A Transfer Morphism for Hermitian K-Theory of Schemes with Involution 3

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b X is a finite morphism of schemes with involution, then the direct image functor π∗ : Chc(Q(Z)) → b Chc(Q(X)) induces a map of spectra [i] ♭ • [i] • TZ/X : GW (Z, σZ , (π I , σπ♭I )) → GW (X, σX , (I , σI )). In light of the work of Balmer-Walter [BW02] and Gille [Gil07a], we also prove a version of the dévissage theorem for Hermitian K-theory of schemes with involution. The following result is proved in Theorem 5.1. 1 Theorem 1.2. Let (X, σX ) and (Z, σZ ) be schemes with involution and with 2 in their global sections. Suppose that (X, σX ) has a minimal dualizing complex with involution (I, σI ) (cf. Definition 2.13). If π : Z ֒→ X is a closed immersion which is invariant under involutions, then the direct image functor b b π∗ : Chc(Q(Z)) → Chc,Z (Q(X)) induces an equivalence of spectra [i] ♭ • [i] • DZ/X : GW (Z, σZ , (π I , σπ♭I )) → GWZ (X, σX , (I , σI )). Note that a Gersten-Witt complex for Witt groups of schemes with involution was constructed by Gille [Gil09], but the above formula seems not to appear anywhere in the literature. If X and Z are both regular, we have the following more precise formulation of the dévissage theorem (cf. Theorem 6.1).

Theorem 1.3 (Dévissage). Let (X, σX ) be a regular scheme with involution. Let (L, σL) be a dualizing coefficient with L a locally free OX -module of rank one. If Z is a regular scheme regularly embedded in X of codimension d which is invariant under σ, then there is an equivalence of spectra [i−d] [i] DZ/X,L : GW (Z, σZ , (ωZ/X ⊗OX L, σωZ/X ⊗ σL)) −→ GWZ (X, σX , (L, σL)). It turns out that if the involution is non-trivial, the canonical double dual identification has a certain sign varying according to the data of dualizing coefficients (cf. Definition 2.8). This sign is not important if the involution is trivial, but it is crucial if the involution is non-trivial. As an application, we use the dévissage theorem to obtain the following result (cf. Theorem 7.1). 1 Theorem 1.4. Let S be a regular scheme with 2 ∈ OX . Then, we have an equivalence of spectra [i] 1 [i] [i+1] GW (P , τP1 ) =∼ GW (S) ⊕ GW (S) S S 1 1 where τP1 : P → P :[x : y] 7→ [y : x]. In particular, we have the following isomorphism on Witt S S S groups i 1 i i+1 W (P , τP1 ) =∼ W (S) ⊕ W (S). S S i 1 ∼ i i−1 1 It is well-known that W (PS) = W (S) ⊕ W (S) if the involution on PS is trivial (cf. [Ne09] and 1 1 [Wal03]). By our dévissage theorem, the involution σ : PS → PS :[x : y] 7→ [y : x] induces a sign (−1) on the double dual identification, and we have i 1 ∼ i i−1 W (PS, σ) = W (S) ⊕ W (S, −can) but W i−1(S, −can) =∼ W i+1(S). 1 We also use the dévissage to prove the following C2-equivariant A -invariance (cf. Theorem 7.5). 1 1 1 Theorem 1.5. Let S be a regular scheme with involution σS and with ∈ OS. Let σA1 : A → A 2 S S S 1 be the involution on A with the indeterminant fixed by σA1 and such that the following diagram S S commutes. σA1 1 S 1 AS −−−−→ AS

p p    σS  yS −−−−→ Sy The pullback ∗ [i] [i] 1 p : GW (S, σS) → GW (A , σA1 ) S S is an isomorphism. A 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]), but it was not known for regular schemes with involution. If S is quasi-projective, then it can be covered by invariant affine open subschemes, and we can use the Mayer-Vietoris of Schlichting ([Sch17]) to reduce the quasi-projective case to the affine case. However, the method of Meyer-Vietoris does not work if S is not quasi-projective because there exist non-quasi-projective regular schemes which can not be covered by invariant affine opens. It is easy to see that schemes with involution can be identified with schemes with C2-action where C2 is the cyclic group of order two considered as an algebraic group. Together with a variant of Nisnevich excision [Sch17, Theorem 9.6] adapted to schemes with involution, we obtain the C2-representability C2 of Hermitian K-theory in the C2-equivariant motivic homotopy category H• (S) of Heller, Krishna and Østvær [HKO14]. The following result is proved in Theorem 7.6.

1 C2 Theorem 1.6. Let S be a regular scheme with 2 ∈ OS, and let (X, σ) ∈ SmS . Then, there is a bijection of sets n [i] [i] [S ∧ (X, σ) , GW ] C2 = GW (X, σ). + H• (S) n

2. Notations and prerequisites 1 In this paper, we assume that every scheme is Noetherian and has 2 in its global sections, and 1 b every ring is commutative Noetherian with identity and 2 . If E is an exact category, we write D (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 following notations associated with R. - R-Mod is the category of (left) R-modules. - M(R) is the category of finitely generated R-modules. - Mfl(R) is the category of finite length R-modules. - MJ (R) is the category of finitely generated R-modules supported in an ideal J of R. Recall that Supp(M) = {p ∈ Spec(R)|Mp 6= 0} and V (J) = {p ∈ Spec(R)|p ⊃ J}. More precisely, MJ (R) is the full subcategory of M(R) consisting of those modules M such that Supp(M) ⊂ V (J). b - Dc(R-Mod) is the derived category of bounded complexes of R-modules with coherent cohomology. b b - Dfl(R) is the full subcategory of Dc(R-Mod) of complexes, whose cohomology modules are finite length R-modules. b b - DJ (R) is the full triangulated subcategory of Dc(R-Mod) of complexes whose cohomology modules are annihilated by some power of the ideal J.

2 Deﬁnition 2.1. An involution σ on a ring R is a ring homomorphism σ : R → R such that σ = idR.

Let (R, σ) be a ring with involution and M a left R-module. Define M opR (or simply M op) to be the same as M as a set. For an element in M op, we use the symbol mop to denote the element m coming from the set structure of M. Consider M op as a right R-module via mopa = (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 → Iop such that i2 = id. Let σ : R → R′ be a ring homomorphism. Let M ′ be a module over R′. We can form the restriction of scalars ′ ′ σ∗M := M |R. which can also be considered as an R-module.

op Remark 2.3. Note that if (R, σ) is a ring with involution and M is an R-module, then σ∗M = M . 4 HENG XIE

I I Remark 2.4. For a duality coeﬃcient (I,i), one can form a category with duality (R-Mod, #σ, canσ) where I op #σ : (R-Mod) → R-Mod : M 7→ HomR(σ∗M, I) is a functor and I I I #σ#σ σ op op canσ,M : M → M : canI,M (x)(f )= i(f(x )) 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−1 and τ ′ := τ −1. Suppose furthermore that one has the following commutative diagram of rings.

′ R σ / R′ σ / R

′ g g g ′ 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 ′ σI : 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 σM : M → σ∗M. Then, the map ′ ′ ǫM,I : HomR(M, I) → σ∗HomR′ (M , I ): f 7→ σI fσM ′ 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 ′ ′ ′ σN : N → τ∗N. The set HomR(N, I) has an S-module structure and the map ′ ′ ǫN,I : HomR(N, I) → τ∗HomR′ (N , I ): f 7→ σI fσN ′ is a well-deﬁned S-module isomorphism. 2.2. Notations on schemes. Let X be a scheme. We ﬁx the following notations that will be needed in the remaining sections.

- OX -Mod is the category of OX -modules. - Q(X) is the category of quasi-coherent OX -modules. - VB(X) is the category of ﬁnite rank locally free coherent modules over X. - MZ (X) is the category of ﬁnitely generated OX -modules supported in a closed scheme Z of X. Recall that Supp(F) = {x ∈ X|Fx 6= 0}. More precisely, MZ (X) is the full subcategory of M(X) consisting of those modules F such that Supp(F) ⊂ Z. b - Kc (Q(X)) is the homotopy category of bounded complexes of quasi-coherent sheaves on X with coherent cohomology. b - Dc(Q(X)) is the derived category of bounded complexes of quasi-coherent sheaves on X with coherent cohomology. b b • - Dc,Z (Q(X)) is the full triangulated subcategory of Dc(Q(X)) consisting of complexes F such i • that {x ∈ X|H (F )x 6= 0 for some i ∈ Z }⊆ Z. - Db(X) is the derived category Db(VB(X)). b b • - DZ (X) is the full triangulated category of D (X) whose objects are the complexes F such i • that {x ∈ X|H (F )x 6= 0 for some i ∈ Z }⊆ Z. b - Chc(Q(X)) is the dg category of bounded complexes of quasi-coherent OX -modules with coherent cohomology. b b • - Chc,Z (Q(X)) is the full dg subcategory of Chc(Q(X)) consisting of complexes F such that i • {x ∈ X|H (F )x 6= 0 for some i ∈ Z }⊆ Z. b b - Ch (X) is the dg category Ch (VB(X)) of bounded complexes of locally free coherent OX - modules. A TRANSFER MORPHISM FOR HERMITIAN K-THEORY OF SCHEMES WITH INVOLUTION 5

b b • i • - ChZ (X) is the full dg subcategory Ch (X) of those complexes F such that {x ∈ X|H (F )x 6= 0 for some i ∈ Z }⊆ Z.

Deﬁnition 2.6. An involution σX on a ringed space X is a morphism σX : X → X of ringed spaces 2 such that σ = idX . Remark 2.7. If X = Spec(R), to give a scheme with involution (X, σ) is the same as giving a ring with involution (R, σ).

Definition 2.8. A duality coefficient (I,i) for the category of OX -modules on X with respect to the involution σX consists of an OX -module I equipped with an OX -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 ′ OX -modules α : I → I such that the following diagram commutes I −−−−→α I′

′ i i    σ∗α  ′ σy∗I −−−−→ σ∗yI . 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 I I (OX -Mod, #σ, canσ). I If the double dual identification canσ is clear from the context, we will use the simplified notation I I I (OX -Mod, #σ) instead of the more precise (OX -Mod, #σ, canσ). Lemma 2.10. Let α : (I,i) → (I′,i′) be an isomorphism of duality coefficients. Then, the identity functor induces a duality preserving equivalence of categories with duality ′ I I (id, α∗) : (OX -Mod, #σ) → (OX -Mod, #σ ) Proof. The duality compatibility isomorphism is defined by ′ α∗ : HomOX (σ∗M, I) → HomOX (σ∗M, I ): f 7→ α ◦ f which is natural in M. Now it is straightforward to check the axioms. 2.3. Dualizing complexes with involution. Recall the following definition of a duality complex from [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

m n−1 • m d m+1 n−1 d n b I := (···→ 0 → I −→ I →···→ I −→ I → 0 → · · · ) ∈ Dc(Q(X)) such that the natural morphism of complexes • • • • |f||m| canI,M : M → HomOX (HomOX (M , I ), I ): m 7→ canI,M (m): f 7→ (−1) f(m) • b r is a quasi-isomorphism for any M ∈ Dc(Q(X)). A dualizing complex is called minimal if I is an essential extension of ker(dr) for all r ∈ Z. Remark 2.12. Any dualizing complex is quasi-isomorphic to a minimal dualizing complex [Gil07b, Proposition 1.9]. 1 Definition 2.13. Let (X, σ) be a Noetherian scheme with involution and 2 ∈ OX . A dualizing • • complex with involution on (X, σ) is a pair (I , σI ) consisting of a dualizing complex I and an • • isomorphism of complexes of OX -modules σI : I → σ∗(I ) such that σ∗(σI ) ◦ σI = idI . p Remark 2.14. Note that (I , σIp ) is a duality coefficient for the category OX -Mod. Lemma 2.15. Let (X, σ) be a Gorenstein scheme of finite Krull dimension with involution and 1 2 ∈ OX . Let (L, σL) be a dualizing coefficient with L a locally free OX -module of rank one. Then, • there exists a minimal dualizing complex with involution (J , σJ ), which is quasi-isomorphic to (L, σL). 6 HENG XIE

(p) Proof. Let X := {x ∈ X| dim OX,x = p}. Recall [Ha66, p241] that we have a Cousin complex

−1 0 d 0 d 1 n 0 → OX −→ I −→ I −→ · · · −→ I −→ 0 −→ · · · 0 p t−2 with I := x∈X(0) ix∗(OX,x) and I := x∈X(p) ix∗(coker(d )x) where the map ix : Spec(OX,x) → X is the canonicalL map. If X is Gorenstein,L the Cousin complex provides a minimal dualizing complex • I on X quasi-isomorphic to OX . This can be checked locally on closed points by [Ha66, Corollary V.2.3 p259]. For the case of Gorenstein local rings, see [Sha69, Theorem 5.4]. p p We deﬁne a map σI : I → σ∗I as follows. We consider the following commutative diagram

σX,x Spec(OX,x) / Spec(OX,σ(x))

ix iσ(x) σ X X / X

Assume that M is an OX -module together with a map M → σ∗M. If x = σ(x), the map M → σ∗M ∗ ∗ induces a canonical map ix,∗ixM → σ∗ix,∗ixM of OX -modules by using the natural isomorphism ∗ ∼ ∗ ∼ ∗ σ∗ix,∗ixM = iσ(x),∗σx,∗ixM = iσ(x),∗iσ(x)σ∗M. If x 6= σ(x), the map M → σ∗M still induces a canonical map ∗ ∗ ∗ ∗ ix,∗ixM ⊕ iσ(x),∗iσ(x)M → σ∗iσ(x),∗iσ(x)M ⊕ σ∗ix,∗ixM p p by the functorial isomorphism above. The map σI : I → σ∗I is obtained by applying this construction inductively. (Note that x ∈ X(p) if and only if σ(x) ∈ X(p)). By our construction it is clear that σ∗(σI ) ◦ σI = idI . If L is a line bundle on X, then J • := L⊗ I• is a dualizing complex, cf. [Ha66, Proof (1) of Theorem V.3.1, p. 266]. Moreover, the complex J • is quasi-isomorphic to L. Note that we have a • • canonical isomorphism σ∗(L⊗ I ) → σ∗L⊗ σ∗I . These observations reveal a canonical involution • • σJ : J → σ∗J . 2.4. Setup for Hermitian K-theory of schemes with involution. Let (X, σ) be a scheme with b involution. In this paper, we work within the framework of Schlichting [Sch17]. Recall that Chc(Q(X)) • • is a closed symmetric monoidal category under the tensor product of complexes M ⊗OX N given by • • n i j |m||n| (M ⊗OX N ) := M ⊗OX N , d(m ⊗ n)= dm ⊗ n + (−1) m ⊗ dn i+Mj=n • • and internal homomorphism complexes HomOX (M ,N ) given by • • n i j |f| HomOX (M ,N ) := HomOX (M ,N ), df = d ◦ f − (−1) f ◦ d. j−Yi=n • Let (I , σI ) be a dualizing complex with involution. Then, we deﬁne the duality functor • I b op b • • • #σ : (Chc(Q(X))) → Chc(Q(X)) : E 7→ HomOX (σ∗E , I ), • • • I I I #σ #σ and the double dual identiﬁcation canσ,E : E → E given by the composition

canE E −→ HomOX (HomOX (E, I), I) −→ HomOX (HomOX (E, σ∗I), I) −→ HomOX (σ∗HomOX (σ∗E, I), I). |x||f| Here the ﬁrst map canE is deﬁned by canE(x)(f) = (−1) f(x) as in the case of trivial duality, the second is induced by σI : I → σ∗I, and the third is induced by σ∗HomOX (M,N) =

HomOX (σ∗M, σ∗N). Lemma 2.16. The quadruple • • b I I (Chc(Q(X)), quis, #σ , canσ ) is a dg category with weak equivalence and duality. Proof. It is enough to check that • • b I I (Chc(Q(X)), #σ , canσ ) is a category with duality, see [Gil09, Section 3.7]. A 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, σX ) with respect to (I, σI ) as • • [i] [i] b I I GW (X, σX , (I, σI )) := GW (Chc(Q(X)), quis, #σ , canσ ) • • for every i ∈ Z, where GW [i](Chb(Q(X)), quis, #I , canI ) is the Grothendieck-Witt spectrum of the c σ σ • • b I I dg category with weak equivalences and duality (Chc(Q(X)), quis, #σ , canσ ). The Witt group of (X, σX ) with respect to (I, σI ) is defined as • • i i b I I W (X, σX , (I, σI )) := W (Dc(Q(X)), #σ , canσ ), • • • • i b I I b I I where W (Dc(Q(X)), #σ , canσ ) is the Witt group of the triangulated category with duality (Dc(Q(X)), #σ , canσ ). Let Z be an invariant closed subscheme of X. We can also define • • [i] [i] b I I GWZ (X, σX , (I, σI )) := GW (Chc,Z (Q(X)), quis, #σ , canσ ) and • • i i b I I WZ (X, σX , (I, σI )) := W (DZ (X), #σ , canσ ).

Let (L, σL) be a dualizing coeﬃcient with L a locally free OX -module of rank one. Then, we have a dg category with duality b L L (Ch (X), quis, #σ , canσ ), where L b op b • • #σ : (Ch (X)) → Ch (X): E 7→ HomOX (σ∗E , L), L L L #σ #σ and the double dual identiﬁcation canσ,E : E → E is given by the composition

canE E −→ HomOX (HomOX (E, L), L) −→ HomOX (HomOX (E, σ∗L), L) −→ HomOX (σ∗HomOX (σ∗E, L), L). |x||f| Here the ﬁrst map canE is deﬁned by canE(x)(f) = (−1) f(x), the second map is induced by

σL : L → σ∗L, and the third map is induced by σ∗HomOX (M,N)= HomOX (σ∗M, σ∗N).

Definition 2.18. We define the Hermitian K-theory spectrum of schemes with involution of (X, σX ) with respect to (L, σL) as [i] [i] b L L GW (X, σX , (L, σL)) := GW (Ch (X), quis, #σ , canσ ) for every i ∈ Z. The Witt group of (X, σX ) with respect to (L, σL) is defined as i i b L L W (X, σX , (L, σL)) := W (D (X), #σ , canσ ). Let Z be an invariant closed subscheme of X. We can also define [i] [i] b L L GWZ (X, σX , (L, σL)) := GW (ChZ (X), quis, #σ , canσ ) and i i b L L WZ (X, σX , (L, σL)) := W (DZ (X), #σ , canσ ).

2.5. Dualizing complex via a ﬁnite morphism. Let (Z, σZ ) 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 π / X

σZ σX Z π / X. Following Hartshorne [Ha66], we write ♭ • ∗ • π I :=π ¯ HomOX (π∗OZ , I ), • whereπ ¯ : (Z, σZ ) → (X, π∗OZ ) =: X¯ is the canonical ﬂat map of ringed spaces. If I is a dualizing ♭ • complex, then π I is also a dualizing complex [Gil07a, Section 2.4]. Note that σX : X → X induces ¯ ¯ a canonical involution σX¯ : X → X deﬁned on the topological spaces, as well as π∗σZ : π∗OZ → 8 HENG XIE

♭ • σX∗π∗OZ = π∗σZ∗OZ on the sheaves of rings. Then, we can deﬁne an involution σπ♭I• : π I → ♭ • ♭ • σZ∗π I on π I through the following composition

♭ • ∗ • [σZ ,1] ∗ • ∗ • [1,σI ] π I =π ¯ HomOX (π∗OZ , I ) −→ π¯ HomOX (π∗σZ∗OZ , I ) −→ π¯ HomOX (σX∗π∗OZ , I ) −→

∗ • ∗ • ∗ • π¯ HomOX (σX∗π∗OZ , σX∗I ) −→ π¯ σX∗HomOX (π∗OZ , I ) =π ¯ σX¯ ∗HomOX (π∗OZ , I ) −→

∗ • ♭ • σZ∗π¯ HomOX (π∗OZ , I ) = σZ∗π I .

♭ • It follows that we obtain a dualizing complex with involution (π I , σπ♭I• ) on Z. Thus, we have a [i] ♭ • well-defined spectrum GW (Z, σZ , (π I , σπ♭I )) for every i ∈ Z. Note that if Z = Spec(B) and X = Spec(A) are affine, then HomA(B, I) has a structure of B- ♭ • module by defining the action b · f as b · f : m 7→ f(bm) for b,m ∈ B. The involution σπ♭I• : π I → ♭ • σZ∗π I is precisely the map of B-modules

A opB σB,I : HomA(B, I) → HomA(B, I) : f 7→ σI fσB .

3. The transfer morphism 1 Theorem 3.1. Let (X, σX ) and (Z, σZ ) be schemes with involution and with 2 in their global sections. Suppose that (X, σX ) has a dualizing complex with involution (I, σI ) (cf. Definition 2.13). If π : Z → b X is a finite morphism of schemes with involution, then the direct image functor π∗ : Chc(Q(Z)) → b Chc(Q(X)) induces a map of spectra [i] ♭ • [i] • TZ/X : GW (Z, σZ , (π I , σπ♭I )) → GW (X, σX , (I , σI )). [i] ♭ • [i] • Definition 3.2. The map TZ/X : GW (Z, σZ , (π I , σπ♭I )) → GW (X, σX , (I , σI )) in the above theorem is called the transfer morphism. [i] ♭ • [i] • Proof of Theorem 3.1. The transfer map TZ/X : GW (Z, σZ , (π I , σπ♭I )) → GW (X, σX , (I , σI )) is induced by the duality preserving functor given below in Lemma 3.3. Lemma 3.3. Under the hypothesis of Theorem 3.1, the direct image functor induces a duality preserving functor • • b π♭I b I π∗ : (Chc(Q(Z)), #σZ ) −→ (Chc(Q(X)), #σX ) with the duality compatibility natural transformation • • π♭I I η : π∗ ◦ #σZ −→ #σX ◦ π∗, given by the following composition of isomorphisms ∼ ♭ • = ♭ • ηG : π∗[σZ∗G, π I ]OZ −−−−→ [π∗σZ∗G, π∗π I ]OX

[p,1] ♭ • −−−−→ [σX∗π∗G, π∗π I ]OX =∼ [1,ξ] • −−−−→ [σX∗π∗G, I ]OX =∼ b ♭ • • 1 for G ∈ Chc(Q(Z)), where ξ : π∗π I → I is the evaluation at one . b b Proof. We need to show that for any M ∈ Chc(Q(Z)) the following diagram in Chc(Q(X)) is commutative. • • π♭I π♭I #σ #σ π∗M −−−−→ π∗(M Z Z )

 • •  • • I I π♭I I #σ #σ #σ #σ (π∗M)y X X −−−−→ (π∗M yZ ) X

1This map is called trace in [Ha66] and [Gil07a], but on aﬃne schemes this map is in fact the evaluation at one. A 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 the bullet in I• and write I instead.

σ π∗can M ♭ ♭ π∗M / π∗[σZ∗[σZ∗M, π I], π I]

σ η ♭ canπ∗M [σZ∗M,π I] [η ,I] M / ♭ [σX∗[σX∗(π∗M), I], I] / [σX∗ π∗[σZ∗M, π I], I]

We depict all the maps by deﬁnition in Figure 1 below. The diagram 1 in Figure 1 is commutative by the case of trivial duality (since all the maps considered here are natural, we drop the labeling of maps). We check the diagram 2 in Figure 1 is also commutative. This can be checked by the following two commutative diagrams.

♭ ♭ ♭ (3.1) σX∗π∗[σZ∗M, π I] / [σX∗π∗σZ∗M, σX∗π∗π I] / [π∗σZ∗σZ∗M, σX∗π∗π I]

♭ ♭ ♭ π∗σZ∗[σZ∗M, π I] / [π∗σZ∗σZ∗M, π∗σZ∗π I] / [π∗M, π∗σZ∗π I]

(3.2) σ π π♭I / σ I / I qqX8 ∗ ∗ X∗ ✄✄ qq ✄✄✄✄ qq ✄✄✄ qqq ✄✄✄ q ✄✄✄✄ ♭ π∗π σX∗I / σX∗I / I

π σ π♭I / π π♭I / I qq∗8 Z∗ ∗ ✄✄ qq ✄✄✄✄ qq ✄✄✄ qqq ✄✄✄ q ✄✄✄✄ ♭ ♭ π∗π σX∗I / π∗π I / I

Diagram (3.1) is commutative for obvious reasons. To see the commutativity of Diagram (3.2), we observe that the upper diagram is commutative by a variant of [Ha66, Proposition 6.6 p170]. The ♭ ♭ bottom diagram in Diagram (3.2) is also commutative by the deﬁnition of π I → σZ∗π I in Section 2. The left square is commutative by [Ha66, Proposition 6.3]). The front square is commutative by 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 square diagram of Diagram (3.2). Combining these two diagrams gives a new one, which by naturality and functoriality can be identiﬁed with 2. The remaining four small squares in Figure 1 are commutative by naturality.

Remark 3.4. Let π : R → S be a ﬁnite morphism of rings with involution. We have constructed a duality preserving functor

b π♭I b I π∗ : (Ch (S-Mod), #σS ) −→ (Ch (R-Mod), #σR ) with the duality compatibility natural transformation

π♭I I η : π∗ ◦ # −→ # ◦ π∗, which is induced by the isomorphism of R-modules op op op op (3.3) HomS(M S , HomR(S, I)) → HomR(M R , I): f 7→ (ξ ◦ f): m R 7→ f(m S )(1S) for M ∈ S-Mod. 10 HENG XIE ] ] I ♭ ] ] I ♭ π , I , I ∗ ] ] , π I ] I , π ♭ ♭ ] I ♭ I ♭ M, π M, π ∗ ∗ M, π ∗ Z M, π Z ∗ Z σ σ [ [ Z σ ∗ ∗ [ σ ∗ π [ Z ∗ ∗ Z σ Z ∗ σ X [ σ π σ ∗ [ [ ∗ / / π π / [ / ] ] ] I , I ] ♭ ] I I ♭ π ♭ , I ∗ ] π , π ] I ∗ , π ♭ ] I ♭ π I ♭ ∗ π ∗ π Z M, π ∗ Z ∗ Z Z σ M, σ ∗ [ M, σ π ∗ [ M, σ [[ [ π ∗ ∗ [ ∗ / π X π / [ σ / [ / ] ] ] I , I ♭ ] ] I ♭ π , I ∗ ] , π ] I M, I , π ♭ [ ] I 2 ♭ ∗ I ♭ Z σ M, π ∗ [ M, π π ∗ [ M, π [[ [ π ∗ ∗ [ ∗ / π X π / [ σ [ / ] , I ] I ♭ π 1 ∗ M, π ∗ π [[ / ] ] , I ] ] , I ] I , I ] ∗ M, I ∗ X M π ∗ ∗ M, I π X ∗ M, σ σ π [ ∗ [[ ∗ π [[ X σ [

Figure 1. Proof of Lemma 3.3 A TRANSFER MORPHISM FOR HERMITIAN K-THEORY OF SCHEMES WITH INVOLUTION 11

4. The local case and finite length modules

Let (R, m, k) be a local ring with an involution σR. Suppose that (R, σR) has a minimal dualizing • complex with involution (I , σI ). 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. σR(m)= m. Therefore, the σR on R induces an involution σk on k. In the previous section, we have deﬁned a transfer morphism i ♭ • i • Tm/R : W (k, σk, (π I , σπ♭I )) → W (R, σR, (I , σI )) which factors through i ♭ • i • Dm/R : W (k, σk, (π I , σπ♭I )) → Wm(R, σR, (I , σI )). i ♭ • i • Moreover, the map Dm/R : W (k, σk, (π I , σπ♭I )) → Wm(R, σR, (I , σI )) factors as i ♭ • i • i • W (k, σk, (π I , σπ♭I )) → W (Mfl(R), σR, (I , σI )) → Wm(R, σR, (I , σI )) b b where the ﬁrst map is induced by the functor π∗ : D (M(k)) → D (Mfl(R)) and the second map is b b b induced by the inclusion D (Mfl(R)) → Dfl(R)= Dm(R). • Theorem 4.1. Let (R, m, k) be a local ring with an involution σR. Suppose that I is a minimal • dualizing complex of R, σI is an involution of I and m is invariant under the involution of R. Then, there is an isomorphism of groups i ♭ • i • Dm/R : W (k, σk, (π I , σπ♭I )) → Wm(R, σR, (I , σI )). Lemma 4.2. Under the assumption of Theorem 4.1, the natural map i • i • W (Mfl(R), σR, (I , σI )) → Wm(R, σR, (I , σI )) is an isomorphism of groups. Proof. By [Ke99, Section 1.15] the inclusion b b b D (Mfl(R)) → Dfl(R)= Dm(R) is an equivalence of categories. Lemma 4.3. Under the assumption of Theorem 4.1, the map i ♭ • i • π∗ : W (k, σk, (π I , σπ♭I )) → W (Mfl(R), σR, (I , σI )) is an isomorphism. Proof. Let I• := (···→ 0 → Im → Im+1 →···→ In → 0 → · · · ) be a minimal dualizing complex on n ♭ R. We set E := I and let π E be the k-module HomR(k, E). Let

opk σπ♭E : HomR(k, E) → HomR(k, E) : f 7→ σπ♭E(f) ♭ be the involution on π E, where σπ♭E(f)(a) = σE fσk(a). Recall that σπ♭E is well-defined, and ♭ (π E, σπ♭E) is a duality coefficient (cf. Definition 2.2) for k-Mod (see Lemma 2.5). Therefore, it induces π♭E a category with duality (k-Mod, #σk ) (recall Remark 2.4). Moreover, the double dual identification ♭ π E opk ♭ opk ♭ ♭ canσk ,M : M → Homk(Homk(M , π E) , π E): m 7→ (can(m): f 7→ σπ E(f(m))) is an isomorphism, therefore the duality is strong in the sense of [Sch10b]. Now, consider the following commutative diagram ∼ i+n ♭ = i+n W k, σk, (π E, σ ♭ ) / W (Mfl(R), σR, (E, σE )) π E h v1 ∼= v2 ∼= i ♭ • i • ♭ / W (k, σk, (π I , σπ I )) π∗ W (Mfl(R), σR, (I , σI )) . 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 12 HENG XIE

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 or skew-symmetric forms. If i + n is odd by [BW02, Proposition 5.2] one has

i+n ♭ i+n W k, σk, (π E, σπ♭E) = W (Mfl(R), σR, (E, σE ))=0. The map v1 is induced by the duality preserving functor • b π♭E[n] b π♭I (id,θ) : (D k, #σk ) → (D k, #σk ),

♭ • π E[n] π♭I where θ : #σk → #σk is the natural isomorphism given by the isomorphism of complexes op op • θM : Homk(M k , HomR(k, E[n])) → Homk(M k , HomR(k, I )) • which is induced by the inclusion of complexes E[n] → I . Notice that θM is an isomorphism, because as an R-module op j op j Homk(M k , HomR(k, I )) =∼ HomR(M R , I )=0 if j 6= n. The ﬁrst isomorphism follows by the isomorphism η in (3.3) and the second identity can be proved by [Gil02, Lemma 3.3]. Indeed, the local ring R is assumed to be Gorenstein in [Gil02, Lemma 3.3 (1)-(4)]. We need to explain why [Gil02, Lemma 3.3] can be applied to our situation that the local ring R only assumed to have a minimal dualizing complex. Note that there is an isomorphism j ∼ I = ⊕µI (Q)=j ER(R/Q) by [Gil07a, Theorem 1.15] where µI : Spec R → Z is the codimension function 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 µI (Q)= n. For any R-module M such that mM = 0, we conclude that HomR(M, ER(R/Q)) = 0 if Q ( m (i.e. µI (Q)

Remark 4.4. For regular local rings Lemma 4.3 was proved by [Gil09, Theorem 4.5]. The situation in [Gil07a, Section 3.1] is very similar to Lemma 4.3, but in the construction of the transfer map we avoid the choice of an embedding of k into its injective hull E = ER(k) over R.

1 • Lemma 4.5. Let (R, m, k) be a local ring with involution and 2 ∈ R. Assume that I is a minimal • dualizing complex of R, σI 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 π / R ④/= k ④④ p ④④ ④④ q ④ R/J . We have a commutative diagram

i ♭ ♭ • q∗ / i ♭ • (4.1) W (k, σk, (q p I , σq♭p♭I )) Wm/J (R/J, σR/J , (p I , σp♭I ))

=∼ p∗ i ♭ • π∗ / i • W (k, σk, (π I , σπ♭I )) / Wm(R, σR, (I , σI ))

Therefore, the map p∗ is an isomorphism Proof. Deﬁne a map γ : q♭p♭I → π♭I

γ : HomR/J (k, HomR(R/J, I)) → HomR(k, I) by sending f : k → HomR(R/J, I) to γ(f): a 7→ 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 A TRANSFER MORPHISM FOR HERMITIAN K-THEORY OF SCHEMES WITH INVOLUTION 13

commutative diagram of triangulated categories with duality

♭ ♭ q∗ ♭ (Dbk, #q p I ) / (Db (R/J), #p I ) σk m/J σR/J

p∗ b π♭I π∗ / b I (D k, #σk ) (DmR, #σR ) , where the left vertical arrow is induced by the isomorphism γ : q♭p♭I → π♭I.

5. The devissage´ theorem for a closed immersion 1 Let (X, σX ) and (Z, σZ ) be schemes with involution and with 2 ∈ OX . Suppose that (X, σX ) has a minimal dualizing complex with involution (I, σI ) (cf. Deﬁnition 2.13). Assume that π : Z ֒→ X b is a closed immersion which is invariant under involutions. Since π∗G ∈ Chc,Z (Q(X)) for all G ∈ b Chc(Q(Z)), we have a map [i] ♭ • [i] • DZ/X : GW (Z, σZ , (π I , σπ♭I )) → GWZ (X, σX , (I , σI )).

In fact, the transfer morphism TZ/X is the composition

[i] ♭ • DZ/X [i] • [i] • GW (Z, σZ , (π I , σπ♭I )) −→ GWZ (X, σX , (I , σI )) → GW (X, σX , (I , σI )), where the second map is extending the support. 1 Theorem 5.1. Let (X, σX ) and (Z, σZ ) be schemes with involution and with 2 ∈ OX . Suppose that X, σX ) has a a minimal dualizing complex with involution (I, σI ). If π : Z ֒→ X is a closed immersion) b b which is invariant under involutions, then the direct image functor π∗ : Chc(Q(Z)) → Chc(Q(X)) induces an equivalence of spectra [i] ♭ • [i] • DZ/X : GW (Z, σZ , (π I , σπ♭I )) → GWZ (X, σX , (I , σI )). Proof. The result follows by Theorem 5.2 and Karoubi induction [Sch17, Lemma 6.4]. Theorem 5.2. Under the hypothesis of Theorem 5.1, there is an isomorphism of groups i ♭ • i • DZ/X : W (Z, σZ , (π I , σπ♭I )) → WZ (X, σX , (I , σI )). Proof. We borrow the idea of [BW02], [Gil02], [Gil07a] to use the filtration of derived category by codimension of points to reduce the problem to the local case. All we need to check is that this approach is compatible with the duality given by a non-trivial involution. p (p) Let XI := {x ∈ X|m ≤ µI(x) ≤ p − 1} and XI := {x ∈ X|µI(x)= p}. We define p • b • p DZ,X := M ∈ Dc,Z (Q(X))|(M )x is acyclic for all x ∈ XI n o • b I p considered as a full subcategory of Dc,Z (Q(X)). Note that the duality functor #σX maps DZ,X • p I p into itself. Therefore, (DZ,X , #σX ) is a triangulated category with duality. The subcategories DZ,X provide a finite filtration b m m+1 p n Dc,Z (Q(X)) = DZ,X ⊇ DZ,X ⊇···⊇DZ,X ⊇···⊇DZ,X ⊇ (0) which induces exact sequences of triangulated categories with duality p+1 p p p+1 DZ,X −→ DZ,X −→ DZ,X /DZ,X . On the other hand, we define

p • b • p DZ := M ∈ Dc(Q(Z))|(M )z is acyclic for all z ∈ Zπ♭I n o b as a full subcategory of Dc(Q(Z)). By the same reason above, we have a ﬁnite ﬁltration b m m+1 p n Dc(Q(Z)) = DZ ⊇ DZ ⊇···⊇DZ ⊇···⊇DZ ⊇ (0) 14 HENG XIE

which induces exact sequences of triangulated categories with duality p+1 p p p+1 DZ −→ DZ −→ DZ /DZ . p p Since µI (z) = µπ♭I (z) for all z ∈ Z, we have π∗(DZ ) ⊂ DZ,X . It follows that we obtain a map of exact sequences of triangulated categories with duality • • • p+1 π♭I p π♭I p p+1 π♭I (DZ , #σZ ) −−−−→ (DZ , #σZ ) −−−−→ (DZ,X /DZ,X , #σZ )

π∗ π∗ π∗

 •  •  • p+1 I p  I p p+1 I (DZ,Xy, #σX ) −−−−→ (DZ,Xy, #σX ) −−−−→ (DZ,X /DyZ,X , #σX ) which induces a map of long exact sequences of groups • • • ··· −→ i Dp π♭I −→ i Dp Dp+1 π♭I −→ i+1 Dp+1 π♭I −→ · · · W ( Z , #σZ ) W ( Z / Z , #σZ ) W ( Z , #σZ )    (5.1) π∗ π∗ π∗ y y y • • • ··· −→ i Dp I −→ i Dp Dp+1 I −→ i+1 Dp+1 I −→ · · · W ( Z,X , #σX ) W ( Z,X / Z,X , #σX ) W ( Z,X , #σX ) . Lemma 5.3. The localization functors p p+1 b loc : DZ,X /DZ,X → DmX,x (OX,x) Y (p) x∈Z∩XI and p p+1 b loc : DZ /DZ → DmZ,z (OZ,z) z∈YZ(p) π♭I induce equivalences of categories. Proof. This result is well-known in the literature, see [Gil07b, Theorem 5.2] for instance.

• • i p p+1 π♭I i p p+1 I Note that the morphism π∗ : W (DZ /DZ , #σZ ) → W (DZ,X /DZ,X , #σX ) is an isomorphism. This can be concluded by the commutative diagram

• i p p+1 π♭I loc i ♭ W (D /D , # ) −−−−→ (p) W (O , σ , (π I , σ ♭ )) Z Z σZ z=σ(z)∈Z mZ,z Z,z OZ,z z π Iz =∼ ♭I L π ∗ ∼ ⊕ (DO O ) π = z Z,z / X,z

 •  i p  p+1 I loc i  W (DZ,X /yDZ,X , #σ ) −−−−→ (p) Wm y(OX,z, σOX,z , (Iz, σIz )), X =∼ z=σ(z)∈Z∩XI X,z L where the right morphism is an isomorphism by Lemma 4.5. Note that the Witt groups on the right hand side do not contain the component σ(z) 6= z. To explain, we write down a duality preserving functor • p p+1 π♭I b b (loc, η) : (DZ /DZ #σZ ) → DmZ,z (OZ,z) × Dm (OZ,σ(z)), # , Z,σ(z) (π♭I) (π♭I) σ(z) # z #σ where # : (M,N) 7→ (N σz ,M σ(z) ) and the duality compatibility isomorphism

♭ ♭ (π I)σ(z) (π♭I) (π♭I) (π I)z # # # #σ σσ(z) σZ σZ z ηM : ((M )z, (M )σ(z)) −→ (Mσ(z) ,Mz ) is deﬁned to be the canonical isomorphism induced by restrictions of scalars via the local isomorphisms −1 of local rings σ : OZ,z → OZ,σ(z) and σ : OZ,σ(z) → OZ,z. By these local isomorphisms, note that Db (O ) is equivalent to Db (O ), and mZ,z Z,z mZ,σ(z) Z,σ(z) i b b W (DmZ,z (OZ,z) × DmZ,σ(z) (OZ,σ(z)), #)=0 by a version of the additivity theorem (see [Sch17, Proposition 6.8], especially the proof). Finally, by induction on p in Diagram (5.1), we conclude the result. A TRANSFER MORPHISM FOR HERMITIAN K-THEORY OF SCHEMES WITH INVOLUTION 15

6. A geometrical computation of the devissage´ for regular schemes

Let (X, σX ) be a regular scheme with involution. Assume (Z, σZ ) is a regular scheme with involution and assume further that Z is regularly embedded in X of codimension d. We have a normal bundle N := NZ/X of X on Z which is locally free of rank d. If Z is invariant under σX , then we can deﬁne an involution σN : N → σ∗N induced from σX : X → X by the invariant ideal sheaf of Z in X. The involution σN induces an involution det(σN ) : det(N) → σ∗ det(N) by taking determinants. Recall that the canonical sheaf is deﬁned by

ωZ/X := HomOZ (det(N), OZ ).

Now, det(σN ) induces an involution

σωZ/X : ωZ/X → σ∗ωZ/X : f 7→ σZ ◦ (σ∗f) ◦ det(σN ).

Theorem 6.1 (D´evissage). Let (X, σX ) be a regular scheme with involution. Let (L, σL) be a dualizing coeﬃcient with L a locally free OX -module of rank one. If Z is a regular scheme regularly embedded in X of codimension d which is invariant under σ, then there is an equivalence of spectra

[i−d] [i] DZ/X,L : GW (Z, σZ , (ωZ/X ⊗OX L, σωZ/X ⊗ σL)) −→ GWZ (X, σX , (L, σL)).

• Proof. By Lemma 2.15 there exists (L, σL) → (I , σI ), a minimal injective resolution compatible with • 0 1 n σX , where I = I → I →···→I . We have the d´evissage isomorphism on the level of coherent Grothendieck-Witt groups

i ♭ • i • DZ/X : GW (Z, σZ , (π I , σπ♭I )) → GWZ (X, σX , (I , σI )).

≃ ♭ • We want to construct a quasi-isomorphism β : ωZ/X ⊗OX L[−d] −→ π I which is compatible with involutions, then we can use a variant of Lemma 2.10 to conclude the result. We will see that this map stems from the fundamental local isomorphism

i • i 0 if i 6= d H HomOX (OZ , I )= ExtOX (OZ , L)= ωZ/X ⊗OX L if i = d

(see [Ha66, p 179]). Before checking the compatibility of the involutions, we review the fundamental local isomorphism. We take an open affine subscheme U =∼ Spec(R) of X, and ZU := Z ×X U =∼ Spec(R/J) with J defined by a regular sequence (x1, ··· , xd) of length d. Let L := LU and I := IU . Let E = ⊕1≤i≤dRei be a free R-module with basis {e1, ··· ,ed}. Let s : E → R : (yiei)1≤i≤d 7→ d i=1 yixi. There exists a Koszul resolution of R/J P d d−1 0 ···→ 0 → E → E →···→ E → R/J ^ ^ ^ with differentials given by

i i i−1 i t+1 d : E → E : α1 ∧···∧ αi 7→ (−1) s(αt)α1 ∧···∧ αt ∧···∧ αi. ^ ^ Xt=1 b Since (x1, ··· , xd) is a regular sequence, the Koszul complex provides a projective resolution of R/J. −i i Set P = Pi = E. The fundamentalV local isomorphism

i 2 ExtR(R/J,L) → HomR(det(J/J ),L) 16 HENG XIE

of the R/J-modules (see [Ha66, Chapter III.7 p 176]) is deﬁned as follows. We draw the following diagram of R-modules

2 HomR(det(J/J ),L/JL) β˜ x 0  d HomR(P ,L) −→ · · · −→ HomR(P ,L)

  0  0 0  d 0 HomR(R/J, I ) −→ HomRy(P , I ) −→ · · · −→ HomR(yP , I )

    1  0 1  d 1 HomR(yR/J, I ) −→ HomR(yP , I ) −→ · · · −→ HomR(yP , I )

    2  0 2  d 2 HomR(yR/J, I ) −→ HomR(yP , I ) −→ · · · −→ HomR(yP , I )

   . . . . y. y. . y.

    n  0 n  d n HomR(yR/J, I ) −→ HomR(yP , I ) −→ · · · −→ HomR(yP , I ) Note that by this diagram, we have zigzags • • • • HomR(R/J, I ) −→ Tot HomR(P , I ) ←− HomR(P ,L) of quasi-isomorphisms of R-modules. Now, we deduce i i • ∼ i • ExtR(R/J,L) := H (HomR(R/J, I )) = H HomR(P ,L) and we deﬁne a map of R-modules 2 β˜ : HomR(det E,L) → HomR(det(J/J ),L/JL): f 7→ β˜(f), where β˜(f)(¯x1 ∧···∧ x¯d) := f(x1 ∧···∧ xd). One checks that the composition

d−1 d β 2 HomR(P ,L) → HomR(P ,L) → HomR(det(J/J ),L/JL) is zero. Thus, we get a map of R-modules d • 2 H Hom(P ,L) → HomR(det(J/J ),L/JL) which can be considered as a map of R/J-modules, and it is an isomorphism. This morphism extends to a global morphism as it does not depend on the choice of the regular sequence (cf. [Ha66, Chapter III.7 Lemma 7.1]). Now, we check that β is compatible with the involution. This can be checked affine locally. We take any open affine subscheme U =∼ Spec(R) of X, which need not to be compatible with the involution. ′ It follows that the subscheme U := σX (U) is also affine and can be assumed to be isomorphic to ′ ′ Spec(R ) for some ring R . Then the involution σX restricts to a pair of isomorphisms of affine ′ ′ ′ ′ −1 schemes σU : U → U and σU : U → U (with σU = σU ) which induces a pair of isomorphisms of ′ ′ ′ ′ −1 ∼ rings σR : R → R and σR : R → R (with σR = σR ). Recall that ZU = Spec(R/J), where J is ′ an ideal of R defined by a regular sequence (x1,...,xd) of length d. Consider the restriction of U ′ ′ ′ ′ ′ into the closed subscheme ZU which is isomorphic to Spec(R /J ) for J an ideal inside R . Since by A 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 / R / R/J

σJ σR σR/J J ′ / R′ / R′/J ′

′ ′ ′ ′ σJ σR σR /J J / R / R/J,

−1 −1 ′ ′ −1 where the vertical arrows are all isomorphisms and σJ = σJ′ , σR = σR′ , σR /J = σR/J . Therefore, ′ ′ the sequence (σR(x1), ··· , σR(xd)) is a regular sequence in R deﬁning J . ′ ′ ′ ′ ′ ′ ′ As above, let L := LU and I := IU . Let E = ⊕1≤i≤dRei be a free R -module with basis ′ ′ ′• ′ ′ ′ ′ {e1, ··· ,ed}. Similarly, we have a Koszul complex P → R /J associated to the section s : E → ′ d ′−i i ′ R : (yiei)1≤i≤d 7→ i=1 yiσ(xi) with P := Λ E . We now consider the following isomorphism of R-modules P ′ σE : E → σR∗E : (yiei)1≤i≤d 7→ (σR(yi)ei)1≤i≤d It induces isomorphisms of R-modules i i i ′ σE : E → σR∗( E ) ^ ^ ^ on the exterior powers i E. It is straightforward to check that the diagram V s E −−−−→ R

σE σR

 ′   ′ s  ′ σRy∗E −−−−→ σRy∗R • ′• of R-modules commutes. By checking the diﬀerentials, we get an isomorphism σP : P → σR∗P of ′• • complexes of R-modules with inverse σP ′ : P → σR′ ∗P . Moreover, we can verify that the diagram

d β 2 HomR(P ,L) / HomR(det(J/J ),L/JL)

ǫP d,L ǫdet(J/J2),L/JL ′ ′d ′ β ′ ′2 ′ ′ ′ HomR′ (P ,L ) / HomR′ (det(J /J ),L /J L ) is also commutative. Using the same notations as in Lemma 2.5, we form the following commutative diagram

• • • • 2 HomR(R/J, I ) / Tot HomR(P , I ) o HomR(P ,L) / HomR(det(J/J ),L/JL)[−d] ǫR/J,I ǫP,I ǫP,L ǫdet(J/J2),L/JL ′ ′ ′• ′• ′• ′• ′ ′ ′2 ′ ′ ′ HomR′ (R /J , I ) / Tot HomR′ (P , I ) o HomR′ (P ,L ) / HomR′ (det(J /J ),L /J L )[−d] of complexes of R-modules. Since the horizontal maps are all quasi isomorphisms and they induce maps on homology, we get ∼ d • = 2 H HomR(R/J, I ) −−−−→ HomR(det(J/J ),L/JL)

ǫR/J,I ǫdet(J/J2),L/JL

 ∼  d  ′ ′ ′• = ′ ′2 ′ ′ ′ H HomR′y(R /J , I ) −−−−→ HomR′ (det(J y/J ),L /J L ) which is the desired commutative diagram of R/J-modules. 18 HENG XIE

d ♭ • ♭ • b Finally, we still need to ﬁnd a map of complexes H π I [−d] → π I in Chc(Q(Z)) which is compatible with the involution. At this stage, we only have a right roof Hdπ♭I•[−d] →s C• ←t π♭I• b • ♭ • compatible with the involution in Dc(Q(Z)), where C is the canonical truncation τ≤dπ I . By this we mean π♭Id/Im(π♭Id−1 → π♭Id) if i = d Ci = π♭Ii if d

′ d σ σ • :=H σ • π♭I π♭I π♭I σ ∗(rs) d ♭ • Z ♭ • σZ∗H π I [−d] / σZ∗π I

b b is commutative in Chc(Q(Z)), while at this stage it is only known that it commutes in Dc(Q(Z)). By [Wei94, Corollary 10.4.7] and the fact that π♭(I•) is injective, the diagram also commutes in b ′ Kc (Q(Z)). Let us deﬁne f := σπ♭I ◦ (rs) − σZ∗(rs) ◦ σπ♭I• , which is a map of complexes from d ♭ • ♭ • b H π I [−d] to σZ∗π I . Then the commutativity in Kc (Q(Z)) implies that f is null homotopic. Since the complex Hdπ♭I•[−d] is concentrated in degree d, we see that the morphism of complexes f is also concentrated in degree d. By the deﬁnition of null homotopy, there exists a map d ♭ • ♭ d−1 π♭I of OZ -modules u : H π I → σZ∗π I such that u ◦ d = f. We now want to conclude that d ♭ ♭ d−1 ∼ d ♭ ♭ d−1 HomOZ (H π I, π I ) = HomOZ (H π I, σZ∗π I ) vanishes. This can be checked locally, and d−1 by using that ht(J) = d and HomR/J (M, HomR(R/J, I )) = 0. This last fact is proved in [Gil02, Lemma 3.3 (5)]. Therefore, one has u = f = 0 . Corollary 6.2. Under the hypothesis of Theorem 6.1, the direct image map i−d i W (Z, σZ , (ωZ/X ⊗OX L, σωZ/X ⊗ σLZ )) −→ WZ (X, σX , (L, σL)) induces an isomorphism of groups.

7. Some computations 1 1 Let S be a regular scheme with 2 ∈ OS . Let PS be the projective line Proj(OS[X, Y ]).

1 1 1 7.1. P with switching involution. Let τP1 : P → P :[X : Y ] 7→ [Y : X] be the involution which S S S 1 switches the coordinates of PS. 1 Theorem 7.1. Let S be a regular scheme with 2 ∈ OS . Then, we have an equivalence of spectra [i] 1 [i] [i+1] GW (P , τP1 ) =∼ GW (S) ⊕ GW (S) S S In particular, we have the following isomorphism on Witt groups i 1 i i+1 W (P , τP1 ) =∼ W (S) ⊕ W (S) S S − 1 1 Lemma 7.2. Let σA1 : AS → AS be the involution given by t 7→ −t. Then, the pullback S ∗ [i] [i] 1 − p : GW (S) −→ GW (AS, σA1 ) S is an isomorphism. A TRANSFER MORPHISM FOR HERMITIAN K-THEORY OF SCHEMES WITH INVOLUTION 19

1 Proof. Let pt : S → AS be the rational point corresponding to 0. Then, we have the following commutative diagram pt 1 S AS id ∗ p S which induces ∗ [i] p [i] 1 − GW (S) GW (AS, σA1 ) S

id ∗ pt GW [i](S) .

∗ [i] [i] 1 − This implies that the pullback p : GW (S) −→ GW (AS , σA1 ) is split injective. Now, consider the S composition ∗ ∗ [i] 1 − pt [i] p [i] 1 − GW (AS , σA1 ) −→ GW (S) −→ GW (AS, σA1 ). S S We want to show that it is an isomorphism. We use the following C2-equivariant homotopy A1 × A1 A1 (t′,t) t′t

A1 × A1 A1 (t′, −t) −t′t. This diagram tells us that 1 − p pt 1 − (AS , σA1 ) −→ (S, id) −→ (AS, σA1 ) S S 1 − 1 − is homotopic to id : (AS , σA1 ) → (AS , σA1 ). The result follows by Theorem 7.5 below. S S 0 1 0 Let P[1:1] ֒→ PS be the closed point cut out by the homogeneous ideal (X − Y ). Then, P[1:1] is 1 invariant under the involution τP1 on P . S S 1 Lemma 7.3. Let S be a regular scheme with 2 ∈ OS. The d´evissage theorem provides an isomorphism [i+1] =∼ [i] 1 DP0/P1 : GW (S) −→ GWP0 (PS, τP1 ). [1:1] S Proof. By Theorem 6.1, we conclude that

[i−1] 0 =∼ [i] 1 GW (P[1:1], id, (ωP0 /P1 , σω)) −→ GWP0 (PS, τP1 ). [1:1] S [1:1] S 0 1 Let I = (X − Y ) be the homogeneous ideal deﬁning P = S in P . Then, the involution τP1 : [1:1] S S 1 1 PS → PS induces an involution on I which sends X − Y to Y − X, and therefore induces a map 2 2 2 σI/I2 : I/I → I/I : X − Y 7→ Y − X of OS-module. The map µ : I/I → OS deﬁned by sending X − Y to 1 is an isomorphism of OS-modules. Now, we have a commutative diagram

2 µ I/I / OS

σI/I2 −id 2 µ I/I / OS which implies (ωP0 P1 , σω) =∼ (OS, −id) and hence [1:1]/ S

[i−1] 0 =∼ [i−1] ∼= [i−1] GW (P , id, (ωP0 P1 , σω)) −→ GW (S, id, (OS, −id)) −→ GW (S, −can). [1:1] [1:1]/ S Also, GW [i−1](S, −can) =∼ GW [i+1](S). 20 HENG XIE

Proof of Theorem 7.1. By [Sch17], we have the following homotopy ﬁbration sequence

[i] 1 [i] 1 [i] 1 0 GWP0 (PS, τP1 ) −→ GW (PS, τP1 ) −→ GW (PS − P[1:1], τP1 −P0 ). [1:1] S S S [1:1] Note that there is an invariant isomorphism 1 − 1 0 (AS , σA1 ) → (PS − P[1:1], τP1 −P0 ) S S [1:1] 1 1 deﬁned by sending t to [t + 2 : t − 2 ]. Therefore, we get a homotopy ﬁbration sequence

[i] 1 [i] 1 [i] 1 − GWP0 (PS, τP1 ) GW (PS , τP1 ) GW (AS, σA1 ) [1:1] S S S

∗ ∗ =∼ p q GW [i](S) .

∗ [i] [i] 1 − By Lemma 7.2, we have the isomorphism p : GW (S) → GW (AS, σA1 ) which provides a splitting S for the above homotopy ﬁbration sequence. By combining it with Lemma 7.3, we conclude that [i] 1 ∼ [i] 1 [i] ∼ [i+1] [i] GW (PS , τP1 ) = GWP0 (PS, τP1 ) ⊕ GW (S) = GW (S) ⊕ GW (S) . S [1:1] S The result now follows.

1 1 1 7.2. C2-equivariant A -invariance. Consider the involution σP1 : P → P given by the graded S S S morphism OS [X, Y ] → OS [X, Y ] of graded sheaves of OS-algebras, such that a 7→ σS(a) if a ∈ OS, and X 7→ X, Y 7→ Y . There is an element X OP1 (−1) −−−−→ OP1   [1] 1 β := Y Y ∈ GW (P , σP1 ) . 0 S S       X    OP1 −−−−→ OP1 (1)  y y  1 Proposition 7.4. In this result, the base S can be singular, and still 2 ∈ OS. The map of spectra ∗ ∗ [i] [i−1] [i] 1 (q ,β ∪ q (−)) : GW (S, σS ) ⊕ GW (S, σS ) → GW (P , σP1 ) S S is an equivalence. Proof. The proof of [Sch17, Theorem 9.10] can be applied without modiﬁcation.

1 1 1 Theorem 7.5. Let S be a regular scheme with involution σS and with ∈ OS. Let σA1 : A → A 2 S S S 1 be the involution on A with the indeterminant ﬁxed by σA1 and such that the following diagram S S commutes σA1 1 S 1 AS −−−−→ AS

p p    σS  yS −−−−→ S.y Then, the pullback ∗ [i] [i] 1 p : GW (S, σS) → GW (A , σA1 ) S S is an isomorphism. 1 Proof. Let pt : S → PS be the rational point with X = 0 and Y = 1. It follows that there is a commutative diagram σ S −−−−→S S

pt pt    σP1  y1 S y1 PS −−−−→ PS . A TRANSFER MORPHISM FOR HERMITIAN K-THEORY OF SCHEMES WITH INVOLUTION 21

By Schlichting [Sch17, Theorem 6.6], we have the following localization sequence [i] 1 [i] 1 [i] 1 GW (P , σP1 ) −→ GW (P , σP1 ) −→ GW (A , σA1 ) pt S S S S S S We write out the following commutative diagram

0 1 1 0 [i−1]   [i] [i−1] [i] GW (S, σS ) GW (S, σS ) ⊕ GW (S, σS ) GW (S, σS )

∗ ∗ ∗ Dpt/P1 (7.1) S q β∪q (−) p

[i] 1 [i] 1 [i] 1 GW (P , σP1 ) GW (P , σP1 ) GW (A , σA1 ) . pt S S S S S S The main reason for the commutativity of the left square is that we have the locally free resolution X 0 −→ OP1 (−1) −→ OP1 −→ Opt −→ 0.

By d´evissage, we conclude that D P1 is an equivalence. By Proposition 7.4, we know that the pt/ S middle map q∗ β ∪ q∗(−) is an equivalence. It follows that the right arrow ∗ [i] [i] 1 p : GW (S, σS) → GW (A , σA1 ) S S is also an equivalence. 7.3. Representability. It is easy to see that schemes with involution can be identiﬁed with schemes with a C2-action where C2 is the cyclic group of order two considered as an algebraic group. Let C2 H• (S) be the C2-equivariant motivic homotopy category of Heller, Krishna and Østvær [HKO14]. [i] op Consider the presheaf of simplicial sets GW : SmS → sSet by a big vector bundle argument cf. [i] C2 [Sch17, Remark 9.2]. The presheaf GW is an object in H• (S). We prove that the following representability result.

C2 Theorem 7.6. Let (X, σ) ∈ SmS . Then, there is a bijection of sets n [i] [i] [S ∧ (X, σ) , GW ] C2 = GW (X, σ). + H• (S) n 1 Proof. By [HKO14, Corollary 4.9], we need to prove the C2-equivariant Nisnevich excision and A - [i] invariance for Hermitian K-theory. Note that the C2-equivariant Nisnevich excision for GW can be proved by a variant of [Sch17, Theorem 9.6] adapted to schemes with involution since the argument is independent of the duality. Moreover, A1-invariance is proved in Theorem 7.5. Acknowledgements. I want to thank Marco Schlichting for useful discussion. I appreciate the referee for pointing out a gap in an earlier version. I thank Thomas Hudson for the proofreading of the paper. I would like to acknowledge support from the EPSRC Grant EP/M001113/1, the DFG priority programme 1786 and the GRK2240. Part of this work was carried out when I was visiting Hausdorﬀ Research Institute for Mathematics in Bonn and Max-Planck-Institut in Bonn. I would like to 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 punctured affine 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 Mathematics 343, Springer, 1973. [BW02] P. Balmer and C. Walter, A Gersten-Witt spectral sequence for regular schemes, Annales Scientifiques de l’ENS´ 35 (2002) 127-152. 22 HENG XIE

[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 dévissage 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. Pure Appl. Algebra 208 (2007) 391-419. [Gil09] S. Gille, A Gersten-Witt complex for hermitian Witt groups of coherent algebras over schemes II: Involution of the 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 motivic homotopy 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 of Topology 8 (2015) 1202-1236. [Kar80] M. Karoubi, Le théorème fondamental de la K-théorie 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, Birkhäuser, 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, American Mathematical 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 abelian categories, J. Algebra 59 (1979) 264-289. [Ran98] A. Ranicki, High-dimensional Knot Theory: Algebraic Surgery in Codimension 2, Springer Monographs in Mathematics, 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 Universitat¨ Wuppertal, Gaußstraße 20, 42119 Wuppertal, Germany E-mail address: [email protected]