The stable Adams operations on Hermitian K-theory
aa r X i v : . [ m a t h . K T ] M a y THE STABLE ADAMS OPERATIONS ON HERMITIAN K -THEORY JEAN FASEL AND OLIVIER HAUTION
Abstract.
We prove that exterior powers of (skew-)symmetric bundles induce a λ -ringstructure on the ring GW ( X ) ⊕ GW ( X ), when X is a scheme where 2 is invertible. Usingthis structure, we define stable Adams operations on Hermitian K -theory. As a byproductof our methods, we also compute the ternary laws associated to Hermitian K -theory. Contents
Introduction 1Acknowledgments 31. Grothendieck-Witt groups and spectra 32. Exterior powers and rank two symplectic bundles 53. Grothendieck-Witt groups of representations 74. The λ -ring structure 104.1. The pre- λ -ring structure 104.2. The λ -ring structure 125. Stable Adams operations 135.1. Adams operations on universal bundles 145.2. Stable Adams operations 156. Ternary laws for Hermitian K -theory 18Appendix A. Some polynomial identities 20A.1. The polynomials P n Q i,j R n Introduction
From their introduction by Adams in his study of vector fields on spheres [Ada62], Adamsoperations have been extremely useful in solving various problems in topology, algebra andbeyond. One may for instance consider the proof of Serre vanishing conjecture by Gillet-Soul´e[GS87], or their use in intersection theory. In algebraic geometry, the work of several authorsallowed to extend these operations (initially defined at the level of the Grothendieck group K ) to the whole world of K -theory; the most recent and probably most natural extensionbeing due to Riou [Rio10] using the world of (stable) motivic homotopy theory. Date : May 19, 2020.This work was supported by the DFG grant HA 7702/5-1 and Heisenberg fellowship HA 7702/4-1.
Over a scheme X , it is often useful to study vector bundles endowed with some extradecoration, such a symmetric or a symplectic form. The analogues of the Grothendieck group K ( X ) in this context are the so-called Grothendieck-Witt groups GW i ( X ) for i ∈ Z / K -theory can be generalizedin the context of Grothendieck-Witt groups and Hermitian K -theory. For instance, Serre’sVanishing Conjecture make sense in this broader context [FS08].As for Adams operations, Zibrowius [Zib15, Zib18] has proved that the exterior poweroperations on symmetric bundles yield a λ -ring structure on the Grothendieck-Witt groupGW ( X ) of any smooth variety X over a field of characteristic not two. This provides inparticular Adams operations on these groups. It is not very difficult to construct λ -operationsin GW ( X ), and a good part of [Zib15, Zib18] consists in showing this pre- λ -ring is actuallya λ -ring. In other words, it is not so difficult to construct the Adams operations, but muchharder to show that they are multiplicative. To achieve this, Zibrowius followed the strategyused in [BGI71] for the analog problem in K -theory, and reduced the question to provingthat the symmetric representation ring GW ( G ) of an affine algebraic group G (over a fieldof characteristic not two) is a λ -ring. This is done by further reducing to the case when G is the split orthogonal group, and using explicit descriptions of the representations of certainsubgroups in that case.A first purpose of this paper is to extend the construction of Zibrowius in two directions:(1) allow X to be an arbitrary Z [ ]-scheme admitting an ample family of line bundles,(2) replace GW ( X ) with GW ± ( X ) = GW ( X ) ⊕ GW ( X ).The first objective is achieved by showing that the map GW ( G ) → GW ( G Q ) is injective,when G is a split semisimple simply connected algebraic group over Z [ ]. Since the targetis a λ -ring by the results of Zibrowius, so is GW ( G ), and thus also GW ( X ) when X is a Z [ ]-scheme admitting an ample family of line bundles.For (2), a natural strategy is to mimic Zibrowius’s proof, by considering not just symmetricrepresentations of algebraic groups, but also antisymmetric ones. Although we believe thatthis idea might work, we were not able to implement it satisfyingly. Instead we observe thatwe may pass from GW ( X ) to GW ( X ) using the quaternionic projective bundle theorem.The next natural step consists in considering the groups GW i ( X ) for i odd, as well as thehigher Grothendieck-Witt groups GW ij ( X ) for j ∈ N . For this, we focus on Adams operations,and follow the approach of Riou to construct a stable version of those. The fact that GW ± ( X )is a λ -ring ended up being a crucial input, allowing us to understand the behaviour of theAdams operations with respect to stabilisation. This approach is carried out in Section 5,where we build a morphism of ring spectra for any odd integerΨ i : GW → GW h i i where the left-hand side is the spectrum representing Hermitian K -theory and the right-handside is the same after inversion of i . The same procedure could be used to construct operationsΨ i for i even, but these operations turn out to be hyperbolic, and thus less interesting. Inany case, these operations extend the Adams operations on K -theory, in the sense that there HE STABLE ADAMS OPERATIONS ON HERMITIAN K -THEORY 3 is a commutative diagram of ring spectra GW Ψ i / / (cid:15) (cid:15) GW [ i ] (cid:15) (cid:15) BGL Ψ i / / BGL [ i ]in which the vertical morphisms are the forgetful maps and the bottom horizontal morphismis the Adams operation on K -theory defined by Riou (see [Rio10, Definition 5.3.2]).We expect that these operations will be useful in many situations. For instance, Bachmannand Hopkins recently used them in [BH20] to compute the η -inverted homotopy sheaves ofthe algebraic symplectic and special linear cobordism spaces. Their construction of Adamsoperation is quite different in spirit to the one presented here, but we believe that the twoconstructions coincide.Our initial motivation was to solve the vanishing conjecture stated in [FS08], but we areunable to do it at the moment. We still offer an application in the last section of this paperunder the form of the computation of the ternary laws associated to Hermitian K -theory.These laws are the analogue of the formal group laws associated to any oriented ring spectrumin the context of Sp-oriented ring spectra. In short, they express the characteristic classes ofa threefold product of symplectic bundles of rank 2 and are expected to play an importantrole in the classification of Sp-oriented cohomology theories. We refer the interested readerto [DF19] for more information on these laws. Acknowledgments.
The first named author is grateful to Aravind Asok, Baptiste Calm`esand Fr´ed´eric D´eglise for useful discussions. Both authors warmly thank Alexey Ananyevskiyfor sharing a preprint on Adams operations, which has been a source of inspiration for theresults of the present paper, and Tom Bachmann for useful suggestions.1.
Grothendieck-Witt groups and spectra
Let X be a scheme over Z [ ] (with an ample family of line bundles). Following [Sch17,Definition 9.1], we can consider the Grothendieck-Witt groups GW ij ( X ) for any j ∈ Z andany i ∈ Z /
4. For X affine and i = 0, the groups GW j ( X ) are the orthogonal K -theory groupsKO j ( X ) as defined by Karoubi, while for i = 2 (and X still affine) the groups GW j ( X ) are thesymplectic K -theory groups KSp j ( X ) (see [Sch17, Corollary A.2]). The collection of groupsGW ij ( X ) fit in a well-behaved cohomology theory, and the functors X GW ij ( X ) are actuallyrepresentable by explicit (geometric) spaces in the A -homotopy category H ( Z [ ]) of Morel-Voevodsky (see [ST15, Theorem 1.3]). These explicit spaces permit to construct an explicit P -spectrum representing higher Grothendieck-Witt groups (see for instance [PW10a]). Webriefly sketch its construction to fix notation.Recall first that Panin and Walter [PW10b] defined a smooth affine Z [ ]-scheme H P n forany n ∈ N . On H P n , there is a canonical bundle U which is symplectic of rank 2, yieldinga canonical element U ∈ GW (H P n ). We will denote by ϕ the symplectic form on U in thesequel. For any n ∈ N , there are morphisms i n : H P n → H P n +1 JEAN FASEL AND OLIVIER HAUTION such that i ∗ n U = U , whose colimit (say in the category of sheaves of sets) is denoted by H P ∞ .It is a geometric model of BSp . As H P = Spec( Z [ ]), we consider all these schemes aspointed by i . Recall moreover that H P is weak-equivalent to ( P ) ∧ . In fact H P = Q ,where the latter is the affine scheme defined in [ADF16]. Notation 1.1.
We set T := H P , that we consider as pointed by i . We also denote by Ω T the right adjoint of the endofunctor T ∧ ( − ) of H ( Z [ ]).Next, let OGr be the orthogonal Grassmannian of Schlichting-Tripathi [ST15]. As alreadymentioned above, this space represents orthogonal K -theory in the unstable homotopy cate-gory H ( Z [ ]), i.e. we have for any smooth Z [ ]-scheme[ X, Z × OGr] = K O( X ) = GW ( X ) . Moreover, we have identifications for any integer n Ω n T ( Z × OGr) ≃ Z × OGr . On the other hand, we may consider the hyperbolic Grassmannian HGr of Panin-Walter. Ithas the property to represent symplectic K -theory in H ( Z [ ]), in the sense that[ X, Z × HGr] = K Sp( X ) = GW ( X )for any smooth Z [ ]-scheme X . Moreover, we have equivalences Ω T ( Z × HGr) ≃ Z × OGrand Ω T ( Z × OGr) ≃ Z × HGr. This periodicity allows us to define a T -spectrum with terms Z × OGr in degrees 2 n , and Z × OGr in degrees 2 n + 1, where bonding maps given by theabove equivalences. In more precise terms, the bonding maps in degrees 2 n are the adjointsof the periodicity maps Z × OGr ≃ Ω H ( Z × HGr)given for any smooth Z [ ]-scheme X as the isomorphismGW ( X ) → GW ( X + ∧ H )obtained by the multiplication with the class of U − H − , where U is the universal rank twosymplectic bundle on H and H − is the rank two hyperbolic bundle. The description of thebonding maps in odd degrees is obtained in a similar manner, switching OGr and HGr. Definition 1.2.
We denote by GW the T -spectrum defined above. Remark . In fact, the definition we just gave of GW is that of a naive T -spectrum (e.g.[Rio10, § ( Z [ ]) = KO ( Z [ ]) ≃ Z / × Z / T -spectrum lifts to a genuine T -spectrum which is unique up to a unique isomorphism.For a smooth Z [ ]-scheme X , we then haveGW ij ( X ) = [Σ ∞T X + , Σ − jS Σ i T GW ] SH ( Z [ ]) . If now X is a regular Z [ ]-scheme with structural morphism p X : X → Spec( Z [ ]), we canconsider the functor p ∗ X : SH ( Z [ ]) → SH ( X ) and the spectrum p ∗ X GW . On the other hand,one can consider the T X -spectrum GW X representing Grothendieck-Witt groups in the stablecategory SH ( X ). It follows from [PW10a, discussion before Theorem 13.5] that the naturalmap p ∗ X GW → GW X is in fact an isomorphism. Consequently,GW ij ( X ) = [Σ ∞T X + , Σ − jS Σ i P p ∗ X GW ] SH ( X ) and we say that GW is an absolute H -spectrum over Z [ ]. HE STABLE ADAMS OPERATIONS ON HERMITIAN K -THEORY 5 Exterior powers and rank two symplectic bundles
In this section, we fix a Z [ ]-scheme X . When V is a vector bundle on X , we denote itsdual by V ∨ . A bilinear form on V is a morphism ν : V → V ∨ . When x, y ∈ H ( X, V ), wewill sometimes write ν ( x, y ) instead of ν ( x )( y ). We will abuse notation, and denote by V n ν ,for n ∈ N , the bilinear form on V n V given by the composite V n V ∧ n ν −−−→ V n ( V ∨ ) → ( V n V ) ∨ .We will also denote the pair ( V n V, V n ν ) by V n ( V, ν ). Similar conventions will be used forthe symmetric or tensor powers of bilinear forms, or their tensor products.Explicit formulas for symmetric and exterior powers are given as follows. Let n be aninteger, and denote by ǫ : S n → {− , } the signature homomorphism. Then for any opensubscheme U of X and x , . . . , x n , y , . . . , y n ∈ H ( U, V ), we have(2.a) ( V n ν )( x ∧ · · · ∧ x n , y ∧ · · · ∧ y n ) = X σ ∈ S n ǫ ( σ ) ν ( x , y σ (1) ) · · · ν ( x n , y σ ( n ) ) , (2.b) (Sym n ν )( x · · · x n , y · · · y n ) = X σ ∈ S n ν ( x , y σ (1) ) · · · ν ( x n , y σ ( n ) ) , Lemma 2.1.
Let V be a rank n vector bundle over X equipped with a symplectic form ν .Then for every i there is an isometry V i ( V, ν ) ≃ V n − i ( V, ν ) . Proof.
Assuming that X = ∅ , we may write n = 2 m for some integer m (the induced form overresidue field of a closed point is symplectic, and such forms over fields have even dimension).The morphism V ⊗ n ≃ V ⊗ m ⊗ V ⊗ m → V m V ⊗ V m V ∧ m ν ⊗ id −−−−−→ ( V m V ) ∨ ⊗ V m V → O X descends to a morphism λ V : V n V → O X . Let us denote by V ( V, ν ) = ( V V, V ν ) the orthog-onal sum of the forms ( V i V, V i ν ) for i = 0 , . . . , n . The natural morphisms V i V ⊗ V n − i V → V n V yield morphisms V i V → ( V n − i V ) ∨ ⊗ V n V which assemble into a morphism p V : V V → ( V V ) ∨ ⊗ V n V . Consider the morphism s V : V V p V −−→ ( V V ) ∨ ⊗ V n V id ⊗ λ −−−→ ( V V ) ∨ ( ∧ ν ) − −−−−→ V V. If V ′ is another vector bundle of constant rank over X equipped with a symplectic form ν ′ ,the identification V ( V ⊕ V ′ ) ≃ V V ⊗ V V ′ yields an identification s V ⊕ V ′ = s V ⊗ s V ′ .Since s V ( V i V ) ⊂ V n − i V for all i , it will suffice to prove that s V yields an isometry V ( V, ν ) ≃ V ( V, ν ). To do so, we may assume that X is the spectrum of a local ring. In thiscase the symplectic form ( V, ν ) is hyperbolic [MH73, I, (3.5)]. Given the behaviour of s V withrespect to direct sums, we may assume that V is a hyperbolic plane. So there are v , v ∈ V such that ν ( v , v ) = 0 , ν ( v , v ) = 0 and ν ( v , v ) = 1 . For v ∈ V V , the element s V ( v ) is characterized by the condition ν ( s V ( v ) , w ) = λ ( v ∧ w ) forall w ∈ V V . We have λ V ( v ∧ v ) = 1, so that s V (1) = v ∧ v , s V ( v ) = v , s V ( v ) = v , s V ( v ∧ v ) = 1 , and it follows easily that s V is an isometry. (cid:3) JEAN FASEL AND OLIVIER HAUTION
Let V be a vector bundle over X . Consider the involution σ of V ⊗ exchanging the twofactors. Set V ⊗ = ker( σ − id) and V ⊗ − = ker( σ + id). Since 2 is invertible we have a directsum decomposition V ⊗ = V ⊗ ⊕ V ⊗ − .Let now ν be a bilinear form on V . There are induced bilinear forms ν ⊗ on V ⊗ and ν ⊗ − on V ⊗ − . Writing ( V, ν ) ⊗ , resp. ( V, ν ) ⊗ − , instead of ( V ⊗ , ν ⊗ ), resp. ( V ⊗ − , ν ⊗ − ), we have anorthogonal decomposition(2.c) ( V, ν ) ⊗ = ( V, ν ) ⊗ ⊥ ( V, ν ) ⊗ − . Lemma 2.2.
There are isometries ( V, ν ) ⊗ ≃ h i Sym ( V, ν ) and ( V, ν ) ⊗ − ≃ h i V ( V, ν ) . Proof.
It is easy to see that the morphism i : V V → V ⊗ ; v ∧ v v ⊗ v − v ⊗ v , induces an isomorphism V V ≃ V ⊗ − . If U is an open subscheme of X and v , v , w , w ∈ H ( U, V ), we have, using (2.a) ν ⊗ ( i ( v ∧ v ) , i ( w ∧ w ))= ν ⊗ ( v ⊗ v − v ⊗ v , w ⊗ w − w ⊗ w )= ν ( v , w ) ν ( v , w ) − ν ( v , w ) ν ( v , w ) − ν ( v , w ) ν ( v , w ) + ν ( v , w ) ν ( v , w )=2 ν ( v , w ) ν ( v , w ) − ν ( v , w ) ν ( v , w )=2( V ν )( v ∧ v , w ∧ w ) , proving the second statement. The first is proved in a similar fashion, using the morphismSym V → V ⊗ ; v v v ⊗ v + v ⊗ v . (cid:3) Lemma 2.3.
There is an isometry ( V, ν ) ⊗ ≃ h i Sym ( V, ν ) ⊥ h i V ( V, ν ) . Proof.
This follows from Lemma 2.2 and (2.c). (cid:3)
Lemma 2.4.
Let
E, F be vector bundles over X , respectively equipped with bilinear forms ε, ϕ . Then there is an isometry V ( E ⊗ F, ε ⊗ ϕ ) ≃ h i (cid:16) Sym ( E, ε ) ⊗ V ( F, ϕ ) (cid:17) ⊥ h i (cid:16)V ( E, ε ) ⊗ Sym ( F, ϕ ) (cid:17) . Proof.
It is easy to see that there is an isometry( E ⊗ F, ε ⊗ ϕ ) ⊗ − ≃ (cid:16) ( E, ε ) ⊗ − ⊗ ( F, ϕ ) ⊗ (cid:17) ⊥ (cid:16) ( E, ε ) ⊗ ⊗ ( F, ϕ ) ⊗ − (cid:17) , so that the statement follows by five applications of Lemma 2.2. (cid:3) Proposition 2.5.
Let
E, F be rank two vector bundles over X equipped with symplectic forms ε, ϕ . Then we have in GW ( X ) : [ V n ( E ⊗ F, ε ⊗ ϕ )] = [( E, ε ) ⊗ ( F, ϕ )] if n ∈ { , } , [( E, ε ) ⊗ ] + [( F, ϕ ) ⊗ ] − if n = 2 , if n ∈ { , } , otherwise. HE STABLE ADAMS OPERATIONS ON HERMITIAN K -THEORY 7 Proof.
The cases n = 0 , n ≥ n = 3 , n = 2. We have in GW ( X )[ V ( E ⊗ F, ε ⊗ ϕ )] = h i [Sym ( E, ε )] + h i [Sym ( F, ϕ )] by 2.4 and 2.1= [(
E, ε ) ⊗ ] − h i + [( F, ϕ ) ⊗ ] − h i by 2.3 and 2.1and h , i = 2 ∈ GW ( Z [ ]). (cid:3) Grothendieck-Witt groups of representations
Let A be a commutative (unital) ring with 2 ∈ A × and let R A ( G ) be the abelian categoryof representations of G which are of finite type (as A -modules). We let P A ( G ) be the fullsubcategory of R A ( G ) whose objects are projective as A -modules. The latter category isexact. If P is an object of P A ( G ), then its dual P ∨ := Hom A ( P, A ) is naturally endowedwith an action of G and thus can be seen as an object of P A ( G ). The morphism of functors ̟ : 1 ≃ ∨∨ is easily seen to be an isomorphism of functors P A ( G ) → P A ( G ), and it followsthat P A ( G ) is an exact category with duality. If A is a field, then P A ( G ) = R A ( G ), and thiscategory is in fact abelian. Let now C A ( G ) be the derived category of bounded complexesof objects of R A ( G ), and D A ( G ) be the derived category of bounded complexes of objectsof P A ( G ). The latter is a triangulated category with duality in the sense of Balmer andtherefore one can consider its (derived) Witt groups W i ( D A ( G )) that we denote by W i ( A ; G )for simplicity. We can also consider the Grothendieck-Witt groups GW i ( D A ( G )) that wesimilarly denote by GW i ( A ; G ). Proposition 3.1.
Suppose that F is a field. For any i ∈ Z , we have W i +1 ( F ; G ) = 0 . Proof.
In view of the above remark, the category D A ( G ) is the derived category of an abeliancategory. We can thus apply [BW02, Proposition 5.2]. (cid:3) We now suppose that A is a Dedekind domain with quotient field K (we assume that A = K ). Any object of D A ( G ) has a well defined support, and we can consider the (full)subcategory D flA ( G ) of D A ( G ) whose objects are supported on a finite number of closed pointsof Spec( A ). This is a thick subcategory and we can consider the exact sequence of triangulatedcategories D flA ( G ) → D A ( G ) → D A ( G ) /D flA ( G ) . Note that the duality on D A ( G ) induces a duality on D flA ( G ), i.e. the dual of an object in D flA ( G ) is also in D flA ( G ). As a consequence of [Bal05, Theorem 1.4.14], we obtain a 12-termperiodic long exact sequence(3.a) · · · → W i ( D flA ( G )) → W i ( A ; G ) → W i ( D A ( G ) /D flA ( G )) → W i +1 ( D flA ( G )) → · · · We now identify the category D A ( G ) /D flA ( G ). Note that the extension of scalars inducesa functor D A ( G ) → D K ( G ) which is trivial on the subcategory D flA ( G ). We thus obtain afunctor D A ( G ) /D flA ( G ) → D K ( G ) . Lemma 3.2.
The functor D A ( G ) /D flA ( G ) → D K ( G ) is an equivalence. JEAN FASEL AND OLIVIER HAUTION
Proof.
We have a commutative diagram of functors D A ( G ) / / (cid:15) (cid:15) D K ( G ) (cid:15) (cid:15) C A ( G ) / / C K ( G )in which the vertical arrows are equivalences (use [Ser68, § D flA ( G ) → D A ( G ) → C A ( G ) has essential image the subcategory C flA ( G ) of objects of C A ( G )whose homology is of finite length. Now, we have an exact sequence of abelian categories (see[Ser68, Remarque, p.43]) R flA ( G ) → R A ( G ) → R K ( G )where R flA ( G ) is the subcategory of G -modules of finite length. It follows from [Kel99, § C flA ( G ) → C A ( G ) → C K ( G ) . The claim follows. (cid:3)
As a consequence, we see that the exact sequence (3.a) becomes · · · → W i ( D flA ( G )) → W i ( A ; G ) → W i ( K ; G ) → W i +1 ( D flA ( G )) → · · · Next, we identify the category D flA ( G ). For any maximal ideal v ∈ Spec( A ), we have alocalization functor D flA ( G ) → D flA v ( G )and the arguments of [BW02, Proposition 7.1] apply to show that we obtain an equivalence D flA ( G ) → a v ∈ Max( A ) D flA v ( G )which is fact an equivalence of triangulated categories with duality. Now, suppose that M isrepresentation of G over A v that is of finite length. By [Ser68, § → P → P → M → P , P are projective representations (in fact, free). Dualizing, we obtain an exactsequence 0 → P ∨ → P ∨ → Ext A v ( M, A v ) → A v ( M, A v ) is naturally endowed with a structure of a representationover A v . Using the usual argument, we see that M ♯ := Ext A v ( M, A v ) defines a duality on thecategory of finite length representations over A v . Rewriting the arguments of [BW02, § k ( v ) = A/v )W i +1 ( D flA ( G )) ≃ M v ∈ Max( A ) W i ( k ( v ); G ) . In particular, Proposition 3.1 yields W i ( D flA ( G )) = 0 for any i ∈ Z . HE STABLE ADAMS OPERATIONS ON HERMITIAN K -THEORY 9 Theorem 3.3.
For any i ∈ Z , the natural map R A ( G ) → R K ( G ) induces an exact sequence → W i ( A ; G ) → W i ( K ; G ) → M v ∈ Max( A ) W i ( G k ( v ) ) → W i +1 ( A ; G ) → . We will make use of the following theorem, due to Calm`es and Hornbostel [CH04, Theorem0.1] (see also [Zib15, Theorem 1.4]).
Theorem 3.4.
Let G be a split reductive group over a field F of characteristic different from . There is an injective homomorphism W ( F ; G ) ⊕ W ( F ; G ) → W( F ) ⊗ Z Z [( X ) + ] where ( X ) + is the set of dominant weights fixed by the duality. If G is semisimple and simplyconnected, the homomorphism is an isomorphism. Corollary 3.5.
Suppose that G is a split semisimple and simply connected group. Then, W i +1 ( A ; G ) = 0 .Proof. In view of Theorem 3.3, it suffices to prove that the homomorphismW i ( K ; G ) → M v ∈ Max( A ) W i ( k ( v ); G )is surjective. Now, the residue homomorphismW( K ) → M v ∈ Max( A ) W( k ( v ))is surjective, hence so is the induced homomorphismW( K ) ⊗ Z Z [( X ) + ] → M v ∈ Max( A ) W( k ( v )) ⊗ Z Z [( X ) + ] . To conclude, it suffices to check that the two homomorphisms are compatible, which isstraightforward. (cid:3)
Corollary 3.6.
Assume that A is a principal ideal domain (with invertible). Let G be asplit semisimple and simply connected algebraic group over A . For any i ∈ Z , the naturalmap R A ( G ) → R K ( G ) induces a short exact sequence → GW i ( A ; G ) → GW i ( K ; G ) → M v ∈ Max( A ) W i ( k ( v ); G ) → . Proof.
We have a commutative diagram of short exact sequencesGW i − ( A ; G ) / / (cid:15) (cid:15) K ( A ; G ) / / (cid:15) (cid:15) GW i ( A ; G ) / / (cid:15) (cid:15) W i ( A ; G ) / / (cid:15) (cid:15) i − ( K ; G ) / / K ( K ; G ) / / GW i ( K ; G ) / / W i ( K ; G ) / / K ( A ; G ) (resp. K ( K ; G )) denotes the Grothendieck group of the abelian category R A ( G ) (resp. R K ( G )). Consider next the commutative diagram with exact rowsK ( A ; G ) / / (cid:15) (cid:15) GW i − ( A ; G ) / / (cid:15) (cid:15) W i − ( A ; G ) (cid:15) (cid:15) K ( K ; G ) / / GW i − ( K ; G ) / / W i − ( K ; G )The group W i − ( K ; G ) is trivial by Proposition 3.1, while W i − ( A ; G ) = 0 by Corollary 3.5.Thus, the first diagram becomes a diagram whose rows are exactK ( A ; G ) / / (cid:15) (cid:15) K ( A ; G ) / / (cid:15) (cid:15) GW i ( A ; G ) / / (cid:15) (cid:15) W i ( A ; G ) / / (cid:15) (cid:15) ( K ; G ) / / K ( K ; G ) / / GW i ( K ; G ) / / W i ( K ; G ) / / ( A ; G ) → K ( K ; G ) is an isomorphism by [Ser68, Th´eor`eme 5]. We concludeusing Theorem 3.3. (cid:3) The λ -ring structure Let X be a Z [ ]-scheme and G an affine algebraic group over Z [ ]. We denote by GW i ( X ; G )the Grothendieck-Witt groups of the category of G -equivariant vector bundles over X andset GW ± ( X ; G ) = GW ( X ; G ) ⊕ GW ( X ; G ). When no confusion seems likely to arise, weuse the notation GW ± ( X ) = GW ± ( X ; 1) and GW ± ( G ) = GW ± (Spec Z [ ]; G ).4.1. The pre- λ -ring structure.Proposition 4.1. Let X be a Z [ ] -scheme and G an affine algebraic group over Z [ ] . Thenthe exterior powers operations λ i : GW n ( X ; G ) → GW ni ( X ; G ) defined by ( P, ϕ ) ( V i P, V i ϕ ) endow the ring GW ± ( X ; G ) with the structure of a pre- λ -ring.Proof. This is essentially the same proof as [Zib15, Proposition 2.1]. Step 1 and Step 2 areclear. Let us explain Step 3. Let M be a vector bundle over X equipped with a G -action andan equivariant nondegenerate symmetric or antisymmetric bilinear form µ . Assume that L isa Lagrangian in ( M, µ ), i.e. that we have an exact sequence of G -equivariant vector bundlesover X → L i −→ M i ∨ ◦ µ −−−→ L ∨ → . Then V n M is equipped with a filtration by G -invariant subbundles( V n M ) i = im( V i L ⊗ V n − i M → V n M )fitting into commutative squares V i L ⊗ V n − i M / / (cid:15) (cid:15) ( V n M ) i (cid:15) (cid:15) V i L ⊗ V n − i L ∨ / / ( V n M ) i / ( V n M ) i +1HE STABLE ADAMS OPERATIONS ON HERMITIAN K -THEORY 11 where the bottom horizontal arrow is an isomorphism (see e.g. [BGI71, V, Lemme 2.2.1]).This fact, combined with the exact sequences(4.a) 0 → ( V n M ) i / ( V n M ) i +1 → V n M/ ( V n M ) i +1 → V n M/ ( V n M ) i → i that ( V n M ) i is an admissible subbundle of V n M (i.e. the quotient V n M/ ( V n M ) i is a vector bundle). Assuming that L has rank r , then ( V n M ) i / ( V n M ) i +1 has rank (cid:0) ri (cid:1)(cid:0) rn − i (cid:1) . By induction, using the sequences (4.a), we obtainrank( V n M ) i = r X j = i (cid:18) rj (cid:19)(cid:18) rn − j (cid:19) . It follows that(4.b) rank( V n M ) i + rank( V n M ) n +1 − i = rank V n M. Let i, j be integers. We have a commutative diagram V i L ⊗ V n − i M / / / / α (cid:15) (cid:15) ( V n M ) i (cid:31) (cid:127) / / (cid:15) (cid:15) V n M ∧ n µ (cid:15) (cid:15) ( V j L ⊗ V n − j M ) ∨ (( V n M ) j ) ∨ ? _ o o ( V n M ) ∨ o o o o where α is defined by setting, for every open subscheme U of X and x , . . . , x i , y , . . . , y j ∈ H ( U, L ) and x i +1 , . . . , x n , y j +1 , . . . , y n ∈ H ( U, M ), α ( x ∧· · ·∧ x i ⊗ x i +1 ∧· · ·∧ x n )( y ∧· · ·∧ y j ⊗ y j +1 ∧· · ·∧ y n ) = X σ ∈ S n ǫ ( σ ) µ ( x , y σ (1) ) · · · µ ( x n , y σ ( n ) ) . If i + j > n , then for each σ ∈ S n there is e ∈ { , . . . , n } such that x e ∈ H ( U, L ) and y σ ( e ) ∈ H ( U, L ), so that ν ( x e , y σ ( e ) ) = 0, and α = 0. Thus ( V n M ) i ⊂ (( V n M ) j ) ⊥ , and inparticular ( V n M ) i is a totally isotropic subspace when 2 i > n .If n = 2 k − ≥
0, then 2 rank( V n M ) k = rank V n M by (4.b), hence ( V n M ) k ⊂ V n M is alagrangian. Thus V n ( M, µ ) = H (( V n M ) k ) ∈ GW ± ( G )depends only on the class of ( V n M ) k in K ( G ), which by d´evissage depends only on theclasses of V i L ⊗ V n − i L ∨ in K ( G ) for i ≥ k . Therefore V n ( M, µ ) ∈ GW ± ( G ) depends onlyon [ L ] ∈ K ( G ) (the odd integer n being fixed). In particular, V n ( M, µ ) = V n H ( L ), asrequired.Assume now that n = 2 k . Then the inclusion ( V n M ) k ⊂ (( V n M ) k +1 ) ⊥ is an equality byrank reasons (see (4.b)). It follows from the above description of the morphism α that thebilinear form induced by µ on V k L ⊗ V k L ∨ is given by the formula, for every open subscheme U of X and x , . . . , x k , y , . . . , y k ∈ H ( X, L ) and f , . . . , f k , g , . . . , g k ∈ H ( U, L ∨ ), h x ∧ · · · ∧ x k ⊗ f k +1 ∧ · · · ∧ f n , y ∧ · · · ∧ y k ⊗ g k +1 ∧ · · · ∧ g n i = X σ,τ ∈ S k ǫ ( σ ) ǫ ( τ ) g ( x σ (1) ) · · · g k ( x σ ( k ) ) f ( y τ (1) ) · · · f k ( y τ ( k ) ) , and in particular depends only on L . We conclude as above, using [QSS79, Lemma 5.3]. (cid:3) The λ -ring structure. Let us recall a construction from [BGI71, V, § § λ -ring”/“special λ -ring” is used instead of “pre- λ -ring”/“ λ -ring”. Let R be a (commutative unital) ring. Onedefines a ring Λ( R ), whose underlying set is 1 + tR [[ t ]]. The addition in Λ( R ) is given bymultiplication of power series, while multiplication in Λ( R ) is given by the formula (cid:16) X n ∈ N f n t n (cid:17)(cid:16) X n ∈ N g n t n (cid:17) = X n ∈ N P n ( f , . . . , f n , g , . . . , g n ) t n , where P n are certain universal polynomials defined (A.1.a). A pre- λ -ring structure on Λ( R )is defined by setting for j ∈ N − { } λ j (cid:16) X n ∈ N f n t n (cid:17) = X i ∈ N Q i,j ( f , . . . , f ij ) t i , where Q i,j are certain universal polynomials defined (A.2.a).A pre- λ -structure on R yields a morphism of abelian groups λ t : R → Λ( R ) ; r X n ∈ N λ n ( r ) t n . The pre- λ -ring R is called a λ -ring if λ t is a morphism of pre- λ -rings. This amounts to thefollowing relations, for all n, i, j ∈ N − { } :(4.2.c) λ n ( xy ) = P n ( λ ( x ) , . . . , λ n ( x ) , λ ( y ) , . . . , λ n ( y )) for x, y ∈ R, (4.2.d) λ i ( λ j ( z )) = Q i,j ( λ ( z ) , . . . , λ ij ( z )) for z ∈ R. Note that if E is a subset of R such that (4.2.c) and (4.2.d) are satisfied for all x, y, z ∈ E ,then (4.2.c) and (4.2.d) are satisfied for all x, y, z lying in the subgroup generated by E in R .Note also that if R is a λ -ring, and x, y, z ∈ R , it follows from Lemma A.3.1 that(4.2.e) λ n ( xyz ) = R n ( λ ( x ) , . . . , λ n ( x ) , λ ( y ) , . . . , λ n ( y ) , λ ( z ) , . . . , λ n ( z )) , where R n is a polynomial defined in § A.3.
Proposition 4.2.1.
Let G be a split semisimple and simply connected algebraic group over Z [ ] . Then the pre- λ -ring GW ( G ) is a λ -ring.Proof. By [Zib15, Proposition 2.1] the pre- λ -ring GW ( G Q ) is a λ -ring. It follows fromTheorem 3.3 that GW ( G ) is a pre- λ -subring of GW ( G Q ), hence a λ -ring. (cid:3) Corollary 4.2.2.
For every Z [ ] -scheme X , the pre- λ -ring GW ( X ) is a λ -ring.Proof. This follows from Proposition 4.2.1, using the arguments of [BGI71, Expos´e VI,Th´eor`eme 3.3] (see [Zib18, § (cid:3) When x ∈ GW ( X ) is the class of a rank two symplectic bundle, it follows from Lemma 2.1that λ t ( x ) = 1 + tx + t . In other words, in the notation of (A.1.b), we have(4.2.f) λ i ( x ) = ℓ i ( x ) ∈ GW ± ( X ) for all i ∈ N − { } . Lemma 4.2.3.
The relations (4.2.c) and (4.2.d) are satisfied for all x, y, z ∈ GW ( X ) .Proof. By the symplectic splitting principle [PW10b, § x, y, z areeach represented by a rank two symplectic bundle. In view of (4.2.f), the relation (4.2.d)follows from Lemma A.2.1. The relation (4.2.c) has been verified in Proposition 2.5, seeLemma A.1.1. (cid:3) HE STABLE ADAMS OPERATIONS ON HERMITIAN K -THEORY 13 Theorem 4.2.4.
The pre- λ -ring GW ± ( X ) is a λ -ring.Proof. Taking Proposition 4.2.1 and Lemma 4.2.3 into account, it only remains to verify(4.2.c) when x ∈ GW ( X ) and y ∈ GW ( X ). Let i ≥ n , and consider the scheme X × H P i .It is endowed with a universal symplectic bundle of rank two, whose class we denote by u ∈ GW ( X × H P i ). Denote again by x, y ∈ GW ± ( X × H P i ) the pullbacks of x, y ∈ GW ± ( X ).Then using successively Proposition 4.2.1 and Lemma 4.2.3 λ t ( xyu ) = λ t ( x ) λ t ( yu ) = λ t ( x ) λ t ( y ) λ t ( u ) . On the other hand, by Lemma 4.2.3 λ t ( xyu ) = λ t ( xy ) λ t ( u ) . Let us consider the decomposition (see [PW10b, Theorem 8.2])GW ± ( X × H P i ) = GW ± ( X ) ⊕ GW ± ( X ) u ⊕ · · · ⊕ GW ± ( X ) u i . In view of (4.2.f), it follows from Lemma A.1.2 that the u n -component of the t n -coefficient of λ t ( xy ) λ t ( u ) is λ n ( xy ), and that the u n -component of the t n -coefficient of λ t ( x ) λ t ( y ) λ t ( u ) is P n ( λ ( x ) , . . . , λ n ( x ) , λ ( y ) , . . . , λ n ( y )). This proves (4.2.c). (cid:3) We now obtain an explicit formula for the λ -operation on products of three classes of ranktwo symplectic bundles, providing a different proof of [Ana17, Lemma 8.2]. This result will beused in our computation in Section 6 of the ternary laws associated to Hermitian K -theory.It will be convenient to have a basis for the symmetric polynomials in three variables u , u , u . Following [DF19, § σ ( u i u j u k ) = X ( a,b,c ) u a u b u c where the sum runs over the monomials u a u b u c in the orbit of u i u j u k under the action ofthe permutation of the variables. Lemma 4.2.5.
Let u , u , u ∈ GW ( X ) be the classes of rank two symplectic bundles. Then λ i ( u u u ) = u u u if i = 1 . σ ( u u ) − σ ( u ) + 4 if i = 2 . σ ( u u u ) − u u u if i = 3 . σ ( u ) + u u u − σ ( u ) + 6 if i = 4 .Proof. In view of (4.2.e) and (4.2.f), this follows from Lemma A.3.2. (cid:3) Stable Adams operations
The exterior power operations λ i : GW ( X ) → GW ( X ) considered in the previous sectionyield morphisms of spaces λ i : ( Z × OGr) → ( Z × OGr)as well as morphism of spaces λ i : ( Z × HGr) → ( Z × HGr) when i is odd. When i is even,we obtain instead morphism of spaces λ i : ( Z × HGr) → ( Z × OGr). These operations are notadditive, and there is a standard procedure to obtain additive operations out of λ i . Indeed,following [Rio10, Remark 6.2.2.3], we define operations ψ i through the inductive formula ψ := λ = id and ψ n − λ ψ n − + λ ψ n − + · · · + ( − n − λ n − ψ + ( − n nλ n = 0 for n ≥
2. For instance, this yields ψ = λ ψ − λ = (id) − λ , etc.5.1. Adams operations on universal bundles.
In this section, we consider the universalbundle U over H P , and denote again by U its class in GW (H P ). For any Z [ ]-scheme X , wealso denote by H + ∈ GW ( X ) and H − ∈ GW ( X ) the classes of the hyperbolic bundles (ofrank 2). A direct computation shows that ( H − ) ⊗ = 2 H + = ( H + ) ⊗ , and ( H − )( H + ) = 2 H − . Lemma 5.1.1.
We have U = 2 H + ( U ) − H + in GW (H P ) .Proof. The first Borel class of U in GW (H P ) is U − H − . The quaternionic projective bundletheorem [PW10b, Theorem 8.2] asserts that ( U − H − ) = 0 in GW (H P ), and the statementfollows. (cid:3) Note that if X is a Z [ ]-scheme and e ∈ GW ( X ) is the class of a rank two symplecticbundle, it follows from (4.2.f) that ψ ( e ) = e − ψ i ( e ) = eψ i − ( e ) − ψ i − ( e ) for i > . Lemma 5.1.2.
For i ≥ , we have ψ i ( H − ) = H − if i is odd. H + − if i ≡ . if i ≡ .Proof. A direct computation shows that the result is true for i ≤
4. We work by inductionon i . Suppose that i ≥ j ≤ i −
1. Then, using H − (2 H + −
2) = 2 H − , we get ψ i ( H − ) = H − ψ i − ( H − ) − ψ i − ( H − ) = 2 H − − H − = H − . Suppose next that i = 4 r and that the result is true for j ≤ i −
1. Then ψ i ( H − ) = H − ψ i − ( H − ) − ψ i − ( H − ) = 2 H + − (2 H + −
2) = 2 . The last case is obtained via a similar reasoning. (cid:3)
Definition 5.1.3.
For any odd i ∈ N , we set ω ( i ) := h ( − i ( i − / i i · i ǫ in GW ( Z ), where i ǫ = P ij =1 h ( − j − i (see [Mor12, Lemma 3.14]). Lemma 5.1.4.
For i ≥ , we have ψ i ( U − H − ) = ( ( U − H − ) · i H − if i is even. ( U − H − ) · ω ( i ) if i is odd.Proof. We proceed again by induction on i , the case i = 1 being obvious. For i = 2, we obtain ψ ( U − H − ) = ψ ( U ) − ψ ( H − ) = U − − H + + 2 = U − H + . We conclude using Lemma 5.1.1. Now, we have ψ i ( U − H − ) = ψ i ( U ) − ψ i ( H − )= U ψ i − ( U ) − ψ i − ( U ) − H − ψ i − ( H − ) + ψ i − ( H − )= U ψ i − ( U − H − ) + U ψ i − ( H − ) − H − ψ i − ( H − ) − ψ i − ( U − H − )= U ψ i − ( U − H − ) + ( U − H − ) ψ i − ( H − ) − ψ i − ( U − H − ) . HE STABLE ADAMS OPERATIONS ON HERMITIAN K -THEORY 15 Suppose then that i is odd and that the result is proved for j ≤ i −
1. As ( U − H − ) = 0, weobtain U ( U − H − ) = H − ( U − H − ) and U ψ i − ( U − H − ) = U ( U − H − ) · ( i − H − = ( U − H − ) · ( i − H + . On the other hand,( U − H − ) ψ i − ( H − ) = ( U − H − ) if i = 1 (mod 4).(2 H + − U − H − ) if i = 3 (mod 4).Using these computations and the fact that elements of GW ( Z ) are classified by their rankand signature, it is easy to conclude that the formula for i odd holds. Suppose now that i iseven and that the result holds for j ≤ i −
1. As above, we have ψ i ( U − H − ) = U ψ i − ( U − H − ) + ( U − H − ) ψ i − ( H − ) − ψ i − ( U − H − ) . Using again U ( U − H − ) = H − ( U − H − ) and H + H − = 2 H − , we obtain U ψ i − ( U − H − ) = ( U − H − )(( i − H − )and ( U − H − ) ψ i − ( H − ) = ( U − H − ) H − . A direct computation allows us to conclude, usingthe explicit expression of ψ i − ( U − H − ). (cid:3) Remark . We are indebted to Tom Bachmann for suggesting the current expression for ω ( i ), which is much simpler than the one we originally used. Corollary 5.1.6.
Let X be a scheme over Z [ ] and let i be an odd integer. Then, the followingdiagram commutes GW ± ( X ) / / ω ( i ) ψ i (cid:15) (cid:15) GW ∓ ( X + ∧ T ) ψ i (cid:15) (cid:15) GW ± ( X ) / / GW ∓ ( X + ∧ T ) . Here, the horizontal maps are multiplication by ( U − H − ) , where U is the universal symplecticbundle on T .Proof. The Adams operations on a λ -ring are multiplicative (see e.g. [AT69, Proposition 5.1]).Thus Theorem 4.2.4 implies that, for any e ∈ GW ± ( X ), ψ i ( e ( U − H − )) = ψ i ( e ) ψ i ( U − H − ) . Lemma 5.1.4 shows that the right-hand term is ω ( i ) ψ i ( e )( U − H − ), which proves the result. (cid:3) Stable Adams operations.
In this section, we produce the spectrum which will bethe target of the stable Adams operations. We start with an easy lemma whose proof is leftto the reader.
Lemma 5.2.1.
Let i be an odd integer. Then ω ( i ) ∈ GW ( Z [ ]) ⊗ Z [ i ] is invertible. This result shows that we are essentially reduced to invert i in order to define our Adamsoperations. We then consider for any odd integer i the category SH ( Z [ ]) ⊗ Z [ i ] obtainedfrom formally inverting i (which is certainly more familiar than the one obtained inverting ω ( i )). This can be obtained via a formal Bousfield localization process, considering the classof morphisms S p,q ∧ X + i −→ S p,q ∧ X + for any smooth scheme X and any integers p, q ∈ Z .In our situation, we may also follow [Bac18, §
6] to obtain a more concrete model for our localization. This allows in any case to obtain a T -spectrum GW [ i ] over Z [ ] having theproperty h Σ ∞T X + , Σ − jS Σ s T (cid:16) GW h i i(cid:17)i SH ( Z [ ]) = GW sj ( X ) ⊗ Z h i i for any j, s ∈ Z and any smooth Z [ ]-scheme X . In case X is regular, the same property holdsusing the spectrum p ∗ X ( GW [ i ]). We are now in position to follow the procedure describedin [DF19, §
4] to construct the stable Adams operations. We first observe that for any oddinteger i ∈ N , the operation ψ i defines a well-defined additive transformation ψ i : Z × HGr → Ω ∞T (cid:16) T ∧ GW h i i(cid:17) . For any n ∈ N , we defineΨ in : Z × HGr → Ω ∞T (cid:16) T ∧ (1+2 n ) ∧ GW h i i(cid:17) as the composite Z × HGr ω ( i ) − (1+2 n ) ψ i −−−−−−−−−→ Ω ∞T (cid:16) T ∧ GW h i i(cid:17) → Ω ∞T (cid:16) T ∧ (1+2 n ) ∧ GW h i i(cid:17) where the second arrow is the obtained via (repeated) periodicity. By Corollary 5.1.6, weobtain for each n ∈ N a commutative diagram in H ( Z [ ]) of the form Z × HGr / / Ψ in (cid:15) (cid:15) Ω T ( Z × HGr) Ω T (Ψ in +1 ) (cid:15) (cid:15) Ω ∞T ( T ∧ (1+2 n ) ∧ ( GW [ i ]) / / Ω T Ω ∞T ( T ∧ (3+2 n ) ∧ ( GW [ i ])and it follows that the collection (Ψ in ) n ∈ N defines a stable operation in the sense of [DF19,Definition 4.2.2]. In view of [DF19, Proposition 4.2.4] this stable operation lifts to a morphismof T -spectra T ∧ GW → T ∧ GW h i i . Proposition 5.2.2.
The morphism of spectra lifting the stable operation given by the collec-tion (Ψ in ) n ∈ N is unique.Proof. By [DF19, Proposition 4.2.4], it is sufficient to show that R lim n ∈ N Hom Ab (cid:16) KSp , T ∧ (1+2 n ) ∧ GW h i i(cid:17) = 0 , which is the content of [PW10a, Theorem 13.1]. (cid:3) This morphism of spectra can be extended to a morphism of spectra p ∗ X ( T ∧ GW ) → p ∗ X (cid:16) T ∧ GW h i i(cid:17) for any Z [ ]-scheme X . In the case of regular schemes, this gives a morphism of spectrarepresenting Higher Grothendieck-Witt groups (with i inverted). HE STABLE ADAMS OPERATIONS ON HERMITIAN K -THEORY 17 Definition 5.2.3.
For any odd integer i and any regular scheme X over Z [ ], we writeΨ i : GW X → GW X h i i for the T -desuspension of the unique morphism of spectra lifting the stable operation (Ψ in ) n ∈ N .We call it the stable i -th Adams operation .We note that by definition the stable Adams operations are additive. Theorem 5.2.4.
For any odd integer i , the stable Adams operation Ψ i is a morphism of ringspectra.Proof. We first prove that the diagram(5.2.a) GW ∧ GW Ψ i ∧ Ψ i / / µ (cid:15) (cid:15) GW [ i ] ∧ GW [ i ] µ (cid:15) (cid:15) GW Ψ i / / GW [ i ]is commutative in SH ( Z [ ]). To see this, we observe that we have a morphism µ : ( Z × HGr) ∧ ( Z × HGr) → T ∧ ∧ GW classifying the element ( U − H − ) ⊗ ( U − H − ) of GW (( Z × HGr) ∧ ( Z × HGr)). For anyintegers m, n ∈ N , we can consider the morphism µ m + n +1) : T ∧ m ∧ ( Z × HGr) ∧ T ∧ n ∧ ( Z × HGr) → T m + n +1) ∧ GW obtained via switching the second and third factors and then applying µ to the last two fac-tors. The composite ( T m + n +1) ∧ Ψ i ) ◦ µ m + n +1) classifies then the element ω ( i ) − m + n +1) ψ i (( U − H − ) ⊗ ( U − H − )).On the other hand, the composite of the tautological map T ∧ m ∧ ( Z × HGr) ∧ T ∧ n ∧ ( Z × HGr) → T ∧ m +1 ∧ GW ∧T ∧ n +1 ∧ GW with ( T ∧ m + n +1) µ ) ◦ ( T ∧ (2 m +1) Ψ i ∧ T ∧ (2 n +1) Ψ i ) classifies the element ω ( i ) − (2 m +1) ψ i ( U − H − ) ⊗ ω ( i ) − (2 n +1) ψ i ( U − H − ) . By Theorem 4.2.4, we have ω ( i ) − m + n +1) ψ i (( U − H − ) ⊗ ( U − H − )) = ω ( i ) − (2 m +1) ψ i ( U − H − ) ⊗ ω ( i ) − (2 n +1) ψ i ( U − H − )and therefore the morphisms µ ◦ (Ψ i ∧ Ψ i ) and Ψ i ◦ µ have the same image under the canonicalmapHom SH (cid:16) GW ∧ GW , GW h i i(cid:17) → lim ←− n Hom(( Z × OGr) ∧ ( Z × OGr) , T ∧ n +2 ∧ GW )of [PPR09, Corollaries 3.4 and 3.5]. We conclude then from [PW10a, Theorem 13.1] thatboth morphisms are equal, i.e. that Diagram (5.2.a) commutes. Next, we show that the diagram commutes Z [ ] ǫ / / ǫ ●●●●●●●● GW Ψ i (cid:15) (cid:15) GW [ i ]where ǫ are the unit maps. This is equivalent to showing thatΨ i (1) = 1which follows from Corollary 5.1.6 and Theorem 4.2.4. (cid:3) For the following result, we note that the stable operation ψ i : GW → GW h i i induces for any n ∈ N a morphism Ψ i : GW [ n ] → GW [ in ] Proposition 5.2.5.
For any odd integers i, j , the composite GW Ψ i −→ GW h i i Ψ j −−→ GW h ij i is equal to Ψ ij .Proof. Following the arguments of the above theorem, the result is a direct consequenceof Theorem 4.2.4 (the Adams operations in a λ -ring satisfy ψ j ◦ ψ i = ψ ij , see e.g. [AT69,Proposition 5.1]). (cid:3) Ternary laws for Hermitian K -theory Recall from [DF19, § U ⊗ U ⊗ U on H P n × H P n × H P n , where U i are the universal bundles onthe respective factors. The ternary laws permit to compute Borel classes of threefold prod-ucts of symplectic bundles, and are likely to play an important role in the classification ofall Sp-oriented cohomology theories. At present, there are few computations of such laws,including MW-motivic cohomology and motivic cohomology which are examples of the so-called additive ternary laws [DF19, Definition 3.3.3]. In this section, we compute the ternarylaws of Hermitian K -theory (and thus also of K -theory as a corollary). These are examplesof ternary laws which are not additive.We will denote by σ i ( X , . . . , X ) ∈ Z [ X , . . . , X ] the elementary symmetric polynomials. Lemma 6.1.
Let X be a Z [ ] -scheme and let e , . . . , e ∈ GW ( X ) be the classes of rank twosymplectic bundles over X . Then λ i ( e + · · · + e ) = σ ( e , . . . , e ) if i = 1 . σ ( e , . . . , e ) + 4 if i = 2 . σ ( e , . . . , e ) + 3 σ ( e , . . . , e ) if i = 3 . σ ( e , . . . , e ) + 2 σ ( e , . . . , e ) + 6 if i = 4 . HE STABLE ADAMS OPERATIONS ON HERMITIAN K -THEORY 19 Proof.
In view of (4.2.f), it suffices to expand the product(1 + te + t )(1 + te + t )(1 + te + t )(1 + te + t ) . (cid:3) Lemma 6.2.
In the ring Z [ x , x , x , x , y ] , we have the following equalities: σ i ( x − y, . . . , x − y ) = σ − y if i = 1 , σ − yσ + 6 y if i = 2 , σ − σ y + 3 σ y − y if i = 3 , σ − σ y + σ y − σ y + y if i = 4 ,where σ i = σ i ( x , . . . , x ) for any i = 1 , . . . , .Proof. Simple computation. (cid:3)
There are hyperbolic maps H + : K ( X ) → GW ( X ) and H − : K ( X ) → GW ( X ) sendingthe class of a vector bundle E to the class of the corresponding hyperbolic bundle (whoseunderlying vector bundle is E ⊕ E ∨ ). We will again denote by H + and H − the compositesGW ± ( X ) → K ( X ) → GW ± ( X ), where the first map is the forgetful map. As in § H + ∈ GW ( X ) and H − ∈ GW ( X ) instead of H + (1) and H − (1). In the nextstatement b GW i denotes the i -th Borel class [PW10b, Definition 8.3]. Proposition 6.3.
Let X be a Z [ ] -scheme and let E be a symplectic bundle of rank on X .Writing e = [ E ] ∈ GW ( X ) , we have: b GW i ( E ) = e − H − if i = 1 . λ ( e ) − H + (cid:0) − e + 12 (cid:1) if i = 2 . λ ( e ) − e + H − (cid:0) − λ ( e ) + 6 e − (cid:1) if i = 3 . λ ( e ) − λ ( e ) + 2 + H + (cid:0) − λ ( e ) + 2 λ ( e ) − e (cid:1) if i = 4 .Proof. Using the symplectic splitting principle [PW10b, § E splits asan orthogonal sum of rank two symplectic bundles, whose classes in GW ( X ) we denote by e , . . . , e . The Borel classes b GW i ( E ) are then given by the elementary symmetric polynomialsin the elements e − H − , . . . , e − H − , which can be computed using Lemma 6.2. For i = 1,the result is immediate. For i = 2, we have σ ( e − H − , . . . , e − H − ) = σ ( e , . . . , e ) − H − ) σ ( e , . . . , e ) + 6( H − ) . We have σ ( e , . . . , e ) = λ ( e ) − H − ) = 2 H + , and( H − ) σ ( e , . . . , e ) = H + ( σ ( e , . . . , e )) = H + ( e ) , the case i = 2 follows. We now pass to the case i = 3. Using Lemma 6.1, we have b GW3 ( E ) = σ ( e , . . . , e ) − H − ) σ ( e , . . . , e ) + 3( H − ) e − H − ) = λ ( e ) − e − H − )( λ ( e ) −
4) + 6( H + ) e − H − = λ ( e ) − e − H − ( λ ( e )) + 6 H − ( e ) − H − . We finish by considering the case i = 4. We have b GW4 ( E ) = σ ( e , . . . , e ) − ( H − ) σ ( e , . . . , e ) + ( H − ) σ ( e , . . . , e ) − ( H − ) e + ( H − ) . Using Lemma 6.1, we find σ ( e , . . . , e ) = λ ( e ) − σ ( e , . . . , e ) − λ ( e ) − λ ( e ) + 2 , ( H − ) σ ( e , . . . , e ) = ( H − )( λ ( e ) − e ) = H + ( λ ( e ) − e ) , ( H − ) σ ( e , . . . , e ) = 2( H + ) σ ( e , . . . , e ) = 2( H + )( λ ( e ) − . Since ( H − ) e = 4( H − ) e = 4 H + ( e ) and ( H − ) = 8 H + , we conclude summing up the previousexpressions. (cid:3) Finally, we are in position to compute the ternary laws of Hermitian K-theory. The com-putation is obtained by combining Proposition 6.3 and Lemma 4.2.5.
Theorem 6.4.
Let E , E and E be symplectic bundles of rank on a Z [ ] -scheme X . Let u , u , u be their respective classes in GW ( X ) . Then the Borel class b GW i ( E ⊗ E ⊗ E ) ∈ GW i ( X ) equals (using the notation of (4.2.g) ) u u u − H − if i = 1 , σ ( u u ) − σ ( u ) + H + (cid:0) − u u u + 12 (cid:1) if i = 2 , σ ( u u u ) − u u u + H − (cid:0) − σ ( u u ) + 4 σ ( u ) + 6 u u u − (cid:1) if i = 3 , σ ( u ) + σ ( u u u ) − σ ( u u ) + H + (cid:0) − σ ( u u u ) + 4 u u u + 2 σ ( u u ) − σ ( u ) + 8 (cid:1) if i = 4 . Appendix A. Some polynomial identities
When U , . . . , U m is a series of variables we denote by σ n ( U ) ∈ Z [ U , . . . , U m ] the elementarysymmetric functions, defined by the formula, valid in Z [ U , . . . , U m ][ t ],(A.0.a) Y ≤ i ≤ m (1 + tU i ) = X n ∈ N t n σ n ( U ) . A.1.
The polynomials P n . By the theory of symmetric polynomials, there are polynomials P n ∈ Z [ X , . . . , X n , Y , . . . , Y n ] such that(A.1.a) Y ≤ i,j ≤ m (1 + tU i V j ) = X n ∈ N t n P n ( σ ( U ) , . . . , σ n ( U ) , σ ( V ) , . . . , σ n ( V ))holds in Z [ U , . . . , U m , V , . . . , V m ][ t ] for every m .Let R be a commutative ring. For every x ∈ R , let us define elements ℓ i ( x ) ∈ R for eachinteger i ≥ ℓ i ( x ) = x if i = 1 , i = 2 , i > . For elements a , . . . , a r ∈ R × , we consider the polynomial(A.1.c) π a ,...,a r ( t ) = Y ε ,...,ε r ∈{ , − } (1 + ta ε · · · a ε n n ) ∈ R [ t ] . These polynomials can be expressed inductively as(A.1.d) π a ,...,a r ( t ) = π a ,...,a r − ( ta r ) · π a ,...,a r − ( ta − r ) . Note that for any a ∈ R × π a ( t ) = 1 + ( a + a − ) t + t , and for any a, b ∈ R × , setting x = a + a − and y = b + b − ,(A.1.e) π a,b ( t ) = 1 + txy + t ( x + y −
2) + t xy + t . HE STABLE ADAMS OPERATIONS ON HERMITIAN K -THEORY 21 Lemma A.1.1.
Let R be a commutative ring and x, y ∈ R . Then P n ( ℓ ( x ) , . . . , ℓ n ( x ) , ℓ ( y ) , . . . , ℓ n ( y )) = if n ∈ { , } ,xy if n ∈ { , } ,x + y − if n = 2 , if n > . Proof.
Consider the ring S = R [ a, a − , b, b − ] / ( x − a − a − , y − b − b − ). Then S contains R .We have σ i ( a, a − ) = ℓ i ( x ) and σ i ( b, b − ) = ℓ i ( y ) for all i , so that, by (A.1.a) and (A.1.c) π a,b ( t ) = X n P n ( ℓ ( x ) , . . . , ℓ n ( x ) , ℓ ( y ) , . . . , ℓ n ( y )) t n . Thus the statement follows from (A.1.e). (cid:3)
Lemma A.1.2.
Let R be a commutative ring and n ∈ N − . Then for every r , . . . , r n ∈ R ,the element P n ( r , . . . , r n , ℓ ( B ) , . . . , ℓ n ( B )) − B n r n ∈ R [ B ] is a polynomial in B of degree ≤ n − .Proof. The ring S = R [ B, A, A − ] / ( B − A − A − ) contains R [ B ]. Since σ i ( A, A − ) = ℓ i ( B )for all i , we have in S [ U , . . . , U m ] m X n =1 P n ( σ ( U ) , . . . , σ n ( U ) , ℓ ( B ) , . . . , ℓ n ( B )) t n = n Y i =1 (1 + tU i A )(1 + tU i A − ) , and thus, in R [ B ][ U , . . . , U m ], m X n =1 P n ( σ ( U ) , . . . , σ n ( U ) , ℓ ( B ) , . . . , ℓ n ( B )) t n = m Y i =1 (1 + tU i B + t U i ) . Expanding the last product, we see that P n ( σ ( U ) , . . . , σ n ( U ) , ℓ ( B ) , . . . , ℓ n ( B )) has leadingterm B n σ n ( U ) as a polynomial in B (in view of (A.0.a)). (cid:3) A.2.
The polynomials Q i,j . By the theory of symmetric polynomials, there are polynomials Q i,j ∈ Z [ X , . . . , X ij ] such that(A.2.a) Y ≤ α < ··· <α j ≤ m (1 + U α · · · U α j t ) = X i ∈ N t i Q i,j ( σ ( U ) , . . . , σ ij ( U ))holds in Z [ U , . . . , U m ][ t ] for every m . Lemma A.2.1.
Let R be a commutative ring and x ∈ R . Then Q i,j ( ℓ ( x ) , . . . , ℓ ij ( x )) = ℓ i ( x ) if j = 1 , if i = 1 and j = 2 , or if i = 0 , otherwise . Proof.
Let S = R [ a, a − ] / ( x − a − a − ). Then S contains R . Setting w = a , w = a − and w k = 0 in S for k >
2, we have σ k ( w ) = ℓ k ( x ) for all k , and X i ∈ N t i Q i,j ( ℓ ( x ) , . . . , ℓ ij ( x )) (A.2.a) = Y ≤ α < ··· <α j ≤ m (1 + w α · · · w α j t ) = tx + t if j = 1 , t if j = 2 , j > (cid:3) A.3.
The polynomials R n . By the theory of symmetric polynomials, there are polynomials R n ∈ Z [ X , . . . , X n , Y , . . . , Y n , Z , . . . , Z n ] such that Y ≤ i,j,k ≤ m (1 + tU i V j W k ) = X n ∈ N t n R n ( σ ( U ) , . . . , σ n ( U ) , σ ( V ) , . . . , σ n ( V ) , σ ( W ) , . . . , σ n ( W ))holds in Z [ U , . . . , U m , V , . . . , V m , W , . . . , W m ][ t ] for every m . Lemma A.3.1.
For n ≤ m , we have in Z [ X , . . . , X m , Y , . . . , Y m , Z , . . . , Z m ] R n = P n ( X , . . . , X n , P ( Y , Z ) , . . . , P n ( Y , . . . , Y n , Z , . . . , Z n )) . Proof.
Observe that, in Z [ U , . . . , U m , V , . . . , V m ][ t ], Y ≤ i,j ≤ m (1 + tU i V j ) = m Y i =1 m Y j =1 (1 + tU i V j ) (A.0.a) = m Y i =1 (cid:16) X n ∈ N σ n ( V ) U ni t n (cid:17) . Since the elements Y r = σ r ( V ) for r = 1 , . . . , m are algebraically independent, in view of(A.1.a) it follows that we have in Z [ U , . . . , U m , Y , . . . , Y m ][ t ], (writing Y s = 0 for s > m )(A.3.a) X n ∈ N P n ( σ ( U ) , . . . , σ n ( U ) , Y , . . . , Y n ) t n = m Y i =1 (cid:16) X n ∈ N Y n U ni t n (cid:17) . Now in Z [ V , . . . , V m , W , . . . , W m ], set for any n ∈ N , p n = P n ( σ ( V ) , . . . , σ m ( V ) , σ ( W ) , . . . , σ m ( W )) , so that, in Z [ U , . . . , U m , V , . . . , V m , W , . . . , W m ][ t ], Y ≤ i,j,k ≤ m (1 + tU i V j W k ) (A.1.a) = m Y i =1 (cid:16) X n ∈ N p n U ni t n (cid:17) (A.3.a) = X n ∈ N P n ( σ ( U ) , . . . , σ n ( U ) , p , . . . , p n ) t n . Since the elements X r = σ r ( U ) , Y r = σ r ( V ) , Z r = σ r ( W ) for r = 1 , . . . , m are algebraicallyindependent, this yields the statement. (cid:3) Lemma A.3.2.
Let R be a commutative ring and x, y, z ∈ R . Then R n ( ℓ ( x ) , . . . , ℓ n ( x ) , ℓ ( y ) , . . . , ℓ n ( y ) , ℓ ( z ) , . . . , ℓ n ( z ))= if n ∈ { , } ,xyz if n ∈ { , } ,x y + x z + y z − x + y + z ) + 4 if n ∈ { , } ,x yz + xy z + xyz − xyz if n ∈ { , } ,x + y + z + x y z − x + y + z ) + 6 if n = 4 , if n > . HE STABLE ADAMS OPERATIONS ON HERMITIAN K -THEORY 23 Proof.
Consider the ring S = R [ a, a − , b, b − , c, c − ] / ( x − a − a − , y − b − b − , z − c − c − ).Then S contains R . We have σ i ( a, a − ) = ℓ i ( x ) , σ i ( b, b − ) = ℓ i ( y ) , σ i ( c, c − ) = ℓ i ( z ) for all i .Writing r n = R n ( ℓ ( x ) , . . . , ℓ n ( x ) , ℓ ( y ) , . . . , ℓ n ( y ) , ℓ ( z ) , . . . , ℓ n ( z )), we have by definition of R n and (A.1.c) π a,b,c ( t ) = X n ∈ N r n t n ∈ S [ t ] . Since π a,b,c ( t ) = π a,b ( tc ) · π a,b ( tc − ) by (A.1.d), it follows from (A.1.e) that π a,b,c ( t ) equals(1 + txyc + t ( x + y − c + t xyc + t c )(1 + txyc − + t ( x + y − c − + t xyc − + t c − ) . To conclude, we compute the coefficients r n by expanding the above product. We have r = r = 1 and r n = 0 for n >
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