Cdh Descent for Homotopy Hermitian $K$-Theory of Rings with Involution

We provide a geometric model for the classifying space of automorphism groups of Hermitian vector bundles over a ring with involution $R$ such that $\frac{1}{2} \in R$; this generalizes a result of Schlichting-Tripathi \cite{SchTri}. We then prove a periodicity theorem for Hermitian $K$-theory and use it to construct an $E_\infty$ motivic ring spectrum $\mathbf{KR}^{\mathrm{alg}}$ representing homotopy Hermitian $K$-theory. From these results, we show that $\mathbf{KR}^{\mathrm{alg}}$ is stable under base change, and cdh descent for homotopy Hermitian $K$-theory of rings with involution is a formal consequence.


Introduction
Algebraic K-theory is an algebraic invariant introduced in the 1950s by Alexander Grothendieck where it served as the cornerstone of his reformulation of the Riemann-Roch theorem [Gro57].Twenty years previously, Ernst Witt developed the notion of quadratic forms over arbitrary fields and introduced the Witt ring as an object to encapsulate the nature of all the quadratic forms over a given field [Wit37].Combining the ideas of Grothendieck and Witt, Hyman Bass introduced a category of quadratic forms Quad(R) with isometries over a ring R and studied K 1 (Quad(R)) and K 0 (Quad(R)).K 0 (Quad(R)) is what we know today as the Grothendieck-Witt ring, and Bass was able to recover the Witt ring as a quotient of K 0 (Quad(R)) by the image of the hyperbolic quadratic forms.He went on to show that K 1 (Quad(R)) was related to the stable structure of the automorphisms of hyperbolic modules, which complemented the relationship between K 1 (R) and the group GL(R).The K-theory of quadratic forms soon found applications to surgery theory 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 via the + 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 homotopy category 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 motivic cohomology, it also opened the door for applications of topological tools like obstruction theory to more algebraic 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 that Hornbostel defined Hermitian K-theory on schemes by extending the definition on rings using Jouanolou's trick.In 2011 Hu-Kriz-Ormsby showed that Hermitian K-theory on the category of C 2 -schemes over a field is representable in the C 2 -equivariant stable motivic homotopy category [HKO11].Here they used a similar trick to Hornbostel in order to extend Hermitian K-theory from rings with involution to schemes with involution.In the meantime, Schlichting, building off of work of Thomason, Karoubi, and Balmer, defined the higher Hermitian K-theory of a dg-category with weak equivalences and duality and proved the analogues of the fundamental theorems of higher K-theory for these groups [Sch17].Although some of Schlichting's theorems are stated only for schemes (rather than schemes with C 2 action), many of his proofs require only trivial modification to extend to Grothendieck-Witt groups of schemes with C 2 action.See also [Xie18] for the proofs of the equivariant version of some of the theorems together with a new transfer morphism.Another approach is taken by Hesselholt-Madsen, who define real algebraic K-theory of a category with weak equivalences and duality as a symmetric spectrum object in the monoidal category of pointed C 2 -spaces.Schlichting's higher Grothendieck-Witt groups can be recovered from the Hesselholt-Madsen construction by taking homotopy groups of C 2 -fixed points of deloopings of the real algebraic K-theory spaces with respect to the sign representation spheres.We note as well that the Ph.D. thesis of Alejo L ópez-Ávila [Lv18] shows that the motivic spectrum representing Hermitian K-theory in the nonequivariant setting has an E ∞ structure.
Back in K-theory land, Cisinski proved that the six functor formalism in motivic homotopy theory developed by Ayoub [Ayo07] together with the fact that the motivic K-theory spectrum KGL is a cocartesian section of SH(−) yields a simple proof of cdh-descent (descent in the completely decomposed h topology) for homotopy K-theory [Cis13].This in turn yields a 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-theory of schemes with C 2 action.The techniques in [Hoy16] provide our pathway to descent.In order to show that Hermitian K-theory is a cocartesian section of SH C 2 (−), we need to show that the Hermitian K-theory space Ω ∞ GW can be represented by a certain Grassmannian, and we need a periodization theorem in order to pass from the Hermitian K-theory space Ω ∞ GW to the homotopy Hermitian K-theory motivic spectrum L A 1 GW .Schlichting and Tripathi [SST14] show that Ω ∞ GW is representable by a Grassmannian over schemes with trivial action over a regular base scheme with 2 invertible.Their techniques extend to the equivariant 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 Ktheory 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 finite group scheme over a base S which is Noetherian of finite Krull dimension, has an ample family of line bundles, and has 1 2 ∈ Γ(S, O S ).First we review the definition of the equivariant étale and Nisnevich topologies, then we introduce the isovariant étale topology and give some examples of covers.For the reader familiar with non-equivariant motivic homotopy theory, the assumptions we make on G are strong enough so that structural results are mostly the same: • the equivariant Nisnevich topology is generated by a nice cd-structure, • equivariant schemes are locally affine in the equivariant Nisnevich topology, and • to invert G-affine bundles Y → S it suffices to invert A 1 S .
The content in this section is a selection of relevant content from [HKØ15].We end this section with the definition of the unstable and stable equivariant motivic ∞-categories 2.4 a la Hoyois [Hoy17].
Section 3 reviews the definitions and results on Hermitian forms which will be necessary to work with the Grothendieck-Witt spectrum.Section 3.1 contains the basic definitions and examples, while section 3.3 contains the tools necessary to show that Hermitian forms are locally determined by rank in the isovariant or equivariant étale topologies.The final section 3.4 reviews the main definitions of [Sch17] to allow us to talk 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 the techniques of [SST14] and [Hoy16] in order to show that classifying spaces of automorphism groups of Hermitian vectors bundles are representable in the C 2 -equivariant motivic homotopy category.
This section culminates with the representability result, Theorem 4.14, which we note holds over nonregular base schemes: Theorem 1.1.Let S be a Noetherian scheme of finite Krull dimension with an ample family of line bundles and 1 2 ∈ S.There is an equivalence of motivic spaces on Sm With a simple modification to remove the regularity hypothesis, one can follow [SST14] to show that 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 2 (S).The crucial fact is that the localizing version of Hermitian K-theory of rings with involution, denoted GW , is the periodization of GW with respect to a certain Bott map derived from projective bundle formulas for P 1 and P σ (see Corollary 5.8).Here P σ is a copy of P 1 with action [x : y] → [y : x].The fact that the periodization functor is monoidal together with Schlichting's results on monoidality of GW immediately give that the motivic spectrum L A 1 GW ∈ SH C 2 (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 1 2 ∈ S. Then L A 1 GW lifts to an E ∞ motivic spectrum, denoted KR alg , over Sm C 2 S,qp .
The final section 6 follows the recipe given by Cisinski and summarized in [Hoy16] to prove cdh descent for equivariant homotopy Hermitian K-theory on the category of quasi-projective S-schemes.After reviewing 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 1 2 ∈ S. Then the homotopy Hermitian K-theory spectrum of rings with involution L A 1 GW satisfies descent for the equivariant cdh topology on Sch

Acknowledgements
This paper is a condensed and cleaned up version of my doctoral thesis.I owe a debt of gratitude to my advisor Jeremiah Heller for his smooth and continuous support throughout my time in graduate school.I would also like to thank my academic siblings Tsutomu Okano and Brian Shin for the many helpful conversations about motivic homotopy theory.Finally, I want to thank Elden Elmanto for his help and willingness to answer even my most inane questions.
This paper is much better off because of the detailed reading and helpful comments of an anonymous reviewer, and I would like to thank them for their work.

Equivariant Topologies and the Equivariant Motivic Homotopy Category
This section reviews the foundations of equivariant motivic homotopy theory.The key definitions are those of the equivariant étale and Nisnevich topologies -two topologies that play a crucial role in defining the equivariant motivic infinity category H G (S) over a Noetherian base scheme S with finite Krull dimension, with an ample family of line bundles, and with S,qp be the full subcategory of schemes smooth over S.
Notation 2.1.Throughout this section, G will be either a finite group or the group scheme over S associated to 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 Sch S,qp ): Whenever we write down a pullback square involving schemes, we'll tacitly be thinking of G as a group scheme, and X × Y will really mean X × S Y .
We introduce the background definitions from [HKØ15] which will allow us to define the isovariant étale topology.This is a topology which is slightly coarser than the equivariant étale topology, but whose points are still nice enough so that Hermitian vector bundles are locally determined by rank.
Definition 2.2.For a G-scheme X, the isotropy group scheme is a group scheme G X over X defined by the cartesian square By the pasting lemma, this is the same as the pullback 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 described as 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 Galois extension k ֒→ L. Let G = Gal(L/k) be the Galois group.The set-theoretic stabilizer of the unique point in Spec 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 schemes to sets preserves any pullback Z × X Y .The forgetful functor Sch G S,qp → Sch S,qp is a right adjoint, hence preserves pullbacks.
Since the inclusion of a point Spec k(x) ֒→ X × S X will be a monomorphism, the difference between the set-theoretic and scheme-theoretic stabilizers is due to the fact that the underlying space of X × S X is not necessarily the fiber product of the underlying spaces.Indeed, in the example above, SpecL × k SpecL g∈G Spec k, whereas the pullback in spaces is just a single point.

The Equivariant and Isovariant Étale Topologies.
Notation 2.8.Let S be a G-scheme.The equivariant étale topology on Sch G S,qp will denote the site whose covers are étale 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 isomorphism on isotropy groups G Y G X × X Y .A collection {f i : X i → X} i∈I of equivariant maps called an isovariant étale cover if it is an equivariant étale cover such that each f i is isovariant.
Remark 2.10.The isovariant topology is equivalent to the topology whose covers are equivariant, stabilizer preserving, étale maps.We'll use this notion more often in computations.
Remark 2.11.The points in the isovariant étale topology are schemes of the form G × G x Spec(O sh X,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 étale topology are either strictly henselian local or hyperbolic rings will be crucial when we want to describe the isovariant étale sheafification of the category of Hermitian vector bundles.Fortunately Hermitian vector bundles over such rings are well understood, and we'll in fact show that Hermitian vector bundles are up to isometry determined by rank locally in the isovariant étale topology.
) where the induced action on the residue field is trivial.
2.2.The Equivariant Nisnevich Topology.Similarly to the non-equivariant case, the equivariant Nisnevich topology is defined by a particularly nice cd-structure.While there are a few different definitions of this topology in the literature which can give non Quillen equivalent model structures, we use the definition from [HKØ15].Definition 2.14.The equivariant Nisnevich cd-structure on Sch G S,qp is the collection of distinguished equivariant Nisnevich squares in Sch G S,qp .The next remark has the important consequence that to prove a map is an equivariant motivic equivalence, it suffices to check that it's an equivalence on affine G-schemes.
Remark 2.15.By Lemma 2.20 in [HKØ15], for finite groups G, any separated G-scheme of finite type over S is Nisnevich-locally affine.

2.3.
The Equivariant cdh Topology.The completely decomposed h (cdh) topology is, roughly speaking, the coarsest topology satisfying Nisnevich excision and which allows for a theory of cohomology with compact support.Like the Nisnevich topology (and unlike the étale topology) it can be generated by a cd-structure, which gives a convenient 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 where i is a closed immersion and p is a proper map which induces an isomorphism ( X − Z) (X − Z).
Definition 2.17.The cdh topology is the topology generated by the cd-structure whose distinguished squares are • the equivariant Nisnevich distinguished squares; • the abstract blowup squares.
One canonical example of a cdh cover is the map X red → X for an equivariant scheme X → S. Another example is given by resolution of singularities: given a proper birational map p : X → Y , it's an isomorphism over some open set U in Y , so letting Z = Y − U and Z = X − p −1 (U) we get an abstract blowup square Finally we come to the definition of a motivic G-space, namely a presheaf that is both Nisnevich excisive and homotopy invariant.
Definition 2.20.Let S be a G-scheme.A motivic G-space over S is a presheaf on Sm G S,qp that is homotopy invariant and Nisnevich excisive.Denote by H G (S) ⊂ P (Sm G S,qp ) the full subcategory of motivic G-spaces over S. Let denote the motivic localization functor, where the colimit is in the ∞-category of presheaves.
In order to form the stable equivariant motivic homotopy category, we also need to discuss pointed motivic G-spaces.Definition 2.21.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 need only invert the Thom space of the regular representation T ρ .Definition 2.22.Let S be a G-scheme.The symmetric monoidal ∞-category of motivic G-spectra over S is defined by where T ρ is the Thom space of the regular representation A ρ /A ρ − 0 of G.The colimit is taken in the ∞-category of presentable ∞-categories.
2.5.Computations with Equivariant Spheres.Because we'll be using equivariant spheres to index our spectra, we'll record some of their basic properties here.These computations will be important when we investigate periodicity of GW in section 5. Though there are exotic elements of the Picard group even in non-equivariant stable motivic homotopy theory, we'll be concerned with the four building blocks Here G σ m is the C 2 scheme corresponding to S[T , T −1 ] with action T → T −1 .
Lemma 2.24.There is a homotopy pushout square, where f can be taken to map C 2 to {[1 : 1]}: The above square is equivalent to the square By the lemma above, is a homotopy pushout square.But adding a disjoint basepoint is a monoidal functor, so X + ∧ Y + (X × Y ) + and this square is equivalent to the desired square.
Proof.Let Q denote the homotopy cofiber of (C 2 × G σ m ) + → (G σ m ) + , and Q denote the homotopy cofiber of (C 2 × A 1 ) + → P σ + .Then the lemma above implies that implies that the cofiber of S σ → Q is P σ .
The result now follows from the commutativity of the following diagram and homotopy invariance of homotopy cofiber:

Hermitian Forms on Schemes
This section reviews the definitions and properties of Hermitian forms over schemes with involution from [Xie18].After defining the proper notion of the dual of a quasi-coherent module over a scheme with involution, 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 discuss in section 3.3 the structure of Hermitian forms over semilocal rings as this is the fundamental tool for showing that Hermitian forms are locally trivial in the isovariant étale topology.We prove this particular statement in Corollary 3.27.We end this section by recalling Schlichting's definition of a dg categoy with weak equivalence and duality and the Grothendieck-Witt groups of such an object.Now, we generalize the above definitions to schemes.
Definition 3.6.Let X be a scheme, and Definition 3.7.Let X be a scheme with involution σ, and M a right O X -module.Note that there's an induced 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 ), 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 We claim that these two choices of dual are naturally isomorphic.
On the other hand, given id, this is clearly the inverse to the map above.
It's clear that these assignments are natural, since they're just postcomposition with a natural transformation.
Definition 3.9.Define the adjoint module M * to be Hom mod−O X (σ * M, O X ).By the remark above, it doesn't really matter which of the two possible definitions we choose here.From this point, we will also use Hom mod−O X synonymously with Hom O X .Definition 3.10.Given a right O X -module M, we define the double dual isomorphism can M : M → M * * as follows: given an open U ⊆ X, we define a map More globally, there's an evaluation map 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 isomorphism of Definition 3.10.A Hermitian vector bundle over X is a locally free right O X -module V with an O X -module map φ : V → V * such that φ = φ * can V .A Hermitian vector bundle is non-degenerate if φ is an isomorphism.
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(Mˇ)) in one direction and the sheaf of sections in the other.For locally free sheaves, we have Mˇ⊗ Nˇ (M ⊗ N )ˇso that the functor is monoidal.We will use this to think of a Hermitian form as a map of schemes 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 2n S with the Hermitian form (x 1 , . . ., x 2n , y 1 , . . ., y 2n ) → n i=1 x 2i−1 y 2i−1 − x 2i y 2i .Denote this Hermitian form by h diag .As defined this way, the matrix of this Hermitian form is For this definition to give a Hermitian space isometric to other standard definitions of the hyperbolic form, it's crucial that 2 be invertible.
The isometries of H R (where we give it the hyperbolic form above) have the form Example 3.14.Similarly to above, we can define a hyperbolic form h by the matrix 0 I I 0 .
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 the Hermitian 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) Hermitian vector 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 that it's Hermitian.Given the map ω : But pullback commutes with sheaf dual for locally free sheaves of finite rank, so we just need to check that changing the module structure via the involution commutes with pullback; that is, we need to check that f * (V ) = f * (V ).However, this is clear since the structure map on f * (V ) is given by 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 canonical isomorphism g * (M ⊥ ) (g * M) ⊥ .
Proof.Recall that, by definition, It follows that the composite map g * (M ⊥ ) → g * V → g * (M * ) is zero, and hence by universal property of kernel there's a canonical map where we've used the canonical isomorphism g * (M * ) (g * (M)) * for locally free sheaves.
We claim that this map is an isomorphism.It suffices to check on stalks, where the map can be identified with a map ⊗ O Y ,y → 0 is split exact, and the canonical map is an isomorphism.
We record two incredibly useful results for working with Hermitian forms.The first implies that Hermitian forms over fields can be written as an orthogonal sum of rank 1 Hermitian forms, while the second gives a useful characterization of non-degenerate submodules of a Hermitian module.
Theorem 3.17.(Knus [Knu91] 6.2.4)Let (M, b) be a non-degenerate Hermitian vector bundle 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 and char D 2.

Hermitian Forms on Semilocal
Rings.From here on out, all rings are assumed to be commutative.Many of the results of this section can be deduced from [FW17], though we include proofs in an effort to make the document self contained.
The following lemma is a slight generalization of a result from [Bae78] which will allow us to conclude that Hermitian forms diagonalize over semilocal rings with involution.
Lemma 3.19.Let (R, σ) be a commutative 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 where F is a free non-degenerate subspace of E, there exists an orthogonal decomposition E = F ⊥ G of E with F free and non-degenerate, and To wit, since 1 − st ∈ I for some s by assumption (because the determinant is a unit mod the ideal I), then st cannot be contained in any maximal ideal, so st ∈ R × =⇒ t ∈ R × .It follows that the λ i are zero (otherwise we would have a non-zero vector in the kernel of an invertible matrix), so that the x i are independent as desired.The determinant fact also shows that F is non-degenerate, so by the lemma above, it has an orthogonal summand G.By construction Lemma 3.20.Hermitian forms over R 1 ×R 2 (with trivial involution) are in bijection with Herm(R 1 )×Herm(R 2 ).
A Hermitian form we note that, by linearity, it must be the case that e 1 M ⊗ e 2 M → R 1 × R 2 is the zero map; to wit, b(e 1 m 1 , e 2 m 2 ) = e 1 e 2 b(m 1 , m 2 ) = 0. Thus this Hermitian form is determined completely by the Finally, note that, again by linearity, we see that , and e 1 R 2 = 0. Similarly for the other map.Hence the Hermitian form is completely determined by the maps e 1 M ⊗ e 1 M → R 1 and e 2 M ⊗ e 2 M → R 2 .
We claim that Hermitian forms over finite products of fields diagonalize, and then the result will follow from Lemma 3.19.By induction and Lemma 3.20, a Hermitian module M is determined by Hermitian modules M i over s important to note here that the rank of each M i is the same by assumption).Thus a diagonalization of M is given by (a 1,1 , . . ., a Corollary 3.22.Let R be a local ring with trivial involution and with 1 2 ∈ R. Then any Hermitian module (which is necessarily free) over R diagonalizes.
Lemma 3.23.Let R be a ring, and consider the ring R × R with the involution that switches factors.Then any module M can be written as e 1 M ⊕ e 2 M as in the proof of Lemma 3.20.A non-degenerate Hermitian form on this module is determined by a map e 1 M ⊗ e 2 M → R × R. In other words, the matrix representing the map where A is invertible.Corollary 3.24.Let R, M be as in lemma 3.23 and such that 1 2 ∈ R. Then M H(e 1 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 of Knus.Now by Lemma 3.23 b| e 1 M = 0, so M has direct summands e 1 M, e 2 M such that e 1 M = e 1 M ⊥ and M = e 1 M ⊕ e 2 M. Now [Knu91, Corollary 3.7.3]applies to finish the proof.
Corollary 3.25.Let R be a semi-local ring with involution and with 2 invertible.Then any Hermitian module of constant rank over R diagonalizes.
Proof.Using Lemma 3.19 and reducing modulo the Jacobson radical (which is always stable under the involution), it suffices to prove the corollary for R a finite product of fields.Then R = F 1 × • • • × F n is semi-simple, and hence we can index the fields in a particularly nice way (proof is by considering idempotents), writing By Theorem 3.17 and Corollary 3.24, the form when restricted to each M i or N 2i ⊕ N 2i−1 is diagonalizable, so the form is diagonalizable (see the proof of Corollary 3.21).
Lemma 3.26.Non-degenerate Hermitian vector bundles are determined by rank over strictly henselian local rings (R, m) with 1 2 ∈ R such that the residue field R/m has trivial involution.Proof.By Corollary 3.25, any Hermitian vector bundle over R diagonalizes.Thus it suffices to prove that any 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 one dimensional Hermitian matrix).Because R is strictly henselian, there is a square root c of x −1 .We claim that c = c.Assume not.Then because the involution on R/m is trivial, c − c ∈ m.Since 2 is invertible, we It follows that c+c 2 is a unit.Otherwise it would be contained in m which would imply that the unit c was contained in m.
However, we calculate This shows that given any one dimensional Hermitian matrix x, there's a unit c such that cxc = 1 so that all one dimensional Hermitian forms are isometric to the form 1 .
Corollary 3.27.Non-degenerate Hermitian vector bundles are locally determined by rank in the isovariant étale topology.
Proof.The points in the isovariant étale topology are either strictly henselian local rings whose residue field has trivial involution or a ring of the form O sh X,x × O sh X,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 3.26 shows that non-degenerate Hermitian forms are determined by rank.If the ring is hyperbolic, then by Corollary 3.24 all non-degenerate Hermitian forms over the ring are hyperbolic forms of projective modules over a local ring.Since projective modules over a local ring are determined by rank, the corresponding hyperbolic forms are determined by rank.
3.4.Higher Grothendieck-Witt Groups.In [Xie18], the author works with coherent Grothendieck-Witt groups 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 homotopy fixed points of K-theory to Grothendieck-Witt theory), there is no difference between the additive and localizing versions of Grothendieck-Witt spectra in this setting.
Therefore, we work instead with Grothendieck-Witt spectra of sPerf(X) = Ch b Vect(X), the dg category of strictly perfect complexes on X.We review the relevant definitions from [Sch17] now.
Definition 3.28.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 of dg functors called the double dual identification such that can ∨ A • can A ∨ = 1 A ∨ for all objects A in A. Remark 3.29.A dg category with duality has an underlying exact category with duality (Z 0 A ptr , ∨, can), where Z 0 A ptr has the same objects as A ptr but the morphism sets are the zero cycles in the morphism complexes of A ptr .Here A ptr is the pretriangulated hull of A (see [Sch17] definition 1.7).
Definition 3.30.A dg category with weak equivalences is a pair (A, w) where A is a pointed dg category and w ⊆ Z 0 A ptr is a set of morphisms which saturated in A. A map f in w is called a weak equivalence.Definition 3.31.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 3.32.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 the dg subcategory A w ⊂ A of w-acyclic objects is closed under the duality functor ∨ and can A : A → A ∨∨ is a weak equivalence for all objects A of A.
Definition 3.33.Let A = (A, w, ∨, can) be a dg category with weak equivalences and dualiy.A symmetric space in A ptr is a Hermitian object A whose dual map φ : A → A ∨ is a weak equivalence in A ptr .The Grothendieck-Witt group GW 0 (A ) of A is the abelian group generated by symmetric spaces [X, φ] in the underlying category with weak equivalences and duality (Z 0 A ptr , w, ∨, can), subject to the following relations: ( , and (3) if (E • , φ • ) is a symmetric space in the category of exact sequences in Z 0 A ptr , that is, a map Definition 3.34.Given a dg-category with weak equivalences and duality A = (A, w, ∨, can), Schlichting defines [Sch17, Section 4.1] a functorial monoidal symmetric spectrum GW (A ) using a modified version of 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, GW in [Sch17, Section 8.1] as a bispectrum.The reason Schlichting defines GW as an object in bispectra rather than spectra is to get a monoidal structure on GW .We provide an alternative approach to producing GW via periodization in section 5. Definition 3.35.Let X be a Noetherian scheme of finite Krull dimension with an ample family of line bundles, and let σ : X → X be an involution on X.Let sPerf(X) denote the category of strictly perfect complexes on X with the weak equivalences being the quasi-isomorphisms.Define a family of dualities on sPerf(X) indexed by i ∈ N by Note that because σ is an involution, σ * E is a strictly perfect complex.Define the canonical isomorphim can as in Definition 3.10 as the adjoint of the evaluation map Combining all this data we get a collection of dg categories with weak equivalences and duality (sPerf(X), q. iso, * i , can).
The ith shifted Grothendieck-Witt spectrum of (X, σ) is defined as If Z is an invariant closed subset of X, then the duality on sPerf(X) restricts to a duality on the subcategory of 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-theory pulls back nicely.In particular, given f : X → S a map of schemes over S, one can use ind-representability of KGL to show that f * (KGL S ) = KGL X .Together with the formalism of six operations in motivic homotopy theory, one obtains rather formally cdh descent for algebraic K-theory, see [Cis13].
The goal of this section is to define a sheaf on Sm By analogy with the K theory case, the equivariant scheme representing Hermitian K-theory on Sm C 2 S,qp will be a colimit of schemes which parametrize non-degenerate Hermitian sub-bundles of a given Hermitian vector bundle V .The new results here are mostly the definitions, as the proofs in this section are either minor modifications or identical to the proofs in [SST14].The main difference which might cause concern is that stalks in the isovariant étale topology are now semi-local (rather than local) rings.
We combine the techniques of [SST14] with a Morel-Voevodsky style argument to compare RGr 2d (H ∞ ) to the isovariant étale 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 étale topology, Hermitian vector bundles are determined by rank.This will utilize some of the analysis of Hermitian forms over semi-local rings from section 3.3.Note that this is a key difference 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] allows 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 comparing 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 RGr.The definition here describes the sections of the underlying scheme of RGr 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 2 S,qp defined by the formula (X, σ : Fix a (possibly degenerate) Hermitian vector bundle (V , φ) over a base scheme S with 2 invertible and with trivial involution.The canonical example of such a base scheme is S = SpecZ[ 1 2 ].We'll define a presheaf RGr : (Sm C 2 S,qp ) op → Set by first defining a presheaf on Sm S,qp , showing that it's representable, equipping with an action, then taking the corresponding representable functor on Sm C 2 S,qp .We can then extend to an arbitrary equivariant base T with 2 invertible by pulling back along the unique map where W is locally free.
Here by an isomorphism of split surjections we mean a diagram There's a natural action of C 2 on RGr V whose non-trivial natural transformation will be denoted η.Define η as follows: Fix an object X ∈ Sm S,qp .Define We'll define the maps q and t now.Let t ′ denote the canonical map ker p → f * V , let q ′ be the map where we've used the identification im t ′ = im(id − (s • p)).
Recall that and similarly for (ker p) ⊥ .Leaving out the can map for convenience, we get split exact sequences By the splitting lemma for abelian categories, f * V W ⊥ ⊕ W * , and there's a (canonical) split surjection f * V ։ W ⊥ with W ⊥ locally free.Similarly we obtain a canonical surjection q : f * V ։ (ker p) ⊥ split by a map t.
Given an isomorphism we get an isomorphism of (split) diagrams and hence an isomorphism of split surjections , 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 an element By Lemma 3.16, there's a canonical isomorphism g * ((ker(p) ⊥ )) → (g * (ker(p))) ⊥ , and under this isomorphism 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 RGr : To determine its values on a C 2 -scheme (X, σ), we note that a fixed point of the action of Lemma 4.1 is determined by an isomorphism of split surjections Note that because σ is an involution, for any O X -module M, there's a canonical isomorphism of O Xmodules σ * M σ * M. Thus there's a natural isomorphism

It follows that any Hermitian form
can be promoted to a Hermitian form 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 promoted form φ. Said differently, we claim that But using the natural isomorphism between σ * and σ * , together with the natural isomorphisms and σ * f * V f * V , this becomes a question of whether σ * is left exact.In general it isn't, but because σ is an involution, σ * is naturally isomorphic to σ * which is left exact.The claim follows.4.2.Representability of RGr.Fix a Hermitian vector bundle (V , φ) over S where dim(V ) = n and S is a scheme with trivial involution.Then the underlying scheme of RGr(V ) is the pullback where the right vertical map sends p → (p • p, p).In other words, the underlying scheme is the scheme of idempotent 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) → RGr(V ) corresponds to an idempotent p : V X → V X such that φ −1 (γ −1 (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 structure map 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 and the adjoint of p is given by φ −1 (σ * p) * φ.Expanding, this is and so we recover the condition that p † = p, which corresponds to the fact that V X = ker p ⊥ im p, and hence the restriction of the form on V X to im p (and ker p) is non-degenerate.
To summarize, the underlying scheme of RGr(V ) represents idempotents, and equivariant maps pick out those 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 RGr d (V ) to be the closed subscheme of RGr(V ) cut out by rk(p) = d, where rk is the rank map.In other words, RGr d (V ) is the pullback The requirement that V be non-degenerate is necessary so that the action on RGr(V ) sends rank d subspaces to rank d subspaces and hence induces an action on RGr d (V ).
Remark 4.3.Denote by g : RGr d (V ) → S the structure map of RGr d (V ).Because RGr d (V ) is representable by a C 2 -scheme, there's an idempotent g * (V ) → g * (V ) corresponding to the identity map id : RGr d (V ) → RGr d (V ).This idempotent is simply the idempotent which, over a point of RGr d (V ) represented by an idempotent p : V → V , restricts to p. There's an action σ on RGr d (V ) × S V induced by the action on RGr d (V ), and using the fact that σpσ = p † one can see that this idempotent is non-degenerate with respect to the promoted Hermitian form on g * (V ) compatible with the involution on RGr d (V ).
Remark 4.4.Since we've shown that RGr(V ) represents non-degenerate Hermitian subbundles of V , at this point we'll move away from explicitly referring to split surjections and just represent the sections of RGr(V ) by non-degenerate subbundles.
Definition 4.5.Let H S denote the hyperbolic space 3.13 over the base scheme S. Let H ∞ = colim n H n S .Similarly given a non-degenerate Hermitian vector bundle V , let V ⊥ H ∞ = colim n V ⊥ H n .For V ⊂ H ∞ a constant rank 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 2 ,op S,qp , Set) be the functor which on objects sends a subbundle V to

4.3.
The Étale Classifying Space.The content of this section is a straightforward generalization of the work of [SST14] to the C 2 -equivariant setting.Fix a scheme S with 2 invertible, and let (V , φ) be a (possibly degenerate) Hermitian vector bundle over S. For a C 2 -scheme f : X → S over S, let be the category of non-degenerate Hermitian sub-bundles of f * V .A morphism in this category from E 0 to E 1 is an isometry not necessarily compatible with the embeddings E 0 , E 1 ⊆ f * V .Using pullbacks of quasi-coherent modules, we turn S into a presheaf of categories on Sm to be the presheaf which on a C 2 -scheme f : X → S assigns the full subcategory of non-degenerate Hermitian sub-bundles of (f * V , f * φ) which have constant rank d.The associated presheaf of objects is RGr d (V , φ).
Note that the object V = (V , 0) ∈ 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.After isovariant étale sheafification, this inclusion becomes an equivalence; this follows from Corollary 3.27 that on the points in the isovariant étale topology, Hermitian vector bundles are determined by rank.
Upon applying the nerve, we get maps of simplicial presheaves which is a weak equivalence in the isovariant étale topology.Abusing notation, let B isoEt O(V ) denote a fibrant replacement of BS |V | (V ⊥ H ∞ ) in the isovariant étale topology so that we get a sequence of maps which are weak equivalences in the isovariant étale topology.
Lemma 4.6.Let (V , φ) be a non-degenerate Hermitian vector bundle over a scheme S with trivial involution and 1 2 ∈ Γ(S, O S ).Then for any affine C 2 -scheme Spec R over S, the map is a weak equivalence of simplicial sets.In particular, the map is a weak equivalence in the equivariant Nisnevich topology, and hence an equivalence after C 2 motivic localization.
Proof.Each Hermitian vector bundle Note that this is an O(V )-torsor because locally in the isovariant étale topology, W V , so that locally Isom(V , W ) Isom(V , V ) O(V ).Because Hermitian vector bundles are isovariant étale locally determined by rank, the same proof as the ordinary vector bundle case shows that the category of O(V ) torsors is equivalent to the category of Hermitian vector bundles.Because over an affine scheme, every Hermitian vector bundle is a summand of a hyperbolic module, it follows that S,qp → Gpd be the sheaf which assigns to f : X → S the groupoid of O(f * V )-torsors.The construction W → Isom(f * V , W ) described above defines a functor S |V | (V ⊥ H ∞ ) → F which is an equivalence when evaluated at affine C 2 -schemes.It follows that there's a sequence where the first map is a weak equivalence of simplicial sets when evaluated at affine C 2 -schemes, and by [Jar01] Theorem 6, the second map is a weak equivalence of simplicial sets when evaluated at any C 2scheme.
Definition 4.7.Following [SST14], let where similarly to the definition of RGr, for V ⊂ V ′ the functor Definition 4.8.Define the infinite orthogonal group where the colimit is over non-degenerate subbundles of H ∞ .If V is a Hermitian vector bundle, define where W is a non-degenerate subbundle of V ⊥ H ∞ .
Definition 4.9.Let R be a commutative ring.Define ∆R to be the simplicial ring with involution [n] → R[x 0 , . . ., x n ]/( x i − 1), where the involution is inherited from the involution on R.
Lemma 4.10.( [SST14]) Let V be a non-degenerate Hermitian vector bundle over a commutative ring with involution (R, σ) such that 1 2 ∈ R. Then the inclusion Proof.First, assume that V = H.Consider the map j : We claim that this is naïvely A 1 homotopic to the inclusion i : where I n denotes an n × n identity matrix.Then i = gjg −1 = gjg t .Because g corresponds to an even permutation matrix, it can be written as a product of elementary matrices, each of which is naïvely A 1 homotopic to the identity.It follow that g is naïvely A 1 homotopic to the identity, and hence the induced maps i, j : O(H n )(∆R) → O(H 2n+2 )(∆R) are simplicially homotopic via a base-point preserving homotopy.It follows that i, j induce the same map on homotopy groups, so that j * = i * : is the colimit of a map corresponding 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 the claim 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 are weak equivalences, so by 2 out of 6 the first map is a weak equivalence.Because it is a map of simplicial groups it is a homotopy equivalence.
For non-degenerate Hermitian vector bundles (V , φ V ), (W , φ W ) and a commutative R-algebra with involution (A, σ), let St(V , W )(A) be the set of A-linear isometric embeddings f : It follows that there's an isomorphism of presheaves of sets is contractible for a commutative ring (R, σ) with involution and 1 2 ∈ R. Morever, this simplicial set is fibrant because G/H is fibrant for a simplicial group G and subgroup H.We have thus proved: is a contractible Kan set.Now we move to identifying RGr 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 on the right on St(V , U) by precomposition.The map St(V , U) → RGr V (U) : f → im(f ) factors through the quotient St(V , U)/O(V ).The map is clearly surjective, and hence furnishes an isomorphism of sets In particular, there's an isomorphism of presheaves of sets Now, let V be a non-degenerate Hermitian vector bundle over a ring with involution R and let U be a possibly degenerate Hermitian form over R. Define E V (U) to be the category whose objects are R-linear maps 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 There's a natural right action of O(V ) on E V (U) which on objects sends and which on morphisms is the trivial action.Then clearly there's an isomorphism Lemma 4.12.The category Proof.The category is nonempty and every object is initial.
The map of simplicial sets is O(V )(∆R) equivariant and a weak equivalence after forgetting the action.Furthermore, O(V )(∆R) acts freely on both sides, so that the induced map on quotients (3) is also a weak equivalence.As an aside, the inclusion We now show that there's a motivic equivalence RGr • → colim n B isoEt O(H n ) over possibly non-regular Noetherian base rings.
Let X → S be an affine C 2 -scheme over S, and let W be a non-degenerate Hermitian vector bundle over X.Given an isovariant étale O(W ) torsor π : T → X, and an isovariant étale torsor U, let U π denote the twisted 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 étale sheaf of groups on Sm C 2 S,qp acting on an isovariant étale sheaf U. Suppose that, for every X ∈ Sm C 2

S,qp and every isovariant étale torsor
Given an O(W )-torsor π : T → X, we want to check that St(W , To wit, because X is affine there's an embedding V ֒→ H m , and we have , and Lemma 4.11 (which didn't assume regularity of the base) shows that St( is a motivic equivalence.However, we've already shown (2) that ) which after taking colimits gives the desired result.
Theorem 4.14.Let S be a Noetherian scheme of finite Krull dimension with 1 2 ∈ S.There are equivalences of motivic spaces on Sm where the map T −S 2 : O(−1) → O is induced via the tensor-hom adjunction by the composition and the map T −S 2 : O → σ * O(−1) is induced via the tensor-hom adjunction by the composition where µ denotes multiplication.We're abusing notation in the map (6) and using σ Going right first then down yields the composite These are equal since T +S 2 is an invariant global section.Note that the diagram 4 is a map in Fun([1], Vect(P σ )) from to its dual, 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 in detail, the dual and double dual are functors.Applying these two functors, we get the two objects Because can is a natural transformation id → * * , there's a commutative diagram The goal is to show that the vertical maps in the large rectangle are the negative of the vertical maps in diagram 4. Tracing through the definitions, we see that can is the map which sends u ∈ O(−1)(U) to the natural transformation and which is the same thing as On the other hand, T −S 2 : O(−1) → O * is the map 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 functor yields a symplectic form β σ = Cone(φ).We claim that there's an exact sequence where the maps are the maps in diagram 4. The fact that the composite is zero follows from commutativity of that 4. To show that the kernel equals the image, note that any permutation of ( T +S 2 , S−T 2 ) is a regular sequence on k[S, T ].Thus if T +S 2 x + S−T 2 y = 0, reducing mod T +S 2 we see that y ∈ ( T +S 2 ) and reducing mod S−T 2 we see that x ∈ ( S−T 2 ).It follows that the square defining φ is a pushout, and hence the induced map on mapping cones is a quasi isomorphism.Hence β σ is a well-defined, non-degenerate symplectic form in .
Theorem 5.1.Let X be a scheme with involution, an ample family of line bundles, and 1 2 ∈ X, and denote 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 Proof.The proof of Theorem 9.10 in [Sch17] can be easily adapted.Note that our Bott element β σ is a linear change of coordinates from the standard Bott element on P 1 .Keeping in mind that the involution only affects the duality and not the underlying derived category with weak equivalences, it's still true that β σ ⊗ : T sPerf(X) → T sPerf(P 1 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 1 X ) which are isomorphisms in T sPerf(P 1 X )/p * T sPerf(X), we get a sequence (sPerf(X), quis) p * / / (sPerf(P 1 X ), quis) / / (sPerf(P 1 X ), w) which is a Morita exact sequence of categories with duality.That is, the maps are maps of categories with duality, and the underlying sequence of categories is Morita exact.It follows that this sequence induces a homotopy fibration of GW [n] and GW [n] spectra.As remarked above, these fibration sequences split via the exact dg form functors (sPerf(X), quis) β σ ⊗ / / (sPerf(P 1 X ), quis) / / (sPerf(P 1 X ), w) so that the composite induces an equivalence of triangulated categories.Finally, using that GW and GW are invariant under derived equivalences [Sch17, Theorem 6.5] [Sch17, Theorem 8.9], we conclude the theorem.
There's a commutative diagram where f and h are induced by inclusion of the point [Xie18,Theorem 7.5] shows that f is an isomorphism, hence g is an injection.By localization [Sch17, Theorem 6.6], the maps k and g compose to form an exact sequence, and it follows that k is the zero map.5.2.The Periodization of GW .The idea behind the Bass construction in algebraic K-theory is that as a consequence of satisfying localization, there is a Bass exact sequence ending in for all n.This comes from applying K-theory to the pushout square manifesting the usual cover of P 1 together with the projective bundle formula.The map ∂ is split by x → [T ] ∪ p * (x) where p is the projection to the base scheme p : G m → X.It follows that if K exhibits an exact Bass sequence in all degrees n, then K n−1 (X) can be identified with the image of ∂([T ]) ∪ − (i.e. this map is an automorphism of K n−1 (X).In fact, ∂([T ]) ∪ − is the idempotent endomorphism (0, 1) of K 0 (P 1 ) K 0 (X) ⊕ K 0 (X)).The Bass construction can be thought of as defining K B n (X) so that there's an exact sequence n−1 (X) with (0, 1) • K B n−1 (P 1 ).In other words, it can be constructed as the colimit where the pushouts are taken in presheaves and the maps are induced by applying Hom(−, K) in the category of K-modules to the composite Here, loosely speaking, the first map in the composite represents the boundary in the long Bass exact sequence ∂ while the second represents [T ], so that in the category of K modules this map represents cup product with ∂([T ]).
We'll spell out an example a bit more explicitly to give a flavor for the constructions to come.Let W = A 1 G m A 1 , where we emphasize again that the pushout is in the category of presheaves.Because this is a (homotopy) pushout in the category of presheaves, applying Hom(−, K) gives us a homotopy pullback square, and hence a Mayer-Vietoris long exact sequence.In particular, it gives us a map of presheaves of spectra (which can be promoted to a map of K-modules) ΩK(G m ) → K(W ), where we abuse notation and write K(W ) for the internal hom of W into K.Because K B satisfies Nisnevich descent, and K i (−) = K B i (−) for i ≥ 0, it follows by the 5-lemma that K 0 (W ) K 0 (P 1 ) K 0 (X) ⊕ K 0 (X), and that the element ∂([T ]) ∪ − represents projection onto the second factor as an endomorphism of K 0 (W ).Now, we want to explain why ∂([T ]) ∪ K −1 (W ) K B −1 (X).We'll use the fact that K B (W ) K B (P 1 ) and that To begin, because There's a map φ : ) and a commutative diagram which shows that φ restricts to an isomorphism ∂(K 0 (G m )) ∂(K B 0 (G m )), and in particular that φ(∂ where we've crucially used that for Bass K-theory, ∂( and we know that φ| ∂K 0 (G m ) is an isomorphism.Since φ| ∂([T ])∪∂(K 0 (G m )) is surjective by the chain of equalities above, it follows that ∂K 0 (G m ) = ∂([T ]) ∪ K −1 (W ).We've shown that 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 −1 (W , 1) = K B −1 (W , 1), and the map ) is an isomorphism by the projective bundle formula.Now the map p : (W , 1) → 1 is split by inclusion of the base point, and thus p * : 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 1 (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 the localizing GW 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 GW is a consequence of results from [Sch17]. Lemma Because GW is invariant under derived equivalences [Sch17, Theorem 8.9], it follows that p * induces an isomorphism 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 GW (sPerf Z (X)) and GW (sPerf Z (Y )) as the horizontal homotopy fibers.This allows us to conclude the result.
Next, we identify the analogues of the Bass sequence and the splittings therein.From [Sch17, Theorem 9.13], we know that there's a Bass sequence where the last non-trivial map is split by cup product with (the pullback of − −− → GW [1] .Now, we want to find a candidate map Σ σ G σ m → GW [−1] so that we can eventually invert ⊗ GW [1] → GW [0] .Define W σ by the pushout square in the category of presheaves There's an associated homotopy pushout square and taking the homotopy cofiber of the left vertical map yields S σ ∧ G σ m .It follows that the homotopy cofiber of the right vertical map is equivalent to S σ ∧ G σ m , and that there's a long exact sequence (8) Working over the regular ring Z[ 1 2 ], GW (W σ /S 0 ) GW (P σ /S 0 ), and by the projective bundle formula 5.1.The maps in the sequence (8) are maps of GW [0] * -modules, and the sequence is natural in the base scheme.The induced map is an isomorphism of GW [0] 0 (Z[ 1 2 ])-modules, and hence the inverse is uniquely determined by a lift of the element 1 ∈ GW . We stress that this element 1 maps to , 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 X) using functoriality of GW .We summarize in the definition below.
Proof.Because homotopy groups commute with filtered (homotopy) colimits of spectra Fix [m] for now and denote by F i n the image of the map of groups n .Denote by FB i n the same construction as above with GW replaced by GW .For i ≥ −n, we claim that there are exact sequences ).We prove this in conjunction with the statement that, for each n, F i n GW [m] n for i ≥ −n.The proof is induction in i, and we must show that For n ≥ 0, the same argument that we gave for K-theory together with lemma 5.6 works.In more detail, there's an exact sequence and because n ≥ 0, the same argument we gave for K-theory above identifies ∂(GW n−1 (W /S 0 ⊗ W σ /S 0 ) and in turn with GW [m] (X).Then we just use the fact that p * is injective and a module map to conclude that ∂( Now fix an i, and assume by induction that our claim holds for all −n ≤ i.Then there's an exact sequence n−1 (X).Thus, letting p denote the projection W /S 0 ⊗ W σ /S 0 → X to the basepoint,

GW
The meatier part of the argument is producing the exact sequence for F i+1 n−1 , though the proof is essentially the same as the proof of the base case.
First note that for all i and n, there's a chain complex which is just the image of the usual long exact sequence for GW under the map γ * .Depending on n, this sequence may or may not be exact, as the image of an exact sequence is in general not exact.Consider the commutative diagram where the upper two horizontal maps are isomorphisms by what we've already shown.We claim that the left column is exact.The composite is zero since it's a chain complex, and if x ∈ ker(∂), then using the fact that the middle and top maps are isomorphisms we produce a lift of x.Now it remains only to check that the image of ∂ coincides with . This is the part of the proof we adapt from the K-theory case.First, it's clear that ∂( ) restricts to zero in A 1 .For the other containment, by exactness and the fact that the left two vertical arrows are isomorphisms, we know that im We've shown that if the inductive statement holds for i, n, then it holds for i + 1, n − 1.The fact that it holds for i + 1, m for any m < n + 1 is clear by appealing to results for GW .Now, the lemma follows from the explicit description for filtered colimits of groups.
Corollary 5.8.Let γ be the map (9).Then there are weak equivalences of presheaves of spectra Proof.Combining Lemma 5.6 and Lemma 5.7 we see that Q γ GW → Q γ GW induces an isomorphism on stable homotopy groups.Lemma 5.5 shows that Q γ GW ≃ GW .
Recall the definition of β 1+σ from equation (7).Definition 5.9.A GW -module E is called Bott periodic if the map [1] and β σ : P σ /[1 : 1] → GW [−1] lift to P 1 /(P 1 − [−1 : 1]) and P σ /(P σ − [−1 : 1]) respectively by results analogous to Lemma 5.2, and hence there are induced maps . Taking smash products and using that A 1 ⊕ A − A ρ , we get a map (10) When working over a scheme other than the base scheme S, we'll let β ′ X and (β σ ) ′ X denote the analogous constructions with A 1 and G m replaced by A 1 X and (G m ) X .For a vector bundle E, let V 0 (E) denote E/(E −0), the quotient by the complement of the zero section.
Theorem 5.10.Let S be a Noetherian scheme of finite Krull dimension with an ample family of line bundles and 1 2 ∈ S. Then L A 1 GW lifts to an E ∞ motivic spectrum, denoted KR alg S , over Sm C 2 S,qp .Proof.GW is an E ∞ object in presheaves of spectra (it's a commutative monoid in the category of presheaves of symmetric spectra) on Sch C 2 S,qp via the cup product defined in [Sch17, Remark 5.1].By [Hoy16] Lemma 3.3, together with corollary 5.8 above, GW is the periodization of GW with respect to γ.Let T ρ denote the Thom space of the regular representation A ρ .Now GW is Nisnevich excisive, so that GW (W /S 0 ⊗W σ /S 0 ) GW (P 1 ∧ P σ ), and GW is γ periodic if and only if it is Bott periodic.Because L A 1 preserves Nisnevich sheaves of spectra and E ∞ -objects, L A 1 GW is an E ∞ object in the category of homotopy invariant Nisnevich sheaves of spectra.In the notation of Hoyois, let and by forgetting the module structure an E ∞ -algebra in SH C 2 (X).
Lemma 5.11.The A 1 -localization of the Bott element Proof.The proof is identical to Lemma 4.8 in [Hoy16].The main idea is that the identity and the cyclic permutation σ 3 are both induced by matrices in SL 3•2 (Z) acting on A 3ρ , and any two such matrices are (naïvely) A 1 -homotopic so that there's a map h : A 1 × A 3ρ → A 3ρ witnessing the homotopy.We can extend this to a map Letting p : A 1 × S → S denote the projection, φ is an automorphism of the vector bundle p * (A 3ρ ).Now we claim that the automorphisms φ 0 , φ 1 of V 0 (A 3ρ ) induced by the restrictions of φ to 1 and 0 are A 1 -homotopic over L A 1 GW .Let β ′ A 3ρ denote (β ′ X ⊗ (β σ ) ′ X ) ⊗3 .Let β ′ A 3ρ denote (β ′ A 1 ×X ⊗ (β σ ) ′ A 1 ×X ) ⊗3 .To prove the claim, any automorphism φ as above induces a commutative triangle or presheaves of spectra on Sch C 2 A 1 ×X .As in [Hoy16], the diagram ultimately comes from our construction of the Bott elements via the projective bundle formula and the functoriality of the Proj(Sym -) construction with respect to automorpshims of the underlying vector bundle (in particular the fact that there are induced isomorphisms on each twisting sheaf O(d)).By adjunction, this is equivalent to a triangle y y r r r r r r r r r r L A 1 GW X which is an A 1 -homotopy between φ 0 and φ 1 over L A 1 GW as desired.
We've shown that GW is Bott periodic and Nisnevich excisive.Since it's the γ periodization of GW and γ periodicity is equivalent to Bott periodicity for Nisnevich excisive sheaves, GW is in fact the reflection of GW in the subcategory of Nisnevich excisive and Bott periodic GW -modules.Thus by definition, L A 1 GW is the reflection of GW in the subcategory 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, Nisnevich excisive, and Bott periodic GW -module.In particular Proof.Given Lemma 5.11, the proof is identical to Proposition 4.9 in [Hoy16].
Replacing GW with its connective cover GW ≥0 , the same reasoning yields: Porism 5.13.The canonical map GW ≥0 → Q β L mot GW ≥0 is the universal map to a homotopy invariant, Nisnevich excisive, and Bott periodic GW ≥0 -module.In particular Proof.In short, the reason the result extends to the connective cover GW ≥0 is that we have at no point used the negative homotopy groups of GW in our arguments.We'll spell this out more explicitly now.
The connective cover construction is monoidal, and the canonical map GW [m] ≥0 → GW [m] is a ring map.The Bott elements β and β σ live in the zeroth homotopy groups of GW [1] (P 1 ) and GW [1] (P σ , − can).It follows that these Bott elements restrict to well defined elements in the zeroth homotopy groups of GW ≥0 (P σ , − can).The definition of the map γ in (9) extends without modification to GW ≥0 , as all the elements involved in the discussion prior to (9) were in the non-negative homotopy groups of GW .
In particular, there's a canonical map GW [m] ≥0 → GW [m] which exhibits GW as a GW ≥0 module and is an isomorphism on non-negative homotopy groups, and Lemma 5.6 remains true replacing GW with GW ≥0 .Lemma 5.7, which is just an analogue of the Bass construction, is an inductive argument which at no point uses any facts about the negative homotopy groups of GW .The exact sequence involving GW in the proof of Lemma 5.7 is just a formal consequence of the definition of W and W σ (more precisely, that they're pushouts of presheaves of spectra), and remains exact replacing GW with GW ≥0 .Finally, the proof of Lemma 5.11 holds without modification when we replace each instance of GW by GW ≥0 .

cdh Descent for Homotopy Hermitian K-theory
Recall from Definition 2.16 that the cdh topology is the topology generated by the Nisnevich and abstract blow-up squares.Fix a Noetherian scheme of finite Krull dimension S, and a scheme X over S.
Let H C 2 (S) denote the motivic ∞-category on Sm C 2 S,qp .Just as in [Hoy16] section 5, we let H and SH denote the "big" versions of H C 2 and SH C 2 : they can be identified with the ∞-categories of sections of Sp(H C 2 (−)) and SH C 2 (−) over Sch C 2 S,qp that are cocartesian over smooth morphisms.By the results of the previous section, homotopy Hermitian K-theory, L A 1 GW , is a Bott-periodic E ∞ -algebra in Sp(H), and thus by [Hoy16, Proposition 3.2], there is a unique Bott periodic E ∞ -algebra KR alg in SH such that Ω ∞ KR alg ≃ L A 1 GW .More explicitly, by Porism 5.13, we can write KR alg as the image under the localization functor We want to show that L A 1 GW is a cdh sheaf on Sch C 2 S,qp .By first checking that the formalism of six operations holds in equivariant motivic homotopy theory and following the same recipe as the K-theory case, [Hoy17, Corollary 6.25] proves that it suffices to show that for each f : D → X ∈ Sch By [Sch17, Appendix A], there's a map Herm(X) + → Ω ∞ GW (X) where Herm(X) is the E ∞ space of non-degenerate Hermitian vector bundles over X and (−) + denotes group completion.If X is an affine C 2 -scheme, the category of vector bundles is a split exact category with duality, and the above map is an equivalence.It follows that is a motivic equivalence in P (Sm C 2 X,qp ).Just as in [Hoy16] we note that n≥0 B isoEt O( 1 ⊥n ) → Herm exhibits Herm as the equivariant Zariski sheafification of the subgroupoid of non-degenerate Hermitian vector bundles of constant rank (in other words, it "corrects" the sections over non-connected or hyperbolic rings).Since L Zar preserves finite products, by [Hoy16, Lemma 5.5], the map remains a Zariski equivalence after group completion yielding a motivic equivalence Fix a map f : D → X in Sch Definition 2.13.A distinguished equivariant Nisnevich square is a cartesian square in Sch G X where i is an open immersion, p is étale, and p restricts to an isomorphism (Y − B) red → (X − A) red .
3.1.Definitions.Definition 3.1.Let R be a ring with involution − : R → R op .A Hermitian module over R is a finitely generated projective right R-module, M, together with a mapb : 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 promote M to an R-bimodule by introducing the right action m • r = mr.Remark 3.3.Let R be a commutative ring so that R = R op .Given an involution σ : R → R and a right R-module M, we can identify M with σ * M, where σ * M is the module M with R-action restricted through σ.Typically the pushforward would just take the right R-module M to another right R-module.Since we really view σ as landing in R op , we use commutativity of R and the canonical identification of right R op modules with left R-modules to think of σ * M as a left module.Indeed, σ * M is a left R-module via the rule r • m = m • σ(r).Remark 3.4.Another way to define a Hermitian form over a ring R with involution σ is to give a finitely generated 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 1 • r • r 2 = r 1 rr 2 , M as a left R-module via the involution, and such 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 right R-module M together with an R-linear map b : M → Mˇ= M * such that b = b * can M , where b * : M * * → M * is given by (b * (f ))(m) = f (b(m)).Here can M : M → M * * , can M (m)(f ) = f (m) is the canonical double dual isomorphism.

Proof.
The first claim is just that b(e 1 x, e 1 y) = 0 = b(e 2 x, e 2 y) for any x, y ∈ M.This follows because b(e 1 x, e 1 y) = b(e 2 1 x, e 2 1 y) = e 1 e 1 b(e 1 x, e 1 y) = e 2 e 1 b(e 1 x, e 1 y) = 0. Similarly for b(e 2 x, e 2 y).The statement about the matrix follows by identifying the map M ⊗ M → R × R with an isomorphism M → M * and using the direct sum decomposition.

C 2 S
,qp , denoted RGr, which represents Hermitian Ktheory in the motivic homotopy category H C 2 S .We first check that over a regular base S with 2 invertible (e.g.Z[ 1 2 ]), Hermitian K-theory is representable in the category of C 2 -schemes over S, Sm C 2 S,qp .To extend this result to non-regular bases S, we utilize the Morel-Voevodsky approach to classifying spaces and obtain representability of homotopy Hermitian K-theory in the motivic homotopy category H C 2 S .

C 2 S
,qp , the restriction map f * (KR alg X ) → KR alg D in SH C 2 (D) is an equivalence.We show this now.

C 2 S
,qp .Again by[Hoy16, Lemma 5.5], since the pullback f * : P (SchC 2 X,qp ) → P (Sch C 2X,qp ) preserves finite products, it commutes with group completion of E ∞ -monoids.The same is true for L mot .It follows that there are motivic equivalences
S,qp denote the category of quasi-projective C 2 -schemes which are separated and finite type over S, and let Sm 1 2 ∈ Γ(S, O S ).Throughout we'll work with two categories of schemes.Let Sch an equivariant map which is a nonequivariant homotopy equivalence.The simplicial group O(H ∞ )(∆R) acts freely on both the domain and codomain, so that the quotientsO(H ∞ )(∆R)\O(V ⊥ H ∞ )(∆R) and O(H ∞ )(∆R)\O(H ∞ )(∆R) are homotopy equivalent.Together with the isomorphism of simplicial sets 5.3.GW is Nisnevich excisive on the category of schemes with an ample family of line bundles over S.