( r N − ( − 1) N − 1 y N ) = M r N . Since the rank of x is r N − 1 > 0 and since the stable ra nge of R is finite, the elemen t M x in K ( R ) is the class of a mo dule Q for sufficien tly large M . If follows that P ⊗ Q is stably free and therefore free if M is a g ain larg e enough. Finally , the case of Z / 2-graded mo dules follows b y the same argumen t, considering gra ded R -mo dules as R [ Z / 2]-mo dules.  CLIFF OR D MOD ULES AND IN V AR IANTS OF QUADRA TIC FORMS 33 Pr o of. (of the theorem). Let us first consider the non graded case. Without restriction of generality , we ma y assume that A is of constan t rank a nd that P and Q are also of constan t rank suc h tha t A ⊗ End( P ) is isomorphic to End( Q ) . According to the pr evious lemma, w e ma y also assume that P is free of constant rank, say n. Therefore, we hav e an alg ebra isomorphism A ⊗ M n ( R ) ∼ = End( Q ) . Let us now consider the fundamen tal idemp otents in the matrix al- gebra M n ( R ) defined by the diagona l mat r ices with all elemen ts = 0 except one whic h is 1 . Thanks to the previous isomorphism, w e may use these idem p oten ts to split Q a s the direct sum of n copies of E . Since the commutan t of M n ( R ) in A ⊗ M n ( R ) is A, it f o llo ws that the represen tation of A in End( Q ) is the orthog onal sum of n copies of a represen tation ρ from A to End( E ) . F rom the previous algebra isomorphism, w e therefore deduce the required identit y A ∼ = End( E ) . Finally , w e make the ob vious mo difications of the previous arg umen t in the Z / 2- graded case by writing the previous algebra isomor phism in the form A b ⊗ M 2 n ( R ) ∼ = End( Q ) , with the o b vious grading on M 2 n ( R ) . W e use again the fundamen tal idemp oten ts in M 2 n ( R ) in order to split Q as a direct sum E n , where E is Z / 2-graded.  9. Reference s [1] M.F. Atiy ah. 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