Chow-Witt rings of split quadrics
aa r X i v : . [ m a t h . K T ] A ug CHOW-WITT RINGS OF SPLIT QUADRICS
JENS HORNBOSTEL, HENG XIE, AND MARCUS ZIBROWIUS
Abstract.
We compute the Chow-Witt rings of split quadrics over a field of characteristic not two.We even determine the full bigraded I -cohomology and Milnor-Witt cohomology rings, includingtwists by line bundles. The results on I -cohomology corroborate the general philosophy that I -cohomology is an algebro-geometric version of singular cohomology of real varieties: our explicitcalculations confirm that the I -cohomology ring of a split quadric over the reals is isomorphic to thesingular cohomology ring of the space of its real points. Contents
1. Introduction 12. Conventions and notation 33. Singular cohomology of real quadrics 44. The blow-up setup of Balmer-Calmès 125. I -cohomology: additive structure 156. I -cohomology: multiplicative structure 197. Geometric bidegrees and real realization 268. Chow-Witt rings 279. Milnor-Witt cohomology 29References 311. Introduction
Chow-Witt groups were introduced by Barge and Morel [BM00] as a cohomology theory containingan
Euler class , detecting if a vector bundle over a smooth affine scheme over a field of characteristic = 2 splits off a trivial line bundle in the critical range. This generalized a previous result of Murthy[Mur94] for Chow groups of smooth affine schemes over algebraically closed fields. In recent years,subsequent work of Morel [Mor12], Fasel [Fas08], Fasel-Srinivas [FS09] and Asok-Fasel [AF16] hascompleted the picture if two is invertible. As Chow-Witt groups are a quadratic refinement of Chowgroups, one might hope for generalizations of other results using Chow-Witt groups in the future.In this paper, we determine the Chow-Witt ring of a split quadric Q n of dimension n ≥ F of characteristic = 2. Our computation follows the general strategies laid out by Fasel, Wendtand the first author in the computations for projective spaces, Grassmannians and classifying spaces[Fas13, HW19, Wen18]. The main step (in both descriptions) is thus the computation of I -cohomology.We compute these cohomology groups in all bidegrees, and over an arbitrary smooth base (not just afield). That is, we compute the group H i ( Q n , I j , O ( l )) Research for this publication was conducted in the framework of the DFG Research Training Group 2240: Algebro-Geometric Methods in Algebra, Arithmetic and Topology.The second author was supported by EPSRC Grant EP/M001113/1 and by the DFG Priority Programme 1786:Homotopy Theory and Algebraic Geometry. for all values of i , j and l , where Q n is a split quadric over a smooth scheme, and where O ( l ) denotesthe l th tensor power of O Q (1) (see Section 2 for precise definitions). This additive result is summarizedin Theorem 5.5 below.For the computation of the Chow-Witt ring, we need to understand the I -cohomology groups in “geo-metric” bidegrees. With untwisted coefficients, their direct sum yields the graded ring H • ( Q n , I • ) := L i ≥ H i ( Q n , I i ), which is a graded commutative algebra over the Witt ring W( F ) of the base field.Adding twisted coefficients, this extends to a Z ⊕ Z / F )-algebraH • ( Q n , I • , O ⊕ O (1)) := M i ≥ (cid:0) H i ( Q n , I i , O ) ⊕ H i ( Q n , I i , O (1)) (cid:1) . The explicit ring structure can often be written down more concisely when twisted coefficients are takeninto account. Indeed, this is already true for projective spaces. The following result is Proposition 4.5in a recent preprint of Matthias Wendt [Wen18], based on the additive computations of Fasel [Fas13].
Theorem (Fasel/Wendt) . Let F be field of characteristic = 2 , with Witt ring W( F ) and fundamentalideal I( F ) ⊂ W( F ) . Let P p denote the p -dimensional projective space over F . For any p ≥ , wehave a Z ⊕ Z / -graded ring isomorphism H • ( P p , I • , O ⊕ O (1)) = W( F )[ ξ, α ] / (I( F ) ξ, ξ p +1 , ξα, α ) with ( | ξ | = (1 , | α | = ( p, p + 1)We generalize this result for projective spaces to the full Z ⊕ Z ⊕ Z / Theorem A.
For any n ≥ , we have a Z ⊕ Z / -graded ring isomorphism H • ( Q n , I • , O ⊕ O (1)) ∼ = W( F )[ ξ, α, β ] / (I( F ) ξ, ξ p +1 , ξα + ξβ, α + β , αβ ) if n = 2 p and p is even W( F )[ ξ, α, β ] / (I( F ) ξ, ξ p +1 , ξα + ξβ, α , β ) if n = 2 p and p is odd W( F )[ ξ, α, β ] / (I( F ) ξ, ξ p +1 , ξα, α , β ) if n = 2 p + 1 Here, the generators are of degrees | ξ | = (1 , , | α | = ( p, p + 1) , | β | = ( q, q + 1) where q = p if n = 2 p is even and q = p + 1 if n = 2 p + 1 is odd. In fact, Theorems 6.8 and 6.9 again describe the full Z ⊕ Z ⊕ Z / I -cohomology of Q n . Finally,for the Chow-Witt ring, we arrive at two related presentations. The first presentation, discussed inSection 8, looks as follows. Theorem B.
The Z ⊕ Z / -graded GW( F ) -algebra g CH • ( Q n , O ⊕ O (1)) can be described as a fibreproduct of Z ⊕ Z / -graded rings of the following form: g CH • ( Q n , O ⊕ O (1)) ∼ = → H • ( Q n , I • , O ⊕ O (1)) × Ch • ( Q n , O⊕O (1)) (ker ∂ ⊕ ker ∂ O (1) ) . Here, the second factor ker ∂ ⊕ ker ∂ O (1) is described explicitly in Theorem 8.3. The fibre product istaken over the Z ⊕ Z / -graded ring Ch • ( Q n , O ⊕ O (1)) := Ch • ( Q n )[ τ ] / ( τ − , where Ch • ( Q n ) isthe (well-known) Chow-ring modulo two of Q n , concentrated in degrees ( ∗ , , and τ is an artificialgenerator of bidegree (0 , . The second presentation of the Chow-Witt ring is discussed in the final section. It gives a fulladditive description of the Z ⊕ Z ⊕ Z / tot ( Q n , K MW ) includingall twists, refining [Fas13, Theorem 11.7] from projective spaces to split quadrics. The argumentspresented in Section 6 can also be refined to obtain a description of H tot ( Q n , K MW ) as an algebrain terms of generators and relations over the coefficient ring H tot ( F, K MW ), but we do not make thisexplicit here.For a real variety X , I -cohomology is closely related to the singular cohomology of the real points X ( R ) equipped with the analytic topology. The strongest general result in this direction is thefollowing theorem of Jacobson [Jac17]: HOW-WITT RINGS OF SPLIT QUADRICS 3
Theorem (Jacobson) . For a real variety X , the group H i ( X, I j ) is isomorphic to the singular coho-mology group H i ( X ( R ) , Z ) in all bidegrees with j > dim( X ) . Note that this general result does not apply to the interesting geometric bidegrees i = j with i ≤ dim( X ). However, the known computations for projective spaces and Grassmannians indicatethat the situation for cellular varieties is even better. In forthcoming work with Wendt [HWXZ], wewill show that realization induces a ring isomorphism H • ( X, I • ) ∼ = H • ( X ( R ) , Z ) for all smooth cellularreal varieties X . This isomorphism also extends to twisted coefficients. Here, we verify this result forsplit real quadrics, through explicit computations.Unfortunately, we could not find the integral cohomology ring of a real split quadric in the liter-ature. All we found was the additive structure of cohomology with Z / I -cohomology that follow, thusproviding a baseline. The results are summarized in Theorem 3.20. The computations for split quad-rics extend from Chow groups not only to Milnor cohomology (Corollary 6.4), but also to full bigradedmotivic cohomology [DI07, Prop. 4.3]. See [Ker09] or [RS18, Section 3] for a comparison of thesetwo bigraded theories. We expect that our techniques and results on Milnor-Witt cohomology groupsof split quadrics also extend to the full bigraded “Milnor-Witt motivic cohomology” (also known as“generalized motivic cohomology”) as developed by Fasel et al., see e.g. [DF17, Theorem 4.2.4] and[CF17], but we have not pursued this.There are many partial results on the Chow groups of quadrics by Swan, Karpenko, Merkurjev,Rost, Vishik and others. Partial results on Witt groups of split quadrics of Nenashev [Nen09] canbe completed using the blowup setup of Balmer-Calmès that we also use in our computations here[Cal08]. Witt groups of split quadrics over the complex numbers have been computed by the thirdauthor via another method, see [Zib11]. We hope to return to computations of the I -cohomology ofother kinds of quadrics in future work. The trace map on Clifford algebras has been found very helpfulfor studying Witt groups of general quadrics, cf. the recent work of the second author [Xie19]. Acknowledgements.
We would like to thank the anonymous referee for carefully reading our ma-nuscript and suggesting many improvements.2.
Conventions and notation
Topology.
We fix the following notation: • Q p,q is the ( p + q )-dimensional real quadric defined as a subspace of RP p + q +1 by the equation x + x + · · · + x p = y + y + · · · + y q , (2.1)where [ x : . . . : x p : y : . . . : y q ] are homogeneous coordinates on RP p + q +1 . Note that Q p,q ∼ = Q q,p . We always assume 1 ≤ p ≤ q .Additional notation is introduced at the beginning of Section 3. Algebraic geometry.
We fix a base scheme S which is smooth over a field F of characteristic = 2.More precisely, S is assumed to be noetherian, separated, smooth and of finite type over F . We mayand will moreover assume without loss of generality that S is connected. These are precisely theassumptions on the base scheme S used in [Fas13]. The following schemes and morphisms are alldefined over S : • Q n denotes the n -dimensional split quadric , i.e. the closed subvariety of P n +1 defined by theequation ( x y + . . . + x p y p = 0 when n = 2 px y + . . . + x p y p + z = 0 when n = 2 p + 1 (2.2) JENS HORNBOSTEL, HENG XIE, AND MARCUS ZIBROWIUS
When n = 2 p , Q n is isomorphic to the subvariety Q p,p defined by equation (2.1). When n = 2 p + 1, it is isomorphic to Q p,p +1 . • P px , P py , P px ′ and P py ′ denote the following subvarieties of Q n :When n = 2 p , we define P px , P py ⊂ P p +1 by the equations y = y = . . . = y p = 0and x = x = . . . = x p = 0 respectively. We define P px ′ , P py ′ ⊂ P p +1 by the equations x = y = . . . = y p = 0 and y = x = . . . = x p = 0.When n = 2 p + 1, we define P px , P py ⊂ P p +2 by the equations y = y = . . . = y p = z = 0and x = x = . . . = x p = z = 0, respectively. • q : Q n → S and p : P p → S denote the projection maps • i y : P py ֒ → Q n − P px , ι y : P py ֒ → Q n , j : Q n − P px ֒ → Q n and i : Q n ֒ → P n +1 are the obviousembeddings; likewise for i x , ι x , i x ′ , ι x ′ , i y ′ and ι y ′ • ρ : Q n − P px → P py is the morphism given by ( ρ : [ x : . . . : x p : y : . . . : y p ] [0 : . . . : 0 : y : . . . : y p ] if n = 2 pρ : [ x : . . . : x p : y : . . . : y p : z ] [0 : . . . : 0 : y : . . . : y p : 0] if n = 2 p + 1(Some of this notation is summarized in Figure 1.) • O Q (1) is defined as the restriction along i : Q n ֒ → P n +1 of the canonical line bundle O (1) on P n +1 . When there is no danger of confusion, we simply denote this restriction again by O (1). P py (cid:31) (cid:127) i y / ( (cid:8) ι y ) ) ❊❊❊❊❊❊❊❊❊ ❊❊❊❊❊❊❊❊❊ Q − P pxρ (cid:15) (cid:15) (cid:31) (cid:127) j / / Q nq (cid:15) (cid:15) (cid:31) (cid:127) i / / P n +1 P py p / / S Figure 1.
Various morphisms used in the algebro-geometric calculations Singular cohomology of real quadrics
In this section, we study the real quadrics Q p,q ⊂ RP p + q +1 (with the analytic topology). Note firstthat Q p,q is a two-fold covering space of RP p × RP q , via the obvious map that takes [ x : y ] to ([ x ] , [ y ]).Write π and π for the two components of this map. A two-fold cover of the quadric itself is given by S p × S q : the real quadric Q p,q is the quotient of S p × S q modulo the involution τ := τ p × τ q , where τ n denotes the involution of S n sending x to − x . The composition of the two covering maps is thecanonical four-fold cover of RP p × RP q . The situation is summarized by the central vertical columnof Figure 2. Also displayed there are the “diagonal” embedding ∆ S : S p ֒ → S p × S q that sends x to( x, ( x, RP p → Q p,q and RP p → RP p × RP q , respectively. Note that∆ splits the projection π : Q p,q → RP p . Twisted coefficients.
A free rank one local coefficient system over a path-connected space X canbe defined as an action of the fundamental group on Z , i.e. as a group homomorphism π ( X, x ) → Aut( Z ) = {± } , for any choice of base point x [DK01, § 5.1][Spa81, Ex. I.F]. As is customary, we de-note the trivial rank one local coefficient system corresponding to the constant group homomorphism π ( X, x ) → Aut( Z ) simply by Z . For semi-locally simply connected X , such coefficient systems corres-pond to fibre bundles with typical fibre Z and structure group {± } [DK01, Lemma 4.7]. Equivalently,we may identify them with principal {± } -bundles, i.e. with two-fold coverings of X .We write Z (1) for the rank one local coefficient system ρ : π ( RP p ) → {± } on RP p correspondingto the two-fold cover S p → RP p . For p ≥
2, the two-fold cover is the universal cover, and ρ is anisomorphism, but the system Z (1) also exists for p = 1. The situation for the real quadrics is similar. HOW-WITT RINGS OF SPLIT QUADRICS 5 S p (cid:15) (cid:15) ∆ S / / S p × S q π S / / π (cid:15) (cid:15) S p (cid:15) (cid:15) Q p,q π (cid:31) (cid:31) π × π (cid:15) (cid:15) RP p / / ∆ RP p × RP q / / RP p Figure 2.
The various continuous maps relating spheres, projective spaces and real quadrics
We have a canonical two-fold cover S p × S q → Q p,q , and we again write Z (1) for the correspondinglocal coefficient system π ( Q p,q ) → {± } . Again, S p × S q is the universal cover for q ≥ p ≥
2. Notethat Z (1) pulls back to Z (1) under the maps π and ∆ in Figure 2, so the notation is consistent.More generally, given a coefficient ring R and an integer s , we define R ( s ) := R ⊗ Z Z (1) ⊗ s withthe action of the fundamental group induced by the trivial action on R and the above action on Z (1).Of course, this really only depends on R and the value of s mod 2. Also note that Z / s ) = Z / s . The direct sum R ⊕ R (1) is a Z / R ⊕ R (1) is a Z ⊕ Z / • ( Q p,q , R ⊕ R (1)) = H • ( Q p,q , R ) ⊕ H • ( Q p,q , R (1)) . Elements α ∈ H i ( Q p,q , R ) have degree | α | = ( i, β ∈ H i ( Q p,q , R (1)) have degree | β | = ( i, R ⊕ R as a Z / S p × S q andH • ( S p × S q , R ⊕ R ) = H • ( S p × S q , R ) ⊕ H • ( S p × S q , R )as a Z ⊕ Z / Rational cohomology ring.
The description of Q p,q as a quotient of S p × S q implies that H • ( Q p,q , Q )is isomorphic to the subring of H • ( S p × S q , Q ) fixed by τ ∗ [Hat02, Proposition 3G.1]. Similar argumentsshow that, more generally, H • ( Q p,q , Q ⊕ Q (1)) is isomorphic to the subring of the Z ⊕ {± } -gradedring H • ( S p × S q , Q ⊕ Q ) fixed by the action of τ ∗ on homogeneous elements in the +1-graded partand the action of − τ ∗ on the ( − τ ∗ n : H n ( S n ) → H n ( S n ) is given by multiplicationwith ( − n +1 , we find:H • ( Q p,q , Q ⊕ Q (1)) ∼ = Q [ α, β ] / ( α , β ) with ( | α | = ( p, p + 1) | β | = ( q, q + 1) (3.1)In particular, Q p,q is orientable if and only if p + q is even (use [Hat02, Theorem 3.26]). A (minimal) cell structure.
In order to compute the twisted (co)homology of the real quadric Q p,q with integral coefficients, we need to study its two-fold cover S p × S q in more detail. Proposition 3.2.
Assume p ≤ q . There exists a cell structure on S p × S q consisting of p + 4 cellsfor which the action of τ := τ p × τ q is cellular: we have characteristic maps j ± , , . . . , j ± p, for cellsin dimensions , . . . , p and characteristic maps j ± ,q , . . . , j ± p,q for cells in dimensions q, . . . , p+q suchthat τ ◦ j + k,i = j − k,i . The skeleta of X = S p × S q with respect to this cell structure are given by: X k = ∆ S ( S k ) for ≤ k < pX k = ∆ S ( S p ) for p ≤ k < qX k + q = ∆ S ( S p ) ∪ ( S k × S q ) for q ≤ k + q ≤ p + q JENS HORNBOSTEL, HENG XIE, AND MARCUS ZIBROWIUS
Here, ∆ S : S p → S p × S q denotes the map x ( x, ( x, , and S k is viewed as an “equator” of S p for k ≤ p , embedded via the first k + 1 coordinates.Proof. We will give an explicit description of this cell structure on S p × S q . First, let us recall andmake explicit one common cell structure on a single sphere S n . For k ∈ { , . . . , n } , let j + k denote thefollowing embedding of D k into S n : j + k : D k −−−→ S n ( x , . . . , x k − ) ( x , . . . , x k − , p − k x k , , . . . , D k into S n is given by j − k := τ n ◦ j + k . Together, these maps j + k and j − k for k ∈ { , . . . , n } define a cell structure on S n with two cells in each dimension.Now take n = p . We give ∆ S ( S p ) ⊂ S p × S q the cell structure induced by the cell structure on S p just described. That is, we consider the characteristic maps j ± k, := ∆ S ◦ j ± k : D k → S p × S q . (3.3)Note that τ ◦ j + k, = j − k, as desired.The second family of 2 p + 2 cells lives in dimensions q , . . . , p + q . In order to define these cells, weneed to fix some further notation. Consider the “northern hemisphere” j + q ( D q ) ⊂ S q , i.e. the set ofall points x = ( x , . . . , x q ) ∈ S q with x q ≥
0. Let e q := (0 , . . . , ,
1) be the “north pole”. Choose andfix a continuous map R : j + q ( D q ) → SO( q + 1) x R x with the property that R x · e q = x . Such a map exists by Lemma 3.7 below.Also, we fix for each dimension n a map that “wraps the n -disk around the n -sphere”, i.e. asurjection w n : D n → S n that sends 0 ∈ D n to the north pole e n ∈ S n and every point on theboundary ∂D n to the south pole − e n ∈ S n , and that induces a homeomorphism ¯ w n : D n /∂D n ∼ = S n .We choose w n as follows: w n : D n → S n x ( s ( x ) · x, − k x k ) , (3.4)where s ( x ) is a non-negative scalar determined by the requirement that w n ( x ) ∈ S n . Finally, fixhomeomorphisms φ m,n between products of disks D m × D n and the disks D m + n of the appropriatedimensions: φ m,n : D m × D n ∼ = −→ D m + n ( x, y ) max( k x k , k y k ) k ( x,y ) k ( x, y ) (3.5)In view of the homeomorphisms φ k,q , we may define the characteristic maps for the ( k + q )-cells of S p × S q on the products D k × D q . For every k ∈ { , . . . , p } , we define one such characteristic map asfollows: j + k,q : D k × D q → S p × S q ( x, y ) ( j + k ( x ) , − R ( j + k ( x ) , ( w q y )) (3.6)Here, the entry j + k ( x ) in the first coordinate denotes the embedding j + k : D k → S p , and the entry( j + k ( x ) ,
0) in the second coordinate denotes the composition of j + k with the embedding x ( x,
0) of S p into S q as an “equator”. We define a second characteristic map for a ( k + q )-cell as j − k,q := τ ◦ j + k,q .When restricted to the inner points of D k × D q , the characteristic maps j ± k,q induce homeomorph-isms: ◦ D k × ◦ D q ∼ = −−→ ( j ± k ( ◦ D k ) × S q ) \ ∆ S ( S p )The images of these homeomorphisms clearly constitute a cover of ( S p × S q ) \ ∆ S ( S p ) by disjoint sets.Thus, altogether the maps j ± k,i for i ∈ { , q } and k ∈ { , . . . , p } defined in (3.3) and (3.6) constitute acell structure on S p × S q . (cid:3) HOW-WITT RINGS OF SPLIT QUADRICS 7
Lemma 3.7.
There exists a continuous map R : j + q ( D q ) → SO( q + 1) , x R x such that R x · e q = x for all x .Proof. Consider the evaluation map SO( q + 1) → S q that takes a matrix R to R · e q . This evaluationmap constitutes a principal SO( q )-bundle. Over the contractible space j + q ( D q ) ∼ = D q , this bundle isnecessarily trivial, hence it admits a section. (cid:3) Cellular (co)homology.
Let Z [ τ ] denote the group ring associated with the group with two elements,i.e. Z [ τ ] := Z ⊕ Z τ with τ = 1. Let e C ( n ) (0) denote the chain complex of free Z [ τ ]-modules e C ( n ) (0) : Z [ τ ] − τ ←−− Z [ τ ] τ ←−− Z [ τ ] − τ ←−− Z [ τ ] τ ←−− · · · ← Z [ τ ]concentrated in degrees 0 to n . This is the cellular chain complex of S n with respect to the cellstructure arising from the two-fold covering map S n → RP n and the standard cell structure on RP n .Multiplication by τ on the complex is the chain map induced by the involution τ n of S n . The usualhomology of RP n with coefficients in Z is the homology of e C ( n ) (0) ⊗ Z [ τ ] Z , where Z is viewed as trivial Z [ τ ]-module; if instead we tensor with the Z [ τ ]-module Z (1) on which τ acts as −
1, we obtain thehomology of RP n with twisted coefficients.Let e C ( n ) (1) denote the complex with the same groups in each degree, but with the roles of multi-plication by 1 − τ and multiplication by 1 + τ reversed: e C ( n ) (1) : Z [ τ ] τ ←−− Z [ τ ] − τ ←−− Z [ τ ] τ ←−− Z [ τ ] − τ ←−− · · · ← Z [ τ ] Proposition 3.8.
For p ≤ q , the cellular chain complex associated with the cell structure on S p × S q described in Proposition 3.2 has the form C ( S p × S q ) ∼ = e C ( p ) (0) ⊕ e C ( p ) ( q + 1)[ q ] , where q + 1 indicatesthe value of q + 1 modulo two, and [ q ] indicates that the second summand is shifted q degrees to theright. Multiplication by τ on the complex is the chain map induced by the involution τ on S p × S q .Proof sketch. We clearly have a decomposition C ( S p × S q ) = C (∆ S ( S p )) ⊕ C ′ for some complex C ′ concentrated in degrees q, . . . , p + q , except that, a priori, there might be somedifferentials from C ′ to C (∆( S p )). However, there are no such differentials. This can easily be seenby considering the map on chain complexes induced by the projection π S : S p × S q → S p : this mapis zero on C ′ and maps C (∆ S ( S p )) isomorphically to C ( S p ). Thus, the above decomposition is anhonest decomposition of chain complexes.We may identify C (∆ S ( S p )) with C ( S p ) with respect to the usual cell structure on S p consisting oftwo cells in each dimension. This is precisely the cell structure used in the usual computation of thehomology of RP p (cf. [DK01, 5.2.1]). The complex C ′ can be computed in an analogous fashion. Thenon-zero boundary maps d of C ′ are Z [ τ ]-linear maps between free Z [ τ ]-modules of rank one, eachwith a generator { j + k,q } represented by one of the characteristic maps j + k,q . Write d : C ′ k + q → C ′ k − q as d { j + k,q } = d + { j + k − ,q } + d − τ { j + k − ,q } . Both coefficients d + and d − can easily be seen to be ±
1. The crucial value we need to know for thehomology/cohomology calculations is the relative sign d + /d − ∈ {± } .Recall how the coefficients of the boundary map d in the cellular chain complex of a CW complex X are defined. Let X i denote the i -skeleton. Given a characteristic map j : D i → X i , write π ( j ) : X i ։ S i for the map that sends the complement of the open cell defined by j to the “south pole” − e i andrestricts to the “wrapping map” defined in (3.4) on the open cell itself: π ( j ) ◦ j = w i . Given an( i − j ′ , the coefficient of { j ′ } in d { j } is the degree of the composition π ( j ′ ) ◦ j | ∂D i .So consider X = S p × S q with our chosen cell structure. The coefficients d ± are the degrees of thefollowing two compositions: f ++ : ∂ ( D k × D q ) j + k,q | ∂ ( Dk × Dq ) −−−−−−−−−→ X k − q π ( j + k − ,q ) −−−−−−→ S k − q (3.9) JENS HORNBOSTEL, HENG XIE, AND MARCUS ZIBROWIUS f + − : ∂ ( D k × D q ) j + k,q | ∂ ( Dk × Dq ) −−−−−−−−−→ X k − q π ( j − k − ,q ) −−−−−−→ S k − q (3.10)In order to compare the degrees of these maps, we first describe them more explicitly. Recall from(3.5) our notation φ m,n for the homeomorphism D m × D n ∼ = D m + n , and let ¯ φ m,n denote the inducedhomeomorphism ¯ φ m,n : D m × D n ∂ ( D m × D n ) ∼ = −−−−−→ D m + n ∂D m + n . Also recall that ¯ w n denotes the homeomorphism D n /∂D n → S n induced by the wrapping map w n .Let w m,n : D m × D n → S n + m denote the composition w m,n := w m + n ◦ φ m,n , and let ¯ w m,n denotethe following composition in which the first map is the canonical surjection from the product to thesmash product:¯ w m,n : D m ∂D m × D n ∂D n −−−→ D m ∂D m ∧ D n ∂D n = D m × D n ∂ ( D m × D n ) ∼ = −−−−−→ ¯ φ m,n D m + n ∂D m + n ∼ = −−−−−→ ¯ w m + n S m + n The maps π ( j ± k,q ) : X k + q → S k + q can be described as follows: π ( j + k,q )( x, y ) = ( ¯ w k,q (cid:16) x , . . . , x k − , ¯ w − q (cid:16) − R − x, ( y ) (cid:17)(cid:17) if x k ≥ x, = y − e k + q otherwise π ( j − k,q )( x, y ) = ( ¯ w k,q (cid:16) − ( x , . . . , x k − ) , ¯ w − q (cid:16) − R − − ( x, ( y ) (cid:17)(cid:17) if x k ≤ x, = y − e k + q otherwiseIn either line, the coordinates are x = ( x , . . . , x k , , . . . , ∈ S p and y = ( y , . . . , y q ) ∈ S q . Note thatonly the first k coordinates of x , i.e. the coordinates ( x , . . . , x k − ) ∈ D k , appear as arguments of ¯ w k,q .For the maps f + ± : ∂ ( D k × D q ) → S k + q − in (3.9) and (3.10), we obtain the following description: f ++ ( x, y ) = ( ¯ w k − ,q ( x , . . . , x k − , y ) if x k − ≥ k x k = 1 − e k − q otherwise (3.11) f + − ( x, y ) = ( ¯ w k − ,q (cid:16) − ( x , . . . , x k − ) , ¯ w − q (cid:16) − R − − ( x, R ( x, w q ( y ) (cid:17)(cid:17) if x k − ≤ k x k = 1 − e k − q otherwise (3.12)Here, the coordinates in either line are x = ( x , . . . , x k − ) ∈ D k and y = ( y , . . . , y q − ) ∈ D q .The map f + − we have just computed is homotopic to the following simpler map: e f + − ( x, y ) = ( ¯ w k − ,q (cid:0) − ( x , . . . , x k − ) , ( − q +1 y , y , . . . , y q − (cid:1) if x k − ≤ k x k = 1 − e k − q otherwise (3.13)To construct a homotopy f + − e f + − , first consider the following map: S : j − q − ( D q − ) → O( q + 1) x S x := − R − − x R x Here, x has coordinates ( x , . . . , x q − , ∈ D q with x q − ≤
0. Note that both x and − x lie in j + q ( D q ),so that both R x and R − x are defined. It follows from our description of R in Lemma 3.7 that S x e q = e q for each x . So the map S factors through the inclusion of O( q ) into O ( q + 1) as “upper left block”.Let x S ′ x ∈ O( q ) denote the restricted map: j − q − ( D q − ) S ′ / / S , , O( q ) (cid:31) (cid:127) / / O( q + 1) HOW-WITT RINGS OF SPLIT QUADRICS 9 As j − q − ( D q − ) is contractible, the map S ′ is nullhomotopic. As R x always has determinant one,the determinant of S ′ x can be computed as det( S ′ x ) = det( S x ) = det( − id R q +1 ) = ( − q +1 . We maytherefore choose a homotopy from S ′ to the constant map j − q − ( D q − ) → O( q ) with value the ( q × q )-diagonal matrix T q +1 q , where T iq := ( − i . Consequently, S is homotopic to the map j − q − ( D q − ) → O( q + 1) with constant value ( T q +1 ) q +1 .Denote this homotopy by S ( t ): [0 , × j − q − ( D q − ) → O( q + 1)( x, t ) S x ( t )Then S x (0) = S x , S x (1) = ( T q +1 ) q +1 , and S x ( t ) e q = e q for all x and all t . Now consider the homotopy f : [0 , × ∂ ( D k × D q ) → S k − q defined as follows: f ( t )( x, y ) := ( ¯ w k − ,q ( − ( x , . . . , x k − ) , ¯ w − q ( S ( x, ( t ) · w q ( y ))) if x k − ≤ k x k = 1 − e k − q otherwiseThis is the homotopy from f + − to e f + − that we need. Indeed, the equality f (0) = f + − is obvious,and the equality f (1) = e f + − easily follows once we observe that T q +1 w q ( y ) = w q ( T q y ) for our choiceof wrapping map w q (see (3.4)).It follows in particular that f + − and e f + − have the same degree. On the other hand, we see from theformulas given in (3.11) and (3.13) that e f + − = f ++ σ for the involution σ : ∂ ( D k × D q ) → ∂ ( D k × D q )given by σ ( x , . . . , x k − , y , . . . , y q − ) := ( − x , − x , . . . , − x k − , ( − q +1 y , y , . . . , y q − ) . So altogether we find that d + /d − = deg( f ++ ) / deg( e f + − ) = deg( σ ) = ( − k + q +1 . This shows thatthe cellular chain complex of S p × S q has the form claimed. (cid:3) Now let Z (0) = Z and Z (1) denote the trivial Z [ τ ]-module and the Z [ τ ]-module on which τ actsas multiplication by −
1, respectively. Let C ( n ) ( t ) denote the following chain complexes of abeliangroups: C ( n ) (0) : Z ←− Z ←− Z ←− Z ←− · · · ← Z C ( n ) (1) : Z ←− Z ←− Z ←− Z ←− · · · ← Z These are related to the chain complexes of Z [ τ ]-modules considered above by canonical isomorphisms e C ( n ) ( t ) ⊗ Z [ τ ] Z ( s ) ∼ = C ( n ) ( s + t ). Corollary 3.14.
Assume p ≤ q . There exists a cell structure on Q p,q with p + 2 cells: one i -cell foreach i ∈ { , . . . , p } , and one i -cell for each i ∈ { q, . . . , p + q } . The associated chain complexes withcoefficients in Z ( s ) are given by C ( Q p,q , Z ( s )) ∼ = C ( p ) ( s ) ⊕ C ( p ) ( s + q + 1)[ q ] . This description of C ( Q p,q , Z ( s )) immediately gives us additive descriptions of the homology andcohomology of Q p,q with arbitrary – untwisted and twisted – coefficients. For example, H • ( Q p,q , Z / • ( RP p , Z / Corollary 3.15.
For coefficients R ∈ { Z , Z (1) , Z / } , the embedding ∆ : RP p ֒ → Q p,q and the projec-tion π : Q p,q ։ RP p induce mutually inverse isomorphisms H i ( Q p,q , R ) ∼ = H i ( RP p , R ) in all degrees i < p . In degree p , ∆ ∗ is an epimorphism split by the monomorphism π ∗ . Proof.
This is clear from Figure 2 and the fact that any split monomorphism or epimorphism Z → Z or Z / → Z / (cid:3) Corollary 3.16.
For p < q and s ∈ Z , each cohomology group H i ( Q p,q , Z ( s )) is isomorphic to oneof the groups , Z / or Z . In all degrees i in which H i ( Q p,q , Z ( s )) ∼ = Z , the mod- -reduction map H i ( Q p,q , Z ( s )) → H i ( Q p,q , Z / is a surjection Z ։ Z / . In all degrees i in which H i ( Q p,q , Z ( s )) ∼ = Z / , the mod- -reduction map is an isomorphism.The same assertions are also true in the case p = q in all degrees i = p . The reduction maps H p ( Q p,p , Z ( s )) → H p ( Q p,p , Z / are epimorphisms or monomorphisms as follows: Z ⊕ Z ։ Z / ⊕ Z / when p + s is odd; Z / ֒ → Z / ⊕ Z / when p + s is even.Proof. This is immediate from the cellular chain complexes. (cid:3)
Remark . The additive structure of H • ( Q p,q , Z ⊕ Z (1)) canalso be computed using localization sequences, (twisted) Thom isomorphisms and a topological versionof the blow-up setup described in Section 4. That is, there is a topological variant of the algebro-geometric computations that will follow, which we leave as an exercise to the diligent reader. A thirdapproach would be to derive the topological results from the geometric ones, using the results of theforthcoming article [HWXZ], which will establish an isomorphism given by the real realization functor between the I -cohomology ring of any smooth real cellular variety and the singular cohomology ring ofits real points (see Theorem 7.1 below). The topological computations presented here are intentionallyindependent of these considerations. Mod-two cohomology ring.Proposition 3.18.
The cohomology ring of the real quadric Q p,q with coefficients in Z / has theform H • ( Q p,q , Z /
2) = ( Z / ξ, ζ ] / ( ξ p +1 , ζ + ξ p ζ ) if p = q and p is even Z / ξ, ζ ] / ( ξ p +1 , ζ ) in all other caseswith | ξ | = 1 and | ζ | = q . For p < q , the generators ξ and ζ are the unique non-zero elements of therespective degrees. In the case p = q , ξ is the unique non-zero element of degree 1, and ζ is the uniquenon-zero element in the kernel of ∆ ∗ : H p ( Q p,p , Z / → H p ( RP p , Z / (cf. Figure 2).Proof of 3.18, apart from the computation of ζ when p = q . Write h i for H i ( Q p,q , Z / ξ be thegenerator of h . By Corollary 3.15 and the known cohomology of RP p , each of the powers ξ i generates h i for i ∈ { , . . . , p − } . Moreover, ξ p ∈ h p is non-zero, and ξ p +1 = 0. Let ζ = ζ denote the generatorof the kernel of ∆ ∗ on h q . When p < q , ker(∆ ∗ ) = h q , so ζ is simply a generator of h q . When p = q ,the elements ξ p and ζ together form a basis of h p . Pick generators ζ , . . . , ζ p in the remaining degrees h q , . . . , h p + q , so that | ζ i | = i + q . Poincaré duality implies ξ p − i ζ i = ζ p for all i ∈ { , . . . , p } . Itfollows that ξ i ζ = ζ i for i ∈ { , . . . , p } . In all cases with p < q , it is moreover clear for degree reasonsthat ζ = 0, so in these cases the proof is complete. In the case p = q , it remains to compute ζ . Thiswill be done in the course of the proof of Theorem 3.20 below. (cid:3) Integral cohomology ring.
The integral cohomology ring of RP p with twisted integral coefficientsis as follows: H • ( RP p , Z ⊕ Z (1)) = Z [ ξ, α ] / (2 ξ, ξ p +1 , ξα, α ) with ( | ξ | = (1 , | α | = ( p, p + 1) (3.19)Mod-2-reduction to H • ( RP p , Z /
2) = Z / ξ ] /ξ p +1 is determined by ξ ξ and α ξ p . Theorem 3.20.
The integral cohomology ring of the real quadric Q p,q has the following form: H • ( Q p,q , Z ⊕ Z (1)) ∼ = ( Z [ ξ, α, β ] / (2 ξ, ξ p +1 , ξα, α , β − αβ ) if p = q and p is even Z [ ξ, α, β ] / (2 ξ, ξ p +1 , ξα, α , β ) in all other cases HOW-WITT RINGS OF SPLIT QUADRICS 110 1 2 3 4 5 6 7 8 9 10 11 H • ( Q , , Z ) Z ξ ξ zz ξβ ξ β Z αβ ( β = αβ )H •− ( Q , , Z (1)) 0 zz ξ ξ ⊕ Z α Z β ξ β ξ β h • ( Q , ) zz1 zz ξ zz ξ zz ξ ⊕ zz ξ zz ζ zz ξζ zz ξ ζ zz ξ ζ zz ξ ζ ( ζ = ξ ζ )H • ( Q , , Z ) Z ξ ξ Z β ξ β ξ β H • ( Q , , Z (1)) 0 zz ξ ξ Z α ξβ ξ β Z αβ h • ( Q , ) zz1 zz ξ zz ξ zz ξ zz ξ zz ζ zz ξζ zz ξ ζ zz ξ ζ zz ξ ζ H • ( Q , , Z ) Z ξ ξ ⊕ Z α Z β ξ β ξ β Z αβ ( β = 0)H • ( Q , , Z (1)) 0 zz ξ ξ ξ zz ξβ ξ β ξ β h • ( Q , ) zz1 zz ξ zz ξ zz ξ zz ξ ⊕ zz ξ zz ζ zz ξζ zz ξ ζ zz ξ ζ zz ξ ζ zz ξ ζ ( ζ = 0)H • ( Q , , Z ) Z ξ ξ Z α ξβ ξ β ξ β H • ( Q , , Z (1)) 0 zz ξ ξ ξ Z β ξ β ξ β Z αβ h • ( Q , ) zz1 zz ξ zz ξ zz ξ zz ξ zz ξ zz ζ zz ξζ zz ξ ζ zz ξ ζ zz ξ ζ zz ξ Table 1.
The integral cohomology and the mod-2-cohomology h • := H • ( − , Z /
2) of some real split quadrics,with zz denoting Z / with generators of degrees | ξ | = (1 , , | α | = ( p, p + 1) , | β | = ( q, q + 1) . The generators ξ and α arepullbacks under π (cf. Figure 2) of the generators of H • ( RP p , Z ⊕ Z (1)) that have the same namesin (3.19) . Under mod- -reduction, ξ ξ , α ξ p and β ζ . See Table 1 for some examples.Proof of Theorem 3.20, Part I (integral products and reduction formulas). It is easy to verify that thequotient rings Z [ ξ, α, β ] / ( . . . ) displayed in the theorem have the correct additive structure. To verifythat the ring structure is correct, we begin by computing the products of the non-torsion classes ( α and β ).We first consider the cases with p < q . In these cases, non-torsion classes α and β of the degreesspecified in Theorem 3.20 are unique up to signs, and we already know from the additive structurethat H i ( Q p,q , Z ( s )) = 0 in the bidegrees ( i, s ) = (2 p, ¯0) and ( i, s ) = (2 q, ¯0) in which α and β reside.So α = β = 0. It now follows from (the twisted version of) Poincaré duality (see [DK01, Theorem 5.7and the following remarks]) that the product αβ is a generator.The case p = q requires more thought. We need to understand the productH p ( Q p,p , Z ( p + 1)) × H p ( Q p,p , Z ( p + 1)) → H p ( Q p,p , Z ) , where H p ( Q p,p , Z ( p + 1)) = Z ⊕ Z . Recall from Figure 2 our notation π for the two-fold cover π : S p × S p → Q p,p , the notation π , π : Q p,p ⇒ RP p for the compositions of the two-fold cover Q p,p → RP p × RP p with the projections onto the two factors, and π S , π S : S p × S p ⇒ S p for theprojections onto the two sphere factors. Consider the two-fold cover S p → RP p . The induced mapon the top cohomology group, H p ( S p , Z ) ← H p ( RP p , Z ( p + 1)), is multiplication by ±
2, as can beseen either explicitly from the cellular computations or from degree considerations. We may thereforechoose generators σ ∈ H p ( S p , Z ) and α ∈ H p ( RP p , Z ( p + 1)) such that α maps to 2 σ under thispullback map. Define α i := π ∗ i ( α ) and σ i := ( π Si ) ∗ σ , so that π ∗ ( α i ) = 2 σ i . Choose a generator β of ker(∆ ∗ ) ⊂ H p ( Q p,p , Z ( p + 1)). As π ∗ is split by ∆ ∗ , it follows that α , β forma Z -basis of H p ( Q p,p , Z ( p + 1)). The pullback of β lives in the kernel of (∆ S ) ∗ , hence can be writtenas π ∗ β = a ( σ − σ )for some a ∈ Z .We now analyse what happens in cohomological degree 2 p . As the generators α i are pulled backfrom RP p , we find that α = α = 0. In H • ( S p × S p , Z ), we know that similarly σ = σ = 0, while σ σ is a generator in degree 2 p . The pullback π ∗ : H p ( S p × S p , Z ) ← H p ( Q p,p , Z ) is multiplicationby ±
2. (Indeed, degree considerations show that π ∗ is either ± π ∗ is either ± ± γ ∈ H p ( Q p,p , Z )such that π ∗ γ = 2 σ σ . For this generator, we find α α = 2 γ . For the remaining products of α and β , we obtain: π ∗ ( β ) = a ( − σ σ − ( − p σ σ ) = − a (1 + ( − p ) π ∗ γ, so β = − a (1 + ( − p ) γ,π ∗ ( α β ) = 2 aσ ( σ − σ ) = − aσ σ = − aπ ∗ γ, so α β = − aγ Thus, in the basis of H p ( Q p,p , Z ( p + 1)) given by α , β , the product is described by the followingmatrix: (cid:18) − ( − p a − a − a (1 + ( − p ) (cid:19) By Poincaré duality, this matrix must define a perfect pairing on Z . So a = ±
1, and by changingthe sign of β if necessary, we may as well assume that a = 1. It follows that α β is a generator ofH p ( Q p,p , Z ). For even p , we moreover find that β = α β ; for odd p , we find that β = 0. Altogether,this is precisely the result displayed above, with α written as α . (We also find that 2 β = α − α .)We now determine the images of the various generators under mod-2-reduction. As the elements ξ and α are pulled back from cohomology classes of RP p under π , the formulas for these elements followfrom the corresponding formulas for RP p . The elements β ∈ H p ( Q p,p , Z ( p + 1)) and ζ ∈ H p ( Q p,p , Z / RP p ֒ → Q p,p . As ∆ is split by π , it follows that β ζ , as claimed. (cid:3) End of proof of Proposition 3.18.
We have just observed that β reduces to ζ . The formula for ζ displayed in Proposition 3.18 is therefore immediate from the formula for β that we have alreadyverified. (cid:3) Proof of Theorem 3.20, Part II (torsion products).
As observed in Corollary 3.16, each cohomologygroup H i ( Q p,q , Z ( s )) is either isomorphic to Z / µ ∈ H i ( Q p,q , Z ( s )) and an arbitrary class µ ′ ∈ H i ′ ( Q p,q , Z ( s ′ )). Thenthe product µµ ′ ∈ H i + i ′ ( Q p,q , Z ( s + s ′ )) is again a two-torsion class. If H i + i ′ ( Q p,q , Z ( s + s ′ )) is freeabelian, we deduce µµ ′ = 0. If, on the other hand, H i + i ′ ( Q p,q , Z ( s + s ′ )) ∼ = Z /
2, then by Corollary 3.16the reduction map H i + i ′ ( Q p,q , Z ( s + s ′ )) → H i + i ′ ( Q p,q , Z /
2) is injective. We can therefore computethe product µµ ′ by passing to the mod-2-reductions ¯ µ and ¯ µ ′ and computing the product ¯ µ ¯ µ ′ inH • ( Q p,q , Z / (cid:3) The blow-up setup of Balmer-Calmès
We now turn to algebraic geometry. Recall from Section 2 that Q n denotes the split n -dimensionalquadric over a smooth scheme S over a field F , where n ≥
3. We begin by summarizing some materialfrom [Nen09].
Lemma 4.1.
Let N P px denote the normal bundle of P px in Q n . Then det( N P px ) ∼ = O P px ( n − p − .Proof. See [Nen09, Lemma 6.1 and above Theorem 6.4]. (cid:3)
HOW-WITT RINGS OF SPLIT QUADRICS 13
Definition 4.2.
An affine space bundle of rank r is a Zariski locally trivial fibre bundle f : X → Y with fibres isomorphic to A r . This means that Y = ∪ i ∈ I U i can be covered by open subschemes U i such that f − ( U i ) ∼ = U i × A r over U i for every i ∈ I . Lemma 4.3.
The morphism ρ : Q n − P px → P py described in Section 2 is an affine space bundle ofrank p + 1 with i y : P py ֒ → Q n − P px as a global section. When n is even, ρ is even a vector bundle,and i y is its zero section.Proof. This seems to be well-known (see [Nen09, Section 6]). As we are not aware of a suitablereference, we provide here a proof for the case of odd n for the reader’s convenience. The case ofeven n is similar. Let U ∼ = A p be the open subscheme of P py defined by y i = 0. We will checkthat ρ − ( U ) ∼ = U × A p +1 . We may assume that S = Spec( F ), as all schemes are already definedover F . Let us moreover assume for ease of notation that i = 0. Then ρ − ( U ) is an affine quadricin A p +2 ∼ = P p +2 − V ( y ), defined in terms of the coordinates [ x : . . . : x p : 1 : y : . . . : y p : z ] bythe equation x + x y + · · · + x p y p + z = 0. This affine quadric is isomorphic to U × A p +1 via anisomorphism of coordinate rings as follows: F [ x , x , . . . , x p , y , . . . , y p , z ]( x + x y + · · · + x p y p + z ) → F [ x , . . . , x p , y , . . . , y p , z ] x
7→ − ( x y + · · · + x p y p + z ) x i x i for i = 0 y i y i for all i This isomorphism is clearly compatible with the projections to U . (cid:3) Lemma 4.4.
Let n ≥ . Let ι w : P pw → Q n denote one of the closed embeddings ι x or ι y (or ι x ′ or ι y ′ if n is even). This morphism induces an isomorphism ι ∗ w : Pic( Q n ) ∼ = → Pic( P pw ) under which ι ∗ w O Q ( k ) ∼ = O P pw ( k ) . In particular, each of the mentioned embeddings induces the same isomorphismon Picard groups.Proof. We only deal with P py here, as the other cases are analogous. Recall that Pic( X ) ∼ = CH ( X )when X is smooth over a field. Consider the pullback map induced by the morphism along the toprow of Figure 1: Z ⊕ Pic( S ) ∼ = CH ( P py ) i ∗ y ←− CH ( Q n − P px ) j ∗ ←− CH ( Q n ) i ∗ ←− CH ( P n +1 ) ∼ = Z ⊕ Pic( S )Using Lemma 4.3 and the homotopy invariance of Chow groups, we see that i ∗ y is an isomorphism.Using the assumption n ≥ j ∗ is anisomorphism. As the composition i ◦ j ◦ i y is a linear embedding of P py into P n +1 , the composition ofall these pullback maps is likewise an isomorphism, sending O P n +1 (1) to O P py (1). The claim follows. (cid:3) The closed immersion ι : P px ֒ → Q n is a regular immersion of codimension p (if n = 2 p ) or p + 1(if n = 2 p + 1), since P px and Q n are both smooth (cf. [EGA4, Ch. 4, 17.12.1]). Let Bl denote theblow-up of Q n along P px and let E denote the exceptional fibre. The following proposition shows thatthis setup satisfies [BC09, Hypothesis 1.2]. Proposition 4.5.
There exists a morphism ˜ ρ : Bl → P py making the following diagram commutative: P px ι / / Q n U := Q n − P px ∼ = Bl − E ? _ j o o ρ (cid:15) (cid:15) ˜ j v v ♠♠♠♠♠♠♠♠♠♠♠♠♠♠ E ˜ π O O (cid:31) (cid:127) ˜ ι / / Bl ˜ ρ / / π O O P py (4.6) Here, ι and ˜ ι are closed immersions, π and ˜ π are projections, j and ˜ j are open immersions, and ρ isthe affine space bundle from Lemma 4.3. Proof.
We concentrate on the case n = 2 p ; the case of odd n is similar. Then Bl is contained in (infact equal to) the closed subscheme of Q n × P p = Proj( O S [ x , . . . , x p , y , . . . , y p ] / ( x y + · · · + x p y p )) × Proj( O S [ T , . . . , T p ])defined by the homogeneous polynomials y k T j − y j T k (for k, j = 0 , , . . . , p ) and P pi =0 x i T i = 0. Thisfollows from the universal property of a blow-up, or from the geometric description of the blow-up along P px as the closure of the graph of the rational map Q n → P d , [ x : . . . : x p : y : . . . : y p ] [ y : . . . : y p ],see [Har92, paragraph above Exercise 7.19]. Define ˜ ρ to be the composition Bl ֒ → Q n × P p → P py ,where the first map is the inclusion and the last map is the projection onto the second factor. Therelations y k T j = y j T k guarantee the commutativity of diagram (4.6). (cid:3) Diagram (4.6) descends to the following diagram of Picard groups:Pic( Q n ) π ∗ ( ) (cid:15) (cid:15) j ∗ ≃ / / Pic( U ) ( ρ ∗ ) − ≃ (cid:15) (cid:15) Pic( Q n ) ⊕ Z [ E ] ∼ = Pic( Bl ) ˜ j ∗ ❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧ Pic( P py ) ˜ ρ ∗ ( λ ) o o (4.7)The map λ : Pic( P py ) → Z [ E ] does not vanish in general. Hence, the square does not generally commute(cf. [BC09, Remark 2.2]), but both triangles in this diagram commute. For even n , we now computethe value of λ ( O (1)) as − Proposition 4.8.
Assume n = 2 p with p ≥ . The pullback homomorphism ˜ ρ = (cid:0) λ (cid:1) : Pic( P py ) → Pic( Q n ) ⊕ Z [ E ] sends O P p (1) to ( O Q (1) , − .Proof. All schemes in sight are smooth over a field, so we can identify all Picard groups with codimen-sion one Chow groups as above. The dimensions of our schemes are dim( Bl ) = dim( Q n ) = 2 p anddim( E ) = 2 p −
1. The identification of CH ( Bl ) with CH ( Q n ) ⊕ Z [ E ] is given by π ∗ : CH ( Q n ) → CH ( Bl ) and by ˜ ι ∗ : CH ( E ) = Z [ E ] → CH ( Bl ). This identification is obtained by noting that theusual localization sequence of Chow groups is split exact:0 → CH ( E ) ˜ ι ∗ −→ CH ( Bl ) ˜ j ∗ −→ CH ( U ) → · · · → H , ( Bl ) ˜ j ∗ −→ H , ( U ) ∂ −→ CH ( E ) ˜ ι ∗ −→ CH ( Bl ) ˜ j ∗ −→ CH ( U ) → j ∗ is split surjective via ˜ ρ ∗ ◦ ( ρ ∗ ) − for all degrees. The map ˜ j ∗ : CH ( Bl ) → CH ( U ) in degree one can also be split by π ∗ ◦ ( j ∗ ) − .Next, we describe the maps in diagram (4.7) explicitly in terms of generators. Let us write [ y ] ∈ CH ( Q n ) for the cycle on Q n corresponding to the subscheme defined by y = 0, and analogouslyfor cycles on other schemes. In this notation, each of the groups CH ( P py ), CH ( U ) and CH ( Q n ) isgenerated by the cycle [ y ] corresponding to the line bundle O (1) in each Picard group. The pullbackmaps are determined by ρ ∗ [ y ] = [ y ] = j ∗ [ y ], ˜ ρ ∗ [ y ] = [ T ] and π ∗ [ y ] = [ y ]. By the commutativityof both triangles in diagram (4.7), we see ˜ j ∗ [ T ] = [ y ] = ˜ j ∗ [ y ] in CH ( U ). So the exact sequence 4.9shows that [ T ] − [ y ] = λ [ E ] (4.10)in CH ( Bl ) for some integer λ . This is the integer that we need to compute.To compute λ , first note that for the closed subschemes V ( y ) and V ( T ) of Bl we have an equalityof sets V ( y ) = V ( T ) ∪ E with neither of E or V ( T ) contained in one another. Here, the exceptionaldivisor E is the smooth subscheme defined by { ([ x : . . . : x p ] , [ T : . . . : T p ]) ∈ P p × P p : P x i T i = 0 } .In particular, E is integral, hence an irreducible component of V ( y ). Using [Ful78, Section 1.5], weconclude [ y ] = [ T ] + ℓ [ E ] , (4.11) HOW-WITT RINGS OF SPLIT QUADRICS 15 where ℓ is the length of O V ( y ) ,E as a module over itself. Now note that O V ( y ) ,E is a field: O Bl,E is a discrete valuation ring with maximal ideal given by ( y , y , · · · , y p ); as y i = y T p T , this ideal isprincipal, generated by y . The relation y = 0 in O V ( y ) ,E kills this maximal ideal of O Bl,E , so O V ( y ) ,E is a field as claimed. It follows that ℓ = 1.By comparing (4.10) and (4.11), we conclude that λ = − (cid:3) I -cohomology: additive structure We now embark on our computations of I -cohomology of split quadrics, keeping the notationestablished in Section 2 and in the previous section. For the definition and basic properties of I -and ¯ I -cohomology, we refer to [Fas08], [Mor12], [AF16] and the survey lectures of Jean Fasel in theseProceedings [Fas19]. In particular, these groups may be described using a variant of the Gerstencomplex, in which the entries are powers of the fundamental ideal in the Witt ring of the appropriateresidue fields. We also refer to [Fas08] for twisted coefficients and how they appear when studyingpushforwards, and to Lemmas 4.1 and 4.4 for possible twists that appear for split quadrics. Finally,we note that similarly to Chow groups, I -cohomology groups satisfy homotopy invariance for affinespace bundles over smooth bases: Theorem 5.1.
Let f : X → Y be an affine space bundle (recall Definition 4.2) over a smooth variety Y over a base field F of characteristic = 2 . Then f ∗ : H i ( Y, I j , L ) → H i ( X, I j , f ∗ L ) is an isomorphismfor any line bundle L on Y .Proof. Since Y is quasicompact, we may assume that the open cover in Definition 4.2 is a finite coverby open affines. The pullback along the restriction of f to any affine subset is an isomorphism by[Fas08, Corollaire 11.2.8]. Arguing by induction, we are reduced to the following commutative ladderdiagram with U affine, in which the rows are exact Mayer-Vietoris sequences: · · · / / H i ( Y, I j , L ) f ∗ (cid:15) (cid:15) / / H i ( U, I j , L| U ) ⊕ H i ( V, I j , L| V ) ( f ∗ U ,f ∗ V ) (cid:15) (cid:15) / / H i ( U ∩ V, I j , L| U ∩ V ) f ∗ U ∩ V (cid:15) (cid:15) / / · · ·· · · / / H i ( X, I j , f ∗ L ) / / H i ( X U , I j , f ∗ ( L| U )) ⊕ H i ( X V , I j , f ∗ ( L| V )) / / H i ( X U ∩ V , I j , f ∗ ( L| U ∩ V )) / / · · · The claim now follows by the five lemma.Note that the required Mayer-Vietoris sequence can be deduced from the localization sequence[Fas08, Théorème 9.3.4] and excision along open embeddings for I -cohomology. (More generally, wehave excision along flat morphisms, cf. [CF17, Lemma 3.7].) (cid:3) Theorem 5.2 (Base change formula [AF16, Theorem 2.12]) . Suppose that f : X → Y is a regularcodimension c embedding of smooth schemes that fits into a cartesian diagram of smooth schemes asfollows: X ′ g (cid:15) (cid:15) v / / X f (cid:15) (cid:15) Y ′ u / / Y Suppose that the natural map on normal bundles
Ω : N X ′ Y ′ → v ∗ N X Y induced by u and v is anisomorphism. Then u ∗ ◦ f ∗ = g ∗ ◦ det Ω ∨ ◦ v ∗ , i.e. the following square commutes for any line bundle L over Y : H i ( X ′ , I j , v ∗ f ∗ L ⊗ v ∗ det( N X Y ) ∨ ) det Ω ∨ ∼ = (cid:15) (cid:15) H i ( X, I j , f ∗ L ⊗ det( N X Y ) ∨ ) v ∗ o o f ∗ (cid:15) (cid:15) H i ( X ′ , I j , g ∗ u ∗ L ⊗ det( N X ′ Y ′ ) ∨ ) g ∗ (cid:15) (cid:15) H i + c ( Y ′ , I j + c , u ∗ L ) H i + c ( Y, I j + c , L ) u ∗ o o Fasel’s computations for projective spaces.
Let S be a smooth scheme over a field F of charac-teristic = 2. The following isomorphisms of graded abelian groups are proved by Fasel in [Fas13]: seeCorollary 5.8, Definition 5.9, and Theorems 9.1, 9.2 and 9.4 of loc. cit. (Note that the sequence in 9.4of loc. cit. splits as the bundle E Fasel considers is trivial in our case.)
Theorem 5.3 (Fasel) . H i ( P p , I j , O ( l )) ∼ = (cid:0) L m even2 ≤ m ≤ p H i − m ( S, ¯ I j − m ) (cid:1) ⊕ H i ( S, I j ) if l is even and p is even (cid:0) L m even2 ≤ m ≤ p H i − m ( S, ¯ I j − m ) (cid:1) ⊕ H i ( S, I j ) ⊕ H i − p ( S, I j − p ) if l is even and p is odd (cid:0) L m odd1 ≤ m ≤ p H i − m ( S, ¯ I j − m ) (cid:1) ⊕ H i − p ( S, I j − p ) if l is odd and p is even L m odd1 ≤ m ≤ p H i − m ( S, ¯ I j − m ) if l is odd and p is oddLet p : P p → S be the projection map and s : S → P p a rational point. The isomorphism above onthe component H i − m ( S, I j − m ) → H i ( P p , I j , O ( l )) is given by the pullback p ∗ if l = m = 0 and by thepushforward s ∗ if m = p and l = p −
1. The map µ m L : H i − m ( S, ¯ I j − m ) → H i ( P p , I j , L ( − m )) (5.4)in the isomorphism above is defined in [Fas13, Definition 5.1]; explicitly it is the compositionH i − m ( S, ¯ I j − m ) p ∗ / / H i − m ( P p , ¯ I j − m ) ∂ L ( − / / H i − m +1 ( P p , I j − m +1 , L ( − c ( O (1)) m − / / H i ( P p , I j , L ( − m ))where ∂ L ( − is the connecting homomorphism (or Bockstein homomorphism) [Fas13, § 2.1] and c ( O (1)) is the Euler class homomorphism [Fas13, § 3]. I-cohomology of completely split quadrics.
The aim of this section is to obtain correspondingresults for split quadrics, i. e. to determine the I -cohomology of split quadrics in all bidegrees, withall twists. Over a field, we will later also compute the ring structure; see Section 6.We will always assume n ≥
3. Recall that Q ∼ = P (for which the previous computation applies)and that Q ∼ = P × P (using the Segre embedding in P ). Theorem 5.5.
For the split quadric Q n of dimension n ≥ , we have isomorphisms of groups asfollows: H i ( Q n , I j ) ∼ = (cid:0) L m ∈ T n H i − m ( S, ¯ I j − m ) (cid:1) ⊕ H i ( S, I j ) ⊕ H i − n ( S, I j − n ) if n = 2 p and p is even (cid:0) L m ∈ T n H i − m ( S, ¯ I j − m ) (cid:1) ⊕ H i ( S, I j ) ⊕ H i − p ( S, I j − p ) ⊕ ⊕ H i − n ( S, I j − n ) if n = 2 p and p is odd (cid:0) L m ∈ T n H i − m ( S, ¯ I j − m ) (cid:1) ⊕ H i ( S, I j ) ⊕ H i − p − ( S, I j − p − ) if n = 2 p + 1 and p is even (cid:0) L m ∈ T n H i − m ( S, ¯ I j − m ) (cid:1) ⊕ H i ( S, I j ) ⊕ H i − p ( S, I j − p ) if n = 2 p + 1 and p is odd HOW-WITT RINGS OF SPLIT QUADRICS 17 where T n := (cid:8) ≤ m ≤ n : m even if 1 ≤ m ≤ ⌊ n ⌋ and m odd if ⌈ n ⌉ + 1 ≤ m ≤ n (cid:9) . H i ( Q n , I j , O (1)) ∼ = (cid:0) L m ∈ U n H i − m ( S, ¯ I j − m ) (cid:1) ⊕ H i − p ( S, I j − p ) ⊕ if n = 2 p and p is even (cid:0) L m ∈ U n H i − m ( S, ¯ I j − m ) (cid:1) if n = 2 p and p is odd (cid:0) L m ∈ U n H i − m ( S, ¯ I j − m ) (cid:1) ⊕ H i − p ( S, I j − p ) ⊕ H i − n ( S, I j − n ) if n = 2 p + 1 and p is even (cid:0) L m ∈ U n H i − m ( S, ¯ I j − m ) (cid:1) ⊕ H i − p − ( S, I j − p − ) ⊕ H i − n ( S, I j − n ) if n = 2 p + 1 and p is odd where U n := (cid:8) ≤ m ≤ n : m odd if 1 ≤ m ≤ ⌊ n ⌋ and m even if ⌈ n ⌉ + 1 ≤ m ≤ n (cid:9) . The remainder of this section constitutes a proof of this theorem. The proof relies on homotopyinvariance (Theorem 5.1), localization and dévissage [Fas08, Théorème 9.3.4 and Remarque 9.3.5] for I -cohomology. We need to distinguish cases based on the parities of n , p and the twist, so there areeight different cases to consider. Twisting homomorphisms for quadrics.Definition 5.6.
Let λ m L : H i − m ( S, ¯ I j − m ) → H i ( Q n , I j , O ( − m )) denote the following composition:H i − m ( S, ¯ I j − m ) q ∗ / / H i − m ( Q n , ¯ I j − m ) ∂ L ( − / / H i − m +1 ( Q n , I j − m +1 , L ( − c ( O (1)) m − / / H i ( Q n , I j , L ( − m ))where ∂ L ( − is the connecting homomorphism (or Bockstein homomorphism) [Fas13, § 2.1] and c ( O (1)) is the Euler class homomorphism [Fas13, § 3].This homomorphism is analogous to the map µ m L defined by Fasel on projective spaces (5.4). Lemma 5.7.
The following diagram commutes. H i ( Q n , I j , L ( − m )) j ∗ / / H i ( Q n − P px , I j , L ( − m ))H i − m ( S, ¯ I j − m ) λ m L O O µ m L / / H i ( P py , I j , L ( − m )) ρ ∗ O O Proof.
Similarly to λ m L for Q n and µ m L for P p , we can define twist homomorphisms δ m L : H i − m ( S, ¯ I j − m ) → H i ( Q n − P px , I j , O ( − m )) for Q − P px . The result then follows as the Bockstein homomorphisms andthe Euler class homomorphisms commute with the pullback homomorphisms j ∗ and ρ ∗ , cf. [Fas13,Proposition 2.1 and § 3]. (cid:3) The key short split exact sequence. If n = 2 p (resp. n = 2 p + 1), we defined q := p (resp. q := p + 1). By dévissage, homotopy invariance and localization, we obtain a long exact sequence · · · → H i − q ( P px , I j − q , ω x ⊗ O ( l )) ( ι x ) ∗ / / H i ( Q n , I j , O ( l )) ι ∗ y / / H i ( P py , I j , O ( l )) ∂ / / H i − q +1 ( P px , I j − q , ω x ⊗ O ( l )) → · · · with ω x ∼ = O (1 − q ) (see Lemma 4.1) and q the codimension of P px in Q n . In this section, we provethe following result: Theorem 5.8.
The sequence / / H i − q ( P px , I j − q , ω x ⊗ O ( l )) ( ι x ) ∗ / / H i ( Q n , I j , O ( l )) ι ∗ y / / H i ( P py , I j , O ( l )) / / is split exact.Proof. Case I.
We consider first the case n = dim( Q n ) = 2 p with p even and l even. We will show thatthe map ι ∗ y : H i ( Q n , I j ) → H i ( P py , I j ) is split surjective. By homotopy invariance, the vector bundle projection ρ of Lemma 4.3 induces an isomorphism ρ ∗ : H i ( P py , I j ) → H i ( Q n − P px , I j ). Consider thefollowing commutative diagram: H i ( Q n , I j ) j ∗ / / ι ∗ y * * ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ H i ( Q n − P px , I j ) (cid:0) ≤ m ≤ p L m even H i − m ( S, ¯ I j − m ) (cid:1) ⊕ H i ( S, I j ) α O O β / / H i ( P py , I j ) ρ ∗ O O The commutativity of the upper right triangle follows from the commutative diagram in Figure 1 and i ∗ y = ( ρ ∗ ) − . Here, β is the isomorphism P m µ m + p ∗ of [Fas13, Theorem 9.1], cf. Definitions 5.1 and5.9 and Corollary 5.8 of loc. cit. The map α is defined similarly as β , namely as α := P m λ m + q ∗ .The diagram commutes by Lemma 5.7. Since β and ρ ∗ are both isomorphisms, we obtain a splitting α ◦ β − of ι ∗ y .The cases • n = 2 p + 1, p even, l even • n = 2 p , p odd, l odd • n = 2 p + 1, p odd, l oddare proved similarly. Case II.
Next, we consider the case n = dim( Q n ) = 2 p with p odd and l even. Let Bl be theblow-up of Q n along P px and let E be the exceptional fibre. Recall from Proposition 4.5 above thatthis setup satisfies Hypothesis 1.2 of [BC09]. We use the same notation as in Proposition 4.5. As inthe previous case, we want to show that j ∗ : H i ( Q n , I j ) → H i ( Q n − P px , I j ) is split surjective. We drawthe following diagram:H i P px ( Q n , I j ) / / H i ( Q n , I j ) j ∗ / / H i ( Q n − P px , I j ) ∂ / / H i +1 P px ( Q n , I j )H i ( Bl, I j , ω π ) π ∗ O O ∼ = (cid:15) (cid:15) H i ( P py , I j ) ρ ∗ ∼ = O O ˜ ρ ∗ v v ♠♠♠♠♠♠♠♠♠♠♠♠ H i ( Bl, I j )Using [BC09, Proposition A.11.(iii)], we find that ω π is isomorphic to O ( p − I -cohomology istwo-periodic in the twist, and as p − π by an analogue of [BC09, Proposition 2.1(A)]. Then,arguing as in the proof of [BC09, Theorem 2.3], we see that the middle diagram is commutative: as λ ( O ) = 0 and as p is odd, λ ( O ) ≡ p − n = 2 p , p even and l odd is obtained similarly. In this case, we need to use Proposition 4.8to compute λ ( O (1)) = − Case III.
Now, we consider the case n = 2 p + 1 = dim( Q n ) is odd and p is odd and l is even.We show that ι ∗ : H i − p − ( P px , I j − p − , O ( − p )) → H i ( Q n , I j ) is split injective. Consider the followingdiagram: H i − p − ( P px , I j − p − , O ( − p )) ι ∗ / / red O ( − p ) (cid:15) (cid:15) H i ( Q n , I j ) red (cid:15) (cid:15) H i − p − ( P px , ¯ I j − p − ) ι ∗ / / H i ( Q n , ¯ I j )The diagram is commutative since pushforward commutes with mod-2-reduction. The lower horizontalmap ι ∗ : H i − p − ( P px , ¯ I j − p − ) → H i ( Q n , ¯ I j ) is split injective because ¯ I -cohomology is oriented and HOW-WITT RINGS OF SPLIT QUADRICS 19 because ι ∗ is part of a localization sequence. Let s : H i ( Q n , ¯ I j ) → H i − p − ( P px , ¯ I j − p − ) denote asplitting. The map red O ( − p ) is also split injective: a splitting is given by the commutative diagramH i − p − ( P px , ¯ I j − p − ) H i − p − ( P px , I j − p − , O ( − p )) red O ( − p ) o o L ≤ m ≤ p H i − p − − m ( S, ¯ I j − p − − m ) pr / / ∼ = P ¯ µ m O O ≤ m ≤ p L m odd H i − p − − m ( S, ¯ I j − p − − m ) ∼ = P µ m O O The commutativity and vertical isomorphisms are all established in [Fas13]. Let this splitting bedenoted by s . Now, the compositionH i ( Q n , I j ) red −−→ H i ( Q n , ¯ I j ) s −→ H i − p − ( P px , ¯ I j − p − ) s −→ H i − p − ( P px , I j − p − , O ( − p ))provides a splitting for ι ∗ : H i − p − ( P px , I j − p − , O ( − p )) → H i ( Q n , I j ).The case n = 2 p + 1, p even, l odd is obtained similarly. (cid:3) Proof of Theorem 5.5.
By Theorem 5.8, we can compute the I -cohomology of quadrics from the I -cohomology of projective spaces (Theorem 5.3). (cid:3) I -cohomology: multiplicative structure In this section, our base S is always the spectrum of a field F of characteristic = 2. Given a smoothscheme X over F , we define H tot ( X, I ) := M i,j ∈ Z L∈ Pic( X ) / H i ( X, I j , L )H ⋆, • ( X, ¯ I ) := M i,j ∈ Z H i ( X, ¯ I j )The ¯ I -cohomology ring is a commutative Z ⊕ Z -graded ring. The I -cohomology ring is Z ⊕ Z ⊕ (Pic( X ) / aa ′ = ( − ii ′ a ′ a for homogeneous elements of degrees | a | = ( i, j, ¯ l ) and | a ′ | = ( i ′ , j ′ , ¯ l ′ ), respectively. For X = P n or X = Q n , we have Pic( X ) ∼ = Z , so that the I -cohomology ring is a Z ⊕ Z ⊕ Z / Z ⊕ Z ⊕ Z / I := M i,j ∈ Z ¯ l ∈ Z / H i ( F, I j , ¯ l )concentrated in degrees (0 , ∗ , ¯0):H i ( F, I j , l ) := (cid:26) I j ( F ) if i = 0 and l ≡ I ′ := (H ( F, I , · I ⊂ I . A reader worried about the indexing set Pic( X ) / The ring H tot ( P p , I ) .Theorem 6.1. Consider P p = P pF , where F is a field of characteristic = 2 . For any p ≥ , we havea Z ⊕ Z ⊕ Z / -graded ring isomorphism H tot ( P p , I ) ∼ = I [ ξ, α ] / ( I ′ ξ, ξ p +1 , ξα, α ) with ( | ξ | = (1 , , | α | = ( p, p, p + 1) Proof.
From the additive result of Fasel quoted as Theorem 5.3 above, we conclude:H i ( P p , I j , O ( l )) ∼ = ¯I j − i ( F ) if 1 ≤ i ≤ p and l ≡ i mod 2I j − i ( F ) if i = 0 and l is evenI j − i ( F ) if i = p and l ≡ − p − ξ be the generator of H ( P p , I , O ( − ∼ = Z /
2, and let more generally ξ m ∈ H m ( P p , I m , O ( − m ))be the additive generator for 1 ≤ m ≤ p . This generator is the image of 1 under the map µ m (cf.(5.4)).We first check that the product φξ m = p ∗ φ ∪ ξ m vanishes for an element φ ∈ I j ( F ) if and only if¯ φ = 0 ∈ ¯ I j ( F ), i.e. if and only if φ ∈ I j +1 ( F ). To see this, consider the following diagram, in whichthe horizontal maps in the right square are the mod-2-reductions:H ( F, I j ) × H ( F, ¯ I ) ∪ (cid:15) (cid:15) ( p ∗ ,µ m ) / / H ( P p , I j ) × H m ( P p , I m , O ( − m )) ∪ (cid:15) (cid:15) red × red / / H ( P p , ¯ I j ) × H m ( P p , ¯ I m ) ∪ (cid:15) (cid:15) H ( F, ¯ I j ) µ m / / H m ( P p , I j + m , O ( − m )) red / / H m ( P p , ¯ I j + m )The whole diagram commutes. Indeed, Lemma 5.2 of [Fas13] identifies the compositions red ◦ µ m withmultiplication with ξ m . (Note that we are applying the lemma in the case when i = 0 and Fasel’s basescheme X is Spec( F ), the spectrum of a field. The second summand on the right side of the formulagiven in the lemma is therefore zero since ∂ L ( α ) lies in the trivial group H ( F, I j +1 ).) The commut-ativity of the outer square of the diagram therefore follows from the ring structure on H ⋆, • ( P p , ¯ I ). Thecommutativity of the square in the right half of the diagram is clear. Moreover, we know from theadditive computations that both factors in the lower horizontal composition are isomorphisms. Inparticular, the commutativity of the square on the left follows from the commutativity of the othertwo squares.Now suppose p ∗ φ ∪ ξ m = 0. As ξ m = µ m (1), the commutativity of the left square in the abovediagram shows that this is equivalent to the condition µ m ( ¯ φ ) = 0. As µ m is an isomorphism, this inturn is equivalent to ¯ φ = 0 ∈ ¯ I j ( F ), as claimed.A similar argument as above also shows the commutativity of the following diagram:H ( F, ¯ I ) × H ( F, ¯ I ) ∪ (cid:15) (cid:15) ( µ l ,µ m ) / / H l ( P p , I l , O ( − l )) × H m ( P p , I m , O ( − m )) ∪ (cid:15) (cid:15) H ( F, ¯ I ) µ l + m / / H l + m ( P p , I l + m , O ( − l − m ))This shows that ξ l ξ m = ξ l + m in the appropriate degrees, and hence that ξ m = ξ m is an additivegenerator for 1 ≤ m ≤ p . Note that ξ p +1 = 0 for degree reasons.Finally, let s : S → P p be a rational point, and let α be the image of 1 ∈ W( F ) = H ( F, I ) underthe isomorphism s ∗ : H ( F, I ) → H p ( P p , I p , O ( − p − ξα = 0 and α = 0.It remains to check that αφ = α ∪ p ∗ φ vanishes for an element φ ∈ H i ( F, I j , k ) if and only if φ = 0.This is immediate from the identity α ∪ p ∗ φ = s ∗ ( φ ), which in turn follows from the projection formulaof [CF17]: for any φ ∈ H ( F, I j ), this formula gives s ∗ ψ ∪ p ∗ φ = s ∗ ( ψ ∪ s ∗ p ∗ φ ), and we conclude bytaking ψ = 1 and noting that p ◦ s = id. (cid:3) HOW-WITT RINGS OF SPLIT QUADRICS 21
Milnor cohomology and ¯ I-cohomology.Theorem 6.3 (Karpenko-Merkurjev, Dugger-Isaksen) . The Chow rings of the split quadrics Q n overa field F can be described as follows: CH • ( Q n ) ∼ = Z [ x, y ] / ( x p +1 − xy, y − x p y ) if n = 2 p with p even Z [ x, y ] / ( x p +1 − xy, y ) if n = 2 p with p odd Z [ x, y ] / ( x p +1 − y, y ) if n = 2 p + 1 where | x | = 1 and | y | = p if n is even, | y | = p + 1 if n is odd.Proof. See [DI07, Theorems A.4 and A.10]. (cid:3)
Corollary 6.4.
The bigraded Milnor sheaf cohomology rings H ⋆, • ( Q n , K M ) := L i,j ∈ Z H i ( Q n , K Mj ) of the split quadrics Q n over a field F can be described as follows: H ⋆, • ( Q n , K M ) ∼ = H ⋆, • ( F, K M )[ x, y ] / ( x p +1 − xy, y − x p y ) if n = 2 p with p evenH ⋆, • ( F, K M )[ x, y ] / ( x p +1 − xy, y ) if n = 2 p with p oddH ⋆, • ( F, K M )[ x, y ] / ( x p +1 − y, y ) if n = 2 p + 1 where | x | = (1 , and | y | = ( p, p ) if n is even, | y | = ( p + 1 , p + 1) if n is odd.Proof. In the computation of the bigraded motivic cohomology ring of quadrics [DI07, Proposition 4.3]we can replace motivic cohomology with Milnor sheaf cohomology. To pass from the geometric bide-grees to the full bigraded ring, note that H ⋆, • ( Q n , K M ) is a free module over H ⋆, • ( F, K M ) withgenerators in geometric bidegrees (0 , , n, n ) when n is odd, and in geometric bidegrees(0 , , n, n ) plus an extra generator in bidegree ( n , n ) when n is even. This is proved byan analogous argument as in [DI07, Proposition 4.1], using localization and homotopy invariance ofbigraded Milnor cohomology. (cid:3) Corollary 6.5.
We have isomorphisms of graded commutative rings as follows: H ⋆, • ( Q n , ¯ I ) ∼ = (cid:26) H ⋆, • ( F, ¯ I )[ ¯ ξ, ¯ β ] / ( ¯ ξ p +1 , ¯ β − ¯ ξ p ¯ β ) if n = 2 p with p evenH ⋆, • ( F, ¯ I )[ ¯ ξ, ¯ β ] / ( ¯ ξ p +1 , ¯ β ) if n = 2 p + 1 or ( n = 2 p with p odd) Here, (cid:12)(cid:12) ¯ ξ (cid:12)(cid:12) = (1 , and (cid:12)(cid:12) ¯ β (cid:12)(cid:12) = ( q, q ) where q = p if n = 2 p is even and q = p + 1 if n = 2 p + 1 is odd.Proof. By the Milnor conjecture, we can identify K M • ( F ) / I • ( F ) via the Pfister norm map, andwe can further identify the ring H ⋆, • ( Q n , ¯ I ) with H ⋆, • ( Q n , K M / (cid:3) Remark . If n is odd, the generators ¯ ξ and ¯ β are uniquely defined by the explicit computation of¯ I -cohomology above, so there is no ambiguity in choosing the generators. If n is even, the generator¯ ξ is still uniquely defined, but the generator ¯ β has an ambiguity because H p ( Q p , ¯ I p ) ∼ = Z / ⊕ Z / / / H ( P px , ¯ I ) ( ι x ) ∗ / / H p ( Q p , ¯ I p ) ( ι y ) ∗ / / H p ( P py , ¯ I p ) / / ( P px ′ , ¯ I ) ( ι x ′ ) ∗ O O ( s x ′ ) ∗ ∼ = / / H ( F, ¯ I ) ( s y ) ∗ ∼ = O O The subschemes P py and P px ′ of Q p intersect transversally in a point, so we have a cartesian squareas in diagram (6.10) below. The induced pullback and pushforward morphisms fit into a square asin the diagram above, and the base change formula shows that this square commutes. We choose ¯ β and ¯ α to be ( ι x ) ∗ (1) and ( ι x ′ ) ∗ (1), respectively. Note that the pushforward maps of ¯ I -cohomologyand K M / β and ¯ α map to the mod-2-reductions of Dugger-Isaksen’s generators α and β in [DI07, Section A.5], respectively. Therefore, we have also the relations ¯ α + ¯ β = ¯ ξ p , ¯ β = ¯ ξ p ¯ β if p is even, and¯ β = 0 if p is odd in ¯ I -cohomology, as in the proof of [DI07, Theorem A.10]. The ring H tot ( Q n , I ) . In the following computations, we use the base change theorem for I -cohomology(Theorem 5.2) and the projection formula [Fas13, § 2.2] several times without explicitly mentioningthem. This requires even more care than in the additive case when dealing with twists by line bundles,so we will be even more explicit about these.Let q = p if n = 2 p is even and q = p + 1 if n = 2 p + 1 is odd. Recall the split exact sequence0 / / H i − q ( P px , I j − q , ω x ⊗ O ( l )) ( ι x ) ∗ / / H i ( Q n , I j , O ( l )) ι ∗ y / / H i ( P py , I j , O ( l )) / / ω x ∼ = O (1 − q ) of Theorem 5.8. Note that, strictly speaking, we have identified O ( l ) with( ι y ) ∗ O ( l ) in this exact sequence. This should not cause any confusion: for any closed immersion ι : X → Y of smooth closed subschemes of a fixed projective space, we have a canonical isomorphism ι ∗ O Y ( l ) ∼ = O X ( l ). We will make this simplification of notation throughout this section wheneverpossible, without further comment. Theorem 6.8.
Let F be a field of characteristic = 2 . For p ≥ , we have a Z ⊕ Z ⊕ Z / -graded ring(and even I -algebra) isomorphism H tot ( Q p +1 , I ) ∼ = I [ ξ, α, β ] / ( I ′ ξ, ξ p +1 , ξα, α , β ) Here, the generators are of degrees | ξ | = (1 , , , | α | = ( p, p, p − , and | β | = ( p + 1 , p + 1 , p ) . Thereduction map H tot ( Q p +1 , I ) → H ⋆, • ( Q p +1 , ¯ I ) , which collapses the Z / -grading, is given by ξ ¯ ξ , α ¯ ξ p and β ¯ β .Proof. From the split exact sequence (6.7) for n = 2 p + 1, or by specializing Theorem 5.5 to the case S = Spec( F ), we obtain:H i ( Q p +1 , I j , O ( l )) = ¯I j − i ( F ) if 1 ≤ i ≤ p and l ≡ i mod 2¯I j − i ( F ) if p + 2 ≤ i ≤ p + 1 and l i mod 2I j − i ( F ) if ( i = p or i = p + 1) and l i mod 2I j − i ( F ) if ( i = 0 or i = 2 p + 1) and l ≡ i mod 20 otherwiseWe claim that the reduction map H i ( Q n , I j , O ( l )) → H i ( Q n , ¯ I j ) ∼ = ¯I j − i ( F ) is an isomorphism for1 ≤ i ≤ p and l ≡ i mod 2, and also for p + 2 ≤ i ≤ p + 1 and l i mod 2. Indeed, in the first case,it follows from degree considerations and (6.2) that the map ι ∗ y in sequence (6.7) is an isomorphism; inthe second case, the map ( ι x ) ∗ in this sequence is an isomorphism for degree reasons again. Considerthe following commutative diagrams:H i ( Q n , I j , O ( l )) / / ι ∗ x (cid:15) (cid:15) H i ( Q n , ¯ I j ) ι ∗ x (cid:15) (cid:15) H i − p − ( P py , I j − p − , ω y ⊗ O ( l )) ( ι y ) ∗ (cid:15) (cid:15) / / H i − p − ( P py , ¯ I j − p − ) ( ι y ) ∗ (cid:15) (cid:15) H i ( P px , I j , O ( l )) / / H i ( P px , ¯ I j ) H i ( Q n , I j , O ( l )) / / H i ( Q n , ¯ I j )Using Fasel’s additive computations for projective spaces (6.2) once again, we see that the lower arrowof the left square is an isomorphism for 1 ≤ i ≤ p and l ≡ i mod 2, and that the upper arrow of theright square is an isomorphism for p + 2 ≤ i ≤ p + 1 and l i mod 2. The claim follows.Let ξ ∈ H ( Q n , I , O (1)) ∼ = Z / α and β ofH p ( Q n , I p , O ( p − ∼ = W( F ) and H p +1 ( Q n , I p +1 , O ( p )) ∼ = W( F ), respectively. We see that thereduction map sends ξ to ¯ ξ , α to ¯ ξ p and β to ¯ β . The relations I ′ ξ , ξ p +1 and ξα in the ring structurefollow from the additive results and by considering reduction to H ⋆, • ( Q n , ¯ I ). The relation β = 0 isclear for degree reasons (H p +2 ( Q p +1 , I p +2 , O ( l )) = 0 since 2 p + 2 > dim Q ), and similarly α = 0since H p ( Q p +1 , I p , O (2 p − (cid:3) HOW-WITT RINGS OF SPLIT QUADRICS 23
We now consider the even-dimensional split quadrics over F , i.e. Q n with n = 2 p . In this case p = q in sequence (6.7). We choose the following additive generators: • Let ξ ∈ H ( Q p , I , O ( − ∼ = ¯I ( F ) ∼ = Z / • Let β ∈ H p ( Q p , I p , O ( p − h i ∈ W( F ) under the following morphismof W( F )-modules:W( F ) p ∗ −→ H ( P px , I ) t x −→ H ( P px , I , ω x ⊗ O ( p − ( ι x ) ∗ −−−→ H p ( Q p , I p , O ( p − O → ω x ⊗ O ( p − x for theelement 1 x := t x p ∗ ∈ H ( P px , I , ω x ⊗ O ( p − ι x ′ , ι y , etc. • Let α ∈ H p ( Q p , I p , O ( p − α := ( ι x ′ ) ∗ (1 x ′ ) = ( ι x ′ ) ∗ t x ′ p ∗ x ′ (1):W( F ) p ∗ −→ H ( P px ′ , I ) t x ′ −−→ H ( P px ′ , I , ω x ′ ⊗ O ( p − ( ι x ′ ) ∗ −−−−→ H p ( Q p , I p , O ( p − , where t x ′ is defined by an isomorphism O → ω x ′ ⊗ O ( p − t x and t ′ x more carefully in the proof of the following theorem.However, these choices have no effect on the final result. Theorem 6.9.
Let F be a field of characteristic = 2 . For any p ≥ , we have a Z ⊕ Z ⊕ Z / -gradedring (and even I -algebra) isomorphism H tot ( Q p , I ) ∼ = ( I [ ξ, α, β ] / ( I ′ ξ, ξ p +1 , ξα + ξβ, α + β , αβ ) if p is even I [ ξ, α, β ] / ( I ′ ξ, ξ p +1 , ξα + ξβ, α , β ) if p is oddAs can be seen from the explicit definitions above, the generators here have degrees | ξ | = (1 , , and | α | = | β | = ( p, p, p − . The reduction map H tot ( Q p , I ) → H ⋆, • ( Q p , ¯ I ) , which collapses the Z / -grading, is given by ξ ¯ ξ , β ¯ β and α ¯ ξ p − ¯ β .Proof. From the split exact sequence (6.7) or by specializing Theorem 5.5, we obtain:H i ( Q p , I j , O ( l )) = ¯I j − i ( F ) if 1 ≤ i ≤ p and i ≡ l mod 2¯I j − i ( F ) if p + 1 ≤ i ≤ p and i l mod 2I j − i ( F ) ⊕ I j − i ( F ) if i = p and i l mod 2I j − i ( F ) if ( i = 0 or i = 2 p ) and l ≡ i n = 2 p +1 is odd, we note that the reduction map H i ( Q p , I j , O ( l )) → H i ( Q p , ¯ I j ) ∼ = ¯I j − i ( F ) is an isomorphism for 1 ≤ i ≤ p and l ≡ i , and also for p + 1 ≤ i ≤ p and l i .The reduction map sends ξ to ¯ ξ , α to ¯ ξ p − ¯ β and β to ¯ β .Let us now compute the ring structure of H tot ( Q p , I ). The relations I ′ ξ, ξ p +1 and ξ ( α + β ) followby using the reduction map as before.Let s x , s y , s x ′ and s y ′ denote the inclusions of a point into P p determined by the following cartesiandiagrams: Spec( F ) s x ′ (cid:15) (cid:15) s y / / P pyι y (cid:15) (cid:15) Spec( F ) s y ′ (cid:15) (cid:15) s x / / P pxι x (cid:15) (cid:15) P px ′ ι x ′ / / Q p P py ′ ι y ′ / / Q p (6.10)Let s and ˜ s denote the rational points Spec( F ) → Q p defined by the diagonal of the left and rightdiagram, respectively. For i = p and l = 1 − p , we can use the maps in the left cartesian diagram to make the splitting in Theorem 5.8 explicit:0 / / H ( P px , I j − p ) ( ι x ) ∗ / / H p ( Q p , I j , O ( p − ( ι y ) ∗ / / H p ( P py , I j , O ( p − / / ( P px ′ , I j − p ) ( ι x ′ ) ∗ O O ( s x ′ ) ∗ ∼ = / / H ( F, I j − p ) ( s y ) ∗ ∼ = O O It follows that α and β form a basis of the degree p part H p ( Q p , I p , O ( p − F )-module. In order to compute the various products of the generators α and β , let usconsider a general element( a, b ) ∈ H ( P px ′ , I , ω x ′ ⊗ O ( p − × H ( P py , I , ω y ⊗ O ( p − . Using the base change formula 5.2 for the left cartesian diagram in (6.10) and the projection formula,we find: ( ι x ′ ) ∗ a ∪ ( ι y ) ∗ b = ( ι x ′ ) ∗ (cid:0) a ∪ ι ∗ x ′ ( ι y ) ∗ b (cid:1) = ( ι x ′ ) ∗ (cid:0) a ∪ ( s x ′ ) ∗ s ∗ y b (cid:1) = ( ι x ′ ) ∗ ( s x ′ ) ∗ (cid:0) s ∗ x ′ a ∪ s ∗ y b (cid:1) = s ∗ ( s ∗ x ′ a ∪ s ∗ y b ) (6.11)Similarly, using the right diagram in (6.10), we find( ι x ) ∗ a ∪ ( ι y ′ ) ∗ b = ˜ s ∗ ( s ∗ x a ∪ s ∗ y ′ b ) . When n = 2 p with odd p , we consider the A -homotopy h : P p × A → Q p × A given by h ([ a , a , . . . , a p ] , t ) := ([ ta , − ta , . . . , ta p , − ta p − ; (1 − t ) a , (1 − t ) a , . . . , (1 − t ) a p − , (1 − t ) a p ] , t ) . This homotopy fits into the following commutative diagram consisting of two cartesian squares: P p h / / id × (cid:15) (cid:15) Q p id × (cid:15) (cid:15) P p × A h / / Q p × A P p h / / id × O O Q p id × O O (6.12)Note that h = ι y and h = ι x φ for some linear change of coordinates φ : P p → P p . Using the basechange formula and homotopy invariance, we find that the pushforward maps along h and h coincideup to an identification of the respective normal bundles. Thus, up to an identification of the respectivenormal bundles, ( ι x ) ∗ and ( ι y ) ∗ also agree up to some unit in H ( P p , I ) ∼ = W( F ) depending on φ . Tobe more precise, choose an isomorphism t h : O → ω h ⊗ O (1 − p ). Define t i := (id × i ) ∗ t h for i ∈ { , } ,and choose the isomorphisms t x and t y in the definitions of α and β above Theorem 6.9 such that t = t y and t = φ ∗ t x . Then ( ι x ) ∗ t x and ( ι y ) ∗ t y agree up to multiplication with a unit in W( F ).By combining the identification of ( ι x ) ∗ t x with ( ι y ) ∗ t y and formula (6.11), we find that αβ is agenerator in degree 2 p . (Note that s ∗ in (6.11) is an isomorphism by the previous additive computa-tions.) The following identities moreover prove the relation α = 0. We write “ ∼ ” for “equal up tomultiplication by a unit”:( ι x ) ∗ x ∪ ( ι x ) ∗ x ∼ ( ι x ) ∗ x ∪ ( ι y ) ∗ y by the A -homotopy= ( ι x ) ∗ (1 x ∪ ι ∗ x ( ι y ) ∗ y ) by the projection formula= 0 since ι ∗ x ( ι y ) ∗ y = 0 by sequence (6.7) (6.13)Similarly, we obtain the relation β = 0. HOW-WITT RINGS OF SPLIT QUADRICS 25
When n = 2 p with even p , we consider the following two A -homotopies P p × A → Q p × A : h ([ a , a , . . . , a p ] , t ) := ([ a , ta , ta , . . . , ta p ; 0 , (1 − t ) a , ( t − a , . . . , (1 − t ) a p , ( t − a p − ] , t )˜ h ([ a , a , . . . , a p ] , t ) := ([0 , ta , ta , . . . , ta p ; a , (1 − t ) a , ( t − a , . . . , (1 − t ) a p , ( t − a p − ] , t )Each homotopy fits into a commutative diagram of the form (6.12). Note that h = ι x , h = ι y ′ φ ,˜ h = ι x ′ and ˜ h = ι y ˜ φ for some some linear changes of coordinates φ, ˜ φ : P p → P p .As in the previous case, we choose isomorphisms t : O → ω h ⊗ O (1 − p ) and ˜ t : O → ω ˜ h ⊗ O (1 − p )and we define t i := (id × i ) ∗ t h and ˜ t i := (id × i ) ∗ ˜ t for i ∈ { , } . Then ( h ) ∗ t = ( h ) ∗ t and(˜ h ) ∗ ˜ t = (˜ h ) ∗ ˜ t as maps H ( P p , I ) → H p ( Q p , I p , O ( p − t x , t y , t x ′ and t y ′ suchthat t = t x , t = φ ∗ t y ′ , ˜ t = t x ′ and ˜ t = ˜ φ ∗ t y . Then we find that ( ι x ) ∗ t x and ( ι y ′ ) ∗ t y ′ agree up tomultiplication by a unit in W( F ), and likewise ( ι x ′ ) ∗ and ( ι y ) ∗ agree up to multiplication by a unit.From a computation analogous to (6.13) and the above A -homotopies, we may now deduce therelation αβ = 0. Similarly, using the homotopy h and diagram (6.11), we find that α is the additivegenerator s ∗ ( t ∪ t
1) in degree 2 p . Using the homotopy ˜ h and (6.11) with x ′ and y replaced by x and y ′ , respectively, we find that β is the additive generator ˜ s ∗ (˜ t ∪ ˜ t
1) in degree 2 p . (To beprecise, in these expressions for α and β the isomorphisms “ t i ” and “˜ t i ” really denote the pullbacksof the respective trivializations t i from P p to the base. For example, for t = t x we use the followingcommutative diagram: W( F ) s ∗ x t x (cid:15) (cid:15) p ∗ / / H ( P p , I ) t x (cid:15) (cid:15) W( F, s ∗ x ω x ( p − p ∗ / / H ( P p , ω x ⊗ O ( p − s ∗ x from our notation for simplicity.)To compare α = s ∗ ( t ∪ t
1) and β = ˜ s ∗ (˜ t ∪ ˜ t τ : P p +1 → P p +1 : τ ([ x : x : · · · : x p ; y : y · · · : y p ]) := [ y : x : · · · : x p ; x : y : · · · : y p ]This involution can be restricted to an involution of Q p , which we still denote by τ . It fits into thefollowing commutative diagram of cartesian squares:Spec( F ) ˜ s ' ' ❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖rrrrrrrrrrrrrrrrrrrr / / (cid:15) (cid:15) P ph (cid:15) (cid:15) ④④④④④④④④④④④④④④④④④④ Spec( F ) s ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗ / / (cid:15) (cid:15) P p ˜ h (cid:15) (cid:15) P p rrrrrrrrrrrrrrrrrrrrrrrr h / / Q pτ } } ③③③③③③③③ P p ˜ h / / Q p As an automorphism of P p +1 , τ induces a canonical isomorphism of line bundles τ ∗ O (1) ∼ = O (1).This isomorphism of line bundles restricts to a canonical isomorphism between the respective linebundles O (1) and τ ∗ O (1) on Q p , and we implicitly use this canonical identification in the following. The product τ × id relates the homotopies h and ˜ h , in the sense that we also have the followingcommutative diagram: P p × A h / / ˜ h (cid:15) (cid:15) Q p × A τ × id x x rrrrrrrrrr Q p × A In particular, τ × id induces a canonical isomorphism of the relative canonical line bundles of h and ˜ h ,Ω τ × id : ω h → ω ˜ h . Changing our (previously arbitrary) choices of t and ˜ t if necessary, we may assumethat the following diagram of bundles over P p × A is commutative: O t / / ˜ t (cid:15) (cid:15) ω h ⊗ O (1 − p ) Ω τ v v ♠♠♠♠♠♠♠♠♠♠♠♠♠ ω ˜ h ⊗ O (1 − p )Consider the map τ ∗ : H p ( Q p , I p , O (2 p − → H p ( Q p , I p , O (2 p − τ and thecanonical identification τ ∗ O (1) ∼ = O (1). By the base change formula, we find that τ ∗ s ∗ ( t ∪ t
1) = ˜ s ∗ id ∗ (˜ t ∪ ˜ t p ( Q p , I p , O (2 p − τ ∗ α = β . We claim that the map τ ∗ is precisely − id, so that weobtain α = − β .To prove the claim, we first note that the pushforward map along the inclusion i : Q p ֒ → P p +1 induces an isomorphism i ∗ : H p ( Q p , I p , ω i ⊗ i ∗ L ) → H p +1 ( P p +1 , I p +1 , L )for any even twist L . This can be seen from the additive computation above and the identificationof the relative canonical line bundle ω i of the inclusion as ω i ∼ = O ( − τ ∗ on Q p cantherefore be translated to the study of τ ∗ on P p +1 . Consider a fixed point of τ on P p +1 :Spec( F ) = (cid:15) (cid:15) pt / / P p +1 τ (cid:15) (cid:15) Spec( F ) pt / / P p +1 The base change formula 5.2 implies that τ ∗ pt ∗ differs from pt ∗ by multiplication with the determinantof the map on normal bundles N pt → N pt induced by τ . It is not hard to see that this determinantis just the determinant of the matrix of the linear change of coordinates τ , which is − (cid:3) Geometric bidegrees and real realization
We now compare Theorems 6.8 and 6.9 with Theorem 3.20. They both apply to the split quadrics Q n with n = ( p, p ) or n = ( p + 1 , p ) and n ≥
3, while the latter theorem also holds more generally. The Z ⊕ Z ⊕ Z / tot ( X, I ) contains the Z ⊕ Z / F )-algebra) given bythe condition that the two Z -degrees coincide. This corresponds to motivic bidegrees (2 i, i ) and henceto Chow and Chow-Witt groups. Following [HW19], [HWXZ] and others, we denote this importantsub-W( F )-algebra by H • ( X, I • , O ⊕ O (1)). The computations of Sections 5 and 6 easily restrict tothese W( F )-submodules/subalgebras. For example, restricting Theorem 6.8 to H • ( Q n , I • , O ⊕ O (1))means replacing I by W( F ) and I ′ by I( F ).Now if W( F ) ∼ = Z , e.g. if F = R , then there is an explicit ring isomorphism from this graded commut-ative ring H • ( Q n , I • , O ⊕ O (1)) in algebraic geometry to the graded commutative ring H • ( X ( R ) an , Z ⊕ Z (1)) in topology, given by mapping α to α − β if n = 2 p , and the identity on all other generators. HOW-WITT RINGS OF SPLIT QUADRICS 27
Moreover, there is another obvious isomorphism between the corresponding rings with Z / I -cohomology of the algebro-geometric Q n also lends itself to a computation of the integralsingular cohomology H • ( Q n , Z ⊕ Z (1)) of the topological real quadric. Even more is true. We recallfrom [Jac17] that there is a real realization functorH • ( X, I • ) → H • ( X ( R ) an , Z )for a smooth variety X over R , which is known to be an isomorphism in some degrees in special cases,see loc. cit. and previous work by Fasel and others. By recent work of the authors and MatthiasWendt [HWXZ], we have a stronger result for smooth cellular varieties. Theorem 7.1.
For any smooth variety X over R , the real realization morphism H • ( X, I • ) → H • ( X ( R ) an , Z ) is a ring homomorphism. If X is cellular, then this is an isomorphism.Proof. See [HWXZ]. (cid:3)
This applies in particular to the split quadrics Q n . The preprint [HWXZ] also studies twistedcoefficients and establishes a generalization of Theorem 7.1, replacing I • and Z by ( I • , O ⊕ O (1)) and Z ⊕ Z (1), respectively. We deduce that over F = R the description of Theorems 6.8 and 6.9 extendsto all quadrics covered by Theorem 3.20, in particular to P × P . However, the Picard group is largerin this case, so we have not computed cohomology with respect to all possible twists for P × P . Ifmore generally F is a subfield of R , then the base change composed with real realization is still a ringhomomorphism. 8. Chow-Witt rings
This section contains a short self-contained presentation and computation of the Chow-Witt ringof Q n , arising from the fiber product description first used in [HW19] that emphasizes the relationshipwith Chow groups. This description relies on the isomorphism Ch • ( X ) ∼ = H • ( X, ¯ I • ), and hence onthe proof of the Milnor conjecture of Voevodsky et al.Recall that the Z ⊕ Z ⊕ Z / tot ( Q n , I ) restricts to Chow-Witt groups in bidegrees( i, i ), corresponding to motivic bidegrees (2 i, i ). By Theorem 6.3, we know that CH • ( Q p,p ) andCH • ( Q p,p +1 ) have no two-torsion. Hence to compute g CH • ( Q p,p ) and g CH • ( Q p,p +1 ), we may applypart (1) of Proposition 2.11 of [HW19], which we briefly recall. Proposition 8.1.
Let X be a smooth scheme over a field F of characteristic = 2 . The canonicalring homomorphism g CH • ( X ) → H • ( X, I • ) × Ch • ( X ) ker ∂ induced from the cartesian square defining Milnor-Witt K -theory is always surjective. It is injectiveif CH • ( X ) has no non-trivial two-torsion. Here, ∂ : CH • ( X ) → H • +1 ( X, I • +1 ) is defined as in loc. cit. The kernel of ∂ is a subring of CH • ( X )because the Bockstein homomorphism satisfies a Leibniz formula [Fas07, proof of Proposition 4.7].Note that the general assumption of [HW19] that the base field is perfect is not necessary in Propos-ition 8.1: the proof of [HW19, Proposition 2.11] applies verbatim if we use Zariski sheaf cohomologyin the “key diagram” considered in loc. cit., and Zariski sheaf cohomology can be computed from theGersten complex [Fas07, Theorem 3.26]. Moreover, Proposition 8.1 also applies to twisted coefficients:the proof of loc. cit. goes through when twisting with a line bundle L . Lemma 8.2.
In terms of our chosen generators, the isomorphism of commutative Z / -algebras Ch • ( Q n ) ∼ = −→ H • ( Q n , ¯ I • ) has the form ¯ x ¯ ξ ¯ y ( β + δξ p if n is even, where δ ∈ { , } β if n is odd Proof.
This is obvious, given the degree constraints and that we are working over Z / (cid:3) Putting everything together, we obtain the following result:
Theorem 8.3.
Let Q n be the split quadric over a field F of characteristic = 2 as above, and L a linebundle over Q n . Then the graded GW( F ) -module g CH • ( Q n , L ) is given by g CH • ( Q n , L ) ∼ = → H • ( Q n , I • , L ) × Ch • ( Q n ) ker ∂ L with ker ∂ L given as follows:for L = O : Z h x, y, xy, x , y i when n = 2 p , and p is even Z h x, x , x p , y i when n = 2 p , and p is odd Z h x, x , y i when n = 2 p + 1 , and p is even Z h x, y, xy, x , x p i when n = 2 p + 1 , and p is oddwhere Z h elements i denotes the subring of CH • ( Q n ) generated by the specified elements (see Theo-rem 6.3). The following four twisted cases are described as submodules of CH • ( Q n ) over the respectivefour rings above:for L = O (1) : x · Z h x , xy i + y · Z + ker(mod 2) when n = 2 p , and p is even x · Z h x , x p , y i + ker(mod 2) when n = 2 p , and p is odd x · Z h x , y, x p − y i + ker(mod 2) when n = 2 p + 1 , and p is even x · Z h x , x p , xy, x p − y i + y · Z + ker(mod 2) when n = 2 p + 1 , and p is oddIn this description, mod 2 denotes the mod- -reduction map CH • ( Q n ) → Ch • ( Q n ) .Proof. Everything except the computation of ker ∂ L has been established already. For the latter,one uses that in the notations of the diagram in [HW19, Section 2.4] we have ∂ L = β L ◦ mod 2 andker β L = Im ρ L . (Strictly speaking, the diagram in [HW19, Section 2.4] does include twists by linebundles, but line bundles can be fitted into the diagram without any difficulty.) Our computation ofthe W( F )-algebra H • ( Q p,q , I • ) and the reduction map ρ L (using Lemma 8.2) allow us to completelycompute the image of the latter. Note that for n = 2 p and p even this is independent of the valueof δ , using the relation ¯ ξ p +1 = 0 both in the untwisted and twisted case and moreover the relation y − x p y in the twisted case. From this we easily deduce the kernels of ∂ L and β L both for L = O and L = O (1). (cid:3) For a more concise description of the Z ⊕ Z / F )-algebra g CH • ( Q n , O ⊕ O (1)), weintroduce an artificial Z / • ( Q n ) and Ch • ( Q n ) as follows:CH • ( Q n , O ⊕ O (1)) := CH • ( Q n )[ τ ] / ( τ − • ( Q n , O ⊕ O (1)) := Ch • ( Q n )[ τ ] / ( τ − g CH • ( Q n , O ⊕ O (1)) ∼ = H • ( Q n , I • , O ⊕ O (1)) × Ch • ( Q n , O⊕O (1)) Z h τ, x, y, xτ, x p τ p +1 , yτ q +1 i , where Z h elements i denotes the subring of CH • ( Q n , O ⊕ O (1)) generated by the specified elements.
HOW-WITT RINGS OF SPLIT QUADRICS 29 Milnor-Witt cohomology
The additive computations of I -cohomology can be transferred to Milnor-Witt cohomology. Again,we begin by recalling the corresponding result for projective spaces: Theorem 9.1 ([Fas13, Theorem 11.7]) . Let P n = P nF , where F is a field of characteristic = 2 . Fordegrees i ∈ { , . . . , n } , the Milnor-Witt cohomology groups of P n are as follows: H i ( P n , K MWj , O ( l )) ∼ = K MWj − i ( F ) if ( i = 0 and l ≡ i mod 2)or ( i = n and l i mod 2)K Mj − i ( F ) if i = 0 and l ≡ i mod 22K Mj − i ( F ) if i = n and l i mod 2 In degrees i
6∈ { , . . . , n } , these groups vanish. Now consider once again our split quadric Q n over a field. Write n = p + q with q = p if n is evenand q = p + 1 if n is odd, so that q is the codimension of ι x : P px ֒ → Q n . Recall from Lemma 4.1 thatdet N ∼ = O ( q −
1) for the normal bundle N of this inclusion. As in I -cohomology, localization anddévissage yield a long exact sequence as follows [Fas08, Corollaire 10.4.10]: · · · → H i − q ( P px , K MWj − q , L (1 − q )) ( ι x ) ∗ −−−→ H i ( Q n , K MWj , L ) → H i ( Q n − P px , K MWj , L ) → · · · Homotopy invariance of I -cohomology (Theorem 5.1) together with the corresponding result for Mil-nor cohomology and the usual five lemma argument implies homotopy invariance for Milnor-Wittcohomology. Applying this to the affine space bundle ρ : Q n − P px → P py , we obtain the following exactsequence: · · · → H i − q ( P px , K MWj − q , L (1 − q )) ( ι x ) ∗ −−−→ H i ( Q n , K MWj , L ) ι ∗ y −→ H i ( P py , K MWj , L ) → · · · (9.2) Theorem 9.3.
Let Q n be the n -dimensional split quadric over a field F of characteristic = 2 , with n ≥ . Let L be a line bundle over Q n . The exact sequence (9.2) above splits, so that we haveisomorphisms H i ( Q n , K MWj , L ) ∼ = H i − q ( P px , K MWj − q , L (1 − q )) ⊕ H i ( P py , K MWj , L ) Proof.
Suppose n = 2 p + 1. Then for degree reasons the map ι ∗ y in sequence (9.2) is an isomorphismin the range 0 ≤ i < p , and the map ( ι x ) ∗ is an isomorphism in the range p + 1 < i ≤ n . In degrees i = p and i = p + 1, we have an exact sequence0 → H p ( Q n , K MWj , L ) ι ∗ y → H p ( P py , K MWj , L ) ∂ → H ( P px , K MWj − q , L ⊗ ω x ) ( ι x ) ∗ −→ H p +1 ( Q n , K MWj , L ) → L = O ( p + 1). Consider the following commutative diagram induced by theshort exact sequences of sheaves 0 → I j +1 → K MWj → K Mj → L are invisible to K Mj :H p ( Q n , I j +1 , L ) / / ι ∗ y ∼ = (cid:15) (cid:15) H p ( Q n , K MWj , L ) / / ι ∗ y (cid:15) (cid:15) H p ( Q n , K Mj ) / / ι ∗ y (cid:15) (cid:15) (cid:15) (cid:15) H p +1 ( Q n , I j +1 , L ) (cid:15) (cid:15) H p ( P py , I j +1 , L ) / / H p ( P py , K MWj , L ) / / H p ( P py , K Mj ) / / p +1 ( Q n , I j +1 , O ( p + 1)) in the top right corner vanishes.Our proof of Theorem 5.5 moreover shows that ι y induces an isomorphism on I -cohomology in degree p , so the vertical map on the far left is an isomorphism. Recall from [Fas13, Theorem 11.1] thatH p ( P py , K Mj ) = K Mj − p ( F ) · e ( O (1)) p . By considering the Euler class of the bundle O (1) over Q n (andusing Lemma 4.4), we find that ι ∗ y is surjective on K M -cohomology in degree p . Altogether, it follows from a version of the five lemma that the second vertical map in the diagram is also surjective. So insequence (9.4), ∂ = 0 and both ι ∗ y and ( ι x ) ∗ are isomorphisms.Now consider the case L = O ( p ). Recall from Lemma 4.1 that det( N ) ∨ = O ( − p ). Consider thefollowing commutative diagram induced by the same short exact sequence of sheaves as in the previouscase: 0 / / (cid:15) (cid:15) (cid:15) (cid:15) H ( P px , I j − q +1 ) / / ( ι x ) ∗ ∼ = (cid:15) (cid:15) H ( P px , K MWj − q ) / / ( ι x ) ∗ (cid:15) (cid:15) H ( P px , K Mj − q ) ( ι x ) ∗ ∼ = (cid:15) (cid:15) H p ( Q n , K Mj ) δ / / H p +1 ( Q n , I j +1 , O ( p )) / / H p +1 ( Q n , K MWj , O ( p )) / / H p +1 ( Q n , K Mj )We have isomorphisms on the left and right in this diagram by the computation of I -cohomologyand Milnor cohomology of quadrics above. (One sees this directly from the localization sequenceand degree considerations.) We claim that the boundary map δ vanishes, and that therefore ( ι x ) ∗ is injective on K MW cohomology. To see this, we compare δ with the boundary map in the longexact sequences associated with the coefficient sequence 0 → I j +1 → I j → ¯ I j →
0, via the followingcommutative diagram (see [Fas13, Lemma 11.3]): . . . / / H p ( Q n , K Mj ) δ / / (cid:15) (cid:15) H p +1 ( Q n , I j +1 , O ( p )) / / . . . H p ( Q n , I j , O ( p )) / / H p ( Q n , ¯ I j ) ∂ / / H p +1 ( Q n , I j +1 , O ( p )) / / . . . Our computations of I - and ¯ I -cohomology show that the reduction map on the left of the lowersequence is surjective for odd-dimensional quadrics. So the boundary map ∂ is zero, and hence so is δ . Suppose now that n = 2 p . In this case, the map ι ∗ y in sequence (9.2) is an isomorphism for degreereasons in the range 0 ≤ i < p −
1, and the map ( ι x ) ∗ is an isomorphism for degree reasons in therange p + 1 < i ≤ n . It remains to show that ι ∗ y is also an isomorphism in degree p −
1, that ( ι x ) ∗ isan isomorphism in degree p + 1, and that the following sequence in degree p is split exact:0 / / H ( P px , K MWj − p , ω x ⊗ L ) ( ι x ) ∗ / / H p ( Q n , K MWj , L ) ι ∗ y / / H p ( P py , K MWj , L ) / / p − p + 1, then follow. (cid:3) Corollary 9.5.
The Milnor-Witt cohomology groups of a split quadric Q n over a field F of charac-teristic = 2 in degrees i ∈ { , . . . , n } are as follows:Case n = 2 p : H i ( Q p , K MWj , O ( l )) ∼ = K MWj − i ( F ) if ( i = 0 and l ≡ i mod 2)or ( i = n and l ≡ i mod 2)K MWj − i ( F ) ⊕ K MWj − i ( F ) if i = p and l i mod 2K Mj − i ( F ) ⊕ Mj − i ( F ) if i = p and l ≡ i mod 2K Mj − i ( F ) if (0 < i < p and l ≡ i mod 2)or ( p < i ≤ n and l i mod 2)2K Mj − i ( F ) if (0 ≤ i < p and l i mod 2)or ( p < i < n and l ≡ i mod 2) HOW-WITT RINGS OF SPLIT QUADRICS 31
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Jens Hornbostel, Bergische Universität Wuppertal, Gaußstraße 20, 42119 Wuppertal, Germany
E-mail address : hornbostel at math.uni-wuppertal.de Heng Xie, Bergische Universität Wuppertal, Gaußstraße 20, 42119 Wuppertal, Germany
E-mail address : sysuxieheng at gmail.com Marcus Zibrowius, Heinrich-Heine-Universität Düsseldorf, Universitätsstr. 1, 40225 Düsseldorf, Ger-many
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