THE HOPF CONDITION FOR BILINEAR FORMS OVER ARBITRARY FIELDS DANIEL DUGGER AND DANIEL C. ISAKSEN Abstract.We settle an old question about the existence of certain `sums- of-squares' formulas over a field F, related to the composition problem * *for quadratic forms. A classical theorem says that if such a formula exists * *over a field of characteristic 0, then certain binomial coefficients must van* *ish. We prove that this result also holds over fields of characteristic p > 2. 1.Introduction Fix a field F . A classical problem asks for what values of r, s, and n do th* *ere exist identities of the form i Xr j i Xs j Xn (1.1) x2i. y2i= z2i i=1 i=1 i=1 where the zi's are bilinear expressions in the x's and y's. Equation (1.1) is t* *o be interpreted as a formula in the polynomial ring F [x1, . .,.xr, y1, . .,.ys]; w* *e call it a sum-of-squares formula of type [r, s, n]. The question of when such formulas exist has been extensively studied: [La] a* *nd [S1] are excellent survey articles, and [S2] is a detailed sourcebook. In this * *paper we prove the following result, solving Problem C of [La]: Theorem 1.2. If F is a field of characteristic not equal to 2, and a sum-of-squ* *ares formula of type [r, s, n] exists over F , then nimust be even for n - r < i < * *s. We now give a little history. It is common to let r *F s denote the smallest * *n for which a sum-of-squares formula of type [r, s, n] exists. Many papers have studi* *ed lower bounds on r*F s, but for a long time such results were known only for fie* *lds of characteristic 0: one reduces to a geometric problem over R, and then topologic* *al methods are used to obtain the bounds (see [La] for a summary). In this paper we begin the process of extending such results to characteristic p, replacing t* *he topological methods by those of motivic homotopy theory. The most classical result along these lines is Theorem 1.2 for the particular* * case F = R, which leads to lower bounds for r *R s. It seems to have been proven in three places, namely [B ], [Ho ], and [St]; but in modern times the given condi* *tion on binomial coefficients is usually called the `Hopf condition'. The paper [S1]* * gives some history, and explains how K. Y. Lam and T. Y. Lam deduced the condition for arbitrary fields of characteristic 0. Problem C of [La, p. 188] explicitly * *asked whether the same condition holds over fields of characteristic p > 2. Work on t* *his question had previously been done by Adem [A1 , A2] and Yuzvinsky [Y ] for spec* *ial values of r, s, and n. In [SS] a weaker version of the condition was proved for arbitrary fields and arbitrary values of r, s, and n. 1 2 DANIEL DUGGER AND DANIEL C. ISAKSEN Stiefel's proof of the condition for F = R used Stiefel-Whitney classes; Behr* *end's (which worked over any formally real field) used some basic intersection theory* *; and Hopf deduced it using singular cohomology. Our proof of the general theorem uses a variation of Hopf's method and motivic cohomology. It can be regarded as pure* *ly algebraic_at least, as `algebraic' as things like group cohomology and algebraic K-theory. These days it is perhaps not so clear that there exists a point where topology ends and algebra begins. We now explain Hopf's proof, and our generalization, in more detail. Given a sum-of-squares formula of type [r, s, n], one has in particular a bilinear map OE: F rx F s! F ngiven by (z1, . .,.zn). If we let q be the quadratic form on F* * k given by q(w1, . .,.wk) = w21+ . .+.w2k, then we have q(OE(x, y)) = q(x)q(y). W* *hen F = R one has that q(w) = 0 only when w = 0, and so OE restricts to a map (Rr - 0) x (Rs - 0) ! (Rn - 0). The bilinearity of OE tells us in particular th* *at we can quotient by scalar-multiplication to get RP r-1x RP s-1! RP n-1. On mod 2 cohomology this gives Z=2[x]=xn ! Z=2[a]=ar Z=2[b]=bs, and the bilinearity of OE shows that x 7! a+b. Since xn = 0 and we have a ring map, it * *follows that (a + b)n = 0 in the target ring. The Hopf condition falls out immediately. This proof used, in a seemingly crucial way, the fact that over R a sum of sq* *uares is 0 only when all the numbers were zero to begin with. This of course does not work over fields of characteristic p (or over C, for that matter). Our bilinear* * form gives us a map of schemes OE: Arx As ! An, but we cannot say that it restricts * *to (Ar - 0) x (As - 0) ! (An - 0) as we did above. To remedy the situation, let Qk denote the projective quadric in Pk+1 defined by the equation w21+ . .+.w2k+2= 0. The bilinear map OE induces (Pr-1 - Qr-2) x (Ps-1- Qs-2) ! (Pn-1 - Qn-2). In effect, we have removed all possible numbers whose sum-of-squares would give us zero. Let DQk denote the deleted quadric Pk - Qk-1 (our convention is that the subscript on a scheme always denotes its dimension). We will compute the mod 2 motivic cohomology of DQk (Theorem 2.3), find that it is close to being a truncated polynomial algebra, and repeat Hopf's argument in this new context. As an amusing exercise (cf. [Ln , 6.3]) one can show that over the field C the * *space DQk_with the complex topology_has the same homotopy type as RP k; so our argument is in some sense `the same' as Hopf's in this case. The idea of using deleted quadrics to deduce the Hopf condition first appeared in [SS]. In that paper the Chow groups of the deleted quadrics were computed, but these are only enough to deduce a weaker version of the Hopf condition (one that is approximately half as powerful). This is explained further in Remark 2.* *7. On the other hand, we should point out that the full power of motivic cohomology is not completely necessary in this paper: one can also derive the Hopf conditi* *on using 'etale cohomology, by the same arguments (see Remark 2.8). Since in this * *case computing 'etale cohomology involves exactly the same steps as computing motivic cohomology, we have gone ahead and computed the stronger invariant (to be used in future work). 1.3. Organization. Section 2 shows how to deduce the Hopf condition from a few easily stated facts about motivic cohomology. Section 3 outlines in more detail* * the basic properties of motivic cohomology needed in the rest of the paper. This li* *st is somewhat extensive, but our hope is that it will be accessible to readers no* *t yet THE HOPF CONDITION FOR BILINEAR FORMS OVER ARBITRARY FIELDS 3 acquainted with the motivic theory_most of the properties are analogs of famili* *ar things about singular cohomology. Finally, Section 4 carries out the necessary calculations. We also include an appendix on the Chow groups of quadrics, as several facts about these play a large role in the paper. 2. The basic argument Because of the nature of the computations that we will make, we use slightly different definitions for the varieties Qn and DQn than those in Section 1. The* *se definitions will remain in effect for the entire paper. Unfortunately, the usef* *ulness of these choices will not become clear until Section 4. From now on the field F is always assumed not to have characteristic 2. Definition 2.1. When n = 2k, let Qn be the projective quadric in Pn+1 defined by the equation a1b1 + a2b2 + . .+.ak+1bk+1 = 0. When n = 2k + 1, let Qn be the projective quadric in Pn+1 defined by the equation a1b1+a2b2+. .a.k+1bk+1+c2 = * *0. In either case, let DQn+1 be Pn+1 - Qn. Note that Q0 is isomorphic to SpecF q SpecF , and Q1 ~=P1. One possible isomorphism P1 ! Q1 sends [x, y] to [-x2, y2, xy]. Occasionally we will need to equip DQn+1 with a basepoint, in which case we'll always choose [1, 1, 0, 0, . .,.0] (although the choice turns out not to matter* *). Lemma 2.2. Suppose that the ground field F has a square root of -1 (call it i). Then Qn is isomorphic to the projective quadric in Pn+1 defined by the equation w21+ . .+.w2n+2= 0. Proof.When n = 2k, use the change of coordinates aj = w2j-1+ iw2j, bj = w2j-1- iw2j. When n = 2k + 1, use the same formulas as above for 1 j k + 1 and also let c = wn+2. We regard P2k ,! P2k+1as the subscheme defined by ak+1 = bk+1, and we regard P2k-1,! P2k as the subscheme defined by c = 0. These choices have the advantage that they give us inclusions Qn-2 ,! Qn-1 and DQn-1 ,! DQn. The following theorem states the computation of the motivic cohomology ring H*,*(DQn; Z=2). In order to understand the statement, the reader needs to know just a few basic facts about motivic cohomology; a more complete account of the* *se facts appears in Section 3. First, H*,*(-; Z=2) is a contravariant functor defi* *ned on smooth F -schemes, taking its values in bi-graded commutative rings of characte* *r- istic 2. If we set M2 = H*,*(SpecF ; Z=2), the map induced by X ! SpecF makes H*,*(X; Z=2) into an M2-algebra. It is known that M0,02~=Z=2, M0,12~=Z=2, and the generator ø 2 M0,12is not nilpotent. Theorem 2.3. Assume that every element of F is a square and that char(F ) 6= 2. (a)If n = 2k + 1 then H*,*(DQn; Z=2) ~=M2[a, b]=(a2 = øb, bk+1), where a has degree (1, 1) and b has degree (2, 1). (b)If n = 2k then H*,*(DQn; Z=2) ~=M2[a, b]=(a2 = øb, bk+1, abk) where a and b are as in part (a). (c)The map H*,*(DQn+1; Z=2) ! H*,*(DQn; Z=2) sends a to a, and b to b. In fact, b is the unique nonzero class in H2,1, and a is the unique nonzero c* *lass in H1,1which becomes zero when restricted to the basepoint SpecF ! DQn. These 4 DANIEL DUGGER AND DANIEL C. ISAKSEN facts are needed below in the proof of Proposition 2.5. See the comments before Proposition 4.6 for more details. Note that if ø were equal to 1 then the above rings would be truncated polyno* *mial algebras (in analogy with the singular cohomology of RP n). A more general version of this theorem, without any assumptions on F , appears as Theorem 4.9. The proof is slightly involved, and so deferred until Section * *4. However, let us at least record how the above statements follow from the more general version: Proof.If every element of F is a square, then M1,12= 0 (see Section 3.2). There* *fore, in Theorem 4.9 both æ and ffl are zero. This gives us the formulas in part (a) * *and (b). Part (c) is Proposition 4.6. For us, the most important consequence of the theorem is the following: Corollary 2.4. In H*,*(DQn; Z=2) we have an+1 = 0 and ai6= 0 for i n. Proof.The claims are immediate from the calculation since all the powers of ø a* *re nonzero. Proof of Theorem 1.2.Suppose we have a sum-of-squares formula of type [r, s, n] over F . This remains true if we extend F , so we may as well assume that every element of F is a square. Therefore, Theorem 2.3 applies. As explained in Section 1, the sum-of-squares formula gives a map p: DQr-1 x DQs-1 ! DQn-1 (this uses Lemma 2.2) and we will consider the induced map on motivic cohomology. There is a Künneth formula for computing motivic cohomol- ogy of products of certain `cellular' varieties (see Proposition 3.9), and the * *deleted quadrics belong to this class by Proposition 4.2. In order to apply Proposition* * 3.9, we also have to observe that H*,*(DQr-1; Z=2) is free over M2, which is apparent from Theorem 2.3. Therefore p* is a map H*,*(DQn-1; Z=2) ! H*,*(DQr-1; Z=2) M2 H*,*(DQs-1; Z=2). We will use the letters a and b to denote the generators of H*,*(DQn-1; Z=2), a1 and b1 for the generators of H*,*(DQr-1; Z=2), and a2 and b2 for the generators* * of H*,*(DQs-1; Z=2). We show in the following proposition that p*(a) = a1 + a2. Since the above corollary says that an = 0, it will follow that (a1 + a2)n = 0. Using the corol* *lary again, this can only happen if niis even for n - r < i < s. Proposition 2.5. Suppose that F is a field of characteristic not 2 in which eve* *ry element is a square. If p*, a, a1, and a2 are as in the above proof, then p*(a)* * = a1 + a2. Before we can give the proof, we need to state a few more properties of motiv* *ic cohomology. Once again, more details are given in Section 3. First, Mp,q2is non* *zero only in the range q 0. Second, when every element of F is a square one has M1,12= 0. Finally, motivic cohomology is A1-homotopy invariant in the following sense. Let i0 and i1 denote the inclusions {0} ,! A1 and {1} ,! A1, respectivel* *y. If H :X x A1 ! Y is a map of smooth schemes, then the composites H(Id x i0) and H(Id x i1) induce the same map H*,*(Y ; Z=2) ! H*,*(X; Z=2). Such a map H is called an `A1-homotopy' from H(Id x i0) to H(Id x i1). THE HOPF CONDITION FOR BILINEAR FORMS OVER ARBITRARY FIELDS 5 Proof.Because p*(a) has degree (1, 1), it must be of the form ffl1a1 + ffl2a2 +* * m . 1, where m belongs to M1,12and ffl1 and ffl2 belong to M0,02~=Z=2. Since M1,12= 0 under our assumptions on F , we can ignore m. To show that ffl1 = 1, in light of Theorem 2.3(c) it would suffice to verify that the map DQ1 x {*} ! DQr-1 x DQs-1 ! DQn-1 is A1-homotopic to the standard inclusion DQ1 ,! DQn-1. (A similar argument will show that ffl2 = 1.) Actually we will not quite do this, but instead verif* *y that the composition j :DQ1 x {*} ! DQr-1 x DQs-1 ! DQn-1 ,! DQn+1 is A1-homotopic to the standard inclusion. By Theorem 2.3(c) again, this is eno* *ugh. For the rest of this section we will use the coordinates w1, . .,.wn+2 on Pn+* *1 given in Lemma 2.2. Recall that OE is our bilinear map F rxF s! F n. Let e1, . .,.ekb* *e the standard basis for F k, and let OE(e1, e1) = (u1, . .,.un) and OE(e2, e1) = (v1* *, . .,.vn). Then the map j :DQ1 ! DQn+1 has the form [a, b] 7! [u1a + v1b, u2a + v2b, . .,.una + vnb, 0, 0], and the sum-of-squares formula satisfied by OE tells us that u21+ . .+.u2n= 1, v21+ . .+.v2n= 1, and u1v1 + . .+.unvn = 0. Note that the standard inclusion DQ1 ,! DQn+1 has the same description but where (u1, . .,.un) = (1, 0, . .,.0) and (v1, . .,.vn) = (0, 1, 0, . .,.0). The* * following lemma gives the desired A1-homotopy, since both the map j and the standard inclusion are homotopic to the map [a, b] 7! [0, 0, . .,.0, a, b]. For the following statement, recall that we are still using the coordinates o* *n Pn+1 given by Lemma 2.2. Lemma 2.6. Suppose that F contains a square root of -1. Let u and v be vectors * *in F nsuch that ju2j= 1 = jv2jand jujvj = 0. Then the map f : DQ1 ! DQn+1 given by [a, b] 7! [u1a + v1b, u2a + v2b, . .,.una + vnb, 0, 0] is A1-homotopic to the map [a, b] 7! [0, 0, . .,.0, a, b]. Proof.Let i be a square root of -1. First define a homotopy DQ1x A1 ! DQn+1 by the formula ([a, b], t) 7! [u1a + v1b, u2a + v2b, . .,.una + vnb, ta - tib, tia + tb* *]. This shows that f is homotopic to g, where g is the map [a, b] 7! [u1a + v1b, u2a + v2b, . .,.una + vnb, a - ib, ia + b]. Now define another homotopy DQ1 x A1 ! Pn+1 by the formula ([a, b], t) 7! [tu1a + tv1b, tu2a + tv2b, . .,.tuna + tvnb, a - tib, tia * *+ b]. The assumptions on the u's and v's imply that the sum of the squares in the ima* *ge is exactly equal to a2 + b2, which is nonzero because [a, b] lies in DQ1. So t* *his is actually a homotopy DQ1 x A1 ! DQn+1, showing that g is homotopic to the desired map. 6 DANIEL DUGGER AND DANIEL C. ISAKSEN Remark 2.7. In [SS] a weaker version of the Hopf condition was obtained by computing the Chow ring CH *(DQn), which essentially corresponds to the subring of H*,*(DQn; Z=2) generated by b (see Property (A) in Section 3). This amounts to seeing about half of what motivic cohomology sees. Remark 2.8. When F has a square root of -1, a theorem of [Lv] says that the 'etale cohomology ring H*et(DQn; ~2*) is isomorphic to H*,*(DQn; Z=2)[ø-1 ]* * ~= H*,*(DQn; Z=2) M2 M2[ø-1 ] (see Property (I) below). Since H*,*(DQn; Z=2) is free over M2, this localization is particularly simple: it is precisely a trunc* *ated poly- nomial algebra M2[ø-1 ][a]=an+1. So the Hopf condition could have been proven using 'etale cohomology. Remark 2.9. When every element of F is a square, it follows from the proof of the Milnor conjecture [V3 ] that M2 ~=Z=2[ø]. We never needed this, but it's us* *eful to keep in mind. 3.Review of motivic cohomology The theory now called motivic cohomology was first developed in two main places, namely [Bl1] and [VSF ] (together with many associated papers). The pap* *er [V4 ] proved that the two approaches give isomorphic theories. Below we recall * *the basic properties of motivic cohomology needed in the paper. For various reasons it is difficult to give simple references to [VSF ] so most of our citations wi* *ll be to [SV , Sec. 3] and the lecture notes [MVW ]. 3.1. Basic properties. For every field F , motivic cohomology is a contravariant functor H*,*(-) from the category of smooth schemes of finite type over F to the category of bi-graded commutative rings. Commutativity means that if a 2 Hp,q(X) and b 2 Hs,t(X) then ab = (-1)psba. For the basic construction we refer the reader to [SV , Sec. 3] or [MVW , Sec. 3]. The list of properties below i* *s far from complete, and in some cases we only give crude versions of more interesting properties_but this is all we will need in the present paper. The scheme SpecF will often be denoted by "pt", and we denote H*,*(pt) by M. The ring M can be very complicated (and is, in general, unknown). The motivic cohomology of a scheme is naturally a graded-commutative algebra over M. Property A. The graded subring nH2n,n(X) is naturally isomorphic to the Chow ring CH *(X). [Bl1, p. 268], [MVW , p. 4; Lect. 17]. In particular, M0,0= Z. In general, H*,*(X) is isomorphic to the higher Chow groups of X [V4 , Cor. 1.2]. Property B. For a closed inclusion j :Z ,! X of smooth schemes of codimension c, there is a long exact sequence of the form . .!.H*-2c,*-c(Z) -j!!H*,*(X) ! H*,*(X - Z) ! H*-2c+1,*-c(Z) ! . . . The map j! is called the `Gysin map' or the `pushforward', and it is a map of M-modules. The long exact sequence is called the Gysin, localization, or purity sequence. [Bl1, Sec. 3], [Bl2]. Property C. Let i0 and i1 denote the inclusions {0} ,! A1 and {1} ,! A1, respectively. If H :X x A1 ! Y is a map of smooth schemes, then the composites H(Id x i0) and H(Id x i1) induce the same map H*,*(Y ) ! H*,*(X). Such a map THE HOPF CONDITION FOR BILINEAR FORMS OVER ARBITRARY FIELDS 7 H is called an A1-homotopy from H(Id x i0) to H(Id x i1). [Bl1, Sec. 2], [SV , Prop. 4.2]. Property D. H*,*(Pn) = M[t]=(tn+1), where t has degree (2, 1). [SV , Prop. 4.4]. Property E. If E ! B is an algebraic fiber bundle (i.e., a map which is locally* * a product in the Zariksi topology) whose fiber is an affine space An, then H*,*(B* *) ! H*,*(E) is an isomorphism. Property (E) is easy to prove by inducting on the size of a trivializing cove* *r, and using the Mayer-Vietoris sequence [SV , Prop. 4.1] together with Property (C). Property F. Mp,q= 0 if q < 0, if p > q 0, or if q = 0 and p < 0. [MVW , p. 4; Th. 3.5] Property G. M1,1= F *and M0,1= 0. [Bl1, Th. 6.1], [MVW , p. 4,(2)]. 3.2. Finite coefficients. For every n 2 Z there is also a theory H*,*(-; Z=n) which is related to H*,*(-) by a natural long exact sequence of the form (3.3.).!.H*,*(X) xn-!H*,*(X) ! H*,*(X; Z=n) ! H*+1,*(X) xn-!. . . For the definition see [MVW , Def. 3.4]. The theory satisfies the analogs of P* *roperties (B) through (F) above. Let M2 denote H*,*(pt; Z=2). Since M may contain 2-torsion, M2 is not neces- sarily the same as M=(2)_rather, there is a long exact sequence of the form . .!.Mp,qx2-!Mp,q! Mp,q2! Mp+1,q! . . . This sequence, together with Property (F) and the fact that M0,0= Z, tells us t* *hat M0,02= Z=2. Note that H*,*(X; Z=2) is naturally a commutative algebra over M2. Since M1,1= F *and M0,1= M2,1= 0, we get the exact sequence (3.4) 0 ! M0,12! F *-x2!F *! M1,12! 0 where the map F *! F *sends x to x2. The usual notation is to let ø 2 M0,12 denote the class which maps to -1, and to let æ 2 M1,12denote the image of -1. * *If F has a square root of -1 then æ = 0. Moreover, if every element of F is a squa* *re then M1,12= 0. 3.5. The Bockstein. The Bockstein map fi :H*,*(-; Z=2) ! H*+1,*(-; Z=2) is defined in the usual manner from the maps in the sequence (3.3). A direct conse- quence of the definition (as in topology) is that fi2 = 0. Note that fi(ø) = æ. Property H. For all a, b 2 H*,*(X; Z=2), fi(ab) = fi(a)b + afi(b). [Lv, Lem. 6.* *1]. 3.6. Relation with 'etale cohomology. There is a natural map of bi-graded rings j :H*,*(X; Z=n) ! H*et(X; ~n*) (cf. [MVW , Th. 10.2], for example). In the case n = 2, the element ø maps to the class of -1 in H0et(pt; ~2) ~={1, -1}, and multiplication by this class is an isomorphism on 'etale cohomology. Note in particular that this implies that the powers of ø are all nonzero in H0,*(pt; Z* *=2). Property I. The induced map H*,*(X; Z=2)[ø-1 ] ! H*et(X; ~2*) is an isomor- phism for any smooth scheme X, provided that F has a square root of -1. [Lv]. 8 DANIEL DUGGER AND DANIEL C. ISAKSEN The construction of the map j from [MVW ] makes it clear that the Bockstein on H*,*(-; Z=2) (which can be regarded as induced by the extension 0 ! Z=2 ! Z=4 ! Z=2 ! 0) is compatible with the Bockstein on 'etale cohomology induced by 0 ! ~2 ! ~4 ! ~2 ! 0. If the field contains a square root of -1 then we can identify ~4 with Z=4, and of course ~2 with Z=2. These observations will be used in the proof of Theorem 4.9. 3.7. Reduced cohomology. Given any basepoint of a scheme X (i.e., a map pt ! X), the kernel of the induced map H*,*(X) ! H*,*(pt) is the reduced cohomology of X and is denoted by H~*,*(X). A similar definition applies with Z=n-coefficients. The above map has a splitting (induced by X ! pt), and thus H*,*(X) ~=M ~H*,*(X). Similarly, H*,*(X; Z=2) ~=M2 ~H*,*(X; Z=2). 3.8. A Künneth theorem. Let C denote the smallest class of smooth schemes satisfying the following properties: (1)C contains the affine spaces Ak. (2)If Z ,! X is a closed inclusion of smooth schemes and C contains two of X, Z, and X - Z, then it also contains the third. (3)If E ! B is an algebraic fiber bundle whose fiber is an affine space, then E* * 2 C if and only if B 2 C. The following result is a modest generalization of [J, Th. 4.5], and can be p* *roven using the same techniques. A complete proof, for a more general class of schemes than C, is given in [DI2]. Proposition 3.9. Suppose X and Y are smooth schemes, with at least one of them belonging to C. If either H*,*(X) or H*,*(Y ) is free as an M-module, then one * *has a Künneth isomorphism of bi-graded rings H*,*(X) M H*,*(Y ) ~=H*,*(X x Y ). Similarly, if either H*,*(X; Z=n) or H*,*(Y ; Z=n) is free as an H*,*(pt; Z=n)- module, then one has a Künneth isomorphism of bi-graded rings H*,*(X; Z=n) H*,*(pt;Z=n)H*,*(Y ; Z=n) ~=H*,*(X x Y ; Z=n). 4.Computations In this section F is an arbitrary ground field not of characteristic 2. We w* *ill study the quadrics Qn and DQn. Note that in this generality Lemma 2.2 does not apply; therefore, Qn and DQn cannot necessarily be redefined in terms of sums of squares. We assume char(F ) 6= 2 so that Qn is smooth for all n, not just even * *n. Proposition 4.1. If n is odd, H*,*(Qn) is a free module over M with generators in degrees (0, 0), (2, 1), (4, 2), . .,.(2n, n). If n is even, H*,*(Qn) is a fr* *ee M-module with generators in degrees (0, 0), (2, 1), . .,.(2n, n) plus an extra generator* * in degree (n, n_2). Proof.The proof is by induction. The result for Q0 is obvious, and the result f* *or Q1 ~=P1 is Property (D). Except for the base cases in the previous paragraph, the argument for the odd and even cases is identical. We give details only for the even case, so let n =* * 2k. Let Z be the (n-1)-dimensional subscheme defined by a1 = 0, and let U = Qn-Z. Note that Z is singular (it's the projective cone on Qn-2), and that U ~=An. Let THE HOPF CONDITION FOR BILINEAR FORMS OVER ARBITRARY FIELDS 9 Q0= Qn - {[0, 1, 0, 0, . .,.0]}, and let Z0 = Z - {[0, 1, 0, 0, . .,.0]}. Then * *Z0 ,! Q0 is a smooth pair, with complement An. So the localization sequence for Z0 ,! Q0 gives us an isomorphism ~H*,*(Q0) ~=H*-2,*-1(Z0). The projection map Z0! Qn-2 which forgets the first two homogeneous coordinates is a fiber bundle with fibe* *r A1, hence H*,*(Z0) ~=H*,*(Qn-2) by Property (E). Taking the computations of the previous paragraph together, we conclude that H*,*(Q0) ~= H*-2,*-1(Qn-2) M. By induction, this is free over M with one generator in each degree (0, 0), (2, 1), . .,.(2n - 2, n - 1) plus an extra gen* *erator in degree (n, n_2). Finally, we consider the localization sequence for {[0, 1, 0, . .,.0]} ,! Qn.* * This has the form . . .H*-2n+1,*-n(pt) -ffiH*,*(Q0) H*,*(Qn) H*-2n,*-n(pt) . . . The generators for H*,*(Q0) (as an M-module) must map to zero under ffi for di- mension reasons. It follows that 0 H*,*(Q0) H*,*(Qn) H*-2n,*-n(pt) 0 a short exact sequence of M-modules, in which the outer terms are known to be f* *ree. So the middle term is a direct sum of the outer terms. The right term provides a generator of degree (2n, n), and the left term provides the rest of the generat* *ors. The above proof also shows the following: Proposition 4.2. The schemes Qn and DQn belong to the class C from Section 3.8. Proof.If one knows by induction that Qn-2 belongs to C then so do Z0, Q0, and Qn, in that order. The fact that projective spaces belong to the class C is trivial: one uses t* *he standard algebraic cell decomposition (cf. [F , 1.9.1]). Then since Qn-1 and Pn* * are both in C, so is DQn. By the above result, in order to understand the ring structure on H*,*(Qn) it suffices just to understand the subring H2*,*(Qn) ~= CH *(Qn), because the M- algebra generators lie in degrees (2*, *). The computation of this Chow ring is well-known; the additive computation can be found in [Sw , 13.3], for instance,* * and the ring structure is stated in [KM ]. For the reader's convenience, and beca* *use we need several of the auxiliary facts, we give a complete account in Appendix * *A. These ideas lead to the following result, whose proof is essentially the conten* *t of Theorem A.4 and Theorem A.10. Proposition 4.3. (a)If n = 2k + 1 then as a ring H*,*(Qn) = M[x, y]=(xk+1 - 2y, y2) where x has degree (2, 1) and y has degree (2k + 2, k + 1). (b)If n = 2k and k is odd, then H*,*(Qn) = M[x, y]=(xk+1 - 2xy, y2) where x has degree (2, 1) and y has degree (2k, k). (c)If n = 2k and k is even, then H*,*(Qn) = M[x, y]=(xk+1-2xy, xk+1y, y2-xky) where x has degree (2, 1) and y has degree (2k, k). 10 DANIEL DUGGER AND DANIEL C. ISAKSEN We will now consider the motivic cohomology of the deleted quadrics DQn. The idea is to use the localization sequence (4.4) * j! . .o.oH*-1,*-1(Qn-1)_ooH*,*(DQn)_oioH*,*(Pn)_ooH*-2,*-1(Qn-1)_oo_. . . By Proposition 4.1 the cohomology of Qn-1 has generators as an M-module in degrees (2*, *), so we can completely determine the M-module map j!just by un- derstanding the pushforward map CH *-1(Qn-1) ! CH *(Pn) of Chow groups. For the quadrics, this is discussed in detail in the appendix: all maps are either * *the iden- tity or multiplication by 2. However, a problem now occurs. Because the ground ring M might have 2-torsion, the kernel and cokernel of j!will not necessarily * *be free over M_so we run into complicated extension problems. As a result, we haven't been able to compute the integral motivic cohomology of DQn. The problem goes away if we work with Z=2 coefficients. Proposition 4.5. If F is a field with char(F ) 6= 2, then H*,*(DQn; Z=2) is a f* *ree M2-module with one generator in degree (i, d_i2e) for each 0 i n, where d_i* *2e is the smallest integer that is at least i_2. Proof.The argument from Proposition 4.1 shows that H*,*(Qn-1; Z=2) is free over M2 on the same set of generators as before, and the map of subrings H2*,*(Qn-1)* * ! H2*,*(Qn-1; Z=2) is just quotienting by the ideal (2). By Lemma A.6, we know that the Gysin map j!:H2i,i(Qn-1) ! H2i+2,i+1(Pn) is multiplication by 2 for 0 i < n-1_2, and is an isomorphism for n-1_2< i * *n - 1. If n is odd, then it is the fold map Z Z ! Z for i = n-1_2. The goal is to use the Z=2-analog of (4.4), so we first have to understand the Gysin map j! with Z=2-coefficients. Since H2*,*(Qn-1; Z=2) and H2*,*(Pn; Z=2) are both obtained from integral cohomology simply by quotienting by the ideal (2), it follows that the Gysin map with Z=2-coefficients is an isomorphism, zer* *o, or the fold map in all degrees (2*, *). Since the generators (as M2-modules) li* *ve in these degrees, we find that the kernel and cokernel of j!:H*,*(Qn-1; Z=2) ! H*,*(Pn; Z=2) are both free over M2. If n = 2k, then the generators for cokerj! are in degrees (0, 0), * * (2, 1), (4, 2), . .,.(2k, k), and the generators for kerj!are in degrees (0, 0), (2, 1)* *, . .,.(2k - 2, k - 1). If n = 2k + 1, then the generators are the same, except that kerj! h* *as another generator in degree (2k, k). From the Z=2-analog of (4.4), we have the short exact sequence 0 kerj! H*,*(DQn; Z=2) cokerj! 0. It follows that the middle group is also free over M2. Be aware that the left m* *ap shifts degrees by (-1, -1). We know M0,02~=Z=2. From Property (F) it follows that Mp,q2= 0 if q < 0, if q* * = 0 and p < 0, or if p > q 0. So the above calculation shows that H1,1(DQn; Z=2* *) ~= M1,12 M0,02, where the first summand comes from the motivic cohomology of SpecF* * . Hence, there is a unique nonzero element a 2 ~H1,1(DQn; Z=2). When n > 1 the calculation gives H2,1(DQn; Z=2) ~=Z=2, and we let b denote the unique nonzero element. For n = 1 we have DQ1 ~=A1-0, and it is known that H2,1(A1-0; Z=2) = 0 (see, for instance, [V2 , Lem. 6.8]). In this case we define b = 0 by convent* *ion. THE HOPF CONDITION FOR BILINEAR FORMS OVER ARBITRARY FIELDS 11 Proposition 4.6. The map H*,*(DQn+1; Z=2) ! H*,*(DQn; Z=2) induced by the inclusion takes a to a and b to b. Proof.In light of the definitions of a and b in the previous paragraph, we just* * need to show that Hi,1(DQn+1; Z=2) ! Hi,1(DQn; Z=2) is surjective for i = 1 or i = 2. Consider the diagram Hi+1,1(Pn+1; Z=2)oo_Hi-1,0(Qn; Z=2)ooHi,1(DQn+1;_Z=2)ooHi,1(Pn+1;_Z=2) | | | | | | | | fflffl| fflffl| fflffl| fflffl| Hi+1,1(Pn; Z=2)oo_Hi-1,0(Qn-1; Z=2)ooHi,1(DQn;_Z=2)oo__Hi,1(Pn; Z=2) in which the rows are localization sequences. The map between the groups on the left is the identity. A similar remark applies to the groups on the right. Fina* *lly, the cohomology groups of the quadrics are both isomorphic to Mi-1,02, and the m* *ap between them is also the identity. It follows from a diagram chase that the des* *ired map is surjective. Lemma 4.7. fi(a) = b in H*,*(DQn; Z=2). Proof.For brevity write DQ = DQn. We look at the long exact sequence . .!.H1,1(DQ; Z) x2-!H1,1(DQ; Z) ! H1,1(DQ; Z=2) -ffi!H2,1(DQ; Z) ! . ... The localization sequence (4.4) for integral cohomology, together with the iden* *ti- fication of j! in Lemma A.6, shows that DQ ,! Pn induces an isomorphism on H1,1(-; Z). It follows that if a were the mod 2 reduction of an integral class* *, it would also be the image of a class in H1,1(Pn; Z=2). But * ,! Pn induces an iso- morphism on H1,1(-; Z=2), whereas the class a in H1,1(DQ; Z=2) restricts to zero on the basepoint. We conclude that a can't be the mod 2 reduction of an integral class, and therefore ffi(a) is nonzero. The sequence (4.4) (again with our knowledge of j!) also shows that H2,1(DQ; * *Z) is isomorphic to Z=2, with the generator being in the image of H2,1(Pn; Z) ! H2,1(DQ; Z). It follows that ffi(a) is the unique nonzero element of H2,1(DQ; Z* *), and the mod 2 reduction of ffi(a) is b. We need one more lemma before stating the final result. Lemma 4.8. H2k+1,k+1(DQ2k+2; Z=2) ! H2k+1,k+1(DQ2k+1; Z=2) is injective. Proof.Consider the diagram H2k,k(Q2k+1; Z=2)ooH2k+1,k+1(DQ2k+2;_Z=2)oo_H2k+1,k+1(P2k+2; Z=2)oo_ | | | | | | fflffl| fflffl| fflffl| f H2k,k(Q2k; Z=2)oo_H2k+1,k+1(DQ2k+1; Z=2)oo_H2k+1,k+1(P2k+1; Z=2)oo_ in which the rows are portions of localization sequences. From Proposition 4.1 and Properties (D) and (F), f must be isomorphic to a map M1,12! M1,12, but Lemma A.6 tells us that this map is the zero map. Also, the right-most vertical map is an isomorphism. A diagram chase would give us the desired result, if 12 DANIEL DUGGER AND DANIEL C. ISAKSEN we knew that the left vertical map was injective. This map is equal to the map CH k(Q2k+1) ! CH k(Q2k) after reducing modulo 2. We look at the diagram Z ______CHk(Q2k+1)_____//_CHk(Q2k)______ZOOOZO ~=|| || | ~= k | Z ______CHk(P2k+2)____//_CH(P2k+1)_______Z Lemma A.6 shows that the left vertical map is an isomorphism. Lemma A.9 iden- tifies the right vertical map as the diagonal, and from that information the re* *sult follows at once. Theorem 4.9. Let F be a field with char(F ) 6= 2. (a)If n = 2k + 1 then H*,*(DQn; Z=2) ~=M2[a, b]=(a2 = æa + øb, bk+1) where a has degree (1, 1) and b has degree (2, 1). (b)If n = 2k, there exists an element ffl in M1,12such that H*,*(DQn; Z=2) ~= M2[a, b]=(a2 = æa + øb, bk+1, abk = fflbk) where a and b are as in (a). Remark 4.10. We haven't been able to identify the class ffl in any nontrivial c* *ase. This is not important for proving the Hopf condition, but it would be satisfyin* *g to resolve the issue of whether ffl is equal to 0, or æ, or some other element. Proof.For convenience we will drop subscripts and superscripts: Q = Qn-1, P = Pn, and DQ = DQn. We know H*,*(DQ; Z=2) additively, we just need to determine the ring structure. Note that the map H2i,i(P) ! H2i,i(DQ) is surjective because it is the map CH i(P) ! CH i(DQ). Therefore, the nonzero element t of H2,1(P; Z=2) goes to the nonzero element b of H2,1(DQ; Z=2). Then ti maps to bi, and surjectivity implies that bi must be the unique nonzero element in H2i,i(DQ; Z=2) for 1 i n_2. Lemma 4.7 showed that fi(a) = b. Since fi2 = 0 one has fi(b) = 0, so Property (H) implies that fi(abi) = bi+1. In particular abi is nonzero for 0 i n_2- * *1. Now ~H1,1(DQ; Z=2) ~=M0,02a and H2i-1,i(DQ; Z=2) ~=M0,02 M1,12bi-1 for 1 i n+1_2, where the first factor arises from the generator in degree (2i - 1,* * i). Property (H) and the fact that M2,12= 0 implies that fi(x) = 0 for any x 2 M1,1* *2bi-1. So we cannot have abi2 M1,12bi-1. Based on our knowledge of H*,*(DQ; Z=2) as an M2-module, we can now con- clude that when n = 2k the classes 1, b, b2, . .,.bk and a, ab, ab2, . .,.abk-1* * are a free basis for H*,*(DQ; Z=2) over M2. The argument is slightly harder when n = 2k + 1, because we must show that abk is nonzero (even though its Bockstein is zero). However, we already know that abk is nonzero in H*,*(DQn+1; Z=2). The map H2k+1,k+1(DQn+1; Z=2) ! H2k+1,k+1(DQn; Z=2) is an injection by Lemma 4.8 and takes abk to abk by Propo- sition 4.6. It follows that 1, b, . .,.bk, a, ab, . .,.abk is a free basis for * *H*,*(DQ; Z=2) when n = 2k + 1. We next identify a2. This part of the argument exactly parallels [V2 , pp. 22* *-23]. The class a2 2 ~H2,2(DQ; Z=2) must be a linear combination over M2 of the eleme* *nts a and b: a2 = Aa + Bb where A 2 M1,12and B 2 M0,12~=Z=2. To identify A it's sufficient to look at the image of a2 under H*,*(DQ; Z=2) ! H*,*(DQ1; Z=2), sin* *ce Aa + Bb goes to Aa under this map by Proposition 4.6 and the fact that b = 0 THE HOPF CONDITION FOR BILINEAR FORMS OVER ARBITRARY FIELDS 13 in H*,*(DQ1; Z=2). Note that DQ1 is isomorphic to A1 - 0, and one knows that H*,*(A1 - 0; Z=2) ~=M2[a]=(a2 = æa) by [V2 , Lem. 6.8]. So A = æ. To identify B, let K be the field consisting of F with a square root of -1 ad* *joined (unless F already has a square root of -1, in which case K = F ). Let DQK be the base change of DQ along the map SpecK ! SpecF . Under the induced map H*,*(DQ; Z=2) ! H*,*(DQK ; Z=2), æ maps to zero; so æa+Bb maps to Bb. Hence it suffices to assume that F contains a square root of -1 and show that a2 = øb. Under the map H*,*(DQ; Z=2) ! H*et(DQ; ~2*) the element ø becomes in- vertible (cf. Property (I)), and so we can write a = øa0 (in H*et(DQ; ~2*)), for some a0 2 H1et(pt; ~02). This group is sheaf cohomology with coefficients * *in the constant sheaf Z=2; if fiet is the Bockstein on 'etale cohomology induced by 0 ! Z=2 ! Z=4 ! Z=2 ! 0 one has that fiet(a0) = (a0)2 by a standard property of the Bockstein on sheaf cohomology (the proof is the same as the one in topology* *). Our remarks in Section 3.6 show that the Bocksteins in motivic and 'etale coho- mology are compatible, because F has a square root of -1. So we now compute that (4.11) a2 = ø2(a0)2 = ø2fiet(a0) = ø . fi(øa0) = ø . fi(a) = ø . b in H*et(DQ; ~2*). Note that the third equality uses the analog of Property (H) * *for 'etale cohomology, together with the fact that fi(ø) = æ = 0 (by our assumption* * on F ). As a consequence of (4.11), we have in particular that a2 is nonzero in H*,*(DQ; Z=2)[ø-1 ]. But a2 = Bb, so B must be nonzero. From the sequence (3.4) we recall that M0,12= {0, ø}, so B = ø. We have therefore shown that a2 = æa + øb 2 H2,2(DQ; Z=2). This finishes part (a) of the theorem. For part (b) we just observe that abk 2 H2k+1,k+1(DQ; Z=2), and H2k+1,k+1(DQ; Z=2) ~=M1,12bk. So for some ffl 2 M1,12we have fflbk = abk. This finishes part (b). Remark 4.12. When n is odd, the cohomology of DQn is the same as the cohomol- ogy of the scheme (An - 0)= 1, which was essentially computed by Voevodsky in [V2 , Th. 6.10]. It seems likely that these two schemes are A1-homotopy equival* *ent, but we haven't proven this. Appendix A. Chow groups of quadrics This appendix contains a calculation of the Chow rings of the quadrics Qn, as well as various pushforward and pullback maps. This is classical, but the detai* *ls are useful and we don't know a suitable reference. We assume a basic familiarity with the Chow ring; see [F ] or [H , App. A]. Let CH i(X) be the Chow group of dimension i cycles on X. If Z ,! X is a closed subscheme there is an exact sequence CH i(Z) ! CH i(X) ! CH i(X - Z) ! 0 where the first map is pushforward and the second map is restriction. If X Pn is a closed subscheme, we let =X Pn+1 denote the projective cone on X. Let =:CH i(X) ! CH i+1(= X) be the map sending a cycle to the projective cone on the cycle, and recall that this is an isomorphism for i 0. Also note * *that CH 0(= X) = Z no matter what X is. Finally, recall that CH i(An) = 0 if i 6= n, whereas CH n(An) = Z. 14 DANIEL DUGGER AND DANIEL C. ISAKSEN When X is nonsingular one defines CH i(X) = CH dimX-i(X). The following discussion is modeled on [Sw , 13.3]. A.1. The odd-dimensional case. Consider the quadric Q2k+1,! P2k+2 defined by a1b1 + . .+.ak+1bk+1 + c2 = 0. We let j be the inclusion. Lemma A.2. For all 0 i 2k + 1, the Chow group CH i(Q2k+1) is isomorphic to Z. The pushforward map j* : CH i(Q2k+1) ! CH i(P2k+2) is an isomorphism if 0 i k, and is multiplication by 2 (as a map Z ! Z) if k + 1 i 2k + 1. Proof.The first claim follows immediately from Proposition 4.1 and Property (A). The proof of the second statement is by induction. The base case Q1 is isomor- phic to P1, and Q1 is imbedded in P2 as a degree two hypersurface. So j* is an isomorphism for i = 0 and is multiplication by 2 for i = 1. If Z is the closed subscheme defined by a1 = 0, we know Q2k+1- Z ~=A2k+1 and Z ~== Q2k-1. The resulting localization sequence gives us a diagram CH i(= Q2k-1)___//_CHi(Q2k+1)___//_CHi(A2k+1) | | | |j* fflffl| fflffl| CH i(= P2k)_____//CHi(P2k+2) in which the top row is exact. Since =P2k is isomorphic to P2k+1, the bottom horizontal arrow is an isomorphism for all 0 i 2k + 1, and both groups in t* *he bottom row are isomorphic to Z. For 0 i 2k, the first two groups in the top row are also isomorphic to Z.* * For 0 i k, the left vertical arrow is known by induction to be an isomorphism. * *The only possibility is that the map j* is an isomorphism in this range. Now for k + 1 i 2k, the left vertical arrow is known by induction to be multiplication by 2. Since the upper left horizontal arrow is a surjection, the* * only possibility is that the map j* is multiplication by 2. Finally, for the case i = 2k + 1 note that Q2k+1 is a degree 2 hypersurface in P2k+2. Thus, the fundamental class [Q2k+1] maps to twice the generator of CH 2k+1(P2k+2). By analyzing the above proof, one can give explicit generators for CH i(Q2k+1* *). If 0 i k, the generator is the class of the cycle determined by setting all coordinates equal to zero except for b1, . .,.bi+1. Note that this cycle is iso* *morphic to Pi. On the other hand, if k + 1 i 2k + 1, then the generator is the clas* *s of the cycle determined by setting a1, . .,.a2k+1-iequal to zero. Note that this c* *ycle is the iterated projective cone on Q2i-2k+1, and also the intersection of Q2k+1* *with a copy of Pi+1. We next want to compute the ring structure on CH *(Q2k+1) as well as the pullback map j* : CH i(P2k+2) ! CH i-1(Q2k+1). It is easier to do the latter fi* *rst. Proposition A.3. The map j*: CH i(P2k+2) ! CH i-1(Q2k+1) is an isomorphism if k + 2 i 2k + 2 and is multiplication by 2 if 1 i k + 1. Proof.The projection formula j*(a . j*b) = (j*a) . b gives us j*(j*[Pi]) = j*([Q2k+1] . j*[Pi]) = j*([Q2k+1]) . [Pi] = 2[P2k+1] . [Pi] = 2[* *Pi-1]. In other words the composition j*j* : CH i(P2k+2) ! CH i-1(P2k+2) is multiplica- tion by 2. When 1 i k + 1, the map j* is an isomorphism, so j* must be THE HOPF CONDITION FOR BILINEAR FORMS OVER ARBITRARY FIELDS 15 multiplication by 2. When k + 2 i 2k + 2, the map j* is multiplication by 2, so j* must be an isomorphism. It is now easy to deduce the ring structure on CH *(Q2k+1), using the map from CH *(P2k+2). Note that when k = 0 we are looking at Q1 ~=P1, and so CH *(Q1) is isomorphic to Z[x]=x2, where x has degree 1. Theorem A.4. If k 0, then CH *(Q2k+1) ~=Z[x, y]=(xk+1 - 2y, y2), where x has degree 1 and y has degree k + 1. Proof.The map j*: CH i(P2k+2) ! CH i(Q2k+1) (which now preserves the grading because we are grading by codimension) is an isomorphism if 0 i k and is multiplication by 2 if k + 1 i 2k + 1. This follows immediately from the previous proposition simply by regrading. Let t be the generator of CH 1(P2k+2), and let x = j*(t). Then xk+1 = j*(tk+1) is twice a generator of CH k+1(Q2k+1), and we let y be this generator. The de- sired isomorphism of rings follows immediately from our knowledge of the groups CH *(Q2k+1) and the description of j* in the previous paragraph. A.5. The even-dimensional case. This case is a little harder. The quadric Q2kis defined by a1b1+. .+.ak+1bk+1 = 0. As before, let j be the inclusion Q2k ! P2k+* *1. Many of the results from the previous section carry over to this section with i* *dentical proofs. The base case is Q0, which is pt q pt. Note that j* : CH 0(Q0) ! CH 0(P1) is * *the fold map Z Z ! Z. We already know from Proposition 4.1 that the Chow group CH i(Q2k) is iso- morphic to Z for all 0 i 2k, except that CH n(Q2k) is isomorphic to Z Z. The same arguments as in the proof of Lemma A.2 allow us to conclude that the pushforward map j* : CH i(Q2k) ! CH i(P2k+1) is an isomorphism if 0 i k - 1, is multiplication by 2 if k + 1 i 2k, and is the fold map if i = k. We summ* *arize these facts (with cohomological grading) in the following lemma, which we state because it is critical for the computations in Section 4. Lemma A.6. For any n, the map j*: CH i(Qn-1) ! CH i+1(Pn) is multiplication by 2 for 0 i < n-1_2, and is an isomorphism for n-1_2< i n - 1. If n is odd* *, then it is the fold map Z Z ! Z for i = n-1_2. Once again, one can give explicit generators for CH i(Q2k). For i 6= k, the description of these generators is the same as in the odd case. For i = k, one generator is determined by b1 = b2 = . .=.bk+1 = 0, and the other generator is determined by a1 = b2 = . .=.bk+1 = 0. We let ff and fi represent these two codimension k cycles. Lemma A.7. Let ff0be the cycle determined by a1 = a2 = . .=.ak+1 = 0, and let fi0 be the cycle determined by b1 = a2 = . .=.ak+1 = 0. If k is odd, then ff = * *ff0 and fi = fi0 in CH n(Q2k). If k is even, then ff = fi0 and fi = ff0in CH n(Q2k). Proof.If k is odd, then ff and ff0do not meet. Then [HP , Th. XII.4.III] says t* *hat ff and ff0are rationally equivalent. Similarly, fi = fi0. The same argument applies to the even case. Lemma A.8. Let [*] be the fundamental class of a point in CH 2k(Q2k). If k is odd, then ff . ff = 0 = fi . fi and ff . fi = [*] in the Chow ring CH *(Q2k). I* *f k is even, then ff . ff = [*] = fi . fi and ff . fi = 0. 16 DANIEL DUGGER AND DANIEL C. ISAKSEN Proof.When k is odd, ff . ff = ff . ff0. However, ff and ff0 do not intersect,* * so ff . ff0= 0. Similarly, fi . fi = 0. Now ff and fi0 intersect transversely at a* * point, so ff . fi = ff . fi0= [*]. Similar arguments apply to the even case. As in Proposition A.3, the map j*: CH i(P2k+1) ! CH i-1(Q2k) is an isomor- phism if k + 2 i 2k + 1 and is multiplication by 2 if 1 i k. After regrading by codimension, this says j*: CH i(P2k+1) ! CH i(Q2k) is an isomor- phism for 0 i < k and multiplication by 2 for k < i 2k. The same argument with the projection formula also shows that when i = k, j* takes the generator * *to uff + (2 - u)fi for some u 2 Z. Lemma A.9. The map j*: CH k(P2k+1) ! CH k(Q2k) sends the generator tk to ff + fi. Proof.We already know that j*(t2k) = 2[*], where [*] is the fundamental class of a point in Q2k and is also the generator of CH 2k(Q2k). Therefore, 2[*] = j*(t2k) = (j*(tk))2 = (uff + (2 - u)fi)2 = u2ff2 + 2u(2 - u)fffi + (2 - * *u)2fi2. If k is odd, Lemma A.8 lets us rewrite this equation as 2[*] = 2u(2-u)[*], so u* * = 1. If k is even Lemma A.8 gives 2[*] = (u2 + (2 - u)2)[*], so again u = 1. Theorem A.10. If k is odd, then there is an isomorphism of rings CH *(Q2k) ~= Z[x, y]=(xk+1-2xy, y2), where x has degree 1 and y has degree k. If k is even, * *then CH *(Q2k) ~=Z[x, y]=(xk+1 - 2xy, xk+1y, y2 - xky), where x has degree 1 and y h* *as degree k. Proof.Let t be the generator [P2k] of CH 1(P2k+1), and let x = j*(t). Then j*(t* *i) = xi. As we know that j* takes generators to generators for 0 i k - 1, it fol* *lows that xi is a generator for CH i(Q2k) in these dimensions. Now xk = j*(tk) = ff+fi by the previous lemma. If we let y equal ff, then xk * *and y are two generators for CH k(Q2k). Note that Lemma A.8 implies that xky = [*] since ff(ff + fi) = [*] in both the even and odd cases. 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Not. 2002, no. 7, 351-355. 18 DANIEL DUGGER AND DANIEL C. ISAKSEN [VSF]V. Voevodsky, A. Suslin, and E. M. Friedlander, Cycles, transfers, and mot* *ivic homology theories, Annals of Mathematics Studies 143, Princeton University Press, P* *rinceton, NJ, 2000. [Y] S. Yuzvinsky, On the Hopf condition over an arbitrary field, Bol. Soc. Mat* *. Mexicana 28 (1983), 1-7. [Z] O. Zariski, Algebraic surfaces, Springer-Verlag: Berlin Heidelberg New Yor* *k, 1971. Department of Mathematics, University of Oregon, Eugene, OR 97403 Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556 E-mail address: ddugger@math.uoregon.edu E-mail address: isaksen@math.wayne.edu