, and
=

, <1> - = <-p, -p>. N*
*ote
that
= = <1, -1, -1, 1> = <1, 1, 1, 1>
and so 4(<1> -

) = 0. Also,

=

= <1, -1, -p> and <1, 1, 1> = <1, -1, -1>. So 3(<1> -

) = <-1> - <-p>. Of course GW (Qp) is generated by <1>, <1> - <-1* *>, <1> -

, and <1> - <-p>, and the previous computation shows the last generator is not needed. So we have a surjective map Z Z=2 Z=4 ! GW (Qp) sending the standard generators to <1>, <1> - <-1>, and <1> -

. This is readily chec* *ked to be injective once one knows that <1, 1> 6~=

. If these forms were isomo* *rphic it would follow by reduction mod some power of p that <1, 1> was isotropic over NOTES ON THE MILNOR CONJECTURES 25 some Fpe; that is, we would have <1, 1> ~=<1, -1>. But we've already computed GW (Fpe), and know this is not the case. The Witt ring is W (Qp) ~=Z=4 Z=4 with generators <1> and <1> -

. The ideal I is generated by 2<1> and <1> -

; I2 is generated by 2(<1> -

); I3* * = 0. Again we have GrIW ~=Z=2 (Z=2 Z=2) Z=2. (g) Return to F = Q. Our understanding of the higher Milnor K-groups of Q is based on passing to the various completions Qp and R. A computation of Bass and Tate [Mi , Lem. A.1] gives an exact sequence i j 0 ! KM2(Q)=2 ! KM2(R)=2 pKM2(Qp)=2 ! Z=2 ! 0, and we already know KM2(Qp)=2 ~=KM2(R)=2 ~=Z=2. A computation of Tate [Mi , Th. A.2, Ex. 1.8] shows that for * 3 one has KM*(Q)=2 ~= pKM*(Qp)=2 KM*(R)=2 ~=0 Z=2. To compute H*(Q; Z=2) we again work one completion at a time. A theorem of Tate [Se, Section II.6.3, Th. B] says that for i 3 one has Y Hi(Q; Z=2) ~=Hi(R; Z=2) x Hi(Qp; Z=2) ~=Hi(R; Z=2) ~=Z=2. p Our computationQof Q*=(Q*)2 ~=H1(Q; Z=2) shows that the map H1(Q; Z=2) ! H1(R; Z=2)x p H1(Qp; Z=2) is injective. More of Tate's work [Se, Sec. II.6.3, * *Th. A] identifies the dual of the kernel with the kernel of H2(Q; Z=2) ! H2(R; Z=2)* * x ( pH2(Qp; Z=2))_thus, this latter map is also injective. Using this, [Se, Sec. * *II.6.3, Th. C] gives a short exact sequence 0 ! H2(Q; Z=2) ! H2(R; Z=2) ( pH2(Qp; Z=2)) ! Z=2 ! 0. As we have already remarked that H2(Qp; Z=2) = H2(R; Z=2) = Z=2, this com- pletes the calculation of H*(Q; Z=2). The method for computing the Witt group W (Q) proceeds similarly by working one prime at a time. See [S1, Section 5.3]. One has an isomorphism of groups W (Q) ~=Z ( pW (Fp)) [S1, Thm. 5.3.4]. With enough trouble one can compute GrIW (Q), but we will leave this for the reader to consider. Remark A.2. Note that the verification of the Milnor conjectures for F = Q tells us exactly how to classify quadratic forms over Q by invariants. First one needs the invariants over R (which are just rank and signature), and then one needs the invariants over each Qp_but for Qp one has I3 = 0, and so p-adic forms are classified by the three classical invariants e0, e1, and e2. These observation* *s are essentially the content of the classical Hasse-Minkowski theorem. The method we've used above, of working one completion at a time, works for a* *ll global fields; this is due to Tate for Galois cohomology, and Bass and Tate for* * KM*. In this way one verifies the Milnor conjecture for this class of fields [Mi , L* *emma 6.2]. Note in particular that the class includes all finite extensions of Q. 26 DANIEL DUGGER Appendix B. More on the motivic Adams spectral sequence This final section is a supplement to Section 4. I will give some hints on co* *mput- ing the E2-term of the motivic Adams spectral sequence, for the reader who would like to try this at home. The computations are not hard, but there are several * *small issues that are worth mentioning. B.1. Setting things up. H**H is the algebra of operations on mod 2 motivic cohomology. We will write this as A from now on. There is the Bockstein fi 2 A1* *,0 and there are squaring operations Sq2i2 A2i,i. We set Sq2i+1= fiSq2i2 A2i+1,i. Finally, there is an inclusion of rings H** ! A sending an element t to the ope* *ration left-multiplication-by-t. Under our standing assumptions about A (see Section 4* *), it is free as a left H**-module with a basis consisting of the admissible seque* *nces Sqi1Sqi2. .S.qik. There are two main differences between what happens next and what happens in ordinary topology. These are: (a)The vector space H** = H**(pt), regarded as a left A-module, is nontrivial. (b)The image of H** ,! A is not central. The above two facts are connected. Let t 2 H** and let Sq denote some Steenrod operation. It is not true in general that Sq(t . x) = t . Sq(x)_instead there * *is a Cartan formula for the left-hand side [V2 , 9.7], which involves Steenrod opera* *tions on t. So the operations Sq . t and t . Sq are not the same element of A. There * *is one notable exception, which is when all the Steenrod squares vanish on t. This happens for elements in Hn,n, for dimension reasons. So we have (c)Every element of Hn,n is central in A. It is important that we can completely understand H** as an A-module. This will follow from (1) the fact that H** ~= nHn,n [ø] (see Remark 2.10); (2) all Steenrod operations vanish on Hn,n for dimension reasons; (3) all Sqi's vanish * *on ø except for Sq1, and Sq1(ø) = æ = {-1} 2 H1,1; (4) the Cartan formula. In particular we note the following two facts about H**, which are all that will be needed later (the second fact only needs Remark 2.10): (d)The map Sq2: Hn-1,n! Hn+1,n+1is zero for all n 1. (e)The map Hp,q Hi,j! Hp+i,q+jis surjective for q p 0 and j i 0. We are aiming to compute ExtaA(H**, b,0H**). In ordinary topology we could use the normalized bar construction to do this, but one has to be careful here because H**, as a left A-module, is not the quotient of A by a two-sided ideal. One way to see this is to use the fact that Sq1(ø) = æ. Under the quotient map A ! H** sending ` to `(1), Sq1 maps to zero but Sq1ø does not (it maps to æ). So instead of the normalized bar construction we must use the unnormalized on* *e. This can be extremely annoying, but for the most part it turns out not to influ* *ence the öl w-dimensional" calculations we're aiming for. It is almost certainly an * *issue when computing past column two of the Adams E2 term, though. Anyway, let Bn = A H** A H** . . .H**A H** H** (n + 1 copies of A). The final H** can be dropped off, of course, but it's usef* *ul to keep it there because the A-module structure on H** is nontrivial and enters NOTES ON THE MILNOR CONJECTURES 27 into the definition of the boundary map. If we denote the generators of Bn as x = a[`1|`2| . .|.`n]t then the differential is d(x) = (a`1)[`2| . .|.`n]t + a[`1`2|`3| . .|.`n]t + . .+.a[`1| . .|.`n-1]`n* *(t). The good news is that our coefficients have characteristic 2, and so we don't h* *ave to worry about signs. Note that Bn, as a left H**-module, is free on generators 1[`1| . .|.`n]1 where each `i is an admissible sequence of Steenrod operations * *(and we must include the possibility of the null sequence Sq0 = 1). We will often dr* *op the 1's off of either end of the bar element, for convenience. Generators of Hom A(Bn, H**) can be specified by giving a bar eleme* *nt [`1| . .|.`n] together with an element t 2 H**. This data defines a homomorphism Bn ! H** sending the generator [`1| . .|.`n] to t and all other generators of B* *n to zero. Let's denote this homomorphism by t[`1| . .|.`n]*. These elements generate Hom A(Bn, H**) as an abelian group. The last general point to make concerns the multiplicative structure in the c* *obar construction. If we were working with ExtA(k, k) where k is commutative and A is an augmented k-algebra, multiplying two of the above generators in the cobar complex just amounts to concatenating the bar elements_the labels t 2 k commute with the `'s, and so can be grouped together: e.g. t[`1| . .|.`n] . u[ff1| . .* *|.ffk] = tu[`1| . .|.`n|ff1| . .|.ffk]. In our case, the fact that H** is not central in* * A immensely complicates the product on the cobar complex: very roughly, the u has to be commuted across each `i, and in each case a resulting Cartan formula will intro* *duce new terms into the product. Luckily there is one case where these complications aren't there, which is when u 2 Hn,n_for then u is in the center of A, and the product works just as above. We record this observation for future use: (f)t[`1| . .|.`n]* . u[ff1| . .|.ffk]* = tu[`1| . .|.`n|ff1| . .|.ffk]* when u * *2 Hq,q. B.2. Computations. We are trying to compute the groups ExtaA(H**, b,0H**), and from here on everything is fairly straightforward. As an example let's look* * at b = 1. Since Hp,q6= 0 only when 0 p q, one sees that Hom A(B0, H**) = 0 and Hom A(B1, 1,0H**) ~=H0,0 H1,1. The generators for this group are elements of the form s[Sq1]* and t[Sq2]*, where s 2 H0,0and t 2 H1,1. We likewise find that Hom A(B2, 1,0H**) ~=H0,1 H0,1 H0,1 H0,1, generated by elements s[Sq1|1]*, s[1|Sq1]*, t[Sq2|1]*, and t[1|Sq2]*. A similar analysis * *shows that Hom A(Bn, 1,0H**) only has such `degenerate' terms for n 2. No degenera* *te terms like these contribute elements to Ext (at worst they can contribute relat* *ions to Ext). So the Extn's vanish for n 2. An analysis of the coboundary shows th* *at everything in dimension 1 is a cycle. So we find that 0 = Ext0(H**, 1,0H**) = Extn(H**, 1,0H**), for n 2 and Ext1(H**, 1,0H**) ~=H0,0 H1,1 with a typical element in the latter group having the form s[Sq1]*+ t[Sq2]* (wh* *ere s 2 H0,0and t 2 H1,1). In general, one sees for degree reasons that the `non-degenerate' terms in Hom A(Bn, n,0H**) all have the form t[`1| . .|.`n]* where each `i is either Sq1 28 DANIEL DUGGER or Sq2. In Hom A(Bn-1, n,0H**) one has non-degenerate terms u[`1| . .|.`n-1]* * *of the following types: (i)Each `i2 {Sq1, Sq2}, and at least one Sq2 occurs. Here u 2 Hj-1,jwhere j is the number of Sq2's. (ii)Each `i 2 {Sq1, Sq2, Sq3}, and exactly one Sq3 occurs. Here u 2 Hj+1,j+1 where j is the number of Sq2's. (iii)Each `i 2 {Sq1, Sq2, Sq2Sq1}, and exactly one Sq2Sq1 occurs. Here one has u 2 Hj+1,j+1where j is the number of Sq2's. (iv)Each `i 2 {Sq1, Sq2, Sq4}, and exactly one Sq4 occurs. Here u 2 Hj+2,j+2 where j is the number of Sq2's. To analyze the part of the boundary Bn ! Bn-1 that we care about, one only needs to know the Adem relations Sq1Sq2 = Sq3 and Sq2Sq2 = øSq3Sq1. (In fact, since Sq3Sq1 doesn't appear in any of the bar elements relevant to Hom (Bn-1, n,0H**), one may as well pretend Sq2Sq2 = 0.) From this it's easy to compute that Extn(H**, n,0H**) ~= H0,0 Hn,n where a typical element has the form s[Sq1|Sq1| . .|.Sq1]* + t[Sq2|Sq2| . .|.Sq2]*. The computation uses r* *e- mark B.1(d). Also, one sees that all elements s[Sq1|Sq2]* and s[Sq2|Sq1]* are zero in Ext2(being the coboundaries of s[Sq3]* and s[Sq2Sq1]*, respectively). U* *s- ing remark (f) from Section B.1, this completely determines n Extn(H**, nH**) as a subring of the whole Ext-algebra. The next step is to compute Ext0(H**, 1,0H*,*), Ext1(H**, 2,0H*,*), and Ext2(H**, 3,0H*,*) completely. The first group is readily seen to vanish. For the second group one has to grind out another term of the bar construction, but it's a very small term. One finds that Ext1(H**, 2,0H*,*) ~=H0,1 H2,2 where the generators have the form s[Sq2]* + (Sq1s)[Sq3]* and t[Sq4]*. To get t* *he Ext2group one will need three more Adem relations, namely Sq2Sq3 = Sq5 + Sq4Sq1, Sq2Sq4 = Sq6 + øSq5Sq1, and Sq3Sq2 = æSq3Sq1. Then the same kind of coboundary calculations (but a few more of them) show that Ext2(H**, 3,0H*,*) ~=H1,2 H2,2 where the generators are s[Sq2|Sq2]* + (Sq1s)[Sq3|Sq2]* and t[Sq1|Sq4]* = t[Sq4|Sq1]* (these last two classes are the same in Ext). It is important to note that all elements u[Sq2|Sq4]* and u[Sq4|Sq2]* are coboundaries (of u[Sq6]* and u[Sq4Sq2]*, respectively). This justifies fact (7) on page 20. To jus- tify fact (6) from that same page (for n = 2), one notices that the cycles s[Sq2|Sq2]* + (Sq1s)[Sq3|Sq2]* and t[Sq4|Sq1]* decompose as a products 2 * 1 3 * 2 * 4 * 1 * s1[Sq ] + (Sq s1)[Sq ] . (s2[Sq ] ) and t1[Sq ] . t2[Sq ] for some s1 2 H0,1, s2 2 H1,1, t1 2 H2,2, and t2 2 H0,0. This uses remarks (e) and (f) from Section B.1, together with the fact that (Sq1s1)s2 = Sq1(s1s2) for s2 2 H2,2(by the Cartan formula). The final step is to analyze the groups Extn-1(H**, n,0H**) for n 4; these complete the E1,*column of the Adams spectral sequence. One doesn't have to NOTES ON THE MILNOR CONJECTURES 29 compute them explicitly, just enough to know that every element is decomposable as a sum of products from Extn-2(H**, n-1,0H**) and Ext1(H**, 1,0H**). The calculations involve nothing more than what we've done so far, except for more sweat. It's fairly easy to write down all the cocycles made up from the cl* *asses of types (i)-(iv) listed previously. All bar elements which have a Sq4 in them * *are cocycles, for instance. But note that such a bar element will either begin or e* *nd with a Sq1 or a Sq2, so that it decomposes as a product of smaller degree cocyc* *les (this again depends on B.1(e,f)). One also finds cocycles of the form s[Sq1|Sq1| . .|.Sq3|Sq1| . .|.Sq1]* + s[Sq1|Sq1| . .|.Sq2Sq1|Sq1| . .|.Sq1]* **, but for each of these a common [Sq1]* can be pulled off of either the left or r* *ight side_again showing it to be decomposable. Certainly there are cocycles which are not decomposable, like ones of the form s[Sq2|Sq1| . .|.Sq1|Sq3]* + s[Sq2Sq1|Sq1| . .|.Sq1|Sq2]*. But this is the coboundary of s[Sq2Sq1|Sq1| . .|.Sq1|Sq3], and so vanishes in E* *xt. Anyway, I am definitely not going to give all the details. But with enough diligence one can see that all elements of Extn-1(H**, n,0H**) for n 3 do indeed decompose into products. Remark B.3. A final note about Adem relations, for those who want to try their hand at further calculations. Every formula I've seen for the motivic Adem relations_in publications or preprints_seems to either contain typos or else is just plain wrong. A good test for a given formula is to see whether it gives Sq3Sq2 = æSq3Sq1 (this formula follows from the smaller Adem relation Sq2Sq2 = øSq3Sq1, the derivation property of the Bockstein, the fact that fi2 =* * 0, and the identity Sq3 = fiSq2). References [AEJ]J. K. Arason, R. Elman, and B. Jacob, The graded Witt ring and Galois coho* *mology I, in Quadratic and Hermetian forms, Canadian Math. Soc. Conference Proceedin* *gs Vol. 4 (1984), 17-50. [BT] H. Bass and J. Tate, The Milnor ring of a global field, Algebraic K-theory* *, II: "Classical" algebraic K-theory and connections with arithmetic (Proc. Conf., Seattle, * *Wash., Battelle Memorial Inst., 1972), pp. 349-446. Lecture Notes in Math., Vol. 342, Spri* *nger, Berlin, 1973. [De] A. Delzant, D'efinition des classes de Stiefel-Whitney d'un module quadrat* *ique sur un corps de charact'eristique diff'erent de 2, C.R. Acad. Sci. Paris 255, 136* *6-1368. [Du] D. Dugger, An Atiyah-Hirzebruch spectral sequence for KR-theory, to appear* * in K-theory. [EL] R. Elman and T.Y. Lam, Pfister forms and K-theory of fields, Jour. of Alge* *bra 23 (1972), 181-213. [Ka] K. Kato, A generalization of local class field theory by using K-groups II* *, J. Fac. Sci. Univ. Tokyo Sect. 1A Math. 27 (1980), no. 3, 603-683. [LZ] J. Lannes and S. Zarati, Invariants de Hopf d'ordre sup'erieur et suite sp* *ectrale d'Adams, C.R. Acad. Sc. Paris 296 (1983), 695-698. [MVW] C. Mazza, V. Voevodsky, and C. Weibel, Lectures on motivic cohomology, pr* *eprint, July 2002. http://www.math.uiuc.edu/K-theory/0486. [M] A. Merkujev, On the norm residue symbol of degree 2, Soviet Math. Doklady * *24 (1981), 546-551. [Mi] J. Milnor, Algebraic K-theory and quadratic forms, Invent. Math. 9 (1970),* * 318-344. [M1] F. Morel, Voevodsky's proof of Milnor's conjecture, Bull. Amer. Math. Soc.* * 35, no. 2 (1998), 123-143. [M2] F. Morel, Suite spectral d'Adams et invariants cohomologiques des formes q* *uadratiques, C.R. Acad. Sci. Ser. 1 Math. 328 (1999), no. 11, 963-968. 30 DANIEL DUGGER [M3] F. Morel, An introduction to A1-homotopy theory, Trieste lectures. Preprin* *t, 2002. Avail- able at http://www.math.jussieu.fr/~morel/. [M4] F. Morel, Milnor's conjecture on quadratic forms and mod 2 motivic complex* *es, preprint, 2004. http://www.math.uiuc.edu/K-theory/0684. [M5] F. Morel, Suites spectrales d'Adams et conjectures de Milnor. Draft availa* *ble online at http://www.math.jussieu.fr/~morel/. [OVV]D. Orlov, A. Vishik, and V. Voevodsky, An exact sequence for KM*=2 with ap* *plications to quadratic forms, preprint, 2000. http://www.math.uiuc.edu/K-theory/0454. [Pf1]A. Pfister, Some remarks on the historical development of the algebraic th* *eory of quadratic forms, in Quadratic and Hermetian forms, Canadian Math. Soc. Conference Pr* *oceedings Vol. 4 (1984), 1-16. [Pf2]A. Pfister, On the Milnor conjectures: history, influence, applications, J* *arhes. Deutsch. Math.-Verein. 102 (2000), 15-41. [R1] M. Rost, Some new results on the Chow groups of quadrics, preprint, 1990. * *Available at http://www.math.uiuc.edu/K-theory/0165/. [R2] M. Rost, Norm varieties and algebraic cobordism, Proceedings of the Intern* *ational Con- gress of Mathematicians, Vol. II (Beijing, 2002), 77-85, Higher Ed. Press,* * Beijing, 2002. [S1] W. Scharlau, Quadratic and Hermitian forms, Grundlehren der mathematischen* * Wis- senschaften 270, Springer-Verlag Berlin Heidelberg, 1985. [S2] W. Scharlau, On the history of the algebraic theory of quadratic forms, in* * Quadratic forms and their applications, Contemp. Math. 272, American Mathematical Society,* * 2000, 229- 259. [Se] J.-P. Serre, Cohomologie Galoisienne, Cinqui`eme 'edition, Lecture Notes i* *n Math. 5, Springer-Verlag Berlin Heidelberg, 1973, 1994. [Su] A. Suslin, Voevodsky's proof of the Milnor conjecture, Current Development* *s in Mathe- matics, 1997 (Cambridge, MA), 173-188. [V1] V. Voevodsky, The Milnor conjecture, preprint, 1996. http://www.math.uiuc.* *edu/K- theory/0170. [V2] V. Voevodsky, Reduced power operations in motivic cohomology, Publ. Math. * *Inst. Hautes 'Etudes Sci., No. 98 (2003), 1-57. [V3] V. Voevodsky, Motivic cohomology with Z=2-coefficients, Publ. Math. Inst. * *Hautes 'Etudes Sci., No. 98 (2003), 59-104. [V5] V. Voevodsky, On motivic cohomology with Z=l coefficients, preprint, 2003.* * Available at http://www.math.uiuc.eud/K-theory/0639. [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. Department of Mathematics, University of Oregon, Eugene, OR 97403 E-mail address: ddugger@math.uoregon.edu