A Transfer morphism for Hermitian K-theory of schemes with involution
aa r X i v : . [ m a t h . K T ] O c t A Transfer morphism for Hermitian K -theory of schemes with involution HENG XIE
Abstract.
In this paper, we consider the Hermitian K -theory of schemes with involution, for whichwe construct a transfer morphism and prove a version of the d´evissage theorem. This theorem is thenused to compute the Hermitian K -theory of P with involution given by [ X : Y ] [ Y : X ]. We alsoprove the C -equivariant A -invariance of Hermitian K -theory, which confirms the representabilityof Hermitian K -theory in the C -equivariant motivic homotopy category of Heller, Krishna andØstvær [HKO14]. Introduction
In the 1970s, researchers found that to understand quadratic forms, it was helpful to study Wittgroups of function fields of algebraic varieties, which has led to substantial progress in quadraticform theory (see [Scha85], [Lam05], [KS80] and [Knu91]). However, for many function fields, Wittgroups are very difficult to understand. Instead of computing the Witt groups of function fields ofalgebraic varieties, Knebusch [Kne77] proposed to study the Witt groups of the algebraic varietiesthemselves, which would certainly reveal some information about their function fields counterparts.In the introduction of loc. cit. , Knebusch suggested developing a version of Witt groups of schemeswith involution, which would not only enlarge the theory of trivial involution but also provides newinsights into Topology. For instance, the L -theory of the Laurent polynomial ring k [ T, T − ] with thenon-trivial involution T T − turned out to be very useful in understanding geometric manifolds[Ran98], and Witt groups can be identified with L -theory if two is invertible.More generally, Witt groups fit into the framework of Hermitian K -theory, [Ba73], [Kar80] and[Sch17]. More precisely, the negative homotopy groups of the Hermitian K -theory spectrum are Wittgroups, cf. [Sch17, Proposition 6.3]. It is also worth mentioning that Hermitian K -theory has beensuccessfully applied to solve several problems in the classification of vector bundles and the theoryof Euler classes (cf. [AF14a], [AF14b] and [FS09]). In light of this, I decided to develop the currentpaper within the framework of the Hermitian K -theory of dg categories of Schlichting [Sch17].The simplest example of Witt groups of schemes with non-trivial involution is W (Spec( C ) , σ ) where σ is complex conjugation (here we consider Spec( C ) over Spec( Z [ ])). We have W (Spec( C ) , σ ) ∼ = Z by taking the rank of positive definite diagonal Hermitian forms over C (cf. [Knu91, I.10.5]). Thiscomputation is different from W (Spec( C )) ∼ = Z / Z in which case every rank two quadratic form ishyperbolic.On the one hand, it is known that the topological Hermitian K -theory of spaces with involution isequivalent to Aityah’s KR -theory ([At66]). On the other hand, the Hermitian K -theory of schemeswith involution is a natural lifting of the topological Hermitian K -theory to the algebraic world. Inlight of this, the Hermitian K -theory of schemes with involution could be regarded as a version of KR -theory in algebraic geometry. By working with Hermitian K -theory of schemes with involution, Hu,Kriz and Ormsby [HKO11] managed to prove the homotopy limit problem for Hermitian K -theory.In this paper, we generalize Gille’s transfer morphism on Witt groups [Gil03] and Schlichting’stransfer morphism on Grothendieck-Witt groups [Sch17, Theorem 9.19] to non-trivial involution.More precisely, in Theorem 3.1 we prove the following: Theorem 1.1.
Let ( X, σ X ) and ( Z, σ Z ) be schemes with involution and with in their global sections.Suppose that ( X, σ X ) has a dualizing complex with involution ( I, σ I ) (cf. Definition 2.13). If π : Z → MSC:
PRIMARY: 19G38; SECONDARY: 19G12 X is a finite morphism of schemes with involution, then the direct image functor π ∗ : Ch bc ( Q ( Z )) → Ch bc ( Q ( X )) induces a map of spectra T Z/X : GW [ i ] ( Z, σ Z , ( π ♭ I • , σ π ♭ I )) → GW [ i ] ( X, σ X , ( I • , σ I )) . In light of the work of Balmer-Walter [BW02] and Gille [Gil07a], we also prove a version of thed´evissage theorem for Hermitian K -theory of schemes with involution. The following result is provedin Theorem 5.1. Theorem 1.2.
Let ( X, σ X ) and ( Z, σ Z ) be schemes with involution and with in their global sections.Suppose that ( X, σ X ) has a minimal dualizing complex with involution ( I, σ I ) (cf. Definition 2.13). If π : Z ֒ → X is a closed immersion which is invariant under involutions, then the direct image functor π ∗ : Ch bc ( Q ( Z )) → Ch bc,Z ( Q ( X )) induces an equivalence of spectra D Z/X : GW [ i ] ( Z, σ Z , ( π ♭ I • , σ π ♭ I )) → GW [ i ] Z ( X, σ X , ( I • , σ I )) . Note that a Gersten-Witt complex for Witt groups of schemes with involution was constructed byGille [Gil09], but the above formula seems not to appear anywhere in the literature. If X and Z areboth regular, we have the following more precise formulation of the d´evissage theorem (cf. Theorem6.1). Theorem 1.3 (D´evissage) . Let ( X, σ X ) be a regular scheme with involution. Let ( L , σ L ) be a dualizingcoefficient with L a locally free O X -module of rank one. If Z is a regular scheme regularly embeddedin X of codimension d which is invariant under σ , then there is an equivalence of spectra D Z/X, L : GW [ i − d ] ( Z, σ Z , ( ω Z/X ⊗ O X L , σ ω Z/X ⊗ σ L )) −→ GW [ i ] Z ( X, σ X , ( L , σ L )) . It turns out that if the involution is non-trivial, the canonical double dual identification has acertain sign varying according to the data of dualizing coefficients (cf. Definition 2.8). This sign is notimportant if the involution is trivial, but it is crucial if the involution is non-trivial. As an application,we use the d´evissage theorem to obtain the following result (cf. Theorem 7.1).
Theorem 1.4.
Let S be a regular scheme with ∈ O X . Then, we have an equivalence of spectra GW [ i ] ( P S , τ P S ) ∼ = GW [ i ] ( S ) ⊕ GW [ i +1] ( S ) where τ P S : P S → P S : [ x : y ] [ y : x ] . In particular, we have the following isomorphism on Wittgroups W i ( P S , τ P S ) ∼ = W i ( S ) ⊕ W i +1 ( S ) . It is well-known that W i ( P S ) ∼ = W i ( S ) ⊕ W i − ( S ) if the involution on P S is trivial (cf. [Ne09] and[Wal03]). By our d´evissage theorem, the involution σ : P S → P S : [ x : y ] [ y : x ] induces a sign ( − W i ( P S , σ ) ∼ = W i ( S ) ⊕ W i − ( S, − can)but W i − ( S, − can) ∼ = W i +1 ( S ).We also use the d´evissage to prove the following C -equivariant A -invariance (cf. Theorem 7.5). Theorem 1.5.
Let S be a regular scheme with involution σ S and with ∈ O S . Let σ A S : A S → A S be the involution on A S with the indeterminant fixed by σ A S and such that the following diagramcommutes. A S σ A S −−−−→ A Sp y p y S σ S −−−−→ S The pullback p ∗ : GW [ i ] ( S, σ S ) → GW [ i ] ( A S , σ A S ) is an isomorphism. TRANSFER MORPHISM FOR HERMITIAN K -THEORY OF SCHEMES WITH INVOLUTION 3 For affine schemes with involution Theorem 1.5 was proved by Karoubi (cf. [Kar74, Part II]), butit was not known for regular schemes with involution. If S is quasi-projective, then it can be coveredby invariant affine open subschemes, and we can use the Mayer-Vietoris of Schlichting ([Sch17]) toreduce the quasi-projective case to the affine case. However, the method of Meyer-Vietoris does notwork if S is not quasi-projective because there exist non-quasi-projective regular schemes which cannot be covered by invariant affine opens.It is easy to see that schemes with involution can be identified with schemes with C -action where C is the cyclic group of order two considered as an algebraic group. Together with a variant of Nisnevichexcision [Sch17, Theorem 9.6] adapted to schemes with involution, we obtain the C -representabilityof Hermitian K -theory in the C -equivariant motivic homotopy category H C • ( S ) of Heller, Krishnaand Østvær [HKO14]. The following result is proved in Theorem 7.6. Theorem 1.6.
Let S be a regular scheme with ∈ O S , and let ( X, σ ) ∈ Sm C S . Then, there is abijection of sets [ S n ∧ ( X, σ ) + , GW [ i ] ] H C • ( S ) = GW [ i ] n ( X, σ ) . Notations and prerequisites
In this paper, we assume that every scheme is Noetherian and has in its global sections, andevery ring is commutative Noetherian with identity and . If E is an exact category, we write D b ( E )the bounded derived category of E .2.1. Notations on rings.
Let R be a ring. Throughout the paper, we will make use of the followingnotations associated with R .- R -Mod is the category of (left) R -modules.- M ( R ) is the category of finitely generated R -modules.- M fl ( R ) is the category of finite length R -modules.- M J ( R ) is the category of finitely generated R -modules supported in an ideal J of R . Recallthat Supp( M ) = { p ∈ Spec( R ) | M p = 0 } and V ( J ) = { p ∈ Spec( R ) | p ⊃ J } . More precisely, M J ( R ) is the full subcategory of M ( R ) consisting of those modules M such that Supp( M ) ⊂ V ( J ).- D bc ( R -Mod) is the derived category of bounded complexes of R -modules with coherent coho-mology.- D bfl ( R ) is the full subcategory of D bc ( R -Mod) of complexes, whose cohomology modules arefinite length R -modules.- D bJ ( R ) is the full triangulated subcategory of D bc ( R -Mod) of complexes whose cohomologymodules are annihilated by some power of the ideal J . Definition 2.1. An involution σ on a ring R is a ring homomorphism σ : R → R such that σ = id R .Let ( R, σ ) be a ring with involution and M a left R -module. Define M op R (or simply M op ) to bethe same as M as a set. For an element in M op , we use the symbol m op to denote the element m coming from the set structure of M . Consider M op as a right R -module via m op a = ( σ ( a ) m ) op . Definition 2.2. [Sch10b, Section 7.4] Let (
R, σ ) be a ring with involution. A duality coefficient ( I, i )on the category R -Mod with respect to the involution σ consists of an R -module I equipped with an R -module isomorphism i : I → I op such that i = id.Let σ : R → R ′ be a ring homomorphism. Let M ′ be a module over R ′ . We can form the restrictionof scalars σ ∗ M ′ := M ′ | R . which can also be considered as an R -module. Remark 2.3.
Note that if (
R, σ ) is a ring with involution and M is an R -module, then σ ∗ M = M op . HENG XIE
Remark 2.4.
For a duality coefficient (
I, i ), one can form a category with duality ( R -Mod , Iσ , can Iσ )where Iσ : ( R -Mod) op → R -Mod : M Hom R ( σ ∗ M, I )is a functor and can
Iσ,M : M → M Iσ Iσ : can σI,M ( x )( f op ) = i ( f ( x op ))is a natural morphism of R -modules (cf. [Sch10b, Section 7.4]).The following lemma will be used in Theorem 6.2. Lemma 2.5.
Let σ : R → R ′ and τ : S → S ′ isomorphisms of rings, and suppose that σ ′ := σ − and τ ′ := τ − . Suppose furthermore that one has the following commutative diagram of rings. R g (cid:15) (cid:15) σ / / R ′ g ′ (cid:15) (cid:15) σ ′ / / R g (cid:15) (cid:15) S τ / / S ′ τ ′ / / S Let I be an R -module and let I ′ be an R ′ -module. Assume that we have an R -module isomorphism σ I : I → σ ∗ I ′ . (1) . Let M be an R -module and let M ′ be an R ′ -module. Assume that we have an R -module isomor-phism σ M ′ : M ′ → σ ′∗ M . Then, the map ǫ M,I : Hom R ( M, I ) → σ ∗ Hom R ′ ( M ′ , I ′ ) : f σ I f σ M ′ is an R -module isomorphism. (2) . Let N be an S -module and let N ′ be an S ′ -module. Assume that we have an S -module isomorphism σ N ′ : N ′ → τ ′∗ N . The set Hom R ( N, I ) has an S -module structure and the map ǫ N,I : Hom R ( N, I ) → τ ∗ Hom R ′ ( N ′ , I ′ ) : f σ I f σ N ′ is a well-defined S -module isomorphism. Notations on schemes.
Let X be a scheme. We fix the following notations that will be neededin the remaining sections.- O X -Mod is the category of O X -modules.- Q ( X ) is the category of quasi-coherent O X -modules.- VB( X ) is the category of finite rank locally free coherent modules over X .- M Z ( X ) is the category of finitely generated O X -modules supported in a closed scheme Z of X . Recall that Supp( F ) = { x ∈ X |F x = 0 } . More precisely, M Z ( X ) is the full subcategoryof M ( X ) consisting of those modules F such that Supp( F ) ⊂ Z .- K bc ( Q ( X )) is the homotopy category of bounded complexes of quasi-coherent sheaves on X with coherent cohomology.- D bc ( Q ( X )) is the derived category of bounded complexes of quasi-coherent sheaves on X withcoherent cohomology.- D bc,Z ( Q ( X )) is the full triangulated subcategory of D bc ( Q ( X )) consisting of complexes F • suchthat { x ∈ X |H i ( F • ) x = 0 for some i ∈ Z } ⊆ Z .- D b ( X ) is the derived category D b (VB( X )).- D bZ ( X ) is the full triangulated category of D b ( X ) whose objects are the complexes F • suchthat { x ∈ X |H i ( F • ) x = 0 for some i ∈ Z } ⊆ Z .- Ch bc ( Q ( X )) is the dg category of bounded complexes of quasi-coherent O X -modules withcoherent cohomology.- Ch bc,Z ( Q ( X )) is the full dg subcategory of Ch bc ( Q ( X )) consisting of complexes F • such that { x ∈ X |H i ( F • ) x = 0 for some i ∈ Z } ⊆ Z .- Ch b ( X ) is the dg category Ch b (VB( X )) of bounded complexes of locally free coherent O X -modules. TRANSFER MORPHISM FOR HERMITIAN K -THEORY OF SCHEMES WITH INVOLUTION 5 - Ch bZ ( X ) is the full dg subcategory Ch b ( X ) of those complexes F • such that { x ∈ X |H i ( F • ) x =0 for some i ∈ Z } ⊆ Z . Definition 2.6. An involution σ X on a ringed space X is a morphism σ X : X → X of ringed spacessuch that σ = id X . Remark 2.7. If X = Spec( R ), to give a scheme with involution ( X, σ ) is the same as giving a ringwith involution (
R, σ ). Definition 2.8. A duality coefficient ( I, i ) for the category of O X -modules on X with respect to theinvolution σ X consists of an O X -module I equipped with an O X -module isomorphism i : I → σ ∗ I such that σ ∗ i ◦ i = id. An isomorphism ( I, i ) → ( I ′ .i ′ ) of duality coefficients is an isomorphism of O X -modules α : I → I ′ such that the following diagram commutes I α −−−−→ I ′ i y i ′ y σ ∗ I σ ∗ α −−−−→ σ ∗ I ′ . Remark 2.9.
Let (
X, σ ) be a scheme with involution. By globalizing the affine case in Remark 2.4,we obtain a category with duality ( O X -Mod , Iσ , can Iσ ) . If the double dual identification can Iσ is clear from the context, we will use the simplified notation( O X -Mod , Iσ ) instead of the more precise ( O X -Mod , Iσ , can Iσ ). Lemma 2.10.
Let α : ( I, i ) → ( I ′ , i ′ ) be an isomorphism of duality coefficients. Then, the identityfunctor induces a duality preserving equivalence of categories with duality (id , α ∗ ) : ( O X -Mod , Iσ ) → ( O X -Mod , I ′ σ ) Proof.
The duality compatibility isomorphism is defined by α ∗ : H om O X ( σ ∗ M, I ) → H om O X ( σ ∗ M, I ′ ) : f α ◦ f which is natural in M . Now it is straightforward to check the axioms. (cid:3) Dualizing complexes with involution.
Recall the following definition of a duality complexfrom [Gil07b, Definition 1.7] or [Ha66].
Definition 2.11.
Let X be a scheme. A dualizing complex I • on X is a complex of injective modules I • := ( · · · → → I m d m −→ I m +1 → · · · → I n − d n − −→ I n → → · · · ) ∈ D bc ( Q ( X ))such that the natural morphism of complexescan I,M : M • → H om O X ( H om O X ( M • , I • ) , I • ) : m (cid:0) can I,M ( m ) : f ( − | f || m | f ( m ) (cid:1) is a quasi-isomorphism for any M • ∈ D bc ( Q ( X )). A dualizing complex is called minimal if I r is anessential extension of ker( d r ) for all r ∈ Z . Remark 2.12.
Any dualizing complex is quasi-isomorphic to a minimal dualizing complex [Gil07b,Proposition 1.9].
Definition 2.13.
Let (
X, σ ) be a Noetherian scheme with involution and ∈ O X . A dualizingcomplex with involution on ( X, σ ) is a pair ( I • , σ I ) consisting of a dualizing complex I • and anisomorphism of complexes of O X -modules σ I : I • → σ ∗ ( I • ) such that σ ∗ ( σ I ) ◦ σ I = id I . Remark 2.14.
Note that ( I p , σ I p ) is a duality coefficient for the category O X -Mod. Lemma 2.15.
Let ( X, σ ) be a Gorenstein scheme of finite Krull dimension with involution and ∈ O X . Let ( L , σ L ) be a dualizing coefficient with L a locally free O X -module of rank one. Then,there exists a minimal dualizing complex with involution ( J • , σ J ) , which is quasi-isomorphic to ( L , σ L ) . HENG XIE
Proof.
Let X ( p ) := { x ∈ X | dim O X,x = p } . Recall [Ha66, p241] that we have a Cousin complex0 → O X d − −→ I d −→ I −→ · · · −→ I n −→ −→ · · · with I := L x ∈ X (0) i x ∗ ( O X,x ) and I p := L x ∈ X ( p ) i x ∗ (coker( d t − ) x ) where the map i x : Spec( O X,x ) → X is the canonical map. If X is Gorenstein, the Cousin complex provides a minimal dualizing complex I • on X quasi-isomorphic to O X . This can be checked locally on closed points by [Ha66, CorollaryV.2.3 p259]. For the case of Gorenstein local rings, see [Sha69, Theorem 5.4].We define a map σ I : I p → σ ∗ I p as follows. We consider the following commutative diagramSpec( O X,x ) i x (cid:15) (cid:15) σ X,x / / Spec( O X,σ ( x ) ) i σ ( x ) (cid:15) (cid:15) X σ X / / X Assume that M is an O X -module together with a map M → σ ∗ M . If x = σ ( x ), the map M → σ ∗ M induces a canonical map i x, ∗ i ∗ x M → σ ∗ i x, ∗ i ∗ x M of O X -modules by using the natural isomorphism σ ∗ i x, ∗ i ∗ x M ∼ = i σ ( x ) , ∗ σ x, ∗ i ∗ x M ∼ = i σ ( x ) , ∗ i ∗ σ ( x ) σ ∗ M . If x = σ ( x ), the map M → σ ∗ M still induces acanonical map i x, ∗ i ∗ x M ⊕ i σ ( x ) , ∗ i ∗ σ ( x ) M → σ ∗ i σ ( x ) , ∗ i ∗ σ ( x ) M ⊕ σ ∗ i x, ∗ i ∗ x M by the functorial isomorphism above. The map σ I : I p → σ ∗ I p is obtained by applying this construc-tion inductively. (Note that x ∈ X ( p ) if and only if σ ( x ) ∈ X ( p ) ). By our construction it is clear that σ ∗ ( σ I ) ◦ σ I = id I .If L is a line bundle on X , then J • := L ⊗ I • is a dualizing complex, cf. [Ha66, Proof (1) ofTheorem V.3.1, p. 266]. Moreover, the complex J • is quasi-isomorphic to L . Note that we have acanonical isomorphism σ ∗ ( L ⊗ I • ) → σ ∗ L ⊗ σ ∗ I • . These observations reveal a canonical involution σ J : J • → σ ∗ J • . (cid:3) Setup for Hermitian K -theory of schemes with involution. Let (
X, σ ) be a scheme withinvolution. In this paper, we work within the framework of Schlichting [Sch17]. Recall that Ch bc ( Q ( X ))is a closed symmetric monoidal category under the tensor product of complexes M • ⊗ O X N • given by( M • ⊗ O X N • ) n := M i + j = n M i ⊗ O X N j , d ( m ⊗ n ) = dm ⊗ n + ( − | m || n | m ⊗ dn and internal homomorphism complexes H om O X ( M • , N • ) given by H om O X ( M • , N • ) n := Y j − i = n H om O X ( M i , N j ) , df = d ◦ f − ( − | f | f ◦ d. Let ( I • , σ I ) be a dualizing complex with involution. Then, we define the duality functor I • σ : (Ch bc ( Q ( X ))) op → Ch bc ( Q ( X )) : E •
7→ H om O X ( σ ∗ E • , I • ) , and the double dual identification can I • σ,E : E → E I • σ I • σ given by the composition E can E −→ H om O X ( H om O X ( E, I ) , I ) −→ H om O X ( H om O X ( E, σ ∗ I ) , I ) −→ H om O X ( σ ∗ H om O X ( σ ∗ E, I ) , I ) . Here the first map can E is defined by can E ( x )( f ) = ( − | x || f | f ( x ) as in the case of trivial du-ality, the second is induced by σ I : I → σ ∗ I , and the third is induced by σ ∗ H om O X ( M, N ) = H om O X ( σ ∗ M, σ ∗ N ). Lemma 2.16.
The quadruple (Ch bc ( Q ( X )) , quis , I • σ , can I • σ ) is a dg category with weak equivalence and duality.Proof. It is enough to check that (Ch bc ( Q ( X )) , I • σ , can I • σ )is a category with duality, see [Gil09, Section 3.7]. (cid:3) TRANSFER MORPHISM FOR HERMITIAN K -THEORY OF SCHEMES WITH INVOLUTION 7 Definition 2.17.
We define the coherent Hermitian K -theory spectrum of schemes with involution of ( X, σ X ) with respect to ( I, σ I ) as GW [ i ] ( X, σ X , ( I, σ I )) := GW [ i ] (Ch bc ( Q ( X )) , quis , I • σ , can I • σ )for every i ∈ Z , where GW [ i ] (Ch bc ( Q ( X )) , quis , I • σ , can I • σ ) is the Grothendieck-Witt spectrum of thedg category with weak equivalences and duality (Ch bc ( Q ( X )) , quis , I • σ , can I • σ ). The Witt group of(
X, σ X ) with respect to ( I, σ I ) is defined as W i ( X, σ X , ( I, σ I )) := W i ( D bc ( Q ( X )) , I • σ , can I • σ ) , where W i ( D bc ( Q ( X )) , I • σ , can I • σ ) is the Witt group of the triangulated category with duality ( D bc ( Q ( X )) , I • σ , can I • σ ).Let Z be an invariant closed subscheme of X . We can also define GW [ i ] Z ( X, σ X , ( I, σ I )) := GW [ i ] (Ch bc,Z ( Q ( X )) , quis , I • σ , can I • σ )and W iZ ( X, σ X , ( I, σ I )) := W i ( D bZ ( X ) , I • σ , can I • σ ) . Let ( L , σ L ) be a dualizing coefficient with L a locally free O X -module of rank one. Then, we havea dg category with duality (Ch b ( X ) , quis , L σ , can L σ ) , where L σ : (Ch b ( X )) op → Ch b ( X ) : E •
7→ H om O X ( σ ∗ E • , L ) , and the double dual identification can L σ,E : E → E L σ L σ is given by the composition E can E −→ H om O X ( H om O X ( E, L ) , L ) −→ H om O X ( H om O X ( E, σ ∗ L ) , L ) −→ H om O X ( σ ∗ H om O X ( σ ∗ E, L ) , L ) . Here the first map can E is defined by can E ( x )( f ) = ( − | x || f | f ( x ), the second map is induced by σ L : L → σ ∗ L , and the third map is induced by σ ∗ H om O X ( M, N ) = H om O X ( σ ∗ M, σ ∗ N ). Definition 2.18.
We define the
Hermitian K -theory spectrum of schemes with involution of ( X, σ X )with respect to ( L , σ L ) as GW [ i ] ( X, σ X , ( L , σ L )) := GW [ i ] (Ch b ( X ) , quis , L σ , can L σ )for every i ∈ Z . The Witt group of (
X, σ X ) with respect to ( L , σ L ) is defined as W i ( X, σ X , ( L , σ L )) := W i ( D b ( X ) , L σ , can L σ ) . Let Z be an invariant closed subscheme of X . We can also define GW [ i ] Z ( X, σ X , ( L , σ L )) := GW [ i ] (Ch bZ ( X ) , quis , L σ , can L σ )and W iZ ( X, σ X , ( L , σ L )) := W i ( D bZ ( X ) , L σ , can L σ ) . Dualizing complex via a finite morphism.
Let (
Z, σ Z ) be a scheme with involution, and let π : Z → X be a finite morphism. Assume that π is compatible with the involutions on Z and X , i.e.,the following diagram commutes Z σ Z (cid:15) (cid:15) π / / X σ X (cid:15) (cid:15) Z π / / X. Following Hartshorne [Ha66], we write π ♭ I • := ¯ π ∗ H om O X ( π ∗ O Z , I • ) , where ¯ π : ( Z, σ Z ) → ( X, π ∗ O Z ) =: ¯ X is the canonical flat map of ringed spaces. If I • is a dualizingcomplex, then π ♭ I • is also a dualizing complex [Gil07a, Section 2.4]. Note that σ X : X → X inducesa canonical involution σ ¯ X : ¯ X → ¯ X defined on the topological spaces, as well as π ∗ σ Z : π ∗ O Z → HENG XIE σ X ∗ π ∗ O Z = π ∗ σ Z ∗ O Z on the sheaves of rings. Then, we can define an involution σ π ♭ I • : π ♭ I • → σ Z ∗ π ♭ I • on π ♭ I • through the following composition π ♭ I • = ¯ π ∗ H om O X ( π ∗ O Z , I • ) [ σ Z , −→ ¯ π ∗ H om O X ( π ∗ σ Z ∗ O Z , I • ) −→ ¯ π ∗ H om O X ( σ X ∗ π ∗ O Z , I • ) [1 ,σ I ] −→ ¯ π ∗ H om O X ( σ X ∗ π ∗ O Z , σ X ∗ I • ) −→ ¯ π ∗ σ X ∗ H om O X ( π ∗ O Z , I • ) = ¯ π ∗ σ ¯ X ∗ H om O X ( π ∗ O Z , I • ) −→ σ Z ∗ ¯ π ∗ H om O X ( π ∗ O Z , I • ) = σ Z ∗ π ♭ I • . It follows that we obtain a dualizing complex with involution ( π ♭ I • , σ π ♭ I • ) on Z . Thus, we have awell-defined spectrum GW [ i ] ( Z, σ Z , ( π ♭ I • , σ π ♭ I )) for every i ∈ Z .Note that if Z = Spec( B ) and X = Spec( A ) are affine, then Hom A ( B, I ) has a structure of B -module by defining the action b · f as b · f : m f ( bm ) for b, m ∈ B . The involution σ π ♭ I • : π ♭ I • → σ Z ∗ π ♭ I • is precisely the map of B -modules σ AB,I : Hom A ( B, I ) → Hom A ( B, I ) op B : f σ I f σ B . 3. The transfer morphism
Theorem 3.1.
Let ( X, σ X ) and ( Z, σ Z ) be schemes with involution and with in their global sections.Suppose that ( X, σ X ) has a dualizing complex with involution ( I, σ I ) (cf. Definition 2.13). If π : Z → X is a finite morphism of schemes with involution, then the direct image functor π ∗ : Ch bc ( Q ( Z )) → Ch bc ( Q ( X )) induces a map of spectra T Z/X : GW [ i ] ( Z, σ Z , ( π ♭ I • , σ π ♭ I )) → GW [ i ] ( X, σ X , ( I • , σ I )) . Definition 3.2.
The map T Z/X : GW [ i ] ( Z, σ Z , ( π ♭ I • , σ π ♭ I )) → GW [ i ] ( X, σ X , ( I • , σ I )) in the abovetheorem is called the transfer morphism . Proof of Theorem 3.1.
The transfer map T Z/X : GW [ i ] ( Z, σ Z , ( π ♭ I • , σ π ♭ I )) → GW [ i ] ( X, σ X , ( I • , σ I ))is induced by the duality preserving functor given below in Lemma 3.3. (cid:3) Lemma 3.3.
Under the hypothesis of Theorem 3.1, the direct image functor induces a duality pre-serving functor π ∗ : (Ch bc ( Q ( Z )) , π ♭ I • σ Z ) −→ (Ch bc ( Q ( X )) , I • σ X ) with the duality compatibility natural transformation η : π ∗ ◦ π ♭ I • σ Z −→ I • σ X ◦ π ∗ , given by the following composition of isomorphisms η G : π ∗ [ σ Z ∗ G , π ♭ I • ] O Z ∼ = −−−−→ [ π ∗ σ Z ∗ G , π ∗ π ♭ I • ] O X [ p, −−−−→ ∼ = [ σ X ∗ π ∗ G , π ∗ π ♭ I • ] O X [1 ,ξ ] −−−−→ ∼ = [ σ X ∗ π ∗ G , I • ] O X for G ∈ Ch bc ( Q ( Z )) , where ξ : π ∗ π ♭ I • → I • is the evaluation at one .Proof. We need to show that for any M ∈ Ch bc ( Q ( Z )) the following diagram in Ch bc ( Q ( X )) is com-mutative. π ∗ M −−−−→ π ∗ ( M π♭I • σZ π♭I • σZ ) y y ( π ∗ M ) I • σX I • σX −−−−→ ( π ∗ M π♭I • σZ ) I • σX This map is called trace in [Ha66] and [Gil07a], but on affine schemes this map is in fact the evaluation at one.
TRANSFER MORPHISM FOR HERMITIAN K -THEORY OF SCHEMES WITH INVOLUTION 9 We write down what it means. To simplify the notation, in the rest of the proof we will drop thebullet in I • and write I instead. π ∗ M π ∗ can σM / / can σπ ∗ M (cid:15) (cid:15) π ∗ [ σ Z ∗ [ σ Z ∗ M, π ♭ I ] , π ♭ I ] η [ σZ ∗ M,π♭I ] (cid:15) (cid:15) [ σ X ∗ [ σ X ∗ ( π ∗ M ) , I ] , I ] [ η M ,I ] / / [ σ X ∗ π ∗ [ σ Z ∗ M, π ♭ I ] , I ]We depict all the maps by definition in Figure 1 below. The diagram (cid:3) in Figure 1 is commutativeby the case of trivial duality (since all the maps considered here are natural, we drop the labelingof maps). We check the diagram (cid:3) in Figure 1 is also commutative. This can be checked by thefollowing two commutative diagrams.(3.1) σ X ∗ π ∗ [ σ Z ∗ M, π ♭ I ] (cid:15) (cid:15) / / [ σ X ∗ π ∗ σ Z ∗ M, σ X ∗ π ∗ π ♭ I ] / / [ π ∗ σ Z ∗ σ Z ∗ M, σ X ∗ π ∗ π ♭ I ] (cid:15) (cid:15) π ∗ σ Z ∗ [ σ Z ∗ M, π ♭ I ] / / [ π ∗ σ Z ∗ σ Z ∗ M, π ∗ σ Z ∗ π ♭ I ] / / [ π ∗ M, π ∗ σ Z ∗ π ♭ I ](3.2) σ X ∗ π ∗ π ♭ I (cid:15) (cid:15) / / σ X ∗ I / / Iπ ∗ π ♭ σ X ∗ I qqqqqqqqqq / / σ X ∗ I / / I ✄✄✄✄✄✄✄✄ ✄✄✄✄✄✄✄✄ π ∗ σ Z ∗ π ♭ I / / π ∗ π ♭ I / / Iπ ∗ π ♭ σ X ∗ I qqqqqqqqqq / / π ∗ π ♭ I / / I ✄✄✄✄✄✄✄✄ ✄✄✄✄✄✄✄✄ Diagram (3.1) is commutative for obvious reasons. To see the commutativity of Diagram (3.2), weobserve that the upper diagram is commutative by a variant of [Ha66, Proposition 6.6 p170]. Thebottom diagram in Diagram (3.2) is also commutative by the definition of π ♭ I → σ Z ∗ π ♭ I in Section2. The left square is commutative by [Ha66, Proposition 6.3]). The front square is commutativeby the naturality of the trace. Therefore, the back square is commutative which is what we need.Now, we apply the functor [ − , I ] to Diagram (3.1) and the functor [[ π ∗ M, − ] , I ] to the back squarediagram of Diagram (3.2). Combining these two diagrams gives a new one, which by naturality andfunctoriality can be identified with (cid:3) . The remaining four small squares in Figure 1 are commutativeby naturality. (cid:3) Remark 3.4.
Let π : R → S be a finite morphism of rings with involution. We have constructed aduality preserving functor π ∗ : (Ch b ( S -Mod) , π ♭ Iσ S ) −→ (Ch b ( R -Mod) , Iσ R )with the duality compatibility natural transformation η : π ∗ ◦ π ♭ I −→ I ◦ π ∗ , which is induced by the isomorphism of R -modules(3.3) Hom S ( M op S , Hom R ( S, I )) → Hom R ( M op R , I ) : f ( ξ ◦ f ) : m op R f ( m op S )(1 S )for M ∈ S -Mod. H E N G X I E π ∗ M / / (cid:15) (cid:15) π ∗ [[ M, π ♭ I ] , π ♭ I ] (cid:15) (cid:15) / / π ∗ [[ M, σ Z ∗ π ♭ I ] , π ♭ I ] / / (cid:15) (cid:15) π ∗ [ σ Z ∗ [ σ Z ∗ M, π ♭ I ] , π ♭ I ] (cid:15) (cid:15) (cid:3) [ π ∗ [ M, π ♭ I ] , π ∗ π ♭ I ] (cid:15) (cid:15) / / [ π ∗ [ M, σ Z ∗ π ♭ I ] , π ∗ π ♭ I ] / / (cid:15) (cid:15) [ π ∗ σ Z ∗ [ σ Z ∗ M, π ♭ I ] , π ∗ π ♭ I ] (cid:15) (cid:15) [[ π ∗ M, I ] , I ] / / (cid:15) (cid:15) [[ π ∗ M, π ∗ π ♭ I ] , I ] / / [ π ∗ [ M, π ♭ I ] , I ] / / [ π ∗ [ M, σ Z ∗ π ♭ I ] , I ] / / [ π ∗ σ Z ∗ [ σ Z ∗ M, π ♭ I ] , I ] (cid:15) (cid:15) [[ π ∗ M, σ X ∗ I ] , I ] (cid:15) (cid:15) (cid:3) [ σ X ∗ [ σ X ∗ π ∗ M, I ] , I ] / / [ σ X ∗ [ π ∗ σ Z ∗ [ M, I ] , I ] / / [ σ X ∗ [ π ∗ σ Z ∗ M, π ∗ π ♭ I ] , I ] / / [ σ X ∗ π ∗ [ σ Z ∗ M, π ♭ I ] , I ] F i g u r e . P r oo f o f L e mm a3 . TRANSFER MORPHISM FOR HERMITIAN K -THEORY OF SCHEMES WITH INVOLUTION 11 The local case and finite length modules
Let ( R, m , k ) be a local ring with an involution σ R . Suppose that ( R, σ R ) has a minimal dualizingcomplex with involution ( I • , σ I ). Note that this is the case if R is Gorenstein local (cf. Lemma 2.15).Suppose also that m is invariant under the involution of R , i.e. σ R ( m ) = m . Therefore, the σ R on R induces an involution σ k on k . In the previous section, we have defined a transfer morphism T m /R : W i ( k, σ k , ( π ♭ I • , σ π ♭ I )) → W i ( R, σ R , ( I • , σ I ))which factors through D m /R : W i ( k, σ k , ( π ♭ I • , σ π ♭ I )) → W i m ( R, σ R , ( I • , σ I )) . Moreover, the map D m /R : W i ( k, σ k , ( π ♭ I • , σ π ♭ I )) → W i m ( R, σ R , ( I • , σ I )) factors as W i ( k, σ k , ( π ♭ I • , σ π ♭ I )) → W i ( M fl ( R ) , σ R , ( I • , σ I )) → W i m ( R, σ R , ( I • , σ I ))where the first map is induced by the functor π ∗ : D b ( M ( k )) → D b ( M fl ( R )) and the second map isinduced by the inclusion D b ( M fl ( R )) → D bfl ( R ) = D b m ( R ). Theorem 4.1.
Let ( R, m , k ) be a local ring with an involution σ R . Suppose that I • is a minimaldualizing complex of R , σ I is an involution of I • and m is invariant under the involution of R . Then,there is an isomorphism of groups D m /R : W i ( k, σ k , ( π ♭ I • , σ π ♭ I )) → W i m ( R, σ R , ( I • , σ I )) . Lemma 4.2.
Under the assumption of Theorem 4.1, the natural map W i ( M fl ( R ) , σ R , ( I • , σ I )) → W i m ( R, σ R , ( I • , σ I )) is an isomorphism of groups.Proof. By [Ke99, Section 1.15] the inclusion D b ( M fl ( R )) → D bfl ( R ) = D b m ( R )is an equivalence of categories. (cid:3) Lemma 4.3.
Under the assumption of Theorem 4.1, the map π ∗ : W i ( k, σ k , ( π ♭ I • , σ π ♭ I )) → W i ( M fl ( R ) , σ R , ( I • , σ I )) is an isomorphism.Proof. Let I • := ( · · · → → I m → I m +1 → · · · → I n → → · · · ) be a minimal dualizing complex on R . We set E := I n and let π ♭ E be the k -module Hom R ( k, E ). Let σ π ♭ E : Hom R ( k, E ) → Hom R ( k, E ) op k : f σ π ♭ E ( f )be the involution on π ♭ E , where σ π ♭ E ( f )( a ) = σ E f σ k ( a ). Recall that σ π ♭ E is well-defined, and( π ♭ E, σ π ♭ E ) is a duality coefficient (cf. Definition 2.2) for k -Mod (see Lemma 2.5). Therefore, it inducesa category with duality ( k -Mod , π ♭ Eσ k ) (recall Remark 2.4). Moreover, the double dual identificationcan π ♭ Eσ k ,M : M → Hom k (Hom k ( M op k , π ♭ E ) op k , π ♭ E ) : m (can( m ) : f σ π ♭ E ( f ( m )))is an isomorphism, therefore the duality is strong in the sense of [Sch10b].Now, consider the following commutative diagram W i + n (cid:0) k, σ k , ( π ♭ E, σ π ♭ E ) (cid:1) ∼ = h / / ∼ = v (cid:15) (cid:15) W i + n ( M fl ( R ) , σ R , ( E, σ E )) ∼ = v (cid:15) (cid:15) W i ( k, σ k , ( π ♭ I • , σ π ♭ I )) π ∗ / / W i ( M fl ( R ) , σ R , ( I • , σ I )) . Here the map h is the transfer morphism, and by [BW02, Proposition 5.1] it is an isomorphism for i + n even, see also [QSS79, Corollary 6.9 and Theorem 6.10] or ([Sch10a, Theorem 6.1]). Note that the categories involved in the diagram above are all derived categories of abelian categories, and if i + n is even, the derived Witt groups can be identified with the usual Witt groups of symmetric orskew-symmetric forms. If i + n is odd by [BW02, Proposition 5.2] one has W i + n (cid:0) k, σ k , ( π ♭ E, σ π ♭ E ) (cid:1) = W i + n ( M fl ( R ) , σ R , ( E, σ E )) = 0 . The map v is induced by the duality preserving functor(id , θ ) : ( D b k, π ♭ E [ n ] σ k ) → ( D b k, π ♭ I • σ k ) , where θ : π ♭ E [ n ] σ k → π ♭ I • σ k is the natural isomorphism given by the isomorphism of complexes θ M : Hom k ( M op k , Hom R ( k, E [ n ])) → Hom k ( M op k , Hom R ( k, I • ))which is induced by the inclusion of complexes E [ n ] → I • . Notice that θ M is an isomorphism, becauseas an R -module Hom k ( M op k , Hom R ( k, I j )) ∼ = Hom R ( M op R , I j ) = 0if j = n . The first isomorphism follows by the isomorphism η in (3.3) and the second identity can beproved by [Gil02, Lemma 3.3]. Indeed, the local ring R is assumed to be Gorenstein in [Gil02, Lemma3.3 (1)-(4)]. We need to explain why [Gil02, Lemma 3.3] can be applied to our situation that thelocal ring R only assumed to have a minimal dualizing complex. Note that there is an isomorphism I j ∼ = ⊕ µ I ( Q )= j E R ( R/Q ) by [Gil07a, Theorem 1.15] where µ I : Spec R → Z is the codimensionfunction of the dualizing complex I • defined after [Gil07a, Lemma 1.12]. By [Gil07a, Lemma 1.14],the maximal ideal m is the only prime ideal Q such that µ I ( Q ) = n . For any R -module M such that m M = 0, we conclude that Hom R ( M, E R ( R/Q )) = 0 if Q ( m (i.e. µ I ( Q ) < n ) by [Gil02, Lemma 3.3(5)] where the local ring is only assumed to be commutative Noetherian.The isomorphism v is defined analogously to the map v . (cid:3) Remark 4.4.
For regular local rings Lemma 4.3 was proved by [Gil09, Theorem 4.5]. The situationin [Gil07a, Section 3.1] is very similar to Lemma 4.3, but in the construction of the transfer map weavoid the choice of an embedding of k into its injective hull E = E R ( k ) over R . Lemma 4.5.
Let ( R, m , k ) be a local ring with involution and ∈ R . Assume that I • is a minimaldualizing complex of R , σ I is an involution of I • and m is invariant under the involution of R . Let J be an invariant ideal of R and consider the following canonical projections R p (cid:15) (cid:15) π / / kR/J q = = ④④④④④④④④ . We have a commutative diagram (4.1) W i ( k, σ k , ( q ♭ p ♭ I • , σ q ♭ p ♭ I )) ∼ = (cid:15) (cid:15) q ∗ / / W i m /J ( R/J, σ
R/J , ( p ♭ I • , σ p ♭ I )) p ∗ (cid:15) (cid:15) W i ( k, σ k , ( π ♭ I • , σ π ♭ I )) π ∗ / / W i m ( R, σ R , ( I • , σ I )) Therefore, the map p ∗ is an isomorphismProof. Define a map γ : q ♭ p ♭ I → π ♭ Iγ : Hom R/J ( k, Hom R ( R/J, I )) → Hom R ( k, I )by sending f : k → Hom R ( R/J, I ) to γ ( f ) : a f ( a )(1). This map is an isomorphism, cf. [Ha66,Proposition 6.2 p166]. The commutativity of Diagram (4.1) follows immediately from the following TRANSFER MORPHISM FOR HERMITIAN K -THEORY OF SCHEMES WITH INVOLUTION 13 commutative diagram of triangulated categories with duality( D b k, q ♭ p ♭ Iσ k ) (cid:15) (cid:15) q ∗ / / ( D b m /J ( R/J ) , p ♭ Iσ R/J ) p ∗ (cid:15) (cid:15) ( D b k, π ♭ Iσ k ) π ∗ / / ( D b m R, Iσ R ) , where the left vertical arrow is induced by the isomorphism γ : q ♭ p ♭ I → π ♭ I . (cid:3) The d´evissage theorem for a closed immersion
Let (
X, σ X ) and ( Z, σ Z ) be schemes with involution and with ∈ O X . Suppose that ( X, σ X ) hasa minimal dualizing complex with involution ( I, σ I ) (cf. Definition 2.13). Assume that π : Z ֒ → X is a closed immersion which is invariant under involutions. Since π ∗ G ∈ Ch bc,Z ( Q ( X )) for all G ∈ Ch bc ( Q ( Z )), we have a map D Z/X : GW [ i ] ( Z, σ Z , ( π ♭ I • , σ π ♭ I )) → GW [ i ] Z ( X, σ X , ( I • , σ I )) . In fact, the transfer morphism T Z/X is the composition GW [ i ] ( Z, σ Z , ( π ♭ I • , σ π ♭ I )) D Z/X −→ GW [ i ] Z ( X, σ X , ( I • , σ I )) → GW [ i ] ( X, σ X , ( I • , σ I )) , where the second map is extending the support. Theorem 5.1.
Let ( X, σ X ) and ( Z, σ Z ) be schemes with involution and with ∈ O X . Suppose that ( X, σ X ) has a a minimal dualizing complex with involution ( I, σ I ) . If π : Z ֒ → X is a closed immersionwhich is invariant under involutions, then the direct image functor π ∗ : Ch bc ( Q ( Z )) → Ch bc ( Q ( X )) induces an equivalence of spectra D Z/X : GW [ i ] ( Z, σ Z , ( π ♭ I • , σ π ♭ I )) → GW [ i ] Z ( X, σ X , ( I • , σ I )) . Proof.
The result follows by Theorem 5.2 and Karoubi induction [Sch17, Lemma 6.4]. (cid:3)
Theorem 5.2.
Under the hypothesis of Theorem 5.1, there is an isomorphism of groups D Z/X : W i ( Z, σ Z , ( π ♭ I • , σ π ♭ I )) → W iZ ( X, σ X , ( I • , σ I )) . Proof.
We borrow the idea of [BW02], [Gil02], [Gil07a] to use the filtration of derived category bycodimension of points to reduce the problem to the local case. All we need to check is that thisapproach is compatible with the duality given by a non-trivial involution.Let X p I := { x ∈ X | m ≤ µ I ( x ) ≤ p − } and X ( p ) I := { x ∈ X | µ I ( x ) = p } . We define D pZ,X := n M • ∈ D bc,Z ( Q ( X )) | ( M • ) x is acyclic for all x ∈ X p I o considered as a full subcategory of D bc,Z ( Q ( X )). Note that the duality functor I • σ X maps D pZ,X into itself. Therefore, ( D pZ,X , I • σ X ) is a triangulated category with duality. The subcategories D pZ,X provide a finite filtration D bc,Z ( Q ( X )) = D mZ,X ⊇ D m +1 Z,X ⊇ · · · ⊇ D pZ,X ⊇ · · · ⊇ D nZ,X ⊇ (0)which induces exact sequences of triangulated categories with duality D p +1 Z,X −→ D pZ,X −→ D pZ,X / D p +1 Z,X . On the other hand, we define D pZ := n M • ∈ D bc ( Q ( Z )) | ( M • ) z is acyclic for all z ∈ Z pπ ♭ I o as a full subcategory of D bc ( Q ( Z )). By the same reason above, we have a finite filtration D bc ( Q ( Z )) = D mZ ⊇ D m +1 Z ⊇ · · · ⊇ D pZ ⊇ · · · ⊇ D nZ ⊇ (0) which induces exact sequences of triangulated categories with duality D p +1 Z −→ D pZ −→ D pZ / D p +1 Z . Since µ I ( z ) = µ π ♭ I ( z ) for all z ∈ Z , we have π ∗ ( D pZ ) ⊂ D pZ,X . It follows that we obtain a map ofexact sequences of triangulated categories with duality( D p +1 Z , π ♭ I • σ Z ) −−−−→ ( D pZ , π ♭ I • σ Z ) −−−−→ ( D pZ,X / D p +1 Z,X , π ♭ I • σ Z ) π ∗ y π ∗ y π ∗ y ( D p +1 Z,X , I • σ X ) −−−−→ ( D pZ,X , I • σ X ) −−−−→ ( D pZ,X / D p +1 Z,X , I • σ X )which induces a map of long exact sequences of groups (5.1) · · · −→ W i ( D pZ , π ♭ I • σ Z ) −→ W i ( D pZ / D p +1 Z , π ♭ I • σ Z ) −→ W i +1 ( D p +1 Z , π ♭ I • σ Z ) −→ · · · π ∗ y π ∗ y π ∗ y · · · −→ W i ( D pZ,X , I • σ X ) −→ W i ( D pZ,X / D p +1 Z,X , I • σ X ) −→ W i +1 ( D p +1 Z,X , I • σ X ) −→ · · · . Lemma 5.3.
The localization functors loc : D pZ,X / D p +1 Z,X → Y x ∈ Z ∩ X ( p ) I D b m X,x ( O X,x ) and loc : D pZ / D p +1 Z → Y z ∈ Z ( p ) π♭ I D b m Z,z ( O Z,z ) induce equivalences of categories.Proof. This result is well-known in the literature, see [Gil07b, Theorem 5.2] for instance. (cid:3)
Note that the morphism π ∗ : W i ( D pZ / D p +1 Z , π ♭ I • σ Z ) → W i ( D pZ,X / D p +1 Z,X , I • σ X ) is an isomorphism.This can be concluded by the commutative diagram W i ( D pZ / D p +1 Z , π ♭ I • σ Z ) loc −−−−→ ∼ = L z = σ ( z ) ∈ Z ( p ) π♭ I W i m Z,z ( O Z,z , σ O Z,z , ( π ♭ I z , σ π ♭ I z )) y π ∗ ∼ = y ⊕ z ( D O Z,z/ O X,z ) W i ( D pZ,X / D p +1 Z,X , I • σ X ) loc −−−−→ ∼ = L z = σ ( z ) ∈ Z ∩ X ( p ) I W i m X,z ( O X,z , σ O X,z , ( I z , σ I z )) , where the right morphism is an isomorphism by Lemma 4.5. Note that the Witt groups on the righthand side do not contain the component σ ( z ) = z . To explain, we write down a duality preservingfunctor ( loc, η ) : ( D pZ / D p +1 Z π ♭ I • σ Z ) → (cid:16) D b m Z,z ( O Z,z ) × D b m Z,σ ( z ) ( O Z,σ ( z ) ) , (cid:17) , where M, N ) ( N ( π♭I ) zσz , M ( π♭I ) σ ( z ) σσ ( z ) ) and the duality compatibility isomorphism η M : (( M ( π♭I ) σZ ) z , ( M ( π♭I ) σZ ) σ ( z ) ) −→ ( M ( π♭I ) zσz σ ( z ) , M ( π♭I ) σ ( z ) σσ ( z ) z )is defined to be the canonical isomorphism induced by restrictions of scalars via the local isomorphismsof local rings σ : O Z,z → O
Z,σ ( z ) and σ − : O Z,σ ( z ) → O Z,z . By these local isomorphisms, note that D b m Z,z ( O Z,z ) is equivalent to D b m Z,σ ( z ) ( O Z,σ ( z ) ), and W i ( D b m Z,z ( O Z,z ) × D b m Z,σ ( z ) ( O Z,σ ( z ) ) , p in Diagram (5.1), we conclude the result. (cid:3) TRANSFER MORPHISM FOR HERMITIAN K -THEORY OF SCHEMES WITH INVOLUTION 15 A geometrical computation of the d´evissage for regular schemes
Let (
X, σ X ) be a regular scheme with involution. Assume ( Z, σ Z ) is a regular scheme with involutionand assume further that Z is regularly embedded in X of codimension d . We have a normal bundle N := N Z/X of X on Z which is locally free of rank d . If Z is invariant under σ X , then we can definean involution σ N : N → σ ∗ N induced from σ X : X → X by the invariant ideal sheaf of Z in X . Theinvolution σ N induces an involution det( σ N ) : det( N ) → σ ∗ det( N ) by taking determinants. Recallthat the canonical sheaf is defined by ω Z/X := H om O Z (det( N ) , O Z ) . Now, det( σ N ) induces an involution σ ω Z/X : ω Z/X → σ ∗ ω Z/X : f σ Z ◦ ( σ ∗ f ) ◦ det( σ N ) . Theorem 6.1 (D´evissage) . Let ( X, σ X ) be a regular scheme with involution. Let ( L , σ L ) be a dualizingcoefficient with L a locally free O X -module of rank one. If Z is a regular scheme regularly embeddedin X of codimension d which is invariant under σ , then there is an equivalence of spectra D Z/X, L : GW [ i − d ] ( Z, σ Z , ( ω Z/X ⊗ O X L , σ ω Z/X ⊗ σ L )) −→ GW [ i ] Z ( X, σ X , ( L , σ L )) . Proof.
By Lemma 2.15 there exists ( L , σ L ) → ( I • , σ I ), a minimal injective resolution compatible with σ X , where I • = I → I → · · · → I n . We have the d´evissage isomorphism on the level of coherentGrothendieck-Witt groups D Z/X : GW i ( Z, σ Z , ( π ♭ I • , σ π ♭ I )) → GW iZ ( X, σ X , ( I • , σ I )) . We want to construct a quasi-isomorphism β : ω Z/X ⊗ O X L [ − d ] ≃ −→ π ♭ I • which is compatible withinvolutions, then we can use a variant of Lemma 2.10 to conclude the result. We will see that thismap stems from the fundamental local isomorphism H i H om O X ( O Z , I • ) = E xt i O X ( O Z , L ) = (cid:26) i = dω Z/X ⊗ O X L if i = d (see [Ha66, p 179]). Before checking the compatibility of the involutions, we review the fundamentallocal isomorphism. We take an open affine subscheme U ∼ = Spec( R ) of X , and Z U := Z × X U ∼ =Spec( R/J ) with J defined by a regular sequence ( x , · · · , x d ) of length d . Let L := L U and I := I U .Let E = ⊕ ≤ i ≤ d Re i be a free R -module with basis { e , · · · , e d } . Let s : E → R : ( y i e i ) ≤ i ≤ d P di =1 y i x i . There exists a Koszul resolution of R/J · · · → → ^ d E → ^ d − E → · · · → ^ E → R/J with differentials given by d i : ^ i E → ^ i − E : α ∧ · · · ∧ α i i X t =1 ( − t +1 s ( α t ) α ∧ · · · ∧ b α t ∧ · · · ∧ α i . Since ( x , · · · , x d ) is a regular sequence, the Koszul complex provides a projective resolution of R/J .Set P − i = P i = V i E .The fundamental local isomorphismExt iR ( R/J, L ) → Hom R (det( J/J ) , L ) of the R/J -modules (see [Ha66, Chapter III.7 p 176]) is defined as follows. We draw the followingdiagram of R -modules Hom R (det( J/J ) , L/JL )˜ β x Hom R ( P , L ) −→ · · · −→ Hom R ( P d , L ) y y Hom R ( R/J, I ) −→ Hom R ( P , I ) −→ · · · −→ Hom R ( P d , I ) y y y Hom R ( R/J, I ) −→ Hom R ( P , I ) −→ · · · −→ Hom R ( P d , I ) y y y Hom R ( R/J, I ) −→ Hom R ( P , I ) −→ · · · −→ Hom R ( P d , I ) y y y ... ... ... ... y y y Hom R ( R/J, I n ) −→ Hom R ( P , I n ) −→ · · · −→ Hom R ( P d , I n )Note that by this diagram, we have zigzagsHom R ( R/J, I • ) −→ Tot (cid:0)
Hom R ( P • , I • ) (cid:1) ←− Hom R ( P • , L )of quasi-isomorphisms of R -modules. Now, we deduceExt iR ( R/J, L ) := H i (Hom R ( R/J, I • )) ∼ = H i Hom R ( P • , L )and we define a map of R -modules˜ β : Hom R (det E, L ) → Hom R (det( J/J ) , L/JL ) : f ˜ β ( f ) , where ˜ β ( f )(¯ x ∧ · · · ∧ ¯ x d ) := f ( x ∧ · · · ∧ x d ). One checks that the compositionHom R ( P d − , L ) → Hom R ( P d , L ) β → Hom R (det( J/J ) , L/JL )is zero. Thus, we get a map of R -modules H d Hom( P • , L ) → Hom R (det( J/J ) , L/JL )which can be considered as a map of R/J -modules, and it is an isomorphism. This morphism extendsto a global morphism as it does not depend on the choice of the regular sequence (cf. [Ha66, ChapterIII.7 Lemma 7.1]).Now, we check that β is compatible with the involution. This can be checked affine locally. We takeany open affine subscheme U ∼ = Spec( R ) of X , which need not to be compatible with the involution.It follows that the subscheme U ′ := σ X ( U ) is also affine and can be assumed to be isomorphic toSpec( R ′ ) for some ring R ′ . Then the involution σ X restricts to a pair of isomorphisms of affineschemes σ U : U → U ′ and σ U ′ : U ′ → U (with σ U ′ = σ − U ) which induces a pair of isomorphisms ofrings σ R : R → R ′ and σ R ′ : R ′ → R (with σ R ′ = σ − R ). Recall that Z U ∼ = Spec( R/J ), where J isan ideal of R defined by a regular sequence ( x , . . . , x d ) of length d . Consider the restriction of U ′ into the closed subscheme Z ′ U which is isomorphic to Spec( R ′ /J ′ ) for J ′ an ideal inside R ′ . Since by TRANSFER MORPHISM FOR HERMITIAN K -THEORY OF SCHEMES WITH INVOLUTION 17 assumption Z is invariant under the involution of X , we have the following commutative diagram of R -modules J / / σ J (cid:15) (cid:15) R / / σ R (cid:15) (cid:15) R/J σ R/J (cid:15) (cid:15) J ′ / / σ J ′ (cid:15) (cid:15) R ′ / / σ R ′ (cid:15) (cid:15) R ′ /J ′ σ R ′ /J ′ (cid:15) (cid:15) J / / R / / R/J , where the vertical arrows are all isomorphisms and σ J = σ − J ′ , σ R = σ − R ′ , σ R ′ /J ′ = σ − R/J . Therefore,the sequence ( σ R ( x ) , · · · , σ R ( x d )) is a regular sequence in R ′ defining J ′ .As above, let L ′ := L U ′ and I ′ := I U ′ . Let E ′ = ⊕ ≤ i ≤ d Re ′ i be a free R ′ -module with basis { e ′ , · · · , e ′ d } . Similarly, we have a Koszul complex P ′• → R ′ /J ′ associated to the section s ′ : E ′ → R ′ : ( y i e i ) ≤ i ≤ d P di =1 y i σ ( x i ) with P ′− i := Λ i E ′ . We now consider the following isomorphism of R -modules σ E : E → σ R ∗ E ′ : ( y i e i ) ≤ i ≤ d ( σ R ( y i ) e i ) ≤ i ≤ d It induces isomorphisms of R -modules ^ i σ E : ^ i E → σ R ∗ ( ^ i E ′ )on the exterior powers V i E . It is straightforward to check that the diagram E s −−−−→ R σ E y σ R y σ R ∗ E ′ s ′ −−−−→ σ R ∗ R ′ of R -modules commutes. By checking the differentials, we get an isomorphism σ P : P • → σ R ∗ P ′• ofcomplexes of R -modules with inverse σ P ′ : P ′• → σ R ′ ∗ P • . Moreover, we can verify that the diagramHom R ( P d , L ) β / / ǫ Pd,L (cid:15) (cid:15)
Hom R (det( J/J ) , L/JL ) ǫ det( J/J ,L/JL (cid:15) (cid:15) Hom R ′ ( P ′ d , L ′ ) β ′ / / Hom R ′ (det( J ′ /J ′ ) , L ′ /J ′ L ′ )is also commutative.Using the same notations as in Lemma 2.5, we form the following commutative diagramHom R ( R/J, I • ) / / ǫ R/J,I (cid:15) (cid:15)
Tot (cid:0)
Hom R ( P • , I • ) (cid:1) ǫ P,I (cid:15) (cid:15)
Hom R ( P • , L ) o o ǫ P,L (cid:15) (cid:15) / / Hom R (det( J/J ) , L/JL )[ − d ] ǫ det( J/J ,L/JL (cid:15) (cid:15) Hom R ′ ( R ′ /J ′ , I ′• ) / / Tot (cid:0)
Hom R ′ ( P ′• , I ′• ) (cid:1) Hom R ′ ( P ′• , L ′ ) o o / / Hom R ′ (det( J ′ /J ′ ) , L ′ /J ′ L ′ )[ − d ]of complexes of R -modules. Since the horizontal maps are all quasi isomorphisms and they inducemaps on homology, we get H d Hom R ( R/J, I • ) ∼ = −−−−→ Hom R (det( J/J ) , L/JL ) y ǫ R/J,I y ǫ det( J/J ,L/JL H d Hom R ′ ( R ′ /J ′ , I ′• ) ∼ = −−−−→ Hom R ′ (det( J ′ /J ′ ) , L ′ /J ′ L ′ )which is the desired commutative diagram of R/J -modules.
Finally, we still need to find a map of complexes H d π ♭ I • [ − d ] → π ♭ I • in Ch bc ( Q ( Z )) which iscompatible with the involution. At this stage, we only have a right roof H d π ♭ I • [ − d ] s → C • t ← π ♭ I • compatible with the involution in D bc ( Q ( Z )), where C • is the canonical truncation τ ≤ d π ♭ I • . By thiswe mean C i = π ♭ I d / Im( π ♭ I d − → π ♭ I d ) if i = dπ ♭ I i if d < i ≤ n s and t are the obvious canonical quasi-isomorphisms. Since π ♭ I • is acomplex of injectives, the quasi-isomorphism t is a split injection in the homotopy category K bc ( Q ( Z ))of complexes, i.e. there is a map of complexes r : C • → π ♭ I • such that r ◦ t ≃ id in K bc ( Q ( Z )), by[Wei94, Lemma 10.4.6]. I claim that any choice of the splitting r will define a quasi-isomorphism ofcomplexes rs : H d π ♭ I • [ − d ] → π ♭ I • in Ch bc ( Q ( Z )) such that the diagram H d π ♭ I • [ − d ] σ ′ π♭ I• := H d σ π♭ I• (cid:15) (cid:15) rs / / π ♭ I • σ π♭ I (cid:15) (cid:15) σ Z ∗ H d π ♭ I • [ − d ] σ Z ∗ ( rs ) / / σ Z ∗ π ♭ I • is commutative in Ch bc ( Q ( Z )), while at this stage it is only known that it commutes in D bc ( Q ( Z )).By [Wei94, Corollary 10.4.7] and the fact that π ♭ ( I • ) is injective, the diagram also commutes in K bc ( Q ( Z )). Let us define f := σ π ♭ I ◦ ( rs ) − σ Z ∗ ( rs ) ◦ σ ′ π ♭ I • , which is a map of complexes from H d π ♭ I • [ − d ] to σ Z ∗ π ♭ I • . Then the commutativity in K bc ( Q ( Z )) implies that f is null homotopic.Since the complex H d π ♭ I • [ − d ] is concentrated in degree d , we see that the morphism of com-plexes f is also concentrated in degree d . By the definition of null homotopy, there exists a mapof O Z -modules u : H d π ♭ I • → σ Z ∗ π ♭ I d − such that u ◦ d π ♭ I = f . We now want to conclude thatHom O Z ( H d π ♭ I , π ♭ I d − ) ∼ = Hom O Z ( H d π ♭ I , σ Z ∗ π ♭ I d − ) vanishes. This can be checked locally, andby using that ht ( J ) = d and Hom R/J ( M, Hom R ( R/J, I d − )) = 0. This last fact is proved in [Gil02,Lemma 3.3 (5)]. Therefore, one has u = f = 0 . (cid:3) Corollary 6.2.
Under the hypothesis of Theorem 6.1, the direct image map W i − d ( Z, σ Z , ( ω Z/X ⊗ O X L , σ ω Z/X ⊗ σ L Z )) −→ W iZ ( X, σ X , ( L , σ L )) induces an isomorphism of groups. Some computations
Let S be a regular scheme with ∈ O S . Let P S be the projective line Proj( O S [ X, Y ]).7.1. P with switching involution. Let τ P S : P S → P S : [ X : Y ] [ Y : X ] be the involution whichswitches the coordinates of P S . Theorem 7.1.
Let S be a regular scheme with ∈ O S . Then, we have an equivalence of spectra GW [ i ] ( P S , τ P S ) ∼ = GW [ i ] ( S ) ⊕ GW [ i +1] ( S ) In particular, we have the following isomorphism on Witt groups W i ( P S , τ P S ) ∼ = W i ( S ) ⊕ W i +1 ( S ) Lemma 7.2.
Let σ − A S : A S → A S be the involution given by t
7→ − t . Then, the pullback p ∗ : GW [ i ] ( S ) −→ GW [ i ] ( A S , σ − A S ) is an isomorphism. TRANSFER MORPHISM FOR HERMITIAN K -THEORY OF SCHEMES WITH INVOLUTION 19 Proof.
Let pt : S → A S be the rational point corresponding to 0. Then, we have the followingcommutative diagram S A S S ptid p ∗ which induces GW [ i ] ( S ) GW [ i ] ( A S , σ − A S ) GW [ i ] ( S ) . p ∗ id pt ∗ This implies that the pullback p ∗ : GW [ i ] ( S ) −→ GW [ i ] ( A S , σ − A S ) is split injective. Now, consider thecomposition GW [ i ] ( A S , σ − A S ) pt ∗ −→ GW [ i ] ( S ) p ∗ −→ GW [ i ] ( A S , σ − A S ) . We want to show that it is an isomorphism. We use the following C -equivariant homotopy A × A A ( t ′ , t ) t ′ t A × A A ( t ′ , − t ) − t ′ t. This diagram tells us that ( A S , σ − A S ) p −→ ( S, id) pt −→ ( A S , σ − A S )is homotopic to id : ( A S , σ − A S ) → ( A S , σ − A S ). The result follows by Theorem 7.5 below. (cid:3) Let P ֒ → P S be the closed point cut out by the homogeneous ideal ( X − Y ). Then, P isinvariant under the involution τ P S on P S . Lemma 7.3.
Let S be a regular scheme with ∈ O S . The d´evissage theorem provides an isomorphism D P / P : GW [ i +1] ( S ) ∼ = −→ GW [ i ] P ( P S , τ P S ) . Proof.
By Theorem 6.1, we conclude that GW [ i − ( P , id , ( ω P / P S , σ ω )) ∼ = −→ GW [ i ] P ( P S , τ P S ) . Let I = ( X − Y ) be the homogeneous ideal defining P = S in P S . Then, the involution τ P S : P S → P S induces an involution on I which sends X − Y to Y − X , and therefore induces a map σ I/I : I/I → I/I : X − Y Y − X of O S -module. The map µ : I/I → O S defined by sending X − Y to 1 is an isomorphism of O S -modules. Now, we have a commutative diagram I/I µ / / σ I/I (cid:15) (cid:15) O S − id (cid:15) (cid:15) I/I µ / / O S which implies ( ω P / P S , σ ω ) ∼ = ( O S , − id) and hence GW [ i − ( P , id , ( ω P / P S , σ ω )) ∼ = −→ GW [ i − ( S, id , ( O S , − id)) ∼ = −→ GW [ i − ( S, − can) . Also, GW [ i − ( S, − can) ∼ = GW [ i +1] ( S ). (cid:3) Proof of Theorem 7.1.
By [Sch17], we have the following homotopy fibration sequence GW [ i ] P ( P S , τ P S ) −→ GW [ i ] ( P S , τ P S ) −→ GW [ i ] ( P S − P , τ P S − P ) . Note that there is an invariant isomorphism( A S , σ − A S ) → ( P S − P , τ P S − P )defined by sending t to [ t + : t − ]. Therefore, we get a homotopy fibration sequence GW [ i ] P ( P S , τ P S ) GW [ i ] ( P S , τ P S ) GW [ i ] ( A S , σ − A S ) GW [ i ] ( S ) . ∼ = p ∗ q ∗ By Lemma 7.2, we have the isomorphism p ∗ : GW [ i ] ( S ) → GW [ i ] ( A S , σ − A S ) which provides a splittingfor the above homotopy fibration sequence. By combining it with Lemma 7.3, we conclude that GW [ i ] ( P S , τ P S ) ∼ = GW [ i ] P ( P S , τ P S ) ⊕ GW [ i ] ( S ) ∼ = GW [ i +1] ( S ) ⊕ GW [ i ] ( S ) . The result now follows. (cid:3) C -equivariant A -invariance. Consider the involution σ P S : P S → P S given by the gradedmorphism O S [ X, Y ] → O S [ X, Y ] of graded sheaves of O S -algebras, such that a σ S ( a ) if a ∈ O S ,and X X, Y Y . There is an element β := O P ( − X −−−−→ O P Y y Y y O P X −−−−→ O P (1) ∈ GW [1]0 ( P S , σ P S ) . Proposition 7.4.
In this result, the base S can be singular, and still ∈ O S . The map of spectra ( q ∗ , β ∪ q ∗ ( − )) : GW [ i ] ( S, σ S ) ⊕ GW [ i − ( S, σ S ) → GW [ i ] ( P S , σ P S ) is an equivalence.Proof. The proof of [Sch17, Theorem 9.10] can be applied without modification. (cid:3)
Theorem 7.5.
Let S be a regular scheme with involution σ S and with ∈ O S . Let σ A S : A S → A S be the involution on A S with the indeterminant fixed by σ A S and such that the following diagramcommutes A S σ A S −−−−→ A Sp y p y S σ S −−−−→ S .
Then, the pullback p ∗ : GW [ i ] ( S, σ S ) → GW [ i ] ( A S , σ A S ) is an isomorphism.Proof. Let pt : S → P S be the rational point with X = 0 and Y = 1. It follows that there is acommutative diagram S σ S −−−−→ S pt y pt y P S σ P S −−−−→ P S . TRANSFER MORPHISM FOR HERMITIAN K -THEORY OF SCHEMES WITH INVOLUTION 21 By Schlichting [Sch17, Theorem 6.6], we have the following localization sequence GW [ i ]pt ( P S , σ P S ) −→ GW [ i ] ( P S , σ P S ) −→ GW [ i ] ( A S , σ A S )We write out the following commutative diagram(7.1) GW [ i − ( S, σ S ) GW [ i ] ( S, σ S ) ⊕ GW [ i − ( S, σ S ) GW [ i ] ( S, σ S ) GW [ i ]pt ( P S , σ P S ) GW [ i ] ( P S , σ P S ) GW [ i ] ( A S , σ A S ) . D pt / P S (cid:16) (cid:17)(cid:16) q ∗ β ∪ q ∗ ( − ) (cid:17) p ∗ The main reason for the commutativity of the left square is that we have the locally free resolution0 −→ O P ( − X −→ O P −→ O pt −→ . By d´evissage, we conclude that D pt / P S is an equivalence. By Proposition 7.4, we know that themiddle map (cid:0) q ∗ β ∪ q ∗ ( − ) (cid:1) is an equivalence. It follows that the right arrow p ∗ : GW [ i ] ( S, σ S ) → GW [ i ] ( A S , σ A S )is also an equivalence. (cid:3) Representability.
It is easy to see that schemes with involution can be identified with schemeswith a C -action where C is the cyclic group of order two considered as an algebraic group. Let H C • ( S ) be the C -equivariant motivic homotopy category of Heller, Krishna and Østvær [HKO14].Consider the presheaf of simplicial sets GW [ i ] : Sm op S → sSet by a big vector bundle argument cf.[Sch17, Remark 9.2]. The presheaf GW [ i ] is an object in H C • ( S ). We prove that the followingrepresentability result. Theorem 7.6.
Let ( X, σ ) ∈ Sm C S . Then, there is a bijection of sets [ S n ∧ ( X, σ ) + , GW [ i ] ] H C • ( S ) = GW [ i ] n ( X, σ ) . Proof.
By [HKO14, Corollary 4.9], we need to prove the C -equivariant Nisnevich excision and A -invariance for Hermitian K -theory. Note that the C -equivariant Nisnevich excision for GW [ i ] can beproved by a variant of [Sch17, Theorem 9.6] adapted to schemes with involution since the argumentis independent of the duality. Moreover, A -invariance is proved in Theorem 7.5. (cid:3) Acknowledgements.
I want to thank Marco Schlichting for useful discussion. I appreciate thereferee for pointing out a gap in an earlier version. I thank Thomas Hudson for the proofreading ofthe paper. I would like to acknowledge support from the EPSRC Grant EP/M001113/1, the DFGpriority programme 1786 and the GRK2240. Part of this work was carried out when I was visitingHausdorff Research Institute for Mathematics in Bonn and Max-Planck-Institut in Bonn. I would liketo express my gratitude for their hospitality.
References [AF14a] A. Asok and J. Fasel, A cohomological classification of vector bundles on smooth affine threefolds, Duke Math.J. 163 (2014) 2561-2601.[AF14b] A. Asok and J. Fasel, Splitting vector bundles outside the stable range and homotopy theory of puncturedaffine spaces, J. Amer. Math. Soc. 28 (2014) 1031-1062.[At66] M. F. Atiyah, K -theory and reality, Quart. J . Math. Oxford 17 (1966) 367-86.[Ba73] H. Bass, Algebraic K -theory III, Hermitian K -theory and geometric applications, Lecture Notes in Mathematics343, Springer, 1973.[BW02] P. Balmer and C. Walter, A Gersten-Witt spectral sequence for regular schemes, Annales Scientifiques de l’´ENS35 (2002) 127-152. [FS09] J. Fasel and V. Srinivas, Chow-Witt groups and Grothendieck-Witt groups of regular schemes, Adv. Math. 221(2009) 302-329.[Gil02] S. Gille, On Witt groups with support, Math. Ann. 322 (2002) 103-137.[Gil03] S. Gille, A transfer morphism for Witt groups, J. Reine Angew. Math. 564 (2003), 215-233.[Gil07a] S. Gille, The general d´evissage theorem for Witt groups, Arch. Math. 88 (2007) 333-343.[Gil07b] S. Gille, A graded Gersten-Witt complex for schemes with a dualizing complex and the Chow group, J. PureAppl. Algebra 208 (2007) 391-419.[Gil09] S. Gille, A Gersten-Witt complex for hermitian Witt groups of coherent algebras over schemes II: Involution ofthe second kind, J. K-theory 4 (2009) 347- 377.[Ha66] R. Hartshorne, Residues and Duality, Lecture Notes in Mathematics 20, Springer-Verlag Berlin Heidelberg 1966.[HKO11] P. Hu, I. Kriz, and K. Ormsby, The homotopy limit problem for Hermitian K-theory, equivariant motivichomotopy theory and motivic Real cobordism, Adv. Math. 228 (2011) 434-480.[HKO14] J. Heller, A. Krishna, and P. A. Østvær, Motivic homotopy theory of group scheme actions, Journal ofTopology 8 (2015) 1202-1236.[Kar80] M. Karoubi, Le th´eor`eme fondamental de la K-th´eorie hermitienne, Ann. Math. 112 (1980) 259-282.[Kar74] Max Karoubi, Localisation de formes quadratiques. II, Ann. Sci. Ecole Norm. Sup. 8 (1975) 99-155.[Ke99] B. Keller, On the cyclic homology of exact categories, Journal of Pure and Applied Algebra, 136 (1999) 1-56.[KS80] M. Knebusch and W. Scharlau, Algebraic Theory of Quadratic Forms, DMW seminar: 1, Birkh¨auser, 1980.[Kne77] M. Knebusch, Symmetric bilinear forms over algebraic varieties, in: Conference on Quadratic Forms, Kingston,1976, in: Queens Papers in Pure and Appl. Math. 46 (1977) 103-283. Queens Univ., Kingston.[Knu91] M. A. Knus, Hermitian Forms and Quadratic Forms, Grund. der math. Wiss. 294, Springer, 1991.[Lam05] T. Y. Lam, Introduction to Quadratic Forms over Fields, Graduate Studies in Mathematics 67, AmericanMathematical Society, 2005.[Ne09] A. Nenashev, On the Witt groups of projective bundles and split quadrics: Geometric reasoning, J. K -theory 3(2009) 533-546.[QSS79] H. G. Quebbemann, W. Scharlau, and M. Schulte. Quadratic and Hermitian forms in additive and abeliancategories, J. Algebra 59 (1979) 264-289.[Ran98] A. Ranicki, High-dimensional Knot Theory: Algebraic Surgery in Codimension 2, Springer Monographs inMathematics, Springer, 1998.[Scha85] W. Scharlau, Quadratic and Hermitian Forms, Grund. der math. Wiss. 270, Springer 1985.[Sch10a] Marco Schlichting, Hermitian K -theory of exact categories, J. K-Theory 5 (2010) 105-165 .[Sch10b] Marco Schlichting, The Mayer-Vietoris principle for Grothendieck-Witt groups of schemes, Invent. Math. 179(2010) 349-433.[Sch17] M. Schlichting, Hermitian K -theory, derived equivalences, and Karoubi’s Fundamental Theorem, J. Pure Appl.Algebra 221 (2017) 1729-1844.[Sha69] R. Y. Sharp, The Cousin complex for a module over a commutative Noetherian ring, Math. Z. 112 (1969)340-356.[Wal03] C. Walter, Grothendieck-Witt groups of projective bundles, K-theory preprint archive (2003).[Wei94] C. Weibel, An introduction to homological algebra, Cambridge Univ. Press, 1994. Heng Xie, Fachgruppe Mathematik and informatik, Bergische Universit¨at Wuppertal, Gaußstraße 20,42119 Wuppertal, Germany
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