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

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A Transfer Morphism for Hermitian K-Theory of Schemes with Involution 3 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 which we construct a transfer morphism and prove a version of the d´evissage theorem. This theorem is then used to compute the Hermitian K-theory of P1 with involution given by [X : Y ] 7→ [Y : X]. We also 1 prove the C2-equivariant A -invariance of Hermitian K-theory, which confirms the representability of Hermitian K-theory in the C2-equivariant motivic homotopy category of Heller, Krishna and Østvær [HKO14]. 1. Introduction In the 1970s, researchers found that to understand quadratic forms, it was helpful to study Witt groups of function fields of algebraic varieties, which has led to substantial progress in quadratic form theory (see [Scha85], [Lam05], [KS80] and [Knu91]). However, for many function fields, Witt groups are very difficult to understand. Instead of computing the Witt groups of function fields of algebraic varieties, Knebusch [Kne77] proposed to study the Witt groups of the algebraic varieties themselves, 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 schemes with involution, which would not only enlarge the theory of trivial involution but also provides new insights into Topology. For instance, the L-theory of the Laurent polynomial ring k[T,T −1] with the non-trivial involution T 7→ T −1 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 Witt groups, cf. [Sch17, Proposition 6.3]. It is also worth mentioning that Hermitian K-theory has been successfully applied to solve several problems in the classification of vector bundles and the theory of Euler classes (cf. [AF14a], [AF14b] and [FS09]). In light of this, I decided to develop the current paper 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 1 ∼ σ is complex conjugation (here we consider Spec(C) over Spec(Z[ 2 ])). We have W (Spec(C), σ) = Z by taking the rank of positive definite diagonal Hermitian forms over C (cf. [Knu91, I.10.5]). This computation is different from W (Spec(C)) =∼ Z/2Z in which case every rank two quadratic form is arXiv:1809.03198v2 [math.KT] 14 Oct 2019 hyperbolic. On the one hand, it is known that the topological Hermitian K-theory of spaces with involution is equivalent to Aityah’s KR-theory ([At66]). On the other hand, the Hermitian K-theory of schemes with involution is a natural lifting of the topological Hermitian K-theory to the algebraic world. In light 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’s transfer morphism on Grothendieck-Witt groups [Sch17, Theorem 9.19] to non-trivial involution. More precisely, in Theorem 3.1 we prove the following: 1 Theorem 1.1. 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 → MSC: PRIMARY: 19G38; SECONDARY: 19G12 1 2 HENG XIE 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 d´evissage 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 d´evissage theorem (cf. Theorem 6.1). Theorem 1.3 (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)). 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 d´evissage 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 d´evissage 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 d´evissage 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 −−−−→ yS 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.
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