Cdh Descent for Homotopy Hermitian K -Theory of Rings with Involution
aa r X i v : . [ m a t h . K T ] S e p CDH DESCENT FOR HOMOTOPY HERMITIAN K -THEORY OF RINGS WITH INVOLUTION DANIEL CARMODY
Abstract.
We provide a geometric model for the classifying space of automorphism groups of Hermitian vectorbundles over a ring with involution R such that ∈ R ; this generalizes a result of Schlichting-Tripathi [SST14].We then prove a periodicity theorem for Hermitian K -theory and use it to construct an E ∞ motivic ring spectrum KR alg representing homotopy Hermitian K -theory. From these results, we show that KR alg is stable under basechange, and cdh descent for homotopy Hermitian K -theory of rings with involution is a formal consequence. Contents
1. Introduction 11.1. Outline 22. Equivariant Topologies and the Equivariant Motivic Homotopy Category 32.1. The Equivariant and Isovariant ´Etale Topologies 42.2. The Equivariant Nisnevich Topology 52.3. The Equivariant cdh Topology 52.4. Computations with Equivariant Spheres 52.5. The Equivariant Motivic Homotopy Category 73. Hermitian Forms on Schemes 83.1. Definitions 83.2. Properties 103.3. Hermitian Forms on Semilocal Rings 113.4. Higher Grothendieck-Witt Groups 134. Representability of Automorphism Groups of Hermitian Forms 144.1. The definition of the Hermitian Grassmannian R Gr 144.2. Representability of R Gr 174.3. The ´Etale Classifying Space 185. Periodicity in the Hermitian K -Theory of Rings with Involution 225.1. A Projective Bundle Formula for P σ GW K -theory 31References 321. Introduction
Algebraic K -theory is an algebraic invariant introduced in the 1950s by Alexander Grothendieck whereit served as the cornerstone of his reformulation of the Riemann-Roch theorem [Gro57]. Twenty yearspreviously, Ernst Witt developed the notion of quadratic forms over arbitrary fields and introduced the Wittring as an object to encapsulate the nature of all the quadratic forms over a given field [Wit37]. Combiningthe ideas of Grothendieck and Witt, Hyman Bass introduced a category of quadratic forms Quad ( R ) withisometries over a ring R and studied K ( Quad ( R )) and K ( Quad ( R )). K ( Quad ( R )) is what we know todayas the Grothendieck-Witt ring, and Bass was able to recover the Witt ring as a quotient of K ( Quad ( R ))by the image of the hyperbolic quadratic forms. He went on to show that K ( Quad ( R )) was related tothe stable structure of the automorphisms of hyperbolic modules, which complemented the relationshipbetween K ( R ) and the group GL ( R ). The K -theory of quadratic forms soon found applications to surgerytheory where the periodic L -groups defined by Wall in 1966 [Wal66] served as obstructions to certain maps being cobordant to homotopy equivalences. When the means to define the higher algebraic K -groups viathe + construction was discovered by Quillen in the 1970s, Karoubi applied it to the orthogonal groups BO in order to define the higher Hermitian K -theory of rings with involution as we know it today [Kar73].Fast forward twenty years into the 1990s when Morel and Voevodsky developed the motivic homotopycategory and proved that algebraic K -theory was representable in the stable motivic homotopy category[MV99]. The development of the stable motivic homotopy category not only gave a new domain to motiviccohomology, it also opened the door for applications of topological tools like obstruction theory to morealgebraic objects. Several subsequent developments inspire our work here.The first set of developments relates to Hermitian K -theory. In 2005 Hornbostel showed that Hermitian K -theory was representable in the stable motivic homotopy category on schemes [Hor05]. We note thatHornbostel defined Hermitian K -theory on schemes by extending the definition on rings using Jouanolou’strick. In 2011 Hu-Kriz-Ormsby showed that Hermitian K -theory on the category of C -schemes over afield is representable in the C -equivariant stable motivic homotopy category [HKO11]. Here they used asimilar trick to Hornbostel in order to extend Hermitian K -theory from rings with involution to schemeswith involution. In the meantime, Schlichting, building o ff of work of Thomason, Karoubi, and Balmer,defined the higher Hermitian K -theory of a dg-category with weak equivalences and duality and provedthe analogues of the fundamental theorems of higher K -theory for these groups [Sch17]. Although someof Schlichting’s theorems are stated only for schemes (rather than schemes with C action), many of hisproofs require only trivial modification to extend to Grothendieck-Witt groups of schemes with C action.See also [Xie18] for the proofs of the equivariant version of some of the theorems together with a newtransfer morphism. Another approach is taken by Hesselholt-Madsen, who define real algebraic K -theoryof a category with weak equivalences and duality as a symmetric spectrum object in the monoidal categoryof pointed C -spaces. Schlichting’s higher Grothendieck-Witt groups can be recovered from the Hesselholt-Madsen construction by taking homotopy groups of C -fixed points of deloopings of the real algebraic K -theory spaces with respect to the sign representation spheres.Back in K -theory land, Cisinski proved that the six functor formalism in motivic homotopy theory de-veloped by Ayoub [Ayo07] together with the fact that the motivic K -theory spectrum KGL is a cocartesiansection of SH( − ) yields a simple proof of cdh-descent for homotopy K -theory [Cis13]. This in turn yieldsa short proof of Weibel’s vanishing conjecture for homotopy K -theory, and inspired work of Kerz, Strunk,and Tamme who solved Weibel’s conjecture by proving pro-cdh descent for ordinary K -theory [KST18].Hoyois in [Hoy16] uses Cisinski’s approach to show cdh descent for equivariant homotopy K -theory.This paper, inspired by the above developments, shows cdh-descent for homotopy Hermitian K -theoryof schemes with C action. The techniques in [Hoy16] provide our pathway to descent. In order to showthat Hermitian K -theory is a cocartesian section of SH C ( − ), we need to show that the Hermitian K -theoryspace Ω ∞ GW can be represented by a certain Grassmannian, and we need a periodization theorem in orderto pass from the Hermitian K -theory space Ω ∞ GW to the homotopy Hermitian K -theory motivic spectrum L A G W . Schlichting and Tripathi [SST14] show that Ω ∞ GW is representable by a Grassmannian overschemes with trivial action over a regular base scheme with 2 invertible. Their techniques extend to theequivariant setting, and with slight modification provide a proof of representability over non-regular bases.The periodization techniques in [Hoy16] extend to Hermitian K -theory by investigating the Hermitian K -theory of T ρ , the Thom space of the regular representation A ρ .1.1. Outline.
Section 2 begins with a review of G -equivariant motivic homotopy theory where G is a finitegroup scheme over a base S which is Noetherian of finite Krull dimension, has an ample family of line bun-dles, and has ∈ Γ ( S, O S ). First we review the definition of the equivariant ´etale and Nisnevich topologies,then we introduce the isovariant ´etale topology and give some examples of covers. For the reader familiarwith non-equivariant motivic homotopy theory, the assumptions we make on G are strong enough so thatstructural results are mostly the same: • the equivariant Nisnevich topology is generated by a nice cd-structure, • equivariant schemes are locally a ffi ne in the equivariant Nisnevich topology, and • to invert G -a ffi ne bundles Y → S it su ffi ces to invert A S .The content in this section is a selection of relevant content from [HKØ15]. We end this section with thedefinition of the unstable and stable equivariant motivic ∞ -categories 2.5 a la Hoyois [Hoy17]. DH DESCENT FOR HOMOTOPY HERMITIAN K -THEORY OF RINGS WITH INVOLUTION 3 Section 3 reviews the definitions and results on Hermitian forms which will be necessary to work withthe Grothendieck-Witt spectrum. Section 3.1 contains the basic definitions and examples, while section 3.3contains the tools necessary to show that Hermitian forms are locally determined by rank in the isovariantor equivariant ´etale topologies. The final section 3.4 reviews the main definitions of [Sch17] to allow us totalk about the Grothendieck-Witt spectra of schemes with involution.Section 4 is where the background material ends and the paper begins in earnest. We combine thetechniques of [SST14] and [Hoy16] in order to show that classifying spaces of automorphism groups ofHermitian vectors bundles are representable in the C -equivariant motivic homotopy category.This section culminates with the representability result, Theorem 4.14, which we note holds non-regularbase schemes: Theorem 1.1.
Let S be a Noetherian scheme of finite Krull dimension with an ample family of line bundles and ∈ S . There is an equivalence of motivic spaces on Sm C S,qp L mot Z × R Gr • ∼ −→ L mot Z × colim n B isoEt O ( H n ) . With a simple modification to remove the regularity hypothesis, one can follow [SST14] to show that L mot Z × colim n B isoEt O ( H n ) ∼ −→ L mot Ω ∞ GW but as this is unnecessary for proving cdh descent, we leave it out of this paper.Section 5 provides a convenient way of passing from the presheaf of Grothendieck-Witt spectra to an E ∞ -motivic spectrum in SH C ( S ). The crucial fact is that the localizing version of Hermitian K -theory ofrings with involution, denoted G W , is the periodization of GW with respect to a certain Bott map derivedfrom projective bundle formulas for P and P σ (see Corollary 5.8). Here P σ is a copy of P with action[ x : y ] [ y : x ]. The fact that the periodization functor is monoidal together with Schlichting’s results onmonoidality of GW immediately give that the motivic spectrum L A G W ∈ SH C ( S ) is an E ∞ object 5.10. Theorem 1.2.
Let S be a Noetherian scheme of finite Krull dimension with an ample family of line bundles and ∈ S . Then L A G W lifts to an E ∞ motivic spectrum, denoted KR alg , over Sm C S,qp . The final section 6 follows the recipe given by Cisinski and summarized in [Hoy16] to prove cdh descentfor equivariant homotopy Hermitian K -theory on the category of smooth quasi-projective S -schemes. Afterreviewing the K -theory case, the section culminates in theorem 6.2. Theorem 1.3.
Let S be a Noetherian scheme of finite Krull dimension with an ample family of line bundles and ∈ S . Then the homotopy Hermitian K -theory spectrum of rings with involution L A G W satisfies descent for theequivariant cdh topology on Sm C S,qp . Equivariant Topologies and the Equivariant Motivic Homotopy Category
This section reviews the foundations of equivariant motivic homotopy theory. The key definitions arethose of the equivariant ´etale and Nisnevich topologies – two topologies that play a crucial role in definingthe equivariant motivic infinity category H( S ) over a Noetherian base scheme S with finite Krull dimension,with an ample family of line bundles, and with ∈ Γ ( S, O S ). Throughout we’ll work with the category ofsmooth, quasi-projective C -schemes over S , denoted Sm C S,qp . Notation 2.1.
Throughout this section, G will be either a finite group or the group scheme over S associatedto a finite group. Recall that to pass between finite groups and group schemes over S , we form the scheme ` G S with multiplication (using that fiber products commute with coproducts in Sm S,qp ): ` G S × S ` G S ∼ / / ` ( g ,g ) ∈ G × G S µ / / ` G S Whenever we write down a pullback square involving schemes, we’ll tacitly be thinking of G as a groupscheme, and X × Y will really mean X × S Y .We introduce the background definitions from [HKØ15] which will allow us to define the isovariant ´etaletopology. This is a topology which is slightly coarser than the equivariant ´etale topology, but whose pointsare still nice enough so that Hermitian vector bundles are locally determined by rank. DANIEL CARMODY
Definition 2.2.
For a G -scheme X , the isotropy group scheme is a group scheme G X over X defined by thecartesian square G X / / (cid:15) (cid:15) G × X ( µ X ,id X ) (cid:15) (cid:15) X ∆ X / / X × X Definition 2.3.
Let X be a G -scheme. The scheme-theoretic stabilizer of a point x in X is the pullback G x / / (cid:15) (cid:15) G X (cid:15) (cid:15) Spec k ( x ) / / X. By the pasting lemma, this is the same as the pullback G x / / (cid:15) (cid:15) G × X (cid:15) (cid:15) Spec k ( x ) / / X × X Definition 2.4.
Let X be a G -scheme, and define the set-theoretic stabilizer S x of x ∈ X to be { g ∈ G | gx = x } . Remark 2.5.
With notation as above, the underlying set of the scheme-theoretic stabilizer G x can be de-scribed as G x = { g ∈ S x | the induced morphism g : k ( x ) → k ( x ) equals id k ( x ) } . The example below shows that set-theoretic and scheme-theoretic stabilizers need not agree.
Example 2.6. (Herrmann [Her13]) Let k be a field, and consider the k -scheme given by a finite Galoisextension k ֒ → L . Let G = Gal ( L/k ) be the Galois group. The set-theoretic stabilizer of the unique point inSpec L is G itself, while the scheme-theoretic stabilzer is { e } ⊂ G . Remark 2.7.
Recall that if Z → X is a monomorphism of schemes, then the forgetful functor from schemesto sets preserves any pullback Z × X Y . The canonical examples of monomorphisms in schemes are closedembeddings, open immersions, and maps induced by localization. Recall as well that the forgetful functor GSch/S → Sch/S is a right adjoint, hence preserves pullbacks.Since the inclusion of a point Spec k ( x ) ֒ → X × S X will be a closed embedding for any separated scheme,the di ff erence between the set-theoretic and scheme-theoretic stabilizers is given by the fact that the under-lying space of X × S X is not necessarily the fiber product of the underlying spaces. Indeed, in the exampleabove, Spec L × k Spec L (cid:27) ` g ∈ G Spec k , whereas the pullback in spaces is just a single point.2.1. The Equivariant and Isovariant ´Etale Topologies.Notation 2.8.
Let S be a G -scheme. The equivariant ´etale topology on Sm GS,qp will denote the site whosecovers are ´etale covers whose component morphisms are equivariant.
Definition 2.9. (Thomason) An equivariant map f : Y → X is said to be isovariant if it induces an iso-morphism on isotropy groups G Y (cid:27) G X × X Y . A collection { f i : X i → X } i ∈ I of equivariant maps called anisovariant ´etale cover if it is an equivariant ´etale cover such that each f i is isovariant. Remark 2.10.
The isovariant topology is equivalent to the topology whose covers are equivariant, stabilizerpreserving, ´etale maps. We’ll use this notion more often in computations.
Remark 2.11.
The points in the isovariant ´etale topology are schemes of the form G × G x Spec( O shX,x ) where x → x → X is a geometric point, and ( − ) sh denotes strict henselization. See [HKØ15] for a proof.The fact that the points in the isovariant ´etale topology are either strictly henselian local or hyperbolicrings will be crucial when we want to describe the isovariant ´etale sheafification of the category of Hermit-ian vector bundles. Fortunately Hermitian vector bundles over such rings are well understood, and we’llin fact show that Hermitian vector bundles are up to isometry determined by rank locally in the isovariant´etale topology. DH DESCENT FOR HOMOTOPY HERMITIAN K -THEORY OF RINGS WITH INVOLUTION 5 Remark 2.12. If G = C , then G x = { e } or G x = C for all x ∈ X . If G x = { e } , then G × G x Spec( O shX,x ) (cid:27) C × Spec( O shX,x ) (cid:27) Spec( O shX,x ) ` Spec( O shX,x ) with a free action. If G x = C , then G × G x Spec( O shX,x ) = Spec( O shX,x ).2.2. The Equivariant Nisnevich Topology.
Similarly to the non-equivariant case, the equivariant Nis-nevich topology is defined by a particularly nice cd-structure. While there are a few di ff erent definitionsof this topology in the literature which can give non Quillen equivalent model structures, we use the defi-nition from [HKØ15]. Definition 2.13.
A distinguished equivariant Nisnevich square is a cartesian square in Sm GS,qp B / / (cid:15) (cid:15) Y p (cid:15) (cid:15) A (cid:31) (cid:127) i / / X where j is an open immersion, p is ´etale, and p restricts to an isomorphism ( Y − B ) red → ( X − A ) red . Definition 2.14.
The equivariant Nisnevich cd -structure on Sm GS,qp is the collection of distinguished equi-variant Nisnevich squares in Sm GS,qp .The next remark has the important consequence that to prove a map is an equivariant motivic equiva-lence, it su ffi ces to check that it’s an equivalence on a ffi ne G -schemes. Remark 2.15.
For finite groups G , any smooth G -scheme is Nisnevich-locally a ffi ne.2.3. The Equivariant cdh Topology.
The cdh topology is, roughly speaking, the coarsest topology satis-fying Nisnevich excision and which allows for a theory of cohomology with compact support. Like theNisnevich topology (and unlike the ´etale topology) it can be generated by a cd-structure, which gives aconvenient way to check whether or not a presheaf is a cdh sheaf.
Definition 2.16.
An abstract blow-up square is a cartesian square in Sm GS,qp e Z / / (cid:15) (cid:15) e X p (cid:15) (cid:15) Z (cid:31) (cid:127) i / / X where i is a closed immersion and p is a proper map which induces an isomorphism ( e X − e Z ) (cid:27) ( X − Z ). Definition 2.17.
The cdh topology is the topology generated by the cd-structure whose distinguishedsquares are(1) The equivariant Nisnevich distinguished squares(2) The abstract blowup squares.One canonical example of a cdh cover is the map X red → X for an equivariant scheme X → S . Another ex-ample is given by resolution of singularities: given a proper birational map p : X → Y , it’s an isomorphismover some open set U in Y , so letting Z = Y − U and e Z = X − p − ( U ) we get an abstract blowup square e Z / / (cid:15) (cid:15) X (cid:15) (cid:15) Z / / Y Computations with Equivariant Spheres.
Because we’ll be using equivariant spheres to index ourspectra, we’ll record some of their basic properties here. These computations will be important when weinvestigate periodicity of GW in section 5. Though there are exotic elements of the Picard group even innon-equivariant stable motivic homotopy theory, we’ll be concerned with the four building blocks S , S σ =colim( ∗ ← ( C ) + → S ) , G m , G σm . Here G σm is the C scheme corresponding to S [ T , T − ] with action T T − . DANIEL CARMODY
Lemma 2.18.
Let P σ denote P with the action defined by [ x : y ] [ y : x ] . There is an equivariant Nisnevichsquare C × G σm / / π (cid:15) (cid:15) P − { } ` P − {∞} f (cid:15) (cid:15) G σm i / / P σ Proof.
Here, we identify G σm with P σ − { , ∞} . The map i is clearly an open immersion. Its complementis { , ∞} , and f maps π − ( { , ∞} ) isomorphically onto { , ∞} . Furthermore, f is a disjoint union of openimmersions, and hence is (in particular) ´etale. (cid:3) Lemma 2.19.
The following square is a homotopy pushout square: ( C ) + ∧ ( G σm ) + / / π (cid:15) (cid:15) ( C ) + f (cid:15) (cid:15) ( G σm ) + i / / P Proof.
The above square is equivalent to the square( C ) + ∧ ( G σm ) + / / π (cid:15) (cid:15) ( C ) + ∧ A f (cid:15) (cid:15) ( G σm ) + i / / P . By the lemma above, ( C × G σm ) + / / π (cid:15) (cid:15) ( C × A ) + f (cid:15) (cid:15) ( G σm ) + i / / P is a homotopy pushout square. But adding a disjoint basepoint is a monoidal functor, so X + ∧ Y + (cid:27) ( X × Y ) + and this square is equivalent to the desired square. (cid:3) Lemma 2.20. P σ ≈ S σ ∧ G σm .Proof. Let Q denote the homotopy cofiber of ( C × G σm ) + → ( G σm ) + , and e Q denote the homotopy cofiber of( C × A ) + → P σ + . Then the lemma above implies that Q ≈ e Q . Q is the homotopy cofiber of ( C ) + ∧ ( G σm ) + → S ∧ ( G σm ) + , which is just S σ ∧ ( G σm ) + . Recall that colim( ∗ ← X → X ∧ Y + ) (cid:27) X ∧ Y since this is X ∧ colim( ∗ ← S → Y + ). Thus the cofiber of S σ → Q is S σ ∧ G σm .The diagram below in which the horizontal rows are cofiber sequences( C ) + / / id (cid:15) (cid:15) S (cid:15) (cid:15) / / S σ (cid:15) (cid:15) ( C ) + / / (cid:15) (cid:15) P σ + / / (cid:15) (cid:15) e Q (cid:15) (cid:15) ⋆ / / P σ / / T implies that the cofiber of S σ → e Q is P σ . DH DESCENT FOR HOMOTOPY HERMITIAN K -THEORY OF RINGS WITH INVOLUTION 7 The result now follows from the commutativity of the following diagram and homotopy invariance ofhomotopy cofiber: S σ id / / (cid:15) (cid:15) S σ (cid:15) (cid:15) Q ∼ / / (cid:15) (cid:15) e Q (cid:15) (cid:15) S σ ∧ G σm / / P σ (cid:3) The Equivariant Motivic Homotopy Category.
In this thesis, we’ll work with a Noetherian schemeof finite Krull dimension and a finite group scheme G over S . Equivariant motivic homotopy theory isdeveloped in somewhat more generality by Hoyois in [Hoy17], though there’s a price to be paid for allowingmore general group schemes in that the motivic localization functor becomes more complicated (one mustin general invert Thom spaces of all G -a ffi ne bundles). Definition 2.21.
A presheaf F on Sm GS,qp is called homotopy invariant if the projection A S → S induces anequivalence F ( X ) ≃ F ( X × A ). Denote by P htp ( Sm GS,qp ) ⊂ P ( Sm GS,qp ) the full subcategory spanned by thehomotopy invariant presheaves. Denote by L htp the corresponding localization endofunctor of P ( Sm GS,qp ).Now we give the usual definition of excision, the condition that guarantees that a presheaf is a Nisnevichsheaf.
Definition 2.22.
A presheaf F on Sm GS,qp is called
Nisnevich excisive if: • F ( ∅ ) is contractible; • for every equivariant Nisnevich square Q in Sm GS,qp , F ( Q ) is cartesian.Denote by P Nis ( Sm GS,qp ) ⊂ P ( Sm GS,qp ) the full subcategory of Nisnevich excisive presheaves. Denote by L Nis the corresponding localization endofunctor.Finally we come to the definition of a motivic G -space, namely a presheaf that is both Nisnevich excisiveand homotopy invariant. Definition 2.23.
Let S be a G -scheme. A motivic G -space over S is a presheaf on Sm GS,qp that is homotopyinvariant and Nisnevich excisive. Denote by H G ( S ) ⊂ P ( Sm GS,qp ) the full subcategory of motivic G -spacesover S .Let L mot = colim n →∞ ( L htp ◦ L Nis ) n ( F )denote the motivic localization functor.In order to form the stable equivariant motivic homotopy category, we also need to discuss pointedmotivic G -spaces. Definition 2.24.
Let S be a G -scheme. A pointed motivic G -space over S is a motivic G -space X over S equipped with a global section S → X . Denote by H G • ( S ) the ∞ -category of pointed motivic G -spaces.The definition of stabilization can in general be complicated. With our assumptions however, we needonly invert the Thom space of the regular representation T ρ . Definition 2.25.
Let S be a G -scheme. The symmetric monoidal ∞ -category of motivic G -spectra over S isdefined by SH G ( S ) = H G • ( S )[( T ρS ) − ] = colim (cid:18) H G • −⊗ T ρ −−−−−→ H G • −⊗ T ρ −−−−−→ · · · (cid:19) where T ρ is the Thom space of the regular representation A ρ / A ρ − G . DANIEL CARMODY Hermitian Forms on Schemes
This section reviews the definitions and properties of Hermitian forms over schemes with involutionfrom [Xie18]. After defining the proper notion of the dual of a quasi-coherent module over a scheme withinvolution, the definition of a Hermitian vector bundle finally appears in Definition 3.11 as a locally free O X -module with a well-behaved map to the dual module. Once the definitions are in place, we discussin section 3.3 the structure of Hermitian forms over semilocal rings as this is the fundamental tool forshowing that Hermitian forms are locally trivial in the isovariant ´etale topology. We prove this particularstatement in Corollary 3.28. We end this section by recalling Schlichting’s definition of a dg categoy withweak equivalence and duality and the Grothendieck-Witt groups of such an object.3.1. Definitions.Definition 3.1.
Let R be a ring with involution − : R → R . A Hermitian module over R is a finitely generatedprojective right R -module, M , together with a map b : M ⊗ Z M → R such that, for all a ∈ R ,(1) b ( xa, y ) = ab ( x, y ),(2) b ( x, ya ) = b ( x, y ) a ,(3) b ( x, y ) = b ( y, x ). Definition 3.2.
Let R be a ring with involution − . Given a right R -module M , define a left R -module,denoted M as follows: M has the same underlying abelian group as M , and the action is given by r · m = m · r .If R is commutative, we can define an R -bimodule by m · r = mr and r · m = mr . Remark 3.3.
Let R be a commutative ring. Given an involution σ : R → R , and an R − R -bimodule M asabove, we can identify M with σ ∗ M . Indeed, σ ∗ M is an R − R -bimodule via the rule r · m = σ ( r ) M , and since R is commutative, we can view this either as a left or right R -module. Remark 3.4.
Another way to define a Hermitian form over a ring R with involution σ is to give a finitelygenerated projective right R -module M together with an R − R -bimodule map b : M ⊗ Z M → R where we view R as a bimodule over itself by r · r · r = r rr , M as a left R -module via the involution, andsuch that b ( x, y ) = σ ( b ( y, x )). If we remove the final condition, we obtain a sesquilinear form.By the usual duality, we have a third definition: Definition 3.5.
A Hermitian module over a ring R with involution is a finitely generated projective R -module M together with an R -linear map b : M → M ˇ = M ∗ such that b = b ∗ can M , where b ∗ : M ∗∗ → M ∗ isgiven by ( b ( f ))( m ) = f ( b ( m )).Now, we generalize the above definitions to schemes. Definition 3.6.
Let X be a scheme, and M a quasi-coherent (locally of finite presentation) O X -module.Define O X ˇ= Hom ( M, O X ). Definition 3.7.
Let X be a scheme with involution σ , and M a right O X -module. Note that there’s aninduced map σ : O X → σ ∗ O X . Define the right (note that we’re working with sheaves of commutative rings,so we can do this) O X -module M to be σ ∗ M with O X action induced by the map σ . That is, if m ∈ σ ∗ M ( U ),and c ∈ O X ( U ), then m · c = m · σ ( c ). Note that this last product is defined, because m ∈ σ ∗ M ( U ) = M ( σ − ( U )), c ∈ σ ∗ O X ( U ) = O X ( σ − ( U )), and M is a right O X -module. Remark 3.8.
We have two choices for the definition of the dual M ∗ . We can either define M ∗ = Hom mod −O X ( σ ∗ M, O X ) , or we can define M ∗ = σ ∗ Hom mod −O X ( M, O X ). We claim that these two choices of dual are naturally isomor-phic. DH DESCENT FOR HOMOTOPY HERMITIAN K -THEORY OF RINGS WITH INVOLUTION 9 Proof.
Let f : σ ∗ M | U → O X | U be a map of right O X | U -modules. Post-composing with the map O X | U → σ ∗ O X | U yields a map f : σ ∗ M | U → σ ∗ O X | U , a.k.a. a map M | σ − U → O X | σ − U . Note that σ ∗ Hom mod −O X ( M, O X )( U ) = Hom mod −O X ( M, O X )( σ − U ), so that f ∈ σ ∗ Hom mod −O X ( M, O X )( U ).On the other hand, given g ∈ σ ∗ Hom mod −O X ( M, O X )( U ), so that g : σ ∗ M | U → σ ∗ O X | U , we can postcomposewith σ ∗ ( σ ) to get a map e g : σ ∗ M | ∗ → σ ∗ σ ∗ O X | U = O X | U . Since σ = id , this is clearly the inverse to the mapabove.It’s clear that these assignments are natural, since they’re just postcomposition with a natural transfor-mation. (cid:3) Definition 3.9.
Define the adjoint module M ∗ to be Hom mod −O X ( σ ∗ M, O X ). By the remark above, it doesn’treally matter which of the two possible definitions we choose here. Definition 3.10.
Given a right O X -module M , we define the double dual isomorphism can M : M → M ∗∗ asfollows: given an open U ⊆ X , we define a map M ( U ) → N at ( σ ∗ N at ( σ ∗ M, O X ) | U , O X | U ) = N at ( N at ( σ ∗ M | σ ( U ) , O X | σ ( U ) ) , O X | U )by u η u , where for an open V ⊆ U , ( η u ) V ( γ ) = ( σ ) − V ( γ σ ( V ) ( u | V )) . Here γ ∈ N at ( σ ∗ M | σ ( U ) , O X | σ ( U ) ) and σ is the morphism of sheaves σ : O X → σ ∗ O X . Note that γ σ ( V ) ( u | V )makes sense because σ ∗ M ( σ ( V )) = M ( V ).More globally, there’s an evaluation map ev σ : M ⊗ σ ∗ N at ( σ ∗ M, O X ) → O X defined by the composition M ⊗ σ ∗ N at ( σ ∗ M, O X ) (cid:27) M ⊗ N at ( M, σ ∗ O X ) ev −−→ σ ∗ O X ( σ ) − −−−−−→ O X which under adjunction yields the above map. Definition 3.11.
Let X be a scheme with involution − : X → X . Let can X be the double dual isomorphismof Definition 3.10. A Hermitian vector bundle over X is a locally free right O X -module V with an O X -modulemap φ : V → V ∗ such that φ = φ ∗ can V . Remark 3.12.
Recall that there’s an equivalence of categories between locally free coherent sheaves on X and geometric vector bundles given by M Spec (Sym( M ˇ)) in one direction and the sheaf of sections inthe other. For locally free sheaves, we have M ˇ ⊗ N ˇ (cid:27) ( M ⊗ N )ˇ so that the functor is monoidal. We will usethis to think of a Hermitian form as a map of schemes V ⊗ V → A .Below we give the key example of a Hermitian vector bundle. Example 3.13.
Define (diagonal) hyperbolic n -space over a scheme ( S, − ) with involution to be A nS withthe Hermitian form ( x , . . . , x n , y , . . . , y n ) P ni =1 x i − y i − − x i y i . Denote this Hermitian form by h diag .As defined this way, the matrix of this Hermitian form is · · · − · · · ... ... ... · · · · · · · · · − the diagonal matrix diag(1 , − , , . . . , − H R (where we give it the hyperbolic form above) have the form " a b ± b ± a with a = ±√ b , b ∈ R (or a − b = 1). The usual identification with R × ⋊ C follows by considering thedecomposition a − b = 1 ⇐⇒ ( a + b )( a − b ) = 1. Example 3.14.
Similarly to above, we can define a hyperbolic form h by the matrix " II . This form is isometric to the above form, and we’ll use both forms below.3.2.
Properties.
We record two unsurprising structural results which will be useful when we define theHermitian Grassmannian in section 4.
Lemma 3.15.
Given a map of schemes with involution f : ( Y , i Y ) → ( X, i X ) and a (non-degenerate) Hermitianvector bundle ( V , ω ) on X , f ∗ ( V ) is a (non-degenerate) Hermitian vector bundle on Y .Proof. The pullback of a locally free O X -module is a locally free O Y -module, so we just need to check thatit’s Hermitian. Given the map ω : V → V ∗ , we get an induced map f ∗ V → f ∗ ( V ∗ ) which is an isomorphismif ω is. Thus we just need to check that f ∗ ( V ∗ ) (cid:27) ( f ∗ V ) ∗ . But pullback commutes with sheaf dual for locallyfree sheaves of finite rank, so we just need to check that changing the module structure via the involutioncommutes with pullback; that is, we need to check that f ∗ ( V ) = f ∗ ( V ). However, this is clear since thestructure map on f ∗ ( V ) is given by O Y × f ∗ V (cid:27) f ∗ O X × f ∗ V f ∗ ( − ) × id −−−−−−−→ f ∗ ( O X ) × f ∗ ( V ) → f ∗ ( V ) . (cid:3) Lemma 3.16.
Let ( V , φ ) be a non-degenerate Hermitian vector bundle over a scheme with trivial involution X ,and let ( M, φ | M ) be a (possibly degenerate) sub-bundle. Given a map of schemes g : Y → X , there is a canonicalisomorphism g ∗ ( M ⊥ ) (cid:27) ( g ∗ M ) ⊥ .Proof. Recall that, by definition, M ⊥ = ker( V φ −→ V ∗ → M ∗ ). Equivalently, M ⊥ is defined by the exactsequence 0 → M ⊥ → V → M ∗ → . It follows that the composite map g ∗ ( M ⊥ ) → g ∗ V → g ∗ ( M ∗ ) is zero, and hence by universal property ofkernel there’s a canonical map g ∗ ( M ⊥ ) → ker( g ∗ V → g ∗ ( M ∗ ) (cid:27) ( g ∗ ( M )) ∗ ) = ( g ∗ ( M )) ⊥ where we’ve used the canonical isomorphism g ∗ ( M ∗ ) (cid:27) ( g ∗ ( M )) ∗ for locally free sheaves.We claim that this map is an isomorphism. It su ffi ces to check on stalks, where the map can be identifiedwith a map M ⊥ g ( y ) ⊗ O Y,y → ker( V g ( y ) ⊗ O Y,y → M ∗ g ( y ) ⊗ O Y,y ) . But V g ( y ) (cid:27) M ⊥ g ( y ) ⊕ M ∗ g ( y ) , so the sequence0 → M ⊥ g ( y ) ⊗ O Y,y → V g ( y ) ⊗ O Y,y → M ∗ g ( y ) ⊗ O Y,y → (cid:3) We record two incredibly useful results for working with Hermitian forms. The first implies that Her-mitian forms over fields can be written as an orthogonal sum of rank 1 Hermitian forms, while the secondgives a useful characterization of non-degenerate submodules of a Hermitian module.
Theorem 3.17. (Knus [Knu91] 6.2.4) Let ( M, b ) be an ǫ -Hermitian space over a division ring D . Then ( M, b ) has an orthogonal basis in the following cases: (1) the involution of D is not trivial (2) the involution of D is trivial, the form is symmetric, and char D , . Lemma 3.18. (Knus) Let ( M, b ) be a Hermitian module, and ( U, b | U ) be a non-degenerate f.g. projective Hermit-ian submodule. Then M = U ⊕ U ⊥ .Proof. Since b | U : U → U ∗ is an isomorphism, given an m ∈ M , there exists u ∈ U such that b ( m, − ) | U = b ( u, − ) | U . But then b ( m − u, − ) | U = 0, so that m − u ∈ U ⊥ , and m = u + m − u . Thus M = U + U ⊥ . Since φ | U isnon-degenerate, U ∩ U ⊥ = 0, so we’re done. (cid:3) DH DESCENT FOR HOMOTOPY HERMITIAN K -THEORY OF RINGS WITH INVOLUTION 11 Hermitian Forms on Semilocal Rings.
The following result from [Bae78] will allow us to concludethat Hermitian forms diagonalize over semilocal rings. We include its proof in order to show that it gener-alizes to semilocal rings with involution – see Corollary 3.22.
Theorem 3.19. (Baeza) Let R be a ring, and let E be a Hermitian module over R . Let I ⊂ Jac ( R ) be an ideal. Forevery orthogonal decomposition E = F ⊥ G of E = E/IE over
R/I , where F is a free non-singular subspace of E ,there exists an orthogonal decomposition E = F ⊥ G of E with F free and non-singular, and F/IF = F, G/IG = G .Proof. Write F = h x i⊕· · ·⊕h x n i with x i ∈ F and det( b ( x i , x j )) ∈ ( R/I ) ∗ . Choose representatives x i ∈ E of x i , andlet F = Rx + · · · + Rx n . We claim that the x i are independent, so that F is free: indeed, if λ x + · · · + λ n x n = 0,then we get n equations λ b ( x , x i ) + · · · + λ n b ( x n , x i ) = 0. But we know that det( b ( x i , x j )) = t ∈ R ∗ , since1 − st ∈ I for some s by assumption (because the determinant is a unit mod the ideal I ), but then st cannotbe contained in any maximal ideal, so st ∈ R ∗ = ⇒ t ∈ R ∗ . It follows that the λ i are zero (otherwise we wouldhave a non-zero vector in the kernel of an invertible matrix), so that the x i are independent as desired. Thedeterminant fact also shows that F is regular, so by the lemma above, it has an orthogonal summand G . Byconstruction F/I = F , so that G = ( F ) ⊥ = ( F/I ) ⊥ = F ⊥ /I = G/I . (cid:3) Lemma 3.20.
Hermitian forms over R × R (with trivial involution) are in bijection with Herm ( R ) × Herm ( R ) .Proof. First, recall that modules over R × R correspond to a module over R and a module over R . Indeed,consider the standard idempotents (1 ,
0) = e , (0 ,
1) = e . Fix a module M over R × R . Then M = e M ⊕ e M .Indeed, any m ∈ M can be written as e m + e m = ( e + e ) m = m . Furthermore, if e m = e m , then e e m = e e m = ⇒ e m .Now, a Hermitian form M ⊗ M → R × R is determined by two maps M ⊗ M → R and M ⊗ M → R .Writing M = e M ⊕ e M , we note that, by linearity, it must be the case that e M ⊗ e M → R × R is the zeromap; to wit, b ( e m , e m ) = e e b ( m , m ) = 0. Thus this Hermitian form is determined completely by themaps e M ⊗ e M → R × R and e M ⊗ e M → R × R . Finally, note that, again by linearity, we see that e M ⊗ e M → R is the zero map: b ( e m , e m ) = b ( e m , e m ) = e b ( e m , e m ), and e R = 0. Similarlyfor the other map. Hence the Hermitian form is completely determined by the maps e M ⊗ e M → R and e M ⊗ e M → R . (cid:3) Corollary 3.21.
Free Hermitian modules diagonalize over rings with finitely many maximal ideals (semi-local rings).
Proof.
By the Chinese Remainder Theorem, R/ ( m ∩· · ·∩ m n ) (cid:27) R/m ×· · ·× R/m n = F ×· · ·× F n . We claim thatHermitian forms over finite products of fields diagonalize, and then the result will follow from the abovetheorem. By induction and the lemma above, a Hermitian module M is determined by Hermitian modules M i over F i , i = 1 , . . . , n as M = M ⊕ M ⊕· · ·⊕ M n with action ( f , . . . , f n ) · ( m , · · · , m n ) = ( f m , . . . , f n m n ). Each M i can be diagonalized into M i = h a ,i i ⊥ · · · ⊥ h a m,i i (it’s important to note here that since M is free, the rankof each M i is the same). Thus a diagonalization of M is given by h ( a , , . . . , a ,n ) i ⊥ · · · ⊥ h ( a ,m , . . . , a m,n ) i . (cid:3) Now, let R be a ring with involution, and I ⊆ Jac ( R ) an ideal. Then C · I ⊆ Jac ( R ) is an ideal fixed by theinvolution.The following corollary has the same proof as the theorem above, the only subtlety is that we need thequotient ring to inherit the involution to make sense of an induced Hermitian module. Corollary 3.22.
Let R be a ring with involution, and let E be a Hermitian module over R . Let I ⊂ Jac ( R )be an ideal fixed by the involution. For every orthogonal decomposition E = F ⊥ G of E = E/IE over
R/I ,where F is a free non-singular subspace of E , there exists an orthogonal decomposition E = F ⊥ G of E with F free and non-singular, and F/IF = F, G/IG = G . Corollary 3.23.
Let R be a local ring with involution (necessarily a map of local rings). Then any Hermitianmodule (which is necessarily free) over R diagonalizes. Lemma 3.24.
Let R be a ring, and consider the ring R × R with the involution that switches factors. Then anymodule M can be written as e M ⊕ e M as above. A non-degenerate Hermitian form on this module is determinedby a map e M ⊗ e M → R × R . In other words, the matrix representing the map e M ⊕ e M → e M ∗ ⊕ e M ∗ has the form " AA t . where A is invertible.Proof. The first claim is just that b ( e x, e y ) = 0 = b ( e x, e y ) for any x, y ∈ M . This follows because b ( e x, e y ) = b ( e x, e y ) = e e b ( e x, e y ) = e e b ( e x, e y ) = 0. Similarly for b ( e x, e y ). The statement about the matrixfollows by identifying the map M ⊗ M → R × R with an isomorphism M → M ∗ and using the direct sumdecomposition. (cid:3) Corollary 3.25.
Let R be as in the lemma additionally with 2 invertible. Then M (cid:27) H ( e M ), where H denotes the hyperbolic module functor. Proof.
The assumption that 2 is invertible implies that M is an even Hermitian space in the notation ofKnus. Now by the corollary above b | e M = 0, so M has direct summands e M, e M such that e M = e M ⊥ and M = e M ⊕ e M . Now [Knu91, Corollary 3.7.3] applies to finish the proof. (cid:3) Corollary 3.26.
Let R be a semi-local ring with involution and with 2 invertible. Then any Hermitianmodule over R diagonalizes. Proof.
Using the theorem above and reducing modulo the Jacobson radical (which is always stable underthe involution), it su ffi ces to prove the corollary for R a finite product of fields. Then R = F ×· · ·× F n is semi-simple, and hence we can index the fields in a particularly nice way (proof is by considering idempotents),writing R = A × · · · × A m × B × . . . B n − m such that A i is fixed by the involution, and σ ( B i ) = B i +1 , σ ( B i +1 ) = B i . Now, any finitely generated module M can be written as a direct sum M = L mi =1 M i L n − m i =1 N i ⊕ N i − .By the two lemmas above, the form when restricted to each M i or N i ⊕ N i − is diagonalizable, so the formis diagonalizable (see the proof of the non-involution case). (cid:3) Lemma 3.27.
Non-degenerate Hermitian vector bundles are determined by rank over strictly henselian local rings ( R, m ) with ∈ R such that the residue field R/m has trivial involution.Proof.
By Corollary 3.26, any Hermitian vector bundle over R diagonalizes. Thus it su ffi ces to prove thatany two non-degenerate Hermitian vector bundles of rank 1 are isometric.A non-degenerate rank 1 Hermitian vector bundle corresponds to a unit x ∈ R × such that x = x (a onedimensional Hermitian matrix). Because R is strictly henselian, there is a square root c of x − . We claimthat c = c . Assume not. Then because the involution on R/m is trivial, c − c ∈ m . Since 2 is invertible, wehave c = c + c + c − c . It follows that c + c is a unit. Otherwise it would be contained in m which would implythat the unit c was contained in m .However, we calculate ( c + c )( c − c ) = c − c . But ( c ) = ( c ) = x − = x − , so that ( c + c )( c − c ) = 0. Because c + c is a unit, it follows that c − c = 0.This shows that given any one dimensional Hermitian matrix x , there’s a unit c such that cxc = 1 so thatall one dimensional Hermitian forms are isometric to the form h i . (cid:3) Corollary 3.28.
Hermitian vector bundles are locally determined by rank in the isovariant ´etale topology.
Proof.
The points in the isovariant ´etale topology are either strictly henselian local rings whose residuefield has trivial involution or a ring of the form O shX,x × O shX,x with involution ( x, y ) ( i ( y ) , i ( x )). Via the map( x, y ) ( x, i ( y )), such rings are isomorphic to hyperbolic rings.If the ring is a stricty henselian local ring whose residue field has trivial involution, Lemma 3.27 showsthat non-degenerate Hermitian forms are determined by rank. If the ring is hyperbolic, then by Corollary3.25 all non-degenerate Hermitian forms over the ring are hyperbolic forms of projective modules over alocal ring. Since projective modules over a local ring are determined by rank, the corresponding hyperbolicforms are determined by rank. (cid:3) DH DESCENT FOR HOMOTOPY HERMITIAN K -THEORY OF RINGS WITH INVOLUTION 13 Higher Grothendieck-Witt Groups.
In [Xie18], the author works with coherent Grothendieck-Wittgroups on a scheme. Because the negative K -theory of the category of bounded complexes of quasi-coherent O X -modules with coherent cohomology vanishes (together with the pullback square relating the homotopyfixed points of K -theory to Grothendieck-Witt theory), there is no di ff erence between the additive andlocalizing versions ofGrothendieck-Witt spectra in this setting.Therefore, we work instead with Grothendieck-Witt spectra of sPerf( X ) = Ch b Vect( X ), the dg category ofstrictly perfect complexes on X . We review the relevant definitions from [Sch17] now. Definition 3.29. A pointed dg category with duality is a triple ( A , ∨ , can) where A is a pointed dg category, ∨ : A op → A is a dg functor called the duality functor, and can : 1 → ∨ ◦ ∨ op is a natural transformation ofdg functors called the double dual identification such that can ∨ A ◦ can A ∨ = 1 A ∨ for all objects A in A . Remark 3.30.
A dg category with duality has an underlying exact category with duality ( Z A ptr , ∨ , can),where Z A ptr has the same objects as A ptr but the morphism sets are the zero cycles in the morphismcomplexes of A ptr . Here A ptr is the pretriangulated hull of A (see [Sch17] definition 1.7). Definition 3.31. A dg category with weak equivalences is a pair ( A , w ) where A is a pointed dg category and w ⊆ Z A ptr is a set of morphisms which saturated in A . A map f in w is called a weak equivalence. Definition 3.32.
Given a pointed dg category with duality ( A , ∨ , can), a Hermitian object in A is a pair( X, φ ) where φ : X → X ∨ is a morphism in A satisfying φ ∨ can X = φ . Definition 3.33. A dg category with weak equivalences and duality is a quadruple A = ( A , w, ∨ , can) where( A , w ) is a dg category with weak equivalences and ( A , ∨ , can) is a dg category with duality such that thedg subcategory A w ⊂ A of w -acyclic objects is closed under the duality functor ∨ and can A : A → A ∨∨ is aweak equivalence for all objects A of A . Definition 3.34.
Let A = ( A , w, ∨ , can) be a dg category with weak equivalences and dualiy. TheGrothendieck-Witt group GW ( A ) of A is the abelian group generated by Hermitian spaces [ X, φ ] in theunderlying category with weak equivalences and duality ( Z A ptr , w, ∨ , can), subject to the following rela-tions:(1) [ X, φ ] + [
Y , ψ ] = [ X ⊕ Y , φ ⊕ ψ ](2) if g : X → Y is a weak equivalence, then [ Y , ψ ] = [
X, g ∨ ψg ], and(3) if ( E • , φ • ) is a symmetric space in the category of exact sequences in Z A ptr , that is, a map E • : ∼ φ • (cid:15) (cid:15) E ∨• : E − / / i / / ∼ φ − (cid:15) (cid:15) E p / / / / ∼ φ (cid:15) (cid:15) E ∼ φ (cid:15) (cid:15) E ∨ / / p ∨ / / E ∨ i ∨ / / / / E ∨ of exact sequences with ( φ − , φ , φ ) = ( φ ∨ can , φ ∨ can , φ ∨− can) a weak equivalence, then[ E , φ ] = " E − ⊕ E , φ φ − ! . Definition 3.35.
Given a dg-category with weak equivalences and duality A = ( A , w, ∨ , can), Schlichtingdefines [Sch17, Section 4.1] a functorial monoidal symmetric spectrum GW ( A ) using a modified versionof the Waldhausen S • construction. For the sake of brevity, we don’t reproduce his construction here.Noting in general that GW doesn’t sit in a localization sequence, Schlichting defines a localizing variant, G W in [Sch17, Section 8.1] as a bispectrum. The reason Schlichting defines G W as an object in bispectrarather than spectra is to get a monoidal structure on G W . We provide an alternative approach to producing G W via periodization in section 5. Definition 3.36.
Let X be a Noetherian scheme of finite Krull dimension with an ample family of linebundles, and let σ : X → X be an involution on X . Let sPerf( X ) denote the category of strictly perfectcomplexes on X with the weak equivalences being the quasi-isomorphisms. Define a family of dualities onsPerf( X ) indexed by i ∈ N by ∗ i : E Hom sPerf( X ) ( σ ∗ E, O X [ i ]) . Note that because σ is an involution, σ ∗ E is a strictly perfect complex. Define the canonical isomorphimcan as in Definition 3.10 as the adjoint of the evaluation map ev : E ⊗ σ ∗ Hom sPerf( X ) ( σ ∗ E, O X [ i ]) → O X [ i ] . Combining all this data we get a collection of dg categories with weak equivalences and duality(sPerf( X ) , q. iso , ∗ i , can) . The i th shifted Grothendieck-Witt spectrum of ( X, σ ) is defined as GW [ i ] ( X, σ ) = GW (sPerf( X ) , q. iso , ∗ i , can) . If Z is an invariant closed subset of X , then the duality on sPerf( X ) restricts to a duality on the subcategoryof complexes supported on Z , sPerf Z ( X ). We define GW [ i ] ( X on Z ) = GW (sPerf Z ( X ) , q. iso , ∗ i , can) . Representability of Automorphism Groups of Hermitian Forms
Representability of K -theory in the stable motivic homotopy category allows one to check that K -theorypulls back nicely. In particular, given f : X → S a map of schemes over S , one can use ind-representabilityof KGL to show that f ∗ (KGL S ) = KGL X . Together with the formalism of six operations in motivic homotopytheory, one obtains rather formally cdh descent for algebraic K -theory, see [Cis13].The goal of this section is to define a sheaf on Sm C S,qp , denoted R Gr, which represents Hermitian K -theory in the motivic homotopy category H C S . We first check that over a regular base S with 2 invertible(e.g. Z [ ]), Hermitian K -theory is representable in the category of C -schemes smooth over S , Sm C S,qp . Toextend this result to non-regular bases S , we utilize the Morel-Voevodsky approach to classifying spacesand obtain representability of homotopy Hermitian K -theory in the motivic homotopy category H C ( S ).By analogy with the K theory case, the equivariant scheme representing Hermitian K -theory on Sm C S,qp will be a colimit of schemes which parametrize non-degenerate Hermitian sub-bundles of a given Hermit-ian vector bundle V . The new results here are mostly the definitions, as the proofs in this section are eitherminor modifications or identical to the proofs in [SST14]. The main di ff erence which might cause concernis that stalks in the isovariant ´etale topology are now semi-local (rather than local) rings.We combine the techniques of [SST14] with a Morel-Voevodsky style argument to compare R Gr d ( H ∞ )to the isovariant ´etale classifying space B isoEt O ( H d ) of the group of automorphisms of hyperbolic d -space.The key to the comparison is that locally in the isovariant ´etale topology, Hermitian vectors bundles aredetermined by rank. This will utilize some of the analysis of Hermitian forms over semi-local rings fromsection 3.3. Note that this is a key di ff erence from the K -theory case where one must pass only to local(rather than strictly henselian local) rings in order for K -theory to be determined by rank.A straightforward generalization of the techniques in [SST14] allow one to compare colim n B isoEt O ( H n )( ∆ R )to the Grothendieck-Witt space defined in section 3.4 by viewing them both as group completions and com-paring their homology. This approach is inspired by the Karoubi-Villamayor definition of higher algebraic K -theory. We don’t carry out this comparison here as it is unnecessary for proving cdh descent.4.1. The definition of the Hermitian Grassmannian R Gr . The definition here describes the sections ofthe underlying scheme of R Gr over a scheme X → S . We advise the hurried reader to skip to section 4.2. Lemma 4.1.
Let F be a presheaf on Sm S,qp and let a : F = ⇒ F be a natural transformation such that a ◦ a = id F .Then there’s an associated presheaf on Sm C S,qp defined by the formula ( X, σ : X → X )
7→ F ( X ) C where the actionof C on F ( X ) is defined by f a X F ( σ )( f ) .Proof. Note that this is indeed a C -action, since a X F ( σ )( a X F ( σ )( f )) = F ( σ ) a X ( a X F ( σ )( f )) = F ( σ )( F ( σ )( f )) = f using naturality. (cid:3) Fix a (possibly degenerate) Hermitian vector bundle (
V , φ ) over a base scheme S with 2 invertible andwith trivial involution. The canonical example of such a base scheme is S = Spec Z [ ]. DH DESCENT FOR HOMOTOPY HERMITIAN K -THEORY OF RINGS WITH INVOLUTION 15 We’ll define a presheaf R Gr : ( Sm C S,qp ) op → Set by first defining a presheaf on Sm S,qp , showing that it’srepresentable, equipping with an action, then taking the corresponding representable functor on Sm C S,qp .We can then extend to an arbitrary equivariant base T with 2 invertible by pulling back along the uniquemap T → Z [ ]. • On objects, R Gr( V )( f : X → S ) for an S -scheme f : X → S is a split surjection ( p, s ) f ∗ V p / / / / W s y y ◆❴♣ , where W is locally free.Here by an isomorphism of split surjections we mean a diagram f ∗ V p / / / / W s y y ◆❴♣ φ (cid:15) (cid:15) f ∗ V p ′ / / / / W ′ s ′ y y ◆❴♣ such that φ is an isomorphism satisfying φ ◦ p = p ′ and s = s ′ ◦ φ . • Given a morphism Y h (cid:31) (cid:31) ❄❄❄❄❄❄❄❄ g / / X f (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧ S over S , define R Gr V ( g )( f ∗ V p / / / / W s y y ◆❴♣ ) = h ∗ V can / / g ∗ f ∗ V g ∗ p / / / / g ∗ W g ∗ s x x ◗❴♠ . There’s a natural action of C on R Gr V whose non-trivial natural transformation will be denoted η . Define η as follows:Fix an object X ∈ Sm S,qp . Define η X ( f ∗ V p / / / / W s y y ◆❴♣ ) = f ∗ V q / / / / (ker p ) ⊥ t w w ◗❴♠ . We claim that this is well-defined.Recall that W ⊥ = ker( f ∗ V f ∗ φ −−−→ f ∗ ( V ∗ ) can −−−→ ( f ∗ V ) ∗ s ∗ −→ W ∗ ) . Leaving out the can map for convenience, we get a split exact sequence0 / / W ⊥ / / f ∗ V s ∗ / / W ∗ p ∗ y y / / . By the splitting lemma for abelian categories, f ∗ V (cid:27) W ⊥ ⊕ W ∗ , and hence there’s a split surjection f ∗ V ։ W ⊥ with W ⊥ locally free.Given an isomorphism f ∗ V p / / / / W s y y ◆❴♣ ψ (cid:15) (cid:15) f ∗ V p ′ / / / / W ′ s ′ y y ◆❴♣ we get an isomorphism of (split) diagrams f ∗ V f ∗ φ / / ( f ∗ V ) ∗ s ∗ / / W ∗ ( ψ − ) ∗ (cid:15) (cid:15) f ∗ V f ∗ φ / / ( f ∗ V ) ∗ ( s ′ ) ∗ / / ( W ′ ) ∗ and hence an isomorphism of split surjections f ∗ V q / / / / W ⊥ t x x P❴♥ δ (cid:15) (cid:15) f ∗ V q ′ / / / / ( W ′ ) ⊥ t ′ x x P❴♥ , so that η X is a well-defined map of sets. Given a map of schemes g : Y → X , such that f ◦ g = h and anelement f ∗ V p / / / / W s y y ◆❴♣ in R Gr V ( X ), R Gr( g ) ◦ η X ( f ∗ V p / / / / W s y y ◆❴♣ ) = R Gr( g )( f ∗ V q / / / / (ker p ) ⊥ t w w ◗❴♠ )= h ∗ V can / / g ∗ f ∗ V g ∗ q / / / / g ∗ ((ker( p )) ⊥ ) g ∗ t u u ❚❩❴❞❥ while η Y ◦ R Gr( g )( f ∗ V p / / / / W s y y ◆❴♣ )) = h ∗ V can / / g ∗ f ∗ V q ′ / / / / ( g ∗ (ker( p ))) ⊥ t ′ u u ❚❩❴❞❥ By Lemma 3.16, there’s a canonical isomorphism g ∗ ((ker( p ) ⊥ )) → ( g ∗ (ker( p ))) ⊥ , and under this isomor-phism q ′ and t ′ correspond to g ∗ q , and g ∗ t , respectively. This concludes the check of naturality.Now by Lemma 4.1, there’s a presheaf R Gr : Sm C S,qp → Set . To determine its values on a C -scheme ( X, σ ),we note that a fixed point of the action of Lemma 4.1 is determined by an isomorphism of split surjections f ∗ V q / / / / σ ∗ (ker( p ) ⊥ ) t w w ❙❴❦ ψ (cid:15) (cid:15) f ∗ V p / / / / ker( p ) s w w ❙❴❦ DH DESCENT FOR HOMOTOPY HERMITIAN K -THEORY OF RINGS WITH INVOLUTION 17 Note that because σ is an involution, for any O X -module M , there’s a canonical isomorphism of O X -modules σ ∗ M (cid:27) σ ∗ M . Thus there’s a natural isomorphismHom mod −O X ( σ ∗ f ∗ V , − ) (cid:27) Hom mod −O X ( σ ∗ f ∗ V , − ) (cid:27) Hom mod −O X ( f ∗ V , − ) . It follows that any Hermitian form φ : f ∗ V → Hom mod −O X ( f ∗ V , O X )can be promoted to a Hermitian form e φ : f ∗ V → Hom mod −O X ( σ ∗ f ∗ V , O X )compatible with an involution σ on X .Let ( M, φ | M ) be a Hermitian sub-bundle of f ∗ V over the scheme X with trivial involution. We claim that σ ∗ ( M ⊥ ) is the orthogonal complement of M viewed as a Hermitian sub-bundle of f ∗ V with the promotedform e φ . Said di ff erently, we claim that σ ∗ (ker( f ∗ V φ | M −−−→ Hom( M, O X )) (cid:27) ker( f ∗ V e φ | M −−−→ Hom( σ ∗ M, O X )) . But using the natural isomorphism between σ ∗ and σ ∗ , together with the natural isomorphisms σ ∗ Hom( M, O x ) (cid:27) Hom( M, O X )and σ ∗ f ∗ V (cid:27) f ∗ V , this becomes a question of whether σ ∗ is left exact. In general it isn’t, but because σ is aninvolution, σ ∗ is naturally isomorphic to σ ∗ which is left exact. The claim follows.4.2. Representability of R Gr . Fix a Hermitian vector bundle (
V , φ ) over S where dim( V ) = n and S is ascheme with trivial involution. Then the underlying scheme of R Gr( V ) is the pullback R Gr( V ) / / (cid:15) (cid:15) Hom O S ( V , V ) × Hom O S ( V , V ) ◦ ,id (cid:15) (cid:15) Hom O S ( V , V ) ∆ / / Hom O S ( V , V ) × Hom O S ( V , V )where the right vertical map sends p ( p ◦ p, p ). In other words, the underlying scheme is the scheme ofidempotent endomorphisms of V . The action corresponds to the map p p † , where p † is the adjoint of p with respect to the form φ .Note that using this description, an equivariant map ( X, σ ) → R Gr( V ) corresponds to an idempotent p : V X → V X such that φ − ( γ − ( σ ∗ p ) γ ) ∗ φ = p , where we’re being cavalier and using ∗ to denote both dual(on the outside) and pullback (by σ ). Here γ is the canonical isomorphism V X γ −→ σ ∗ V X ; if the structuremap of X is f : X → S , then γ arises from the equality σ ◦ f = f .Note that the form on V ( X,σ ) is by definition the composite e φ : V X φ −→ V ∗ X ( γ ∗ ) − −−−−−→ σ ∗ V ∗ X ( η ∗ ) − −−−−−→ σ ∗ V ∗ X , and the adjoint of p is given by e φ − ( σ ∗ p ) ∗ e φ . Expanding, this is φ − ( γ ∗ )( η ∗ )( η ∗ ) − ( σ ∗ p ) ∗ ( η ∗ )( η ∗ ) − ( γ ∗ ) − φ = φ − ( γ − ( σ ∗ p ) γ ) ∗ φ, and so we recover the condition that p † = p , which corresponds to the fact that V X = ker p ⊥ im p , andhence the restriction of the form on V X to im p (and ker p ) is non-degenerate.To summarize, the underlying scheme of R Gr( V ) represents idempotents, and equivariant maps pick outthose idempotents which correspond to orthogonal projections. Definition 4.2.
Now fix a dimension d and a non-degenerate Hermitian vector bundle ( V , φ ) over S . Define R Gr d ( V ) to be the closed subscheme of R Gr( V ) cut out by rk( p ) = d , where rk is the rank map. In other words, the underlying topological space of R Gr d ( V ) is the pullback R Gr d ( V ) / / (cid:15) (cid:15) R Gr( V ) rk (cid:15) (cid:15) { d } / / Z and if S is regular, R Gr d ( V ) is the closed subscheme equipped with the reduced induced scheme structure.If S is non-regular, then let f : S → Z [ ] and note that R Gr d ( V ) over S is just f ∗ ( R Gr d [ ]) since pullbackpreserves rank and non-degeneracy of bundles. The requirement that V be non-degenerate is necessaryso that the action on R Gr( V ) sends rank d subspaces to rank d subspaces and hence induces an action on R Gr d ( V ). Remark 4.3.
Denote by g : R Gr d ( V ) → S the structure map of R Gr d ( V ). Because R Gr d ( V ) is representableby a C -scheme, there’s an idempotent g ∗ ( V ) → g ∗ ( V ) corresponding to the identity map id : R Gr d ( V ) → R Gr d ( V ). This idempotent is simply the idempotent which, over a point of R Gr d ( V ) represented by anidempotent p : V → V , restricts to p . There’s an action σ on R Gr d ( V ) × S V induced by the action on R Gr d ( V ), and using the fact that σpσ = p † one can see that this idempotent is non-degenerate with respectto the promoted Hermitian form on g ∗ ( V ) compatible with the involution on R Gr d ( V ). Remark 4.4.
Since we’ve shown that R Gr( V ) represents non-degenerate Hermitian subbundles of V , atthis point we’ll move away from explicitly referring to split surjections and just represent the sections of R Gr( V ) by non-degenerate subbundles. Definition 4.5.
Let H S denote the hyperbolic space 3.13 over the base scheme S . For V ∈ H ∞ a constantrank non-degenerate subbundle, let | V | denote the rank of V . Order such subbundles of H ∞ by inclusion,and denote the resulting poset by P . Given an inclusion V ֒ → V ′ of non-degenerate subbundles, denote by V ′ − V the complement of V in V ′ . Let H : P → Fun( Sm C ,opS,qp , Set) be the functor which on objects sendsa subbundle V to R Gr | V | ( V ⊥ H ∞ ). Given an inclusion V ֒ → V ′ , the induced map R Gr | V | ( V ⊥ H ∞ ) → R Gr | V ′ | ( V ′ ⊥ H ∞ ) is given by E E ⊥ ( V ′ − V ). Note that because V is non-degenerate, V ⊥ ( V ′ − V ) = V ′ .Define(1) R Gr • = colim H . The ´Etale Classifying Space.
The content of this section is a straightforward generalization of thework of [SST14] to the C -equivariant setting. Fix a scheme S with 2 invertible, and let ( V , φ ) be a (possiblydegenerate) Hermitian vector bundle over S . For a C -scheme f : X → S over S , let S ( V , φ )( X )be the category of non-degenerate Hermitian sub-bundles of f ∗ V . A morphism in this category from E to E is an isometry not necessarily compatible with the embeddings E , E ⊆ f ∗ V . Using pullbacks ofquasi-coherent modules, we turn S into a presheaf of categories on Sm C S,qp . For integer d ≥
0, define S d ( V , φ ) ⊂ S ( V , φ )to be the presheaf which on a C -scheme f : X → S assigns the full subcategory of non-degenerate Hermit-ian sub-bundles of ( f ∗ V , f ∗ φ ) which have constant rank d . The associated presheaf of objects is R Gr d ( V , φ ).Note that the object V = ( V , ∈ S | V | ( V ⊥ H ∞ ) has automorphism group O ( V ). Thus we get an inclusion O ( V , φ ) → S | V | ( V ⊥ H ∞ ), where O ( V ) is the isometry group considered as a category on one object. Afterisovariant ´etale sheafification, this inclusion becomes an equivalence; this follows from Corollary 3.28 thaton the points in the isovariant ´etale topology, Hermitian vector bundles are determined by rank.Upon applying the nerve, we get maps of simplicial presheaves BO ( V ) → B S | V | ( V ⊥ H ∞ ) which is a weakequivalence in the isovariant ´etale topology. Abusing notation, let B isoEt O ( V ) denote a fibrant replacementof B S | V | ( V ⊥ H ∞ ) in the isovariant ´etale topology so that we get a sequence of maps BO ( V ) → B S | V | ( V ⊥ H ∞ ) → B isoEt O ( V ) . which are weak equivalences in the isovariant ´etale topology. DH DESCENT FOR HOMOTOPY HERMITIAN K -THEORY OF RINGS WITH INVOLUTION 19 Lemma 4.6.
Let ( V , φ ) be a non-degenerate Hermitian vector bundle over a scheme S with trivial involution and ∈ Γ ( S, O S ) . Then for any a ffi ne C -scheme Spec R over S , the map B S | V | ( V ⊥ H ∞ )( R ) → B isoEt O ( V )( R ) is a weak equivalence of simplicial sets. In particular, the map B S | V | ( V ⊥ H ∞ ) → B isoEt O ( V ) is a weak equivalence in the equivariant Nisnevich topology, and hence an equivalence after C motivic localiza-tion.Proof. Each Hermitian vector bundle W ∈ S | V | ( V ⊥ H ∞ )( R ) gives rise to an O ( V )-torsor via W Isom ( V , W ).Note that this is an O ( V )-torsor because locally in the isovariant ´etale topology, W (cid:27) V , so that locally Isom ( V , W ) (cid:27) Isom ( V , V ) (cid:27) O ( V ). Because Hermitian vectors bundles are isovariant ´etale locally deter-mined by rank, the same proof as the ordinary vector bundle case shows that the category of O ( V ) torsorsis equivalent to the category of Hermitian vector bundles. Because over an a ffi ne scheme, every Hermitianvector bundle is a summand of a hyperbolic module, it follows that S | V | ( V ⊥ H ∞ )( R ) is equivalent to thecategory of isovariant ´etale O ( V ) torsors.Let F : Sm C S,qp → Gpd be the sheaf which assigns to f : X → S the groupoid of O ( f ∗ V )-torsors. The con-struction W Isom ( f ∗ V , W ) described above defines a functor S | V | ( V ⊥ H ∞ ) → F which is an equivalencewhen evaluated at a ffi ne C -schemes. It follows that there’s a sequence BS | V | ( V ⊥ H ∞ ) → B F → B isoEt O ( V )where the first map is a weak equivalence of simplicial sets when evaluated at a ffi ne C -schemes, andby [Jar01] Theorem 6, the second map is a weak equivalence of simplicial sets when evaluated at any C -scheme. (cid:3) Definition 4.7.
Following [SST14], let S • = colim V ⊂ H ∞ S S | V | ( V ⊥ H ∞ )where similarly to the definition of R Gr, for V ⊂ V ′ the functor S | V | ( V ⊥ H ∞ ) → S | V ′ | ( V ′ ⊥ H ∞ )is defined on objects by E E ⊥ V ′ − V and on morphisms by f f ⊥ V ′ − V . Definition 4.8.
Define the infinite orthogonal group as O = colim W ⊆ H ∞ S O ( W ) . Definition 4.9.
Let R be a commutative ring. Define ∆ R to be the simplicial ring with involution [ n ] R [ x , . . . , x n ] / ( P x i − R . Lemma 4.10. ( [SST14]) Let V be a nondegenerate Hermitian vector bundle over a commutive ring with invo-lution ( R, σ ) such that ∈ R . Then the inclusion H ∞ ⊂ V ⊥ H ∞ induces a homotopy equivalence of simplicialgroups O ( H ∞ )( ∆ R ) → O ( V ⊥ H ∞ )( ∆ R ) A V ⊥ A. Proof.
First, assume that V = H . Consider the map j : O ( H n ) → O ( H n +2 ) sending A to 1 H ⊥ A ⊥ H n +1 .We claim that this is na¨ıvely A homotopic to the inclusion i : O ( H n ) → O ( H n +2 ), i ( A ) = A ⊥ H n +2 whichdefines the colimit O ( H ∞ ). Let g = I n I I n +2 where I n denotes an n × n identity matrix. Then i = gjg − = gjg t . Because g corresponds to an even permutation matrix, it can be written as a product ofelementary matrices, each of which is na¨ıvely A homotopic to the identity. It follow that g is na¨ıvely A homotopic to the identity, and hence the induced maps i, j : O ( H n )( ∆ R ) → O ( H n +2 )( ∆ R ) are simpliciallyhomotopic via a base-point preserving homotopy. It follows that i, j induce the same map on homotopygroups, so that j ∗ = i ∗ : π k O ( H ∞ )( ∆ R ) = colim n π k O ( H n )( ∆ R ) → π k O ( H ∞ )( ∆ R ) is the colimit of a mapcorresponding to a cofinal inclusion of diagrams, and hence is an isomorphism on all simplicial homotopy groups. Because simplicial groups are Kan complexes, it follows that j is a homotopy equivalence, and theclaim is proved when V = H .Now a trivial induction shows that the lemma holds when V = H n . In general, choose an embedding V ⊆ H n , and consider the sequence of maps O ( H ∞ )( ∆ R ) → O ( V ⊥ H ∞ )( ∆ R ) → O ( H n ⊥ H ∞ ∆ R ) → O ( H n ⊥ V ⊥ H ∞ )( ∆ R ) . The composites O ( H ∞ )( ∆ R ) → O ( H n ⊥ H ∞ ) and O ( V ⊥ H ∞ )( ∆ R ) → O ( H n ⊥ V ⊥ H ∞ )( ∆ R ) are weakequivalences, so by 2 out of 6 the first map is a weak equivalence. Because it is a map of simplicial groupsit is a homotopy equivalence. (cid:3) For nondegenerate Hermitian vector bundles (
V , φ V ) , ( W , φ W ) and a commutative R -algebra with invo-lution ( A, σ ), let St(
V , W )( A )be the set of A -linear isometric embeddings f : V A → W A . Given a map A → B of commutative R -algebras with involution, tensoring over R with B makes St( V , W )( − ) a presheaf on commutative R -algebraswith involution. There’s a transitive left action of O ( V ⊥ H ∞ ) on St( V , V ⊥ H ∞ ) given by ( f , g ) f ◦ g .Let i V denote the isometric embedding V ֒ → V ⊥ H ∞ : v ( v, i V is the subgroup O ( H ∞ ) ⊂ O ( V ⊥ H ∞ ) where the inclusion map is A V ⊥ A .It follows that there’s an isomorphism of presheaves of sets O ( H ∞ ) \ O ( V ⊥ H ∞ ) (cid:27) St(
V , V ⊥ H ∞ ) f f ◦ i V . Now Lemma 4.10 shows that the map O ( H ∞ ∆ R ) → O ( V ⊥ H ∞ )( ∆ R ) is an equivariant map which is a non-equivariant homotopy equivalence. The simplicial group O ( H ∞ )( ∆ R ) acts freely on both the domain andcodomain, so that the quotients O ( H ∞ ∆ R ) \ O ( V ⊥ H ∞ )( ∆ R ) and O ( H ∞ ∆ R ) \ O ( H ∞ )( ∆ R ) are homotopy equiva-lent.Together with the isomorphism of simplicial sets O ( H ∞ ∆ R ) \ O ( V ⊥ H ∞ )( ∆ R ) (cid:27) St(
V , V ⊥ H ∞ )( ∆ R )it follows that St( V , V ⊥ H ∞ )( ∆ R ) is a contractible for a commutative ring ( R, σ ) with involution and ∈ R .Morever, this simplicial set is fibrant because G/H is fibrant for a simplicial group G and subgroup H . Wehave thus proved: Lemma 4.11.
Let R be a commutative ring with ∈ R . Then St(
V , V ⊥ H ∞ )( ∆ R ) is a contractible Kan set. Now we move to identifying R Gr V as a quotient of a contractible space by a free group action. Let V be a non-degenerate Hermitian vector bundle over a ring R with involution. Then the group O ( V ) acts onthe right on St( V , U ) by precomposition. The map St(
V , U ) → R Gr V ( U ) : f im( f ) factors through thequotient St( V , U ) /O ( V ). The map is clearly surjective, and hence furnishes an isomorphism of setsSt( V , U ) /O ( V ) (cid:27) R Gr V ( U ) f im( f ) . In particular, there’s an isomorphism of presheaves of sets St(
V , V ⊥ H ∞ ) /O ( H ∞ ) (cid:27) R Gr V ( U ).Now, let V be a non-degenerate Hermitian vector bundle over a ring with involution R and let U be apossibly degenerate Hermitian form over R . Define E V ( U ) to be the category whose objects are R -linearmaps V → U of Hermitian forms (aka isometric embeddings), and whose morphisms from two objects a : V → U and b : V → U are maps c : im( a ) → im( b ) making the diagram V a / / b ! ! ❈❈❈❈❈❈❈❈❈ im( a ) c (cid:15) (cid:15) im( b )commute. DH DESCENT FOR HOMOTOPY HERMITIAN K -THEORY OF RINGS WITH INVOLUTION 21 There’s a natural right action of O ( V ) on E V ( U ) which on objects sends E V ( U ) × O ( V ) → E V ( U ) : ( a, g ) ag and which morphisms is the trivial action. Then clearly there’s an isomorphism E V ( U ) /O ( V ) (cid:27) S V ( U ) a im( a ) . Lemma 4.12.
The category E V ( V ⊥ H ∞ ) is contractible.Proof. The category is nonempty and every object is initial. (cid:3)
The map of simplicial sets St(
V , V ⊥ H ∞ )( ∆ R ) → E V ( V ⊥ H ∞ )( ∆ R )is O ( V )( ∆ R ) equivariant and a weak equivalence after forgetting the action. Furthermore, O ( V )( ∆ R ) actsfreely on both sides, so that the induced map on quotients(2) R Gr V ( V ⊥ H ∞ )( ∆ R ) → S V ( V ⊥ H ∞ )( ∆ R )is also a weak equivalence. As an aside, the inclusion BO ( V ) ⊂ B S V ( V ⊥ H ∞ ) is a weak equivalence since S V ( V ⊥ H ∞ ) is a connected groupoid.We now show that there’s a motivic equivalence R Gr • → colim n B isoEt O ( H n ) over possibly non-regularNoetherian base rings.Let X → S be an a ffi ne C -scheme over S , and let W be a non-degenerate Hermitian vector bundle over X . Given an isovariant ´etale O ( W ) torsor π : T → X , and an isovariant ´etale torsor U , let U π denote thetwisted sheaf ( U × T ) /O ( W ).Our goal is to appy Lemma 2.1 from [Hoy16], which we restate below: Lemma 4.13. (Hoyois) Let Γ be an isovariant ´etale sheaf of groups on Sm C S,qp acting on an isovariant ´etale sheaf U . Suppose that, for every X ∈ Sm C S,qp and every isovariant ´etale torsor π : T → X under Γ , U π → X is a motivicequivalence on Sm C S,qp . Then the map L isoEt ( U/ Γ ) → B isoEt Γ induced by U → ∗ is a motivic equivalence on Sm C S,qp . Given an O ( W )-torsor π : T → X , we want to check that St( W , H ∞ ) π = (St( W , H ∞ ) × T ) /O ( W ) → X is amotivic equivalence on Sm C S,qp . Leting V = W π , this is equivalent to checking that St( V , H ∞ X ) is motivicallycontractible over Sm C X . To wit, because X is a ffi ne there’s an embedding V ֒ → H m , and we have V ⊥ H ∞ X (cid:27) H ∞ X since H ∞ X = colim W ⊂ H ∞ X W . It follows that St( V , H ∞ X ) (cid:27) St(
V , V ⊥ H ∞ X ), and Lemma 4.11 (which didn’tassume regularity of the base) shows that St( V , V ⊥ H ∞ X ) is motivically contractible over Sm C X .It’s a direct consequence of the above lemma that L isoEt (St( W , W ⊥ H ∞ ) /O ( W )) → B isoEt O ( W )is a motivic equivalence. However, we’ve already shown (2) thatSt( W , W ⊥ H ∞ ) /O ( W ) (cid:27) R Gr W ( W ⊥ H ∞ ) , so that L isoEt St(
W , W ⊥ H ∞ ) /O ( W ) (cid:27) L isoEt R Gr W ( W ⊥ H ∞ ) (cid:27) R Gr | W | ( W ⊥ H ∞ )which after taking colimits gives the desired result. Theorem 4.14.
Let S be a Noetherian scheme of finite Krull dimension with ∈ S . There are equivalences ofmotivic spaces on Sm C S,qp Z × R Gr • ∼ −→ Z × colim n B isoEt O ( H n ) Periodicity in the Hermitian K -Theory of Rings with Involution A Projective Bundle Formula for P σ . Let P σ denote P with involution σ defined by [ x : y ] [ y : x ].When necessary we’ll point it at the point [1 : 1]. Throughout this section, we’ll fix the notation O = O P .Consider the square of O -modules(3) O ( − T + S / / T − S (cid:15) (cid:15) O T − S (cid:15) (cid:15) Hom ( σ ∗ O , O ) T + S / / Hom ( σ ∗ O ( − , O )where the map T − S : O ( − → O is induced via the tensor-hom adjunction by the composition(4) O ( − ⊗ (cid:26) T − S (cid:27) ⊗ σ ∗ O id ⊗ i ⊗ id −−−−−−−→ O ( − ⊗ O (1) ⊗ σ ∗ O id ⊗ id ⊗ ( σ ) − −−−−−−−−−−−−→ O ( − ⊗ O (1) ⊗ O µ ⊗ id −−−−→ O ⊗ O µ −→ O and the map T − S : O → σ ∗ O ( −
1) is induced via the tensor-hom adjunction by the composition(5)
O ⊗ (cid:26) T − S (cid:27) ⊗ σ ∗ O ( − σ ⊗ σ ◦ i ⊗ id −−−−−−−−−−−→ σ ∗ O ⊗ σ ∗ O (1) ⊗ σ ∗ O ( − id ⊗ σ ∗ ( µ ) −−−−−−−→ σ ∗ O ⊗ σ ∗ O σ ∗ µ −−−→ σ ∗ O ( σ ) − −−−−−→ O where µ denotes multiplication. We’re abusing notation in the map (5) and using σ to denote both themaps O → σ ∗ O and O (1) → σ ∗ O (1) induced by the graded automorphism of k [ S, T ] given by f ( S, T ) f ( T , S ). The image of 1 ∈ O under the adjoint of the map (5) yields the element S − T as a global section of σ ∗ O (1). The map T + S : Hom ( σ ∗ O , O ) → Hom ( σ ∗ O ( − , O ) is induced by precomposition with σ ∗ ( T + S ), whichis just multiplication by the global section T + S . Equation (4) can be understood similarly.We claim that diagram 3 commutes. Fix an open U ⊆ P σ which need not be invariant, and open V ⊆ U .Going down then right yields the composite map u ( v T − S · u · ( σ ) − (cid:18) T + S · v (cid:19) ) . Going right first then down yields the composite u ( v ( σ ) − ( σ ( T + S · u ) · S − T · v ))These are equal since T + S is an invariant global section. Note that the diagram 3 is a map inFun([1] , Vect( P σ )) from O ( − T + S −−−−→ O to its dual, Hom ( σ ∗ O , O ) T + S −−−−→ Hom ( σ ∗ O ( − , O ) . Thus this diagram defines a (not necessarily non-degenerate) form, which we denote by φ .In order to show that this φ is symplectic, we have to check that φ ∗ ◦ ( − can) = φ . To spell this out indetail, the dual and double dual are functors. Applying these two functors, we get the two objects O ∗ T + S ∗ / / O ( − ∗ and O ( − ∗∗ T + S ∗∗ / / O ∗∗ in Fun([1] , Ch b Vect( P σ )). DH DESCENT FOR HOMOTOPY HERMITIAN K -THEORY OF RINGS WITH INVOLUTION 23 Because can is a natural transformation id → ∗∗ , there’s a commutative diagram O ( − can (cid:15) (cid:15) T + S / / O can (cid:15) (cid:15) O ( − ∗∗ T + S ∗∗ / / T − S ∗ (cid:15) (cid:15) O ∗∗ T − S ∗ (cid:15) (cid:15) O ∗ T + S ∗ / / O ( − ∗ The goal is to show that the vertical maps in the large rectangle are the negative of the vertical maps indiagram 3. Tracing through the definitions, we see that can is the map which sends u ∈ O ( − U ) to thenatural transformation γ ( σ ) − ( γ ( u | V )) , and φ ∗ ◦ can( u ) is the natural transformation v ( σ ) − (cid:18) T − S · v · ( σ ) − ( u ) (cid:19) which is the same thing as v (cid:18) − T − S · ( σ ) − ( v ) · u (cid:19) . On the other hand, T − S : O ( − → O ∗ is the map u ( v T − S · u · ( σ ) − ( v ))which is by what we calculated above equal to − ( φ ∗ ◦ can) = φ ∗ ◦ ( − can).Now just as in [Sch17], taking the mapping cone of φ via the functorCone : Fun([1] , Ch b Vect( P σ )) [0] → (cid:16) Ch b Vect( P σ ) (cid:17) [1] yields a symplectic form β σ = Cone( φ ).We claim that there’s an exact sequence O ( − T + S T − S −−−−−−→ O ⊕ O ∗ (cid:16) T + S − T − S (cid:17) −−−−−−−−−−−−−−→ O ( − ∗ where the maps are the maps in diagram 3. The fact that the composite is zero follows from commutativityof that 3. To show that the kernel equals the image, note that any permutation of ( T + S , S − T ) is a regularsequence on k [ S, T ]. Thus if T + S x + S − T y = 0, reducing mod T + S we see that y ∈ ( T + S ) and reducing mod S − T we see that x ∈ ( S − T ). It follows that the square defining φ is a pushout, and hence the induced mapon mapping cones is a quasi isomorphism. Hence β σ is a well-defined, non-degenerate symplectic form in (cid:16) Ch b Vect( P σ ) (cid:17) [1] . Theorem 5.1.
Let X be a scheme with trivial involution with an ample family of line bundles and ∈ X , anddenote by p : P σ → X the structure map of the equivariant projective line over X , with action [ x : y ] [ y : x ] .Then for all n ∈ Z , the following are natural stable equivalences of (bi-) spectra GW [ n ] ( X ) ⊕ GW [ n − ( X, − can) ∼ −→ GW [ n ] ( P σX ) G W [ n ] ( X ) ⊕ G W [ n − ( X, − can) ∼ −→ G W [ n ] ( P σX )( x, y ) p ∗ ( x ) + β σ ∪ p ∗ ( y ) . Proof.
The proof of Theorem 9.10 in [Sch17] can be easily adapted. Note that our Bott element β σ is alinear change of coordinates from the standard Bott element on P . Keeping in mind that the involutiononly a ff ects the duality and not the underlying derived category with weak equivalences, it’s still true that β σ ⊗ : T sPerf( X ) → T sPerf( P X ) /p ∗ T sPerf( X ) is an equivalence of triangulated categories. As in loc. cit. , if we denote by w the set of morphisms in sPerf( P X ) which are isomorphisms in T sPerf( P X ) /p ∗ T sPerf( X ), weget a sequence (sPerf( X ) , quis) p ∗ / / (sPerf( P X ) , quis) / / (sPerf( P X ) , w )which is a Morita exact sequence of categories with duality. That is, the maps are maps of categories withduality, and the underlying sequence of categories is Morita exact. It follows that this sequence induces ahomotopy fibration of GW [ n ] and G W [ n ] spectra. As remarked above, these fibration sequences split via theexact dg form functors (sPerf( X ) , quis) β σ ⊗ / / (sPerf( P X ) , quis) / / (sPerf( P X ) , w )so that the composite induces an equivalence of triangulated categories. Finally, using that GW and G W are invariant under derived equivalences [Sch17, Theorem 6.5] [Sch17, Theorem 8.9], we conclude thetheorem. (cid:3) Considering GW as a presheaf of spectra on Sm C S,qp it follows from Theorem 5.1 that GW [ n ] ( P σ , [1 : 1]) (cid:27) GW [ n − ( X, − can) (cid:27) GW [ n +1] ( X ), recovering one of the results of [Xie18]. Hence Hom ( Σ ∞ ( P σ , [1 : 1]) , GW [ n ] ) (cid:27) GW [ n +1] as presheaves of spectra on Sm C S,qp . In particular, by the projective bundle formula from [Sch17] and theusual cofiber sequence ([1 : 1] × P σ ) ∨ ( P × [1 : 1]) → P σ × P → P σ ∧ P we obtain the periodicity isomorphism Hom (( P , [1 : 1]) ∧ ( P σ , [1 : 1]) , GW [ n ] ) (cid:27) GW [ n ] induced by the map GW [ n ] ( X ) → GW [ n +1] ( P X ) → GW [ n ] ( P σ P X ) x β ∪ p ∗ ( x ) β σ ∪ q ∗ ( O X [ − ⊗ β ∪ p ∗ ( x ))where p is the projection P X → X , and q is the projection P σ P X → P X . The analogous statements hold forthe presheaf of spectra G W .As notation for later, let β σ denote the induced map(6) β σ : ( P , [1 : 1]) ∧ ( P σ , [1 : 1]) → GW .
Lemma 5.2.
The Bott element β σ restricts to zero in C × A σ = P σ − [1 : 0] ` P σ − [0 : 1] .Proof. As in [Sch17], because the Bott element is natural it su ffi ces to prove that the bott element β σ in P σ Z [ ] restricts to zero. From the definition of the Bott element, it’s clear that it’s supported on [1 : − GW [ n ] ( C × A σ on [1 : − ` [1 : − / / GW [ n ] ( C × A σ ) / / (cid:15) (cid:15) GW [ n ] ( C × A σ − [1 : − t t ✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ GW [ n ] ( C × Spec( Z [ ]))where the vertical maps are induced by inclusion of the point [1 : 1]. Because Z [ ] is regular and C × A σ is equivariantly isomorphic to C × A , [Xie18, Theorem 7.5] shows that the middle vertical map is anisomorphism, hence the upper right map is an injection. By localization [Sch17, Theorem 6.6], the top rowis exact, and it follows that the left horizontal map is the zero map. (cid:3) DH DESCENT FOR HOMOTOPY HERMITIAN K -THEORY OF RINGS WITH INVOLUTION 25 The Periodization of GW . The idea behind the Bass construction in algebraic K -theory is that as aconsequence of satisfying localization, there is a Bass exact sequence ending in · · · → K n ( G m ) ∂ −→ K n − ( X ) → n . This comes from applying K -theory to the pushout square manifesting the usual cover of P together with the projective bundle formula. The map ∂ is split by x [ T ] ∪ p ∗ ( x ) where p is the projectionto the base scheme p : G m → X . It follows that if K exhibits an exact Bass sequence in all degrees n , then K n − ( X ) can be identified with the image of ∂ ([ T ]) ∪ x (i.e. this map is an automorphism of K n − ( X ). In fact, ∂ ([ T ]) ∪ − is the idempotent endomorphism (0 ,
1) of K ( P ) (cid:27) K ( X ) ⊕ K ( X )). The Bass construction canbe thought of as defining K Bn ( X ) so that there’s an exact sequence K Bn ( A ) ⊕ K Bn ( A ) → K Bn ( G m ) → K Bn − ( X ),then identifying K Bn − ( X ) with (0 , · K Bn − ( P ). In other words, it can be constructed as the colimit K B = colim( K → Hom ( A a G m A , K ) → . . . )where the pushouts are taken in presheaves and the maps are induced by applying Hom ( − , K ) in the cate-gory of K -modules to the composite A a G m A → Σ G m T −→ K. Here, loosely speaking, the first map in the composite represents the boundary in the long Bass exactsequence ∂ while the second represents [ T ], so that in the category of K modules this map represents cupproduct with ∂ ([ T ]).We’ll spell out an example a bit more explicitly to give a flavor for the constructions to come. Let W = A ` G m A , where we emphasize again that the pushout is in the category of presheaves. Because this isa (homotopy) pushout in the category of presheaves, applying Hom ( − , K ) gives us a homotopy pullbacksquare, and hence a Mayer-Vietoris long exact sequence. In particular, it gives us a map of presheaves ofspectra (which can be promoted to a map of K -modules) Ω K ( G m ) → K ( W ), where we abuse notation andwrite K ( W ) for the internal hom of W into K . Because K B satisfies Nisnevich descent, and K i ( − ) = K Bi ( − )for i ≥
0, it follows by the 5-lemma that K ( W ) (cid:27) K ( P ) (cid:27) K ( X ) ⊕ K ( X ), and that the element ∂ ([ T ]) ∪ − represents projection onto the second factor as an endomorphism of K ( W ).Now, we want to explain why ∂ ([ T ]) ∪ K − ( W ) (cid:27) K B − ( X ). We’ll use the fact that K B ( W ) (cid:27) K B ( P ) and that K B − ( X ) = ∂ ([ T ]) ∪ K B − ( P ) = ∂ ( K B ( G m )).To begin, because ∂ ([ T ]) is zero in K ( A ), the image of ∂ ([ T ]) ∪ K − ( W ) in K − ( A ) ⊕ K − ( A ) is zero. Byexactness, it follows that ∂ ([ T ]) ∪ K − ( W ) ⊆ ∂K ( G m ).There’s a map φ : K − ( W ) → K B − ( W ) (cid:27) K B − ( P ) and a commutative diagram K ( A ) ⊕ K ( A ) / / (cid:15) (cid:15) K ( G m ) ∂ / / (cid:15) (cid:15) K − ( W ) φ (cid:15) (cid:15) K ( A ) ⊕ K ( A ) / / K ( G m ) ∂ / / K B − ( W )which shows that φ restricts to an isomorphism ∂ ( K ( G m )) (cid:27) ∂ ( K B ( G m )), and in particular that φ ( ∂ ( K ( G m ))) = ∂ ( K B ( G m )). Now φ ( ∂ ([ T ]) ∪ ∂ ( K ( G m )) = ∂ ([ T ]) ∪ φ ( ∂K ( G m )) = ∂ ([ T ]) ∪ ∂K B ( G m ) = ∂K B ( G m )where we’ve crucially used that for Bass K -theory, ∂ ([ T ]) ∪ ∂K B ( G m ) = ∂ ([ T ]) ∪ ∂K B ( P ) = ∂K B ( G m ).But as remarked above, the fact that ∂ ([ T ]) is trivial in K ( A ) implies that ∂ ([ T ]) ∪ ∂ ( K ( G m )) ⊆ ∂ ( K ( G m )),and we know that φ | ∂K ( G m ) is an isomorphism. Since φ | ∂ ([ T ]) ∪ ∂ ( K ( G m )) is surjective by the chain of equalitiesabove, it follows that ∂K ( G m ) = ∂ ([ T ]) ∪ K − ( W ). We’ve shown that ∂ ([ T ]) ∪ K − ( W ) = ∂K ( G m ) (cid:27) ∂K B ( G m ) (cid:27) K B − ( X ) . If we take pointed versions of the above sequences by pointing all the schemes in question at [1 : 1]everything goes through as above with the extra benefit that ∂ ([ T ]) ∪ K B − ( W ,
1) = K B − ( W , K B ( X ) → K B ( W , x p ∗ ( x ) ∪ ∂ ([ T ]) is an isomorphism by the projective bundle formula. Now the map p : ( W , → p ∗ : K ( W , → K (( W , ⊗ ( W , p ∗ ( x ∪ ∂ ([ T ])) = ∂ ([ T ]) ∪ p ∗ ( x ), so that the image of K − ( W ,
1) in K − (( W , ⊗ ( W , x ∂ ([ T ]) ∪ p ∗ ( x ) is, by what we showed above, isomorphic to K B − ( X ). This shows that π − K B = π − colim( K → Hom ( A a G m A , K ) → · · · ) . This argument is mostly formal given a few pieces of structural information: • A map K → K B which respects cup products, • Nisnevich descent for K B , and • A Bass exact sequence split by cup product with an element in K ( G m ).The remainder of this section will show that these three pieces of structure are present for Grothendieck-Witt groups, which will allow us to repeat essentially the same argument to give a construction of thelocalizing G W as a periodization of GW . When the base scheme is a perfect field, a similar construction of GW as a periodic spectrum was given in [HKO11].First, equivariant Nisnevich descent for G W is a consequence of results from [Sch17]. Lemma 5.3. G W is Nisnevich excisive on the category of smooth schemes with an ample family of line bundlesover S .Proof. Recall that the distinguished squares defining the equivariant Nisnevich cd-structure are cartesiansquares in Sm GS,qp B / / (cid:15) (cid:15) Y p (cid:15) (cid:15) A j / / X where j is an open immersion, p is ´etale, and ( Y − B ) red → ( X − A ) red is an isomorphism.As in [Sch17, Theorem 9.6], a result of Thomason [TT07, Theorem 2.6.3] tells us that the map p inducesa quasi-equivalence of dg categories p ∗ : sPerf Z ( X ) → sPerf Z ( Y ) . Because G W is invariant under derived equivalences [Sch17, Theorem 8.9], it follows that p ∗ induces anisomorphism on Grothendieck-Witt groups. Noting that U and the closed subset Z = X − A are G -invariant,the localization sequence [Sch17, Theorem 9.5] generalizes to our setting and identifies G W (sPerf Z ( X )) and G W (sPerf Z ( Y )) as the horizontal homotopy fibers. This allows us to conclude the result. (cid:3) Next, we identify the analogues of the Bass sequence and the splittings therein. From [Sch17, Theorem9.13], we know that there’s a Bass sequence0 / / G W [ n ] i ( X ) / / G W [ n ] i ( A X ) ⊕ G W [ n ] i ( A X ) / / G W [ n ] i ( X [ T , T − ]) / / G W [ n − i − ( X ) / / T ] in G W [1]1 ( Z [ ][ T , T − ]).This gives us a candidate map A ` G m A → Σ G m [ T ] −−−→ GW [1] .Now, we want to find a candidate map Σ σ G σm → GW [ − so that we can eventually invert Σ σ G σm ⊗ Σ G m → GW [ − ⊗ GW [1] → GW [0] . Define W σ by the pushout square in the category of presheaves( C × G σm ) + / / (cid:15) (cid:15) ( C × A σ ) + (cid:15) (cid:15) ( G σm ) + / / W σ . DH DESCENT FOR HOMOTOPY HERMITIAN K -THEORY OF RINGS WITH INVOLUTION 27 There’s an associated homotopy pushout square( C × G σm ) + / ( C ) + / / (cid:15) (cid:15) ( C × A σ ) + / ( C ) + (cid:15) (cid:15) ( G σm ) + /S / / W σ /S and taking the homotopy cofiber of the left vertical map yields S σ ∧ G σm . It follows that the homotopycofiber of the right vertical map is equivalent to S σ ∧ G σm , and that there’s a long exact sequence(7) · · · / / GW [ n ] i ( S σ ∧ G σm ) / / GW [ n ] i ( W σ /S ) / / GW [ n ] i (( C × A σ ) + / ( C ) + ) / / · · · . Here if A σS (cid:27) S , then ( C × A σ ) + / ( C ) + (cid:27) ( C ) + ∧ A σ is contractible and W /S (cid:27) S σ ∧ G σm . Working over theregular ring Z [ ], GW ( W σ /S ) (cid:27) GW ( P σ /S ), and GW [ n ] i ( W σ /S ) (cid:27) GW [ n ] i ( P σ /S ) (cid:27) GW [ n +1] i ( S )by the projective bundle formula 5.1.The maps in the sequence (7) are maps of GW [0] ∗ -modules, and the sequence is natural in the base scheme.The induced map GW [ − ( S σ ∧ G σm ) → GW [0]0 ( Z [ 12 ])is an isomorphism of GW [0]0 ( Z [ ])-modules, and hence the inverse is uniquely determined by a lift of theelement h i ∈ GW [0]0 ( Z [ ]) to GW [ − ( S σ ∧ G σm ). We stress that this element h i maps to β σ ∪ O Z [ ] [ − ∪ h i in GW ( P σ ), and in particular it isn’t the unit of multiplication in GW ( P σ ). We’ll denote this element by[ T σ ] in analogy with the non-equivariant case.Over an arbitrary base scheme X , we denote by [ T σ ] the pullback of [ T σ ] to GW [ − ( S σ ∧ G σm × Z [ ] X )using functoriality of GW . We summarize in the definition below. Definition 5.4.
Let [ T ] denote the class of the element T in G W [1]1 ( Z [ ][ T , T − ]). Let ∂ ([ T ]) denote theimage of [ T ] under the connecting map in the Bass sequence ∂ : GW [1]1 ( Z [ 12 ][ T , T − ]) → GW [1]0 ( P Z [ ] ) . Let [ T σ ] denote the lift of the element h i ∈ GW [0]0 ( Z [ ]) to GW [ − ( S σ ∧ G σm ). Let ∂ ([ T σ ]) denote theimage of [ T σ ] under the connecting map in the long exact sequence 7 ∂ : GW [ − ( S σ ∧ G σm ) / / GW [ − ( W σ /S ) . Over an arbitrary scheme S with ∈ S , let [ T ] and [ T σ ] denote the pullbacks f ∗ ([ T ]), f ∗ ([ T σ ]) under theunique map f : S → Z [ ], and similarly for ∂ ([ T ]) and ∂ ([ T σ ]).Let W = ( A ` G m A ) + . Now (by taking the pointed version of everything) we have a candidate map(8) γ : W σ /S ⊗ W /S → S σ ∧ G σm ⊗ S ∧ G m [ T σ ] ⊗ [ T ] −−−−−−−−→ GW [ − ⊗ GW [1] → GW to invert.Given a presentably symmetric monoidal ∞ -category and a morphism α : x → to the monoidal unit,define Q α E = colim( E α −→ Hom ( x, E ) α −→ Hom ( x ⊗ , E ) α −→ . . . ) . In general Q α E is not the periodization of E with respect to α , one obstruction being that the cyclic per-mutation of α can fail to be homotopic to the identity. This matters because checking periodicity requirespermuting α ⊗ id to id ⊗ α , and these can fail to be homotopic. Lemma 5.5.
The canonical map G W → Q γ G W is an equivalence of (pre)sheaves of spectra on Sm C S,qp . Proof.
We know by the projective bundle formulas that G W ( P σ × P = P σ P ) (cid:27) G W ( P ) ⊕ G W [1] ( P ) (cid:27) G W ( X ) ⊕ G W [ − ( X ) ⊕ G W [1] ( X ) ⊕ G W ( X ) . We claim that under this isomorphism, cup product with ∂ [ T σ ] is projection onto G W [1] ( P ) and cupproduct with ∂ [ T ] on GW [1] ( P ) is projection onto G W ( X ). The latter statement is already known from[Sch17, Theorem 9.10], so we show the former. It su ffi ces to show that cup product with ∂ [ T σX ] ∪ − : G W [ n ] ( X ) ⊕ G W [ n +1] ( X ) → G W ( X ) ⊕ G W [ n +1] ( X ) is projection onto the second factor. But this is preciselyhow [ T σ ] is defined: it’s a lift under ∂ of a generator of G W [1] ( X ), so cup product with it is cup productwith h i on GW [ n +1] ( X ) and it’s necessarily zero on the other factor because it gives a well-defined elementon the pointed G W [ − ( P σ , [1 : 1]).Because G W satisfies equivariant Nisnevich descent, G W ( W /S ) (cid:27) G W ( P , [1 : 1]), and G W ( W σ /S ) (cid:27) G W ( P σ , [1 : 1]). Now we’re essentially done. The maps in the colimit defining Q γ G W first identify G W [ n ] i ( X ) with ∂ ([ T ]) ∪ G W [ n ] i ( P X , [1 : 1]), then identify G W [ n ] i ( P X ) with ∂ ([ T σ ]) ∪ G W [ n ] i ( P σ P X , [1 : 1]). As wenoted above, the projective bundle formulas imply that the image of G W [ n ] i ( X ) under these identificationsis isomorphic to G W [ n ] i ( X ), and hence Q γ G W [ n ] i ( X ) ≃ G W [ n ] i ( X ) as desired. (cid:3) Lemma 5.6.
The canonical map Q γ GW [ m ] → Q γ G W [ m ] (cid:27) G W [ m ] induces isomorphisms π n Q γ GW [ m ] (cid:27) π n G W [ m ] for n ≥ and for all m .Proof. This follows from two out of three and the proof of lemma 5.5 since π n GW [ m ] (cid:27) π n G W [ m ] for n ≥ m . (cid:3) Lemma 5.7.
The canonical map Q γ GW [ m ] → Q γ G W [ m ] (cid:27) G W induces an isomorphism π n Q γ GW [ m ] (cid:27) π n G W [ m ] for n < = 0 and for all m .Proof. Because homotopy groups commute with filtered (homotopy) colimits of spectra π n Q γ GW [ m ] = colim( π n GW α −→ π n Hom ( W /S ⊗ W σ /S , GW ) α ⊗ −−−→ · · · ) . Fix [ m ] for now and denote by F in the image of the map of groups γ ∗ : GW [ m ] n (( W /S ⊗ W σ /S ) ⊗ i ) → GW [ m ] n (( W /S ⊗ W σ /S ) ⊗ i +1 )and note that F n (cid:27) GW [ m ] n . Denote by FB in the same construction as above with GW replaced by G W .For i ≥ − n , we claim that there are exact sequences F in ( A / ⊗ W σ /S ) ⊕ F in ( A / ⊗ W σ /S ) / / F in ( G m / ⊗ W σ /S ) ∂ / / F in − ( W /S ⊗ W σ /S )such that ∂ ( F in ( G m / ⊗ W σ /S )) = ∂ ([ T ]) ∪ ∂ ([ T σ ]) ∪ F in − ( W /S ⊗ W σ /S ). We prove this in conjunction withthe statement that, for each n , F in (cid:27) G W [ m ] n for i ≥ − n . The proof is induction in i , and we must show that ∂ ( F in ( G m / ⊗ W σ /S )) = ∂ ([ T ]) ∪ F in − ( W /S ⊗ W σ /S ). For n ≥
0, the same argument that we gave for K -theorytogether with lemma 5.6 works. In more detail, there’s an exact sequence GW [ m ] n ( A / ⊗ W σ /S ) ⊕ GW [ m ] n ( A / ⊗ W σ /S ) / / GW [ m ] n ( G m / ⊗ W σ /S ) ∂ / / GW [ m ] n − ( W /S ⊗ W σ /S )and because n ≥
0, the same argument we gave for K -theory above identifies ∂ ( GW [ m ] n ( G m / ⊗ W σ /S )) with ∂ ([ T ]) ∪ ∂ ([ T σ ]) ∪ GW [ m ] n − ( W /S ⊗ W σ /S ) and in turn with G W [ m ] ( X ). Then we just use the fact that p ∗ isinjective and a module map to conclude that ∂ ([ T ]) ∪ ∂ ([ T σ ]) ∪ p ∗ ( GW [ m ] n − ( W /S ⊗ W σ /S )) is isomorphic to G W [ m ] ( X ).Now fix an i , and assume by induction that our claim holds for all − n ≤ i . Then there’s an exact sequence G W [ m ] n ( A / ⊗ W σ /S ) ⊕ G W [ m ] n ( A / ⊗ W σ /S ) / / G W [ m ] n ( G m / ⊗ W σ /S ) ∂ / / F in − ( W /S ⊗ W σ /S ) DH DESCENT FOR HOMOTOPY HERMITIAN K -THEORY OF RINGS WITH INVOLUTION 29 which identifies ∂ ( G W [ m ] n ( G m / ⊗ W σ /S )) with ∂ ([ T ]) ∪ ∂ ([ T σ ]) ∪ F in − ( W /S ⊗ W σ /S ), but we know that ∂ ( G W [ m ] n ( G m / ⊗ W σ /S )) is equal to G W [ m ] n − ( W /S ⊗ W σ /S ) (cid:27) G W [ m ] n − ( X ). Thus, letting p denote the projec-tion W /S ⊗ W σ /S → X to the basepoint, G W [ m ] n − ( X ) (cid:27) p ∗ ( ∂ ([ T ]) ∪ ∂ ([ T σ ]) ∪ F in − ( W /S ⊗ W σ /S )) = ∂ ([ T ]) ∪ ∂ ([ T σ ]) ∪ p ∗ ( F in − ( W /S ⊗ W σ /S )) = F i +1 n − since p ∗ is split injective.The meatier part of the argument is producing the exact sequence for F [ i +1] n − , though the proof is essen-tially the same as the proof of the base case.First note that for all i and n , there’s a chain complex F in ( A / ⊗ W σ /S ) ⊕ F in ( A / ⊗ W σ /S ) / / F in ( G m / ⊗ W σ /S ) ∂ / / F in − ( W /S ⊗ W σ /S )which is just the image of the usual long exact sequence for GW under the map γ ∗ . Depending on n , thissequence may or may not be exact, as the image of an exact sequence is in general not exact.Consider the commutative diagram F i +1 n − ( A / ⊗ W σ /S ) ⊕ F i +1 n − ( A / ⊗ W σ /S ) / / (cid:15) (cid:15) F i +1 n − ( G m / ⊗ W σ /S ) ∂ / / (cid:15) (cid:15) F i +1 n − (( W /S ⊗ W σ /S ) ⊗ i +2 ) φ (cid:15) (cid:15) G W ( A / ⊗ W σ /S ⊗ ( W /S ⊗ W σ /S ) ⊗ i +2 ) ⊕ / / G W ( G m / ⊗ W σ /S ⊗ ( W /S ⊗ W σ /S ) ⊗ i +2 ) ∂ B / / G W [ m ] n − (( W /S ⊗ W σ /S ) ⊗ i +3 ) where the left two vertical maps are isomorphisms by what we’ve already shown. We claim that the toprow is exact. The composite is zero since it’s a chain complex, and if x ∈ ker( ∂ ), then using the fact that themiddle and left maps are isomorphisms we produce a lift of x .Now it remains only to check that the image of ∂ coincides with ∂ ([ T ]) ∪ ∂ ([ T σ ]) ∪ F i +1 n − . This is the partof the proof we adapt from the K -theory case. First, it’s clear that ∂ ([ T ]) ∪ ∂ ([ T σ ]) ∪ F i +1 n − ⊆ im( ∂ ), since ∂ ([ T ]) restricts to zero in A . For the other containment, by exactness and the fact that the left two verticalarrows are isomorphisms, we know that im( ∂ ) (cid:27) im( ∂ B ). Now since ∂ ([ T ]) ∪ ∂ ([ T σ ]) ∪ p ∗ ( F i +1 n − ) ⊆ im( ∂ ), it isisomorphic to its image in G W [ m ] n − (( W /S ⊗ W σ /S ) ⊗ i +2 ). But φ is a map of modules, so that φ ( ∂ ([ T ]) ∪ ∂ ([ T σ ]) ∪ F i +1 n − (( W /S ⊗ W σ /S ) ⊗ i +2 )) (cid:27) ∂ ([ T ]) ∪ ∂ ([ T σ ]) ∪ im( φ )But φ is necessarily surjective, and cup product with ∂ ([ T ]) ∪ ∂ ([ T σ ]) is an automorphism of G W . Itfollows that ∂ ([ T ]) ∪ ∂ ([ T σ ]) ∪ F i +1 n − (( W /S ⊗ W σ /S ) ⊗ i +2 ) (cid:27) im( φ ) = im( ∂ B ) (cid:27) im( ∂ )so that ∂ ([ T ]) ∪ ∂ ([ T σ ]) ∪ F i +1 n − (( W /S ⊗ W σ /S ) ⊗ i +2 ) = im( ∂ ).We’ve shown that if the inductive statement holds for i, n , then it holds for i + 1 , n −
1. The fact that itholds for i + 1 , m for any m < n + 1 is clear by appealing to results for G W . Now, the lemma follows from theexplicit description for filtered colimits of groups. (cid:3) Corollary 5.8.
Let γ be the map (8). Then there are weak equivalences of presheaves of spectra Q γ GW ∼ −→ Q γ G W ≃ G W .
Proof.
Combining Lemma 5.6 and Lemma 5.7 we see that Q γ GW → Q γ G W the induces an isomorphismon stable homotopy groups. Lemma 5.5 shows that Q γ G W ≃ G W . (cid:3) Recall the definition of β σ from equation (6). Definition 5.9. A GW -module E is called Bott periodic if the map
Hom ( β σ , E ) : E → Hom (( P , [1 : 1]) ∧ ( P σ , [1 : 1]) , E )is an equivalence. There are zigzags A / G m ֒ → P / ( P − [ − և P / [1 : 1]and A − / G − m ֒ → P σ / ( P σ − [ − և P σ / [1 : 1] . The maps β : P / [1 : 1] → GW [1] and β σ : P σ / [1 : 1] → GW [ − lift to P / ( P − [ − P σ / ( P σ − [ − β ′ : A / G m → GW [1] ( β σ ) ′ : A − / G − m → GW [ − . Taking smash products and using that A ⊕ A − (cid:27) A ρ , we get a map(9) β ′ ⊗ ( β σ ) ′ : A ρ / ( A ρ − → GW [1] ⊗ GW [ − → GW .
For a vector bundle E , let V ( E ) denote E/ ( E − Theorem 5.10.
Let S be a Noetherian scheme of finite Krull dimension with an ample family of line bundles and ∈ S . Then L A G W lifts to an E ∞ motivic spectrum, denoted KR alg , over Sm C S,qp .Proof. GW is an E ∞ object in presheaves of spectra (it’s a commutative monoid in the category of presheavesof symmetric spectra) on Sm C S,qp via the cup product defined in [Sch17, Remark 5.1]. By [Hoy16] Lemma3.3, together with corollary 5.8 above, G W is the periodization of GW with respect to γ . Let T ρ denote theThom space of the regular representation A ρ . Now G W is Nisnevich excisive, so that G W ( W /S ⊗ W σ /S ) (cid:27) G W ( P ∧ P σ ), and G W is γ periodic if and only if it is Bott periodic. By [Hoy16], proposition 3.2, G W liftsto an E ∞ object KR alg in GW mod [( T ρ ) − ]. Because A -localization preserves E ∞ objects, L A G W is an E ∞ object in the subcategory of Nisnevich excisive Bott periodic GW -modules. (cid:3) Lemma 5.11.
The A -localization of the Bott element L A ( β ′ ⊗ ( β σ ) ′ : L A V ( A ρ ) → L A GW , viewed as anelement of Sp ( P A ( Sm C S,qp )) /L A GW is 3-symmetric.Proof. The proof is identical to Lemma 4.8 in [Hoy16]. The main idea is that the identity and the cyclicpermutation σ are both induced by matrices in SL · ( Z ) acting on A ρ , and any two such matrices are(na¨ıvely) A -homotopic so that there’s a map h : A × A ρ → A ρ witnessing the homotopy. We can extendthis to a map A × A ρ π × h −−−−→ A × A ρ . Letting p : A × S → S denote the projection, the displayed map is an automorphism of the vector bundle p ∗ ( A ρ ).Now we claim that given an automorphism φ of p ∗ ( E ) for any vector bundle E , the automorphisms φ , φ of V ( E ) induced by the restrictions of φ to 1 and 0 are A -homotopic over L A GW . Once we’ve shown thisclaim, we can apply it to the automorphism π × h of p ∗ ( A ρ ) to complete the proof of the lemma.To prove the claim, by functoriality of β ′ , any automorphism φ as above induces a triangle V ( p ∗ ( E )) φ / / β ′ p ∗ ( E ) ' ' ❖❖❖❖❖❖❖❖❖❖❖ V ( p ∗ ( E )) β ′ p ∗ ( E ) w w ♦♦♦♦♦♦♦♦♦♦♦ L A GW A × X or presheaves of spectra on Sm C A × X . By adjunction, this is equivalent to a triangle A ⊗ V ( E ) / / β ′ E & & ◆◆◆◆◆◆◆◆◆◆◆ V ( E ) β ′ X y y tttttttttt L A GW X which is an A -homotopy between φ and φ over L A GW as desired. (cid:3) DH DESCENT FOR HOMOTOPY HERMITIAN K -THEORY OF RINGS WITH INVOLUTION 31 We’ve shown that G W is Bott periodic and Nisnevich excisive. Since it’s the Bott periodization of GW ,it is in fact the reflection of GW in the subcategory of Nisnevich excisive and Bott periodic GW -modules.Because C is a finite group (scheme), making A contractible is equivalent to making the regular repre-sentation A ρ contractible. Indeed, there’s an elementary A homotopy A × A ρ → A ρ ( t, s ) t · s between id A ρ and the composite A ρ → S −→ A ρ . Clearly the map S −→ A ρ → S is the identity, so that A ρ → S is a na¨ıve A -homotopy equivalence. Thus by definition, L A G W is the reflection of GW in thesubcategory of homotopy invariant, Nisnevich excisive, and Bott periodic GW -modules. Corollary 5.12.
The canonical map GW → Q β L mot GW is the universal map to a homotopy invariant, Nis-nevich excisive, and Bott periodic GW -module. In particular L A G W = Q β L mot GW Proof.
Given Lemma 5.11, the proof is identical to Proposition 4.9 in [Hoy16]. (cid:3)
Replacing GW with its connective cover GW ≥ , the same reasoning yields: Porism 5.13.
The canonical map GW ≥ → Q β L mot GW ≥ is the universal map to a homotopy invariant,Nisnevich excisive, and Bott periodic GW ≥ -module. In particular L A G W = Q β L mot GW ≥ Proof. (cid:3) cdh Descent for Homotopy Hermitian K -theory Recall from Definition 2.16 that the cdh topology is the topology generated by the Nisnevich and abstractblow-up squares. Fix a Noetherian scheme of finite Krull dimension S , and scheme X which is smooth over S . Let KR alg be the motivic spectrum associated to homotopy Hermitian K -theory L A G W on Sm C S,qp for S a Noetherian scheme of finite Krull dimension with an ample family of line bundles and ∈ S .Let H( S ) denote the motivic homotopy ∞ -category on Sm C S,qp . By Corollary 5.12, we can write KR alg asthe image under the localization functor QL mot : Stab lax T ρ Sp ( P ( Sm C S,qp )) → Stab T ρ Sp ( H ) ≃ SH C of the “constant” T ρ -spectrum c β ′ ⊗ ( β σ ) ′ GW , where the maps T ρ ∧ GW → GW are induced by adjunctionafter applying Hom GW − mod ( − , GW ) to the map β ′ ⊗ ( β σ ) ′ : T ρ → GW with β ′ ⊗ ( β σ ) ′ the map (9). Definition 6.1.
For X ∈ Sm C S,qp , let KR alg X denote the restriction of the motivic spectrum KR alg to Sm C X,qp .We want to show that KR alg is a cdh sheaf on Sm C S,qp . By first checking that the formalism of six op-erations holds in equivariant motivic homotopy theory and following the same recipe as the K -theorycase, [Hoy17, Corollary 6.25] proves that it su ffi ces to show that for each f : D → X ∈ Sm C S,qp , the restrictionmap f ∗ ( KR alg X ) → KR alg D in SH C ( D ) is an equivalence.By [Sch17, Appendix A], there’s a mapHerm( X ) + → Ω ∞ GW ( X )where Herm( X ) is the E ∞ space of non-degenerate Hermitian vector bundles over X and ( − ) + denotes groupcompletion. If X is an a ffi ne C -scheme, the category of Hermitian vector bundles is a split exact categorywith duality, and the above map is an equivalence. It follows thatHerm + → Ω ∞ GW | Sm C X is a motivic equivalence in P ( Sm C X ).Just as in [Hoy16] we note that a n ≥ B isoEt O ( h i ⊥ n ) → Hermexhibits Herm as the equivariant Zariski sheafification of the subgroupoid of non-degenerate Hermitianvector bundles of constant rank (in other words, it fixes the sections over non-connected or hyperbolicrings). Since L Zar preserves finite products, by [Hoy16, Lemma 5.5], the map remains a Zariski equivalenceafter group completion yielding a motivic equivalence a n ≥ B isoEt O ( h i ⊥ n ) + → Ω ∞ GW | Sm C X . Fix a map f : D → X in Sm C S,qp . Again by [Hoy16, Lemma 5.5], since the pullback f ∗ : P ( Sm C X ) →P ( Sm C X ) preserves finite products, it commutes with group completion of E ∞ -monoids. The same is truefor L mot . It follows that there are motivic equivalences f ∗ ( Ω ∞ GW | Sm C X ) → f ∗ a n ≥ B isoEt O ( h i ⊥ n ) + → a n ≥ f ∗ B isoEt O ( h i ⊥ n ) + Because B isoEt O ( h i ⊥ n ) is representable by the results of Section 4, [Hoy16, Proposition 2.9] yields amotivic equivalence a n ≥ f ∗ B isoEt O ( h i ⊥ n ) + → a n ≥ B isoEt f ∗ O ( h i ⊥ n ) + . But f ∗ O ( h i ⊥ n ) | Sm C X = O ( h i ⊥ n ) | Sm C D since f ∗ h i X = h i D . It follows that there’s a motivic equivalence a n ≥ B isoEt f ∗ O ( h i ⊥ n ) + → Ω ∞ GW | Sm C D , and combining everything we get that the restriction map f ∗ ( Ω ∞ GW | Sm C X ) → Ω ∞ GW | Sm C D is a motivic equivalence in the ∞ -category of grouplike E ∞ -monoids in P ( Sm C D ). Moving to the categoryof connective spectra, It follows that pullback agrees with restriction for GW ≥ . Because the localizationfunctor QL mot is also compatible with the base change f ∗ , it follows that each arrow f ∗ ( QL mot GW ≥ | Sm C X ) → QL mot ( f ∗ GW ≥ | Sm C X ) → QL mot ( GW ≥ | Sm C D )is a motivic equivalence. Finally, Porism 5.13 tells us that QL mot GW ≥ ≃ QL mot GW , so we’ve proved Theorem 6.2.
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