The Projective Bundle Formula for Grothendieck-Witt spectra
aa r X i v : . [ m a t h . K T ] A p r the projective bundle formula forgrothendieck-witt spectra herman rohrbach ∗ Abstract
Grothendieck-Witt spectra represent higher Grothendieck-Witt groups andhigher Hermitian K-theory in particular. A description of the Grothendieck-Witt spectrum of a finite dimensional projective bundle P ( E ) over a base scheme X is given in terms of the Grothendieck-Witt spectra of the base, using thedg category of strictly perfect complexes, provided that X is a scheme overSpec Z [1 /
2] and satisfies the resolution property, e.g. if X has an ample familyof line bundles. Orientable cohomology theories E satisfy a straightforward projective bundle for-mula of the form E ( P ( E )) ≃ r M i =0 E ( X ) α i , where P ( E ) is a projective bundle of dimension r over a base scheme X and α issome specific class with a relation α r +1 + a r α r + . . . + a α + a = 0which determines the multiplicative structure, see e.g. [11] and [2, section 3.5]. Wittgroups and Grothendieck-Witt groups parametrize symmetric bilinear forms and areamong the first examples of non-orientable cohomology theories. The Grothendieck-Witt spectrum defined in [12] represents higher Grothendieck-Witt groups, and theunshifted Grothendieck-Witt spectrum GW [0] represents Hermitian K -theory. Thefirst result in the direction of a projective bundle formula is found in [3], where it isproved that W [0] ( P rk ) ∼ = W [0] ( k ) , for k some field. This result already shows that one cannot hope for the projectivebundle formula for oriented cohomology theories to hold in the case of Witt groupsand Grothendieck-Witt groups, even for trivial projective bundles over a point, andsomething more sophisticated is needed. The triangular methods for Witt groups, ∗ The author is supported by the research training group
GRK 2240: Algebro-Geometric Methodsin Algebra, Arithmetic and Topology . he projective bundle formula for grothendieck-witt spectra developed in [4] and [6], and extended to Grothendieck-Witt groups in [16], haveproven to be a fruitful approach to their development as fully fledged cohomologytheories. For example, it is shown in [9] that Hermitian K -theory and Witt groupsare A -representable.In [15], projective bundle formulas are proved for the Grothendieck-Witt groupGW [ n ]0 ( P ( E ) , π ∗ L ⊗ O P ( E ) ( m )) , where E is a locally free sheaf of rank r + 1 on a noetherian scheme X , L is a linebundle on X , m, n ∈ Z , and π : P ( E ) → X is the structure map of the projectivebundle. Subsequently, it is claimed in [12, remark 9.11] that these results immedi-ately generalize to Grothendieck-Witt spectra, but a rigorous proof is only providedfor the case of P X . This paper provides a detailed proof for the general case of aprojective bundle over a general base scheme X , using the techniques developed in[12], thus filling a small hiatus in the literature. The proof contained in this paperuses a combination of the techniques of [15], [12] and [7]. The present results, in theeasier case of trivial projective bundles, have recently been used in the proof of [10,theorem 5.1], which contains a computation of the Grothendieck-Witt spectrum ofa punctured affine space over X .The following two theorems are the main theorems of this paper. Theorem Ais easier to prove than theorem B, due to the felicitous parities of m and r . In thestatements of these theorems, X is a scheme over Spec Z [1 /
2] satisfying the resolutionproperty, E is a locally free O X -module of rank r + 1, s = ⌈ r/ ⌉ , P = P ( E ) is theprojective bundle with projection map π : P → X , L is an invertible O X -module, m, n ∈ Z and π ∗ L ( m ) = O P ( m ) ⊗ π ∗ L is the m -th twist of π ∗ L . Theorem A.
The following statements hold.(i) If m and r are even, then there is a stable equivalence of spectra GW [ n ] ( X, L ) ⊕ K( X ) ⊕ s GW [ n ] ( P , π ∗ L ( m )) . (ii) If m and r are odd, then there is a stable equivalence of spectra K( X ) ⊕ s GW [ n ] ( P , π ∗ L ( m )) . Theorem B.
The following statements hold.(i) If r is even and m is odd, then there is a split homotopy fibration K ( X ) ⊕ s − GW [ n ] ( P , π ∗ L ( m )) GW [ n − r ] ( X, det E ∨ ⊗ L ) . (ii) If r is odd and m is even, then there is a homotopy fibration GW [ n ] ( X, L ) ⊕ K ( X ) ⊕ s − GW [ n ] ( P , π ∗ L ( m )) GW [ n − r ] ( X, det E ∨ ⊗ L ) , which splits if the image in W [ r +1]0 ( P , det π ∗ E ( − r − of the symmetric form ν of proposition 2.2.4 vanishes. The condition for the splitting is satisfied e.g.if E is a trivial bundle. erman rohrbach The present paper is organized as follows. Section 2.1 contains background onthe dg category Perf( X ) of perfect complexes on a scheme X , and the dualities onthis category. In section 2.2, the necessary technical tools for the proofs of the maintheorems are developed; these consist mainly of certain symmetric forms that areconstructed from the Koszul complex. The most involved construction is that oftheorem 2.2.7 and provides the splitting of the homotopy fibration found in theoremB(ii). Section 2.3 recalls some important results from [12] and casts them into aform that is slightly better suited to this paper. Finally, main theorems A and Bare proved in section 2.4 as theorems 2.4.3 and 2.4.7, respectively. Acknowledgements
I am deeply indebted to Marco Schlichting and Heng Xie for providing key ideas be-hind the present results, as well as for helpful discussions to clarify these ideas. Oneidea that stands out in particular is that to consider symmetric forms on the Koszulcomplex. Additionally, I would like to thank Jens Hornbostel, Marcus Zibrowiusand Sean Tilson for many helpful discussions.
The sign conventions used will be the same as those in [12, section 1.11]. Let X bea scheme with the resolution property, that is, satisfying Perf( X ) = sPerf( X ). Animportant class of schemes that satisfy this property is that of schemes with an amplefamily of line bundles. For ease of notation, let O = O X . All tensor products in thissection are taken over O , unless indicated otherwise. Let Perf( X ) be the usual closedsymmetric monoidal pretriangulated dg category of perfect complexes of O -modules,where the monoidal structure is given by the tensor product of complexes and themonoidal unit is the perfect complex O , concentrated in cohomological degree 0. Itis legitimate and instructive to think of Perf( X ) as a subcategory of dg O -modules,analogous to the category of dg k -modules where k is a commutative ring. Definition 2.1.1.
Let A be an object of Perf( X ). The duality defined by A consistsof the following functor ∨ A : Perf( X ) op → Perf( X ) and natural transformationcan A : id → ∨ A ◦ ∨ op A :(i) on objects, ∨ A is given by F [ F, A ];(ii) on mapping complexes, the component [
F, G ] → [[ G, A ] , [ F, A ]] of ∨ A is givenby f ∨ A = ( g ( − | f || g | gf )for homogeneous f ∈ [ F, G ] and g ∈ [ G, A ]; and(iii) the canonical double dual identification can AF : F → [[ F, A ] , A ] on an object F is given by can AF ( x ) = ( f ( − | x || f | f ( x ) . he projective bundle formula for grothendieck-witt spectra The category Perf( X ) equipped with the duality defined by A is denoted(Perf( X ) , ∨ A , can A ) , or shortly Perf( X ) [ A ] . When A is understood, it might be dropped from the notation,especially in the case where A is the monoidal unit O . For n ∈ Z , Perf( X ) [ n ] abbreviates Perf( X ) [ O [ n ]] .Note that, for any A ∈ Perf( X ), one also obtains the dg category with duality(Perf( X ) , ∨ A , − can A ), the duality of which is sometimes called the skew-duality defined by A .The canonical double dual identification can A is in general not a natural isomor-phism, except if A is an invertible complex, that is, if A = L [ n ] is the shift of a linebundle L on X . All the dualities considered here will be of this type. For a linebundle L on X , the duality ∨ L [ n ] on Perf( X ) is closely related to the duality ∨ L onVect( X ), given by F ∨ L = H om ( F , L ) , whose canonical double dual identification is the evaluation map ev F : F → F ∨ L ∨ L .Fix a line bundle L on X , let n ∈ Z and L = L [ n ], and equip Perf( X ) and Vect( X )with the dualities ∨ L and ∨ = ∨ L , respectively. Let A be an object of Perf( X ) withdifferential d . Let B be another object of Perf( X ) and let f ∈ [ A, B ] j . The followingidentities are forced by definition 2.1.1 and recorded for future reference:( A ∨ L ) i = A ∨− i − n ( d ∨ L ) i = ( − i +1 d ∨− i − − n ( f ∨ L ) i = ( − ij f ∨− i − j − n (can LA ) i = ( − i ( n +1) ev A i . (2.1.1)Recall the following standard lemma from algebraic geometry. Lemma 2.1.2.
Let F and G be O -modules and let L be an invertible sheaf on X .If F or G is finite locally free, then the canonical map φ : G ⊗ O H om ( F , L ) → H om ( F , G ⊗ O L ) given by s ⊗ f ( t s ⊗ f ( t )) is an isomorphism.Proof. First assume that F is finite locally free. It suffices to check that φ is anisomorphism locally on X . Let x ∈ X and let U ⊂ X be an open neighborhood of x such that F | U ∼ = O ⊕ nU and L| U ∼ = O U for some n ∈ N . As O ⊕ nU is a coproduct andtherefore a colimit, it holds that H om ( O ⊕ nU , L| U ) ∼ = n Y i =1 H om ( O U , L| U ) ∼ = H om ( O U , L| U ) ⊕ n . Hence (
G ⊗ O H om ( F , L )) | U ∼ = G| U ⊗ O U H om ( O ⊕ nU , O U ) ∼ = G| U ⊗ O U H om ( O U , O U ) ⊕ n ∼ = G| U ⊗ O U O ⊕ nU ∼ = G| ⊕ nU , erman rohrbach as well as H om ( F , G ⊗ L ) | U ∼ = H om ( O ⊕ nU , G| U ) ∼ = H om ( O U , G| U ) ⊕ n ∼ = G| ⊕ nU . Fixing, for an arbitrary O U -module H , the isomorphism H om ( O U , H ) → H as f f (1), the above isomorphisms become natural and the result follows. A similarargument shows that the result holds if G is finite locally free.The finiteness condition in lemma 2.1.2 is necessary. Suppose that F = O ⊕ I isfree but not of finite type. Then it still holds that G ⊗ H om ( O ⊕ I , O ) ∼ = G ⊗ Y I O , but the canonical map G ⊗ Y I O → Y I G given by x ⊗ ( a i ) i ∈ I ( a i x ) i ∈ I is no longer an isomorphism, because the tensorproduct only has finite sums of simple tensors and therefore the inverse ( x i ) i ∈ I P ( x i ⊗ e i ), where e i denotes the element of Q O whose i -th entry is 1 and whoseother entries are zero, is not well-defined. For G not of finite type, the argumentbreaks down even earlier, since an infinite sum of O -modules is not a product andis therefore not necessarily preserved by H om . Corollary 2.1.3.
Let M and N be in Perf( X ) , n ∈ Z , and L a line bundle on X .Then there is a natural isomorphism φ : N ⊗ [ M, L [ n ]] −→ [ M, N ⊗ L [ n ]] given by f ⊗ s ( x s ⊗ f ( x )) .Proof. Note that φ is the composition N ⊗ [ M, L [ n ]] [ L [ n ] , N ⊗ L [ n ]] ⊗ [ M, L [ n ]] [ M, N ⊗ L [ n ]] , ∇⊗ ◦ and therefore a natural morphism. Here, ∇ : N → [ L [ n ] , N ⊗ L [ n ]] is the unit of thetensor-hom adjunction for L [ n ]. Each component φ i : ( N ⊗ [ M, L [ n ]]) i → [ M, N ⊗ L [ n ]] i of φ is a map φ i : M p N i − p ⊗ H om ( M − p − n , L ) → M q H om ( M q − n , N i + q ⊗ L )given by s ⊗ f ( x s ⊗ f ( x )). Thus φ i is a direct sum of isomorphisms as inlemma 2.1.2, which ultimately yields that φ is a natural isomorphism.Let L and L be line bundles on X and m, n ∈ Z . Set L = L [ m ], L = L [ n ]and L L = ( L ⊗ L )[ m + n ], and denote simple tensors of elements similarly.5 he projective bundle formula for grothendieck-witt spectra Remark 2.1.4.
The map O [ m ] ⊗ L ⊗ O [ n ] ⊗ L ⊗ τ ⊗ −→ O [ m ] ⊗ O [ n ] ⊗ L ⊗ L induces a natural isomorphism L ⊗ L → L L given by1 − m ⊗ x ⊗ − n ⊗ y − m − n ⊗ xy, where, for i ∈ Z , 1 − i ∈ O [ i ] − i is the multiplicative unit. Corollary 2.1.5.
Let
M, N ∈ Perf( X ) . Then there is natural isomorphism φ : [ M, L ] ⊗ [ N, L ] → [ M ⊗ N, L L ] given by φ ( f ⊗ g )( x ⊗ y ) = ( − | x || g | ( f ( x ) ⊗ g ( y )) .Proof. It holds that φ is the composition[ M, L ] ⊗ [ N, L ] [ N, L ] ⊗ [ M, L ] [ M, [ N, L ] ⊗ L ][ M, L ⊗ [ N, L ]] [ M, [ N, L ⊗ L ]] [ M ⊗ N, L L ] , τ α [1 ,τ ] β where α and β are natural isomorphisms as in corollary 2.1.3, and the final mapis the tensor-hom adjunction combined with the natural isomorphism of remark2.1.4.The following proposition provides a useful tool for constructing dg form functors(cf. [12, remark 1.32]). Proposition 2.1.6.
Let φ : M → [ M, L ] be a symmetric form in Perf( X ) [ L ] .Then tensoring by ( M, φ ) defines a dg form functor ( M, φ ) ⊗ − : Perf( X ) [ L ] −→ Perf( X ) [ L L ] with duality compatibility morphisms η N : M ⊗ [ N, L ] −→ [ M, L ] ⊗ [ N, L ] −→ [ M ⊗ N, L L ] , where the first arrow is φ ⊗ id and the second arrow is the natural isomorphism ofcorollary 2.1.5.Proof. The assignment N M ⊗ N certainly defines a functor, which leaves to beshown that it defines a form functor with the provided compatibility morphisms, orequivalently, that the square M ⊗ N [[ M ⊗ N, L L ] , L L ] M ⊗ [[ N, L ] , L ] [ M ⊗ [ N, L ] , L L ] can L L ⊗ can L η ∨ L L N η [ N,L (2.1.2)commutes for all N in Perf( X ) [ L ] ; this amounts to a straightforward computationusing the definitions of all the morphisms.6 erman rohrbach Remark 2.1.7.
For the purpose of this remark, let L = L , ǫ = ± i ∈ Z .Consider a symmetric bilinear form φ : M ⊗ M → L in (Perf( X ) , ∨ L , ǫ can L ). Thenit holds that φτ = ǫφ . The multiplication map µ : O [ i ] ⊗ O [ i ] → O [2 i ] given by x ⊗ y satisfies µτ = ( − i µ , as witnessed by the identity µ ( τ ( x ⊗ y )) = ( − i yx = ( − i xy = ( − i µ ( x ⊗ y ) . Thus µτ ⊗ φτ = ( − i ǫ ( µ ⊗ φ ). Since the diagram O [1] ⊗ M ⊗ O [1] ⊗ M O [1] ⊗ O [1] ⊗ M ⊗ M O [1] ⊗ M ⊗ O [1] ⊗ M O [1] ⊗ O [1] ⊗ M ⊗ M ⊗ τ ⊗ τ τ ⊗ τ ⊗ τ ⊗ commutes, as can be seen from a direct computation, it follows that the composition ψ : O [1] ⊗ M ⊗ O [1] ⊗ M ] O [1] ⊗ O [1] ⊗ M ⊗ M O [2] ⊗ L ⊗ τ ⊗ µ ⊗ φ satisfies ψτ = ( − i ǫψ . This yields an equivalence of dg categories with duality( O [ i ] , µ ) ⊗ − : (Perf( X ) , ∨ L , ǫ can L ) −→ (Perf( X ) , ∨ L [2 i ] , ( − i ǫ can L [2 i ] ) , which in particular shows how to turn skew-symmetric forms into symmetric onesby taking ǫ = − i = ± Corollary 2.1.8.
Let φ : M ⊗ M → L and ψ : N ⊗ N → L be symmetric bilinearforms in Perf( X ) [ L ] and Perf( X ) [ L ] , respectively. Then the composition M ⊗ N ⊗ M ⊗ N M ⊗ M ⊗ N ⊗ N L L ⊗ τ ⊗ φ ⊗ ψ is a symmetric bilinear form in Perf( X ) [ L L ] .Proof. Since a dg form functor preserves symmetric forms, the result follows directlyfrom propostion 2.1.6.
Corollary 2.1.9.
Let L = L . Then tensoring by the trivial symmetric form L ⊗ L → L ⊗ induces an equivalence Perf( X ) [0] −→ Perf( X ) [ L ⊗ ] of pretriangulated dg categories with duality.Proof. It is immediate from proposition 2.1.6 that tensoring by the isomorphism L → [ L, L ⊗ ] gives a dg form functor F : Perf( X ) [0] −→ Perf( X ) [ L ⊗ ] whose duality compatibility morphisms are isomorphisms. Furthermore, tensoringby L is an equivalence; an inverse is given by tensoring with [ L, O ]. It follows that F is an equivalence of pretriangulated dg categories with duality.7 he projective bundle formula for grothendieck-witt spectra The results of this section can be seen as a generalization of [7, section 4].Let X be a scheme satisfying the resolution property and let E be a finite locallyfree O X -module of rank r + 1. Let P = P ( E ) be the projective bundle over X associated to E , with projection map π : P → X . Set s = ⌈ r/ ⌉ and O = O P forease of notation. By the construction of the projective bundle, there is a canonicalsurjection π ∗ E −→ O (1) , giving rise to the Koszul complex KK : 0 → . . . → Λ i π ∗ E ( − i ) → . . . → Λ π ∗ E ( − → O → K − i = Λ i π ∗ E ⊗ O ( − i ) in cohomological degree − i , which is acyclic by [14,Tag 0626]. It also holds that Λ r +1 π ∗ E ∼ = π ∗ det E and Λ π ∗ E = π ∗ E . View K as adifferential graded algebra with Λ π ∗ E ( −
1) in degree −
1, so that the cohomologicaldegree and the degree of the grading coincide, and write | x | for the degree of ahomogeneous element x ∈ K . The following proposition is an adaptation of a well-known and very useful result. Proposition 2.2.1.
The bilinear form µ : K ⊗ K → det π ∗ E ( − r − r + 1] givenby x ⊗ y ( x ∧ y ) − r − is symmetric.Proof. This is a straightforward check involving the graded commutativity of K andthe sign change on the twist map τ : K ⊗ K → K ⊗ K .Fix an integer ℓ such that − r − ≤ ℓ ≤ − M = K ≤ ℓ be the naivetruncation of K : M : K − r − K − r . . . K ℓ . . . K : K − r − K − r . . . K ℓ K ℓ +1 . . . K . ι d d dd d d d d d Since K is a differential graded O -algebra, there is a multiplication map ∧ : K ⊗ K → K given by the wedge product. Let ϕ be the composition M ⊗ M ( M ⊗ M ) − r − ( K ⊗ K ) − r − det π ∗ E ( − r − r + 2] , d ⊗ ι ∧ where the first map is the projection map. In a formula, ϕ ( x ⊗ y ) = (cid:26) d ( x ) ∧ y if | x | + | y | = − r −
20 otherwise.The following proposition is instrumental in proving the projective bundle for-mula for Grothendieck-Witt spectra.
Proposition 2.2.2.
The map ϕ : M ⊗ M −→ det π ∗ E ( − r − r + 2] defines a symmetric form φ in Perf( P ) [det π ∗ E ( − r − r ]] . erman rohrbach Proof.
Let τ : M ⊗ M → M ⊗ M be the switch map x ⊗ y ( − | x || y | y ⊗ x . By [12,remark 1.31], ϕ defines a skew-symmetric form φ : M → [ M, det π ∗ E ( − r − r + 2]]if ϕ satisfies ϕ = − ϕτ , which is what will be shown. By construction, ϕ ( τ ( x ⊗ y )) = 0 = − ϕ ( x ⊗ y )if | x | + | y | 6 = − r −
2. Therefore, let x ⊗ y ∈ M ⊗ M with | x | + | y | = − r −
2. Then itholds that ϕ ( τ ( x ⊗ y )) = ( − | x || y | d ( y ) ∧ x. Note that y ∧ x = 0 and that therefore d ( y ∧ x ) = d ( y ) ∧ x + ( − | y | y ∧ d ( x ) = 0by the graded Leibniz rule. Furthermore, y ∧ d ( x ) = ( − | y || d ( x ) | d ( x ) ∧ y . Combiningthese facts yields ϕ ( τ ( x ⊗ y )) = ( − | x || y | ( − | y | +1 ( − | y || d ( x ) | d ( x ) ∧ y. It holds that | x || y | + | y || d ( x ) | = | y | (2 | x | −
1) = | y | mod 2. Thus it holds that ϕ ( τ ( x ⊗ y )) = ( − | y | +1 ( − | y | d ( x ) ∧ y = − d ( x ) ∧ y = − ϕ ( x ⊗ y ) . By remark 2.1.7, tensoring ϕ with the skew-symmetric form O [ − ⊗O [ − → O [ − φ .For the remainder of this section, let φ be the symmetric form of proposition2.2.2. Note that φ is not necessarily a quasi-isomorphism in Perf( P ) and thereforedoes not necessarily define an element of GW [ r ] ( P , det π ∗ E ( − r − ℓ in such a way that the cone of φ becomes theKoszul complex or a related acyclic complex. The results here are a generalizationof [7, section 4] by combining the results there with the technique of the proof of[15, theorem 1.5]. Proposition 2.2.3.
Suppose that r is even and set ℓ = − s − . Then φ is a quasi-isomorphism.Proof. Note that φ becomes isomorphic to the map of complexes K − r − . . . K − s − K − s − . . . K , − d − d d d d concentrated in cohomological degrees [ − r, C ( φ ) of φ isisomorphic to the Koszul complex K , which is acyclic. Consequently, φ is a quasi-isomorphism, as was to be shown.Now suppose that r is odd. Taking ℓ = − s , the cone C ( φ ) becomes isomor-phic to the complex K ⊕ K − s [ s ], where K − s = Λ s π ∗ E ( − s ) is viewed as a complexconcentrated in degree zero. The middle terms of C ( φ ) are . . . K − s − K ⊕ − s K − s +1 . . . , he projective bundle formula for grothendieck-witt spectra where the differential K − s − → K ⊕ − s is given by x ( − dx, dx ) and the differential K ⊕ − s → K − s +1 is given by ( x, y ) dx + dy . The idea presents itself that theremight be a symmetric form whose cone is actually acyclic, as long as K − s “splitsinto two dual parts”. The remainder of this section is dedicated to constructing sucha symmetric form, with a suitable assumption on K − s . Proposition 2.2.4.
The nondegenerate bilinear form ν : K − s [ s ] ⊗ K − s [ s ] −→ det π ∗ E ( − r − r + 1] given by the wedge product x ⊗ y x ∧ y defines an element ν ∈ GW [ r +1]0 ( P , det π ∗ E ( − r − . Proof.
Since ν ( x ⊗ y ) = x ∧ y = ( − s y ∧ x = ν ( τ ( x ⊗ y )), it holds that ντ = ν ,yielding the desired result.It is convenient to temporarily pass to the context of the exact category withduality (Vect( P ) , ♮, ev), where F ♮ = H om ( F , det π ∗ E ), in order to collect intermedi-ate results, after which these result are integrated into the context of Perf( P ) withthe duality defined by det π ∗ E ( − r − r ]. If ν = 0 in W [ r +1] ( P , det π ∗ E ( − r − K − s , ν ) is stably metabolic, whence follows, by [5, remark 29], the existenceof a metabolic space ( N , σ ) such that ( K − s , ν ) ⊥ ( N , σ ) is split metabolic and evenhyperbolic because 2 is invertible; let ( N , σ ) be such a metabolic space and fix asplit exact sequence P K − s ⊕ N P ♮ , ι P ι ♮ P ( ν ⊕ σ )pr P ( ν ⊕ σ ) − pr ♮ P (2.2.1)where P is a split Lagrangian of K − s ⊕ N with orthogonal complement P ♮ . Also fixa short exact sequence L N L ♮ , α α ♮ σ which exists since N is metabolic; oberserve that L stands for “Lagrangian” ratherthan “line bundle” in this case. The following lemma constructs the central squareof what will become a chain map. Lemma 2.2.5.
The square K − s − ⊕ L K − s ⊕ N P K − s ⊕ N K ♮ − s ⊕ N ♮ P ♮ K ♮ − s ⊕ N ♮ K ♮ − s − ⊕ L ♮d ⊕ αd ⊕ α pr P ( ν ⊕ σ ) ι P ι ♮ P ( ν ⊕ σ ) ( d ⊕ α ) ♮ pr ♮ P ( d ⊕ α ) ♮ (2.2.2) anti-commutes. erman rohrbach Proof.
The split exact sequence (2.2.1) yields ι P pr P +( ν ⊕ σ ) − pr ♮ P ι ♮ P ( ν ⊕ σ ) = id (K − s ⊕N ) , which becomes ( ν ⊕ σ ) ι P pr P + pr ♮ P ι ♮ P ( ν ⊕ σ ) = ( ν ⊕ σ )when composed with ( ν ⊕ σ ). Furthermore, note that d ♮ νd : K − s − → K ♮ − s − isgiven by x ( y d ( x ) ∧ d ( y )), but d ( x ) ∧ d ( y ) = d ( x ∧ d ( y )) = d (0) = 0, so d ♮ νd = 0. Thus it holds that the sum of the two paths of the square satisfies( d ⊕ α ) ♮ ( ν ⊕ σ ) ι P pr P ( d ⊕ α ) + ( d ⊕ α ) ♮ pr ♮ P ι ♮ P ( ν ⊕ σ )( d ⊕ α )= ( d ⊕ α ) ♮ (( ν ⊕ σ ) ι P pr P + pr ♮ P ι ♮ P ( ν ⊕ σ ))( d ⊕ α )= ( d ⊕ α ) ♮ ( ν ⊕ σ )( d ⊕ α )= ( d ♮ νd ⊕ α ♮ σα )= 0 , which proves the result.The next lemma applies the previous one in the context of (Perf( P ) , ∨ , can), withthe duality ∨ = ∨ det π ∗ E ( − r − r ] defined by det π ∗ E ( − r − r ]. Firmly imprintingthe identities (2.1.1), the following lemma poses no challenges. Lemma 2.2.6.
The map of complexes ψ given by K − s − ⊕ L PP ♮ K ♮ − s − ⊕ L ♮ pr P ( d ⊕ α ) ι ♮ P ( ν ⊕ σ )( d ⊕ α ) ( − s ( d ⊕ α ) ♮ ( ν ⊕ σ ) ι P ( − s +1 ( d ⊕ α ) ♮ pr ♮ P concentrated in cohomological degrees [ − s, − s +1] is symmetric in the pretriangulateddg category with duality (Perf( P ) , ∨ , can) .Proof. By lemma 2.2.5, the square commutes. For notational convenience, thesubscript of the evaluation map ev F is suppressed. It remains to be shown that ψ = ψ ∨ can. Note that ( ψ ∨ ) − s +1 = ψ ♮s − − r = ψ ♮ − s . Since ( ν ⊕ σ ) is ( − s -symmetric,it holds that ( ι ♮ P ( ν ⊕ σ )( d ⊕ α )) ♮ ev = ( d ⊕ α ) ♮ ( ν ⊕ σ ) ♮ ι ♮♮ P ev= ( d ⊕ α ) ♮ ( ν ⊕ σ ) ♮ ev ι P = ( − s ( d ⊕ α ) ♮ ( ν ⊕ σ ) ι P . A similar computation shows that (cid:16) ( − s ( d ⊕ α ) ♮ ( ν ⊕ σ ) ι P (cid:17) ♮ ev = ι ♮ P ( ν ⊕ σ )( d ⊕ α ) , which concludes the proof. 11 he projective bundle formula for grothendieck-witt spectra Let H (it stands for “half”, since it is, in some sense, half of the Koszul complex)be the following complex, concentrated in degrees [ − r, − s + 1]: K − r − . . . K − s − K − s − ⊕ L P , d d d pr P ( d ⊕ α ) where d : K − s − → K − s − ⊕ L is the composition of the differential d : K − s − → K − s − and the canonical inclusion K − s − → K − s − ⊕L . Piecing together the variousresults obtained thus far yields the following important theorem. Theorem 2.2.7.
Assume that ν = 0 in W [ r +( − s ] ( P , det π ∗ E ( − r − . Let ψ : H → H ∨ be the chain map in Perf( P ) det π ∗ E ( − r − r ] given by H − r . . . H − s − H − s H − s +1 . . . . . . H ♮ − s +1 H ♮ − s H ♮ − s − . . . H ♮ − r , where the central square is that of lemma 2.2.6. Then ψ is symmetric and a quasi-isomorphism.Proof. In this proof, let d ′ denote the differential of H . The central square commutesand is symmetric by lemma 2 . .
6. The square directly left of the central squarecommutes since ψ − s d ′− s − = ι ♮ P ( ν ⊕ σ )( d ⊕ α ) d ′− s − = 0 , and similarly for the square directly right of the central square. It follows that ψ issymmetric and it remains to be shown that ψ is a quasi-isomorphism, or equivalently,that the cone C ( ψ ) of ψ is acyclic. Note that C ( ψ ) is the complex K − r − . . . K − s − ⊕ L K − s ⊕ N K ♮ − s − ⊕ L ♮ . . . K ♮ − r − , which is isomorphic to the Koszul complex away from the middle degrees [ − s − , − s + 2] by proposition 2.2.1. In the middle degrees, C ( φ ) is given as K − s − K − s − ⊕ L K − s ⊕ N K ♮ − s − ⊕ L ♮ K ♮ − s − . d d ⊕ α d ⊕ α ♮ σ d Thus C ( φ ) is the direct sum of the acyclic koszul complex K and the exact sequence L → N → L ♮ , seen as an acyclic complex concentrated in degrees [ − s − , − s + 1].Therefore, C ( φ ) itself is acyclic, as was to be shown.The following lemma provides a sufficient condition for the vanishing of ν . Lemma 2.2.8. If E admits a quotient bundle of odd rank, then ν vanishes in W [ r +( − s ] ( P , det π ∗ E ( − r − .Proof. This is [15, proposition 8.1]. Note that if E is trivial, it certainly admits aquotient bundle of odd rank. 12 erman rohrbach In the case of a trivial projective bundle P rX of odd dimension, it is possible toconstruct an alternative symmetric form in Perf( P rX ) [ r ] , which is more concrete thanthe one obtained in theorem 2.2.7. Let E = r M i =0 O X T i be the free sheaf on X of even rank r + 1. Set P = P ( E ), O = O P and s = ( r + 1) / π : P → E be the projection map. For each i = 0 , . . . , r there is a complex oflocally free sheaves T i : O ( − → O , given by multiplication with T i , with O ( −
1) in cohomological degree −
1. Let M bethe complex M = O ( s − ⊗ r − O i =0 (cid:16) O ( − T i −→ O (cid:17) in Perf( P ). This is the complex O ( − s ) → . . . → O ( − s + i ) ⊕ ( ri ) → . . . → O ( s − ⊕ r → O ( s − O ( − s ) in cohomological degree − r . Furthermore, there is an isomorphism M ∨ ∼ = O ( − s + 1) ⊗ r − O i =0 (cid:16) O T i −→ O (1) (cid:17) , which is a complex O ( − s + 1) → . . . → O ( − s + i + 1) ⊕ ( ri ) → . . . → O ( s − ⊕ r → O ( s )with O ( − s + 1) in cohomological degree 0. Multiplication by T r induces a chain map O ( − s ) O ( − s + 1) ⊕ r . . . O ( s − O ( − s + 1) O ( − s + 2) ⊕ r . . . O ( s ) T r T r T r from M to M ∨ [ r ], which will be shown to be a symmetric form in Perf( P ) [ r ] . Proposition 2.2.9.
The map T r : M → M ∨ [ r ] defines an element µ ∈ GW [ r ]0 ( P ) .Proof. By [14, Tag 0628], the cone of T r is (a twist of) the Koszul complex of E ,which is acyclic. Therefore, T r is a quasi-isomorphism, and µ = ( M, T r ) is an elementof GW [ r ]0 ( P ), as was to be shown. Two of the important results of [12] will be recalled. The first is a combination of[12, theorem 6.6] and [12, theorem 8.10].13 he projective bundle formula for grothendieck-witt spectra
Theorem 2.3.1 (localization for GW and G W) . Let
A → B → C be a Morita exactsequence of dg categories with weak equivalences and duality, all containing . Let n ∈ Z . Then there is a homotopy fibration of Karoubi-Grothendieck-Witt spectra G W [ n ] ( A ) G W [ n ] ( B ) G W [ n ] ( C ) . If, moreover, the sequence
A → B → C of dg categories is quasi-exact, then there isalso a homotopy fibration of Grothendieck-Witt spectra GW [ n ] ( A ) GW [ n ] ( B ) GW [ n ] ( C ) . The following is a slightly more general version of additivity for GW-theory, c.f.[15, theorem 2.5].
Theorem 2.3.2 (additivity for GW) . Let ( A , w, ∨ ) be a pretriangulated dg categorywith weak equivalences and duality equipped with a semi-orthogonal decomposition hA , A , . . . , A r i , such that A ∨ i ⊂ A r − i .(i) The duality ∨ : A op → A induces equivalences A op i ≃ A r − i (ii) Suppose r is odd. Let q = ( r − / . Then the functor q Y i =0 H A i −→ A q Y i =0 ( A i , B i ) q M i =0 A i ⊕ B ∨ i , where H A i is the hyperbolic category with duality associated to A i , induces astable equivalence of spectra q M i =0 K( A i ) −→ GW [ n ] ( A ) . (iii) Suppose r is even. Let q = r/ . Then the functor A q × q − Y i =0 H A i −→ A A q × q − Y i =0 ( A i , B i ) A q ⊕ q − M i =0 A i ⊕ B ∨ i ! induces a stable equivalence of spectra GW [ n ] ( A q ) ⊕ q − M i =0 K( A i ) −→ GW [ n ] ( A ) , which identifies (up to stable equivalence) the homotopy fiber of the forgetfulmaps F : GW [ n ] ( A ) → K( A ) and F ′ : GW [ n ] ( A q ) → K( A q ) . erman rohrbach Proof.
The proof of (i) is easy, since A ∨ i ⊂ A r − i and there is an obvious equivalence ∨ : A op i → A r − i sending a map A → B in A op i to the dual map A ∨ → B ∨ , whichlies in A r − i by assumption.Next, (ii) will be proved. Using the notation in the statement of the theorem,let A − = hA , . . . , A q i and A + = hA q +1 , . . . , A r i . Then A = hA − , A + i is a semi-orthogonal decomposition of A , so there is an exact sequence of pretriangulated dgcategories A − A A + . By additivity [12, proposition 6.8] for GW-theory, it holds that the hyperbolic func-tor of the statement of (ii) induces a stable equivalence of spectra K ( A − ) GW [ n ] ( A ) ∼ Since hA , . . . , A q i is a semi-orthogonal decomposition of A − , additivity for connec-tive K -theory [8, proposition 7.10] yields an equivalence q L i =0 K( A ) K( A − ) ∼ and the proof of (ii) is finished by composing these two equivalences.Finally, (iii) will be proved. Let hA − , A , A + i be the semi-orthogonal decompo-sition of A with A − = hA , . . . , A q − i and A + = hA q +1 , . . . , A r i . Then it holds that A ∨− = A + and A ∨ = A . Hence [17, theorem 3.5.6] yields a stable equivalence ofspectra GW [ n ] ( A q ) ⊕ K( A − ) GW [ n ] ( A ) . ∼ One obtains the desired stable equivalence of spectra with another application of theadditivity of connective K -theory. It remains to show that there is a stable equiva-lence of homotopy fibers hofib( F ) ≃ hofib( F ′ ). There is a commutative diagram ofspectra hofib( F ′ ⊕ id) GW [ n ] ( A q ) ⊕ K( A − ) K( A q ) ⊕ K ( A − )hofib( F ) GW [ n ] ( A ) K( A ) ∼ ∼ F ′ ⊕ id ∼ F where the vertical arrows are stable equivalences. Thus it holds thathofib( F ) ≃ hofib( F ′ ⊕ id) ≃ hofib( F ′ ) ⊕ hofib(id) , but hofib(id) is contractible and the result follows. In this section, formulae for the Grothendieck-Witt spectra of general projectivebundles are stated and proven. Let X be a quasi-compact quasi-separated schemeover Spec Z [1 /
2] with the resolution property, i.e. sPerf( X ) = Perf( X ). As notedbefore, schemes with an ample family of line bundles satisfy the resolution property.Let E be a locally free sheaf of O X -modules of rank r + 1 and write P = P ( E ). Also15 he projective bundle formula for grothendieck-witt spectra set s = ⌈ r/ ⌉ . Let π : P → X be the associated projective bundle and let O = O P .Write A for Perf( P ). Let L be a line bundle on X and let L = ( O ( m ) ⊗ π ∗ L )[0],with m ∈ Z , be the object of Perf( P ) consisting of a single copy of O ( m ) ⊗ π ∗ L concentrated in cohomological degree 0. Let (Perf( P ) [ L ] , w ) be the pretriangulateddg category with weak equivalences the quasi-isomorphisms and duality given bythe mapping complex [ − , L ]; write ♮ for this duality on A and reserve the symbol ∨ for the standard duality on Vect( X ), Vect( P ) and their respective categories ofperfect complexes. By corollary 2.1.9, O ( m ) may be replaced by O ( m + 2 i ) forany i ∈ Z without affecting the Grothendieck-Witt spectrum. Therefore, m can bechosen freely up to parity in the proofs below.The following theorem is contained in the proof of [13, theorem 3.5.1]; a versionof this result in the context of stable ∞ -categories is [1, theorem B]. Theorem 2.4.1.
The following statements hold.(i) For each k ∈ Z , the assignment F 7→ p ∗ F ⊗O ( k ) defines a fully faithful functor Perf( X ) → Perf( P ) whose essential image will be denoted A ( − k ) .(ii) For each i ∈ Z , hA ( i − r ) , . . . , A ( i ) i is a semi-orthogonal decomposition of Perf( P ) . Proposition 2.4.2.
With notation as in theorem 2.4.1, it holds that the essentialimage of A ( k ) under the duality ♮ is A ( i − k ) for all k ∈ Z .Proof. First note that π ∗ can be made into a dg form functor (cf. [12, section9.3], which discusses the functoriality of π ∗ ). Any object of A ( k ) can be written as π ∗ M ⊗ O ( k ) with M ∈ Perf( X ). The dual of such an object satisfies( π ∗ M ⊗ O ( k )) ♮ ∼ = [ π ∗ M ⊗ O ( k ) , L ] ∼ = [ π ∗ M, [ O ( k ) , L ]] ∼ = L ⊗ [ π ∗ M, O ( − k )] ∼ = O ( m ) ⊗ π ∗ L [0] ⊗ π ∗ [ M, O X ] ⊗ O ( − k ) ∼ = π ∗ [ M, L [0]] ⊗ O ( m − k ) , where the isomorphisms are given by various results of section 2.1, as well as standardproperties of the pullback π ∗ . Hence ( π ∗ M ⊗ O ( k )) ♮ is an object of A ( m − k ). Itfollows that A ( k ) ♮ ⊂ A ( m − k ) and A ( m − k ) ♮ ⊂ A ( k ). As ♮ is an equivalence, theproof is done.The following theorem is the important main theorem A. Rather anticlimacti-cally, its proof is a simple application of additivity for Grothendieck-Witt spectra. Theorem 2.4.3 (Theorem A) . Recall that s = ⌈ r/ ⌉ . The following statementshold.(i) If m and r are even, then there is a stable equivalence of spectra GW [ n ] ( X, L ) ⊕ K( X ) ⊕ s GW [ n ] ( P , π ∗ L ( m )) . (ii) If m and r are odd, then there is a stable equivalence of spectra K( X ) ⊕ s GW [ n ] ( P , π ∗ L ( m )) . erman rohrbach Proof.
Without loss of generality, assume that m = − r . By theorem 2.4.1, there isa semi-orthogonal decomposition hA ( − r ) , A ( − r + 1) , . . . , A (0) i of A . The duality maps A ( i ) to A ( − r − i ) for all i ∈ Z by proposition 2.4.2. Hencethe additivity theorem 2.3.2 applies and yields the desired result.This covers the easy cases of the projective bundle formula. Now set m = − r − hA ( − r − , A ( − r ) , . . . , A ( − i of A . By proposition 2.4.2, it holds that A ( i ) ♮ ⊂ A ( − r − − i ) for all i ∈ Z . Let A = hA ( − r ) , . . . , A ( − i . Then the consituents of A are exchanged by the duality, so that GW( A ) may becomputed using the additivity theorem 2.3.2. Furthermore, there is a quasi-exactsequence of pretriangulated dg categories with duality A [ L ]0 A [ L ] ( A / A ) [ L ] . (2.4.1)Thus understanding GW [ n ] (( A / A ) [ L ] ) is paramount to understanding GW [ n ] ( A [ L ] ). Lemma 2.4.4.
There is a quasi-equivalence of pretriangulated dg categories withweak equivalences F : Perf( X ) −→ A / A given by M det π ∗ E ( m )[ r ] ⊗ π ∗ M .Proof. It holds that F is the compositionPerf( X ) A A A / A , π ∗ where the middle arrow is given by tensoring with det π ∗ E ( m )[ r ]. Denote the com-position Perf( X ) → A of the first two maps by F ′ . It holds that F ′ (det E ∨ [ − r ]) = det π ∗ E ( m )[ r ] ⊗ π ∗ det E ∨ [ − r ] ∼ = det π ∗ E [ r ] ⊗ π ∗ det E ∨ [ − r ] ⊗ O ( m ) ∼ = O ( m ) . As tensoring with det E ∨ [ − r ] gives a self-equivalence of Perf( X ), the essential imageof F ′ consists of objects of the form π ∗ M ⊗ O ( m ) and is therefore the subcategory A ( − r −
1) of A of theorem 2.4.1. In particular, F ′ : Perf( X ) → A ( − r −
1) is a quasi-equivalence. Hence F factors through the canonical map F ′′ : A ( − r − → A / A .Since hA ( − r − , A i is a semi-orthogonal decomposition of A , it follows that F ′′ is a quasi-equivalence. Therefore F , being the composition of quasi-equivalences, isalso a quasi-equivalence, as was to be shown.17 he projective bundle formula for grothendieck-witt spectra Lemma 2.4.5.
Fix an − r − ≤ ℓ ≤ − as in section 2.2. The symmetric form φ : M [ − → [ M [ − , det π ∗ E ( m )[ r ]] of 2.2.2 becomes a quasi-isomorphism ¯ φ : det π ∗ E ( m )[ r ] → O in ( A / A ) [det π ∗ E ( m )[ r ]] .Proof. The cone C ( φ ) of φ is given by M ⊕ [ M [ − , det π ∗ E ( m )[ r ]]. Let − r ≤ i ≤ − M i = Λ − i π ∗ E ⊗O ( i ) if i ≤ ℓ and M i = 0 otherwise. As O ( i ) ∈ A ,it follows that C ( φ ) i = 0. Hence the image of C ( φ ) in A / A is C ( φ ) : K − r − . . . K , which corresponds to the image K of the Koszul complex in A / A . As K is acyclicand the quotient functor A → A / A is exact, it follows that C ( φ ) is acyclic as well.Hence the image ¯ φ : det π ∗ E ( m )[ r ] → O of φ in A / A is a quasi-isomorphism, as wasto be shown. In particular, ¯ φ defines an element of GW [ r ]0 (( A / A ) [det π ∗ E ( m )] ). Proposition 2.4.6.
The quasi-equivalence F : Perf( X ) → A / A of proposition2.4.4 can be made into a dg form functor ( F, η ) : Perf( X ) [0] → ( A / A ) [det π ∗ E ( m )[ r ]] . In particular, there is a quasi-equivalence of pretriangulated dg categories with weakequivalences and duality
Perf( X ) [det E ∨ ⊗L [ − r ]] ≃ ( A / A ) [ L ] . Proof.
As in the proof of proposition 2.4.4, write F as a composition F : Perf( X ) [0] A [0] A det π ∗ E [ r ] ( A / A ) det π ∗ E [ r ] , π ∗ φ ⊗ now ornamented with the dualities of each category. Note that π ∗ is a dg formfunctor by [12, section 9.3]. Furthermore, tensoring with the symmetric form φ fromproposition 2.2.2 is a dg form functor by proposition 2.1.6. Finally, the quotientmap A [det π ∗ E ( m )[ r ]] −→ ( A / A ) [det π ∗ E ( m )[ r ]] is a dg form functor by construction. Hence the quasi-equivalence of dg categories F is a composition of dg form functors and therefore itself a dg form functor, as wasto be shown. Finally, twisting the duality in Perf( X ) [0] by the invertible complexdet E ∨ ⊗ L [ − r ] gives the desired quasi-equivalencePerf( X ) [det E ∨ ⊗L [ − r ]] ≃ ( A / A ) [ L ] , and the proof is done.Finally, there are no more obstacles to proving the more interesting and difficultmain theorem B. 18 erman rohrbach Theorem 2.4.7 (Theorem B) . The following statements hold.(i) If r is even and m is odd, then there is a split homotopy fibration K ( X ) ⊕ s − GW [ n ] ( P , π ∗ L ( m )) GW [ n − r ] ( X, det E ∨ ⊗ L ) . (ii) If r is odd and m is even, then there is a homotopy fibration GW [ n ] ( X, L ) ⊕ K ( X ) ⊕ s − GW [ n ] ( P , π ∗ L ( m )) GW [ n − r ] ( X, det E ∨ ⊗ L ) , which splits if the image of the symmetric form ν of proposition 2.2.4 vanishesin W [ r +1]0 ( P , det π ∗ E ( − r − . The condition for the splitting is satisfied e.g.if E is a trivial bundle.Proof. Without loss of generality, one may assume m = − r − [ n ] ( A [ L ]0 ) −→ GW [ n ] ( A [ L ] ) −→ GW [ n ] (( A / A ) [ L ] )by [12, theorem 6.6]. Note that GW [ n ] ( A [ L ] ) is just a different way of writingGW [ n ] ( P , π ∗ L ( m )). As already remarked, the additivity theorem 2.3.2 gives a for-mula for the first term GW [ n ] ( A [ L ]0 ) of both (i) and (ii). Thus it suffices to showthat there is a quasi-equivalence ψ : Perf( X ) [det E ∨ ⊗L [ − r ]] −→ ( A / A ) [ L ] , but this holds by proposition 2.4.6. This yields both claimed homotopy fibrations.If r is even or the image of the symmetric form ν of proposition 2.2.4 vanishesin W [ r +1]0 ( P , det π ∗ E ( − r − ψ : H → [ H, det π ∗ E ( m )[ r ]] which is a quasi-isomorphism by proposition 2.2.3 andtheorem 2.2.7. Therefore, one obtains an element ψ ∈ GW [ r ]0 ( P , det π ∗ E ( m )). Thusthe compositionPerf( X ) [det E ∨ ⊗L [ − r ]] A [det π ∗ E ∨ ⊗ π ∗ L [ − r ]] A [ L ] π ∗ ψ ⊗ yields a compositionGW [ − r ] ( X, det E ∨ ⊗ L ) GW [ − r ] (Perf( P ) , det π ∗ E ∨ ⊗ π ∗ L ) GW [0] ( P , π ∗ L ( m )) π ∗ ψ ∪ of maps of Grothendieck-Witt spectra, which splits the second map in the relevanthomotopy fibration, thus finishing the proof. Note that lemma 2.2.8 provides asufficient condition for ν to vanish, and that this condition holds in particular if E is a trivial bundle. References [1] A. A. Khan. “Descent by quasi-smooth blow-ups in algebraic K-theory”. In:(2018). arXiv: .19 he projective bundle formula for grothendieck-witt spectra [2]
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