Algebraic Morava K-theories and the. higher degree formula

Transcription

1 Algebraic Morava K-theories and the higher degree formula 1

2 Contents Chapter 1. Introduction 1 Chapter 2. The construction of an algebraic Morava K-theory spectrum 5 Chapter 3. The theorem in the general form 18 Chapter 4. The higher degree formula 25 References 32 2

3 CHAPTER 1 Introduction In this thesis we will use homotopy-theoretical methods in algebraic geometry. At the end of 1996 V. Voevodsky published the paper [9] in which he proved the Milnor conjecture. To get such a result he employed standard homotopy theory techniques in an algebraic context. Referring to [9], Theorem 2.25 gives sufficient conditions for the Milnor conjecture to hold; in particular the second one states that H n+1,n B (Č(X a), Z (2) ) = 0 where H B is the Beilinson motivic cohomology and Č(X a ) is a certain simplicial smooth scheme. By ordinary algebraic geometry techniques, which include two theorems of M. Rost ([6], [7]), it is possible to prove that H 2n 1,2 n 1 B (Č(X a), Z) = 0, implying that any class in H 2n 1,2 n 1 B (Č(X a), Z/2) which is a restriction of an integral homology class is 0. On the other hand, by properties of X a and a transfer argument, it is sufficient to prove that the classes in H n+1,n B (Č(X a), Z/2) which are reduction of integral classes, are zero. In algebraic topology, understanding H, B (, Z/2) as ordinary cohomology with Z/2 coefficients, this problem could be solved as follows: we have the cohomological operation Q n 2 Q n 3 Q 1 : H n+1 (X, Z/2) H 2n 1 (X, Z/2) which we want to show to be injective proving that 1

4 S. BORGHESI 2 H n+1 (X, Z (2) ) = 0. The injectivity of such operation can be reduced to the inductive assumption H i (X, Z (2) ) = 0 for i < n + 1 and to the study of the so called Margolis homology groups of H n+1 (X, Z/2) for the X we are interested in. The problem was therefore reduced to creating an environment in which the Beilinson motivic cohomology could be representable by some object. In addition it might be possible to do other useful constructions. In fall 1998 all the details of such construction were published in [5]. The vanishing of the Margolis groups of the cohomology of a space X, can be rephrased as the vanishing of a certain map Φ, 1 (X +) Φ 2d, d 1 (X + ) Q t where Φ 1 is the fiber of H Z/q Σ 2q t 1,q t 1 H Z/q. This can be considered as a first approximation to algebraic Morava K-theory. A few weeks after V. Voevodsky proposed that I investigate a general construction of the Morava K-theories, M. Rost, during a conference in Berkeley in May 1998, announced that he knew of a program that was likely to prove the Milnor conjecture at odd primes, such result goes under the name of Bloch-Kato conjecture. Rost conjectured the existence of an higher degree formula (Voevodsky s degree formula being the r = 1 case): let k be a field of characterisitc 0, q a prime number, X and Y be projective, smooth schemes of dimension r(q t 1) and r and t positive integers. If f : Y X is dominant, then there exists a certain characteristic number t r such that t r (Y ) deg(f) t r (X) mod I r (X) where I r (X) is the reduction modulo q of J r (X) an ideal of Z associated to X. In his program, M. Rost deals with varieties Y cobordant to q fold products of projective, smooth, irreducible varieties Y i of dimension q n 1 and such that s q n 1(Y i ) 0 mod q 2 for

5 S. BORGHESI 3 each i. If the Chern classes c h can be expressed as the h-th elementary symmetric polynomials in the t i, then s k is the characteristic class corresponding to the k-th Newton polynomial in variables t i. The characteristic number t r (Y ) should be nonzero for such varieties and this is exactly the place where algebraic Morava K-theories appear. The point being again that the cohomological operations Q n detect varieties Y i with s q n 1(Y i ) 0 in a precise sense. At this stage, it is hard to formulate a possible definition for what algebraic Morava K-theories should be: in topology these spectra k(t) are characterized by A top H (k(t), Z/q) = or equivalently π (k(t)) = Z/q[v t ] with v t being an homotopy class in dimension 2(q t 1). Currently, in the motivic stable A top Q t homotopy category and for a prime q, two algebraic Morava K-theories are known: one has a canonical MGl-module structure in the sense of [1] and the right cohomology (Theorem 2.1), but the homotopy is not known; the other, considered here in the first chapter, has the same cohomology, admits a resolution of Eilenberg MacLane spectra. No ring structures are known in either cases and very few things can be deduced about their homotopy. In the second chapter the higher degree formula is established by showing that any map MGl Φ r which is nonzero when composed with the resolution morphisms Φ r Φ r 1 H Z/q can be used to define the characteristic number t r. At this point it becomes essential to have a resolution by motivic Eilenberg-MacLane spectra of our model of algebraic Morava K-theory. In the fourth chapter we prove that varieties X satisfying a certain condition (Proposition 4.1) will have the property that I r (X) = 0. This makes the higher

6 S. BORGHESI 4 degree formula a nontrivial result. All the norm varieties are known to satisfy this condition. For any such variety, the higher degree formula provides a surprising evidence of how strongly topological properties (characteristic numbers) can affect algebro-geometrical invariants (degree of maps), showing in particular that for certain varieties the degree of a map between them can be cobordism invariant.

7 CHAPTER 2 The construction of an algebraic Morava K-theory spectrum In this section we are going to define what will be called the algebraic Morava K-theories. Throughout this paper Spec k will always denote the spectrum of a field of characteristic 0, the number q a fixed prime number and t will be a nonnegative integer. By construction, these spectra will come equipped with a H Z/q -pro nilpotent resolution. Po Hu in [2], uses a slightly different definition of spectra from the one proposed by Voevodsky in [11]. Her definition, makes the category of MGl-module spectra closed under colimits. From this she has built a spectrum k (t), which comes equipped with a MGl-module structure, by taking the homotopy colimit of (2.2.1) MGl q E 0 η 1 E 1 η 2 η q t 2 Eq t 2 η q t E q t η q t +1 with E i defined by the exact triangle (2.2.2) Σ 2(i 1),i 1 a i 1 E i 1 Ei 1 η i Ei induced by the MGl-module multiplication of certain homotopy class a i 1 π 2i,i (MGl). Of course, in order for this to make sense we need to define what the homotopy classes a i are. Now, MGl, (P ) is the power series ring MGl, (S 0 )[[x]] with x MGl 2,1 (P ) canonically defined by the identity P MGl 2,1 = P. From 5

8 S. BORGHESI 6 the Kunneth theorem, we have (2.2.3) MGl, (P P ) = MGl, (P ) ˆ MGl, (P ) = MGl, (S 0 )[[x 1, x 2 ]]. Thus the multiplication P P m P induces a map MGl, (S 0 )[[x]] m MGl, (S 0 )[[x 1, x 2 ]]. The power series in two variables m x is defined to be the formal group law associated with MGl. It gives a canonical map MU (S 0 ) = L MGl, (S 0 ) from the Lazard ring L to the motivic homotopy of MGl. We denote by a i the images under this map of the polynomial generators of MU ; notice that a i MGl 2i,i. We need the following result: Theorem 2.1. The spectra k (t), as defined above, has the property that H, (k (t), Z/q) = A A Q t and there exist an exact triangle Σ 2(qt 1),q t 1 k (t) k (t) C 0, with H, (C 0, Z/q) = A. Proof. The motivic cohomology of BGl is known to be a polynomial algebra over the Chern classes with H, (Spec k, Z/q) as coefficient ring ([4], Theorem ). To prove the Theorem, we will need the next result which gives a partial description of the action on the motivic Steenrod algebra on the cohomology of MGl:

9 S. BORGHESI 7 Definition 2.1. Let E and F any two spectra; for a class α [E; Σ p,q F] the invariant is defined as the integer α = p 2q. Proposition 2.1. Let I = (i 1, i 2,, i k ). Then H, A (MGl, Z/q) is a free A (Q 0,, Q t, ) - module over a basis {m I } with m I H2(P i j ),( P i j ) (MGl, Z/q) and i j q n 1 for any positive integer n. Proof. The Chern classes c i have invariant 0 because they are in bidegree (2i, i), hence Q j c i is zero for any i and j because Q j = 1, and there are no classes with invariant 1 in the cohomology of MGl (see Voevodsky [9], Theorem 3.21). Therefore the action of the motivic Steenrod algebra on H, (MGl, Z/q) factors through A. This action is described just by looking at Chern classes and A (Q 0,, Q t, ) then extend by linearity on H, A. A typical element in A (Q 0,, Q t, ) can be written as a(r 1, r 2, ) in the Milnor basis [3], with a H,. The action of that operation on a Chern class c i is (a(r 1, )) c i = a((r 1, )c i ), but (r 1, )c i is a polynomial with Z/q coefficients in the Chern classes, because it has 0 invariant, and these are the only such classes in H, (MGl, Z/q). It follows that, since H, (MGl, Z/q) is a free left H, -module, the only relations in the action A of could possibly arise only topologically. However, it is well A (Q 0,, Q t, ) known that it is not the case. The next step is to study the cohomology of the E i ; to do this we have to understand the behaviour of a i in cohomology. There are two different phenomena happening: the first one is described by

10 S. BORGHESI 8 Lemma 2.1. Assume i q n 1. The cohomology long exact sequence associated to (2.2.4) Σ 2i,i E i a i E i η i+1 Ei+1 splits in short exact sequences of the kind (2.2.5) 0 H +2i, +i (E i, Z/q) ( a i ) H, (E i, Z/q) H, (E i+1, Z/q) 0 Proof. To see what a i induces in cohomology we can argue as follows: let I = (i 1, i 2,, i k ), then, in topology, {m i 1 1 m i 2 2 m i k k } is a basis over Z of H (MU, Z), and their duals are the cohomological classes denoted by the symbol m I. The Hurewicz homomorphism π (MU) h H (MU, Z) is known to send a i to classes m i H 2i (MU, Z) for i p n 1 and a p n 1 to pm p n 1 for any n and any prime p. In our context, let s define m i to be the images of a i for i p n 1 and m p n 1 to be h(a p n 1)/p; the last definition makes sense because of the following: Lemma 2.2. H, (MGl, Z) is a free left H, -module over the m I. Proof. Using similar methods as those of Voevodsky in [10], it is possible to prove that the motive of the Grassmannian G n (A n+m k ) of n-planes in the (n + m)- dimensional affine space splits as Z( )[2 ] with one direct summand for each topological homology class of the same Grassmannian. Since motivic homology

11 S. BORGHESI 9 of a scheme X is represented in the category DM (see [10]) as H i,j (X, Z) = Hom DM (Z(j)[i]; M(X)) where M(X) represents the motive of X, the Lemma follows at once. The previous Lemma implies that the topological realization map H 2i,i (MGl, Z/q) H 2i (MU, Z/q) is an isomorphism for all i, therefore h(a q n 1) must be divisible by q. As in topology, denote with m I the dual class to mi 1 1 m i k k H, (MGl, Z/q), being I = (i 1,, i k ); by brevity we let m i = m (0,, i 1) and assume that i q n 1. Denote by c the generator of H, ((P 1 ) i, Z/q) as H, -module and by a i the map (P 1 ) i MGl; then by duality, 1 = m i, m i = m i; ( a i ) c = ( a i ) m i; c, hence ( a i ) m i H 2i,i ((P 1 ) i, Z/q) is going to be the generator of the reduced cohomology of (P 1 ) i. Using the MGl-module structure of E i we can make a i in a map of spectra Σ 2i,i a E i i Ei. By what just mentioned, m i is going to hit the bottom cohomology class of Σ 2i,i E i, which we call Σ 2i,i τ. From the A linearity of ( a i ) in cohomology, we right away get that A A (Q 0,, Q t, ) Σ2i,i τ ( a i) A A (Q 0,, Q t, ) m i is an isomorphism. ( a i ) preserves the left H, (MGl, Z/q)-module structure, therefore in cohomology m is going to map to Σ 2i,i m (0,, 1,0, i j j because 1)

12 S. BORGHESI 10 a i (m ), Σ 2i,i m (0,, 1,0, i j j = m, (a 1) (0,, 1,0, i j i ) (Σ 2i,i m j ) 1) (2.2.6) = m, m (0,, 1,0, i j j (a i ) (Σ 2i,i 1) = m, m 1) (0,, 1,0, i j j m i = 1 1) Similarly, m J+(0,, i 1) hits Σ 2i,i m J, and this implies that (a i) is surjective and the same is true with finite coefficients. This proves Lemma 2.1. The previous Lemma also shows that H, (E i, Z/q) = H, (E i 1, Z/q)/M i where M i is the H, -submodule generated by the m I with the nonzero i-th entry. On the other hand, when we take the cone E q n of the multiplication by a q n 1 we observe that its motivic cohomology is actually bigger than the one of E q t 1: Lemma 2.3. The cohomology long exact sequence associated to (2.2.7) Σ 2(qn 1),q n 1 E q n 1a qn 1 E q n 1 η q n E q n splits in short exact sequences (2.2.8) 0 H, (E q n 1; Z/q) η q n H, (E q n; Z/q) δ H +1, (Σ 2(qn 1),q n 1 E q n 1; Z/q) 0

13 S. BORGHESI 11 Proof. As before we have the following maps dual one to the other (2.2.9) H, ((P 1 ) qn 1 ; Z/q) (a q n 1 ) H, (E q n 1; Z/q) H, ((P 1 ) qn 1 (a q n 1 ) ; Z/q) H, (E q n 1; Z/q) The Hurewicz homomorphism is defined by the image of the generator of H, ((P 1 ) qn 1 ; Z) through (a q n 1) which we know being 0, when composed with the reduction map modulo Z/q, hence also its dual map in cohomology is zero; therefore the short exact sequences in this case are of the kind stated in the Lemma. A precise description of the action of the motivic Steenrod algebra on H, (E q n, Z/q) can be given. In particular Lemma 2.4. Let τ q n be the bottom cohomology class of E q n; then Q n (τ q n) H, (E q n, Z/q) is nonzero, hence it must be δ(σ 2(qn 1),q n 1 τ q n 1). In other words, going from E q n 1 to E q n we have added the operation Q n on τ q n. Proof. Voevodsky in [9], Lemma 3.6, has proved that any lifting e of the Thom Q n class MGl H Z/q to Φ 1 (the fiber of H Z/q Σ Q n H Z/q ) sends the homotopy class a q n 1 to a nonzero homotopy class of Φ 1. This is equivalent to showing that Q n τ 0

14 S. BORGHESI 12 in the cone of Σ a q n 1 MGl a q n 1 MGl as one can easily see from the commutative diagram (2.2.10) Σ 1,0 Σ 1,0 Q n H Z/q Σ a q n 1 Q n H Z/q Φ 1 H Z/q Σ Qn H Z/q Σ 1,0 τ Σ 1,0 (Q nτ) e τ Q n(τ) Σ 1,0 a q n 1 cone(a q n 1) (P 1 ) 2(qn 1) MGl cone(a q n 1) Using the map MGl E q n 1 we can right away produce a map f : cone(a q n 1) E q n as fill in map between two exact triangles. Consequently, we get f (Q n τ q n) = Q n f (τ q n) = Q n τ 0 proving the Lemma. Corollary 2.1. The class Q n τ q n generates a free copy of H,. Proof. By induction, assume that H, (E q n 1; Z/q) is H, free; if aq n τ q n 1 = b 1 β 1 +b 2 β 2 + +b k β k with b i H, and β i some classes of the cohomology of E q n 1 then not all of the β i are in the image of δ or else the equality aσ 2(qn 1),q n 1 τ q n = b 1 β 1 +b 2 β 2 + +b k β k would hold in H, (Σ 2(qn 1),q n 1 E q n 1; Z/q), contraddicting the inductive assumption. On the other hand, applying η q n to that very same equality we would get 0 = b i1 β i1 + b i2 β i2 + + b ih β ih for those β ij that are not in the image of δ, contraddicting again the inductive assumption. In conclusion, after having taken the colimit of the diagram 2.2.1, the cohomology is going to be as stated by the theorem and the cone of Σ 2(qt 1),q t 1 k (t) a q t 1 k (t)

15 S. BORGHESI 13 will have the motivic Steenrod algebra as cohomology ring. This finishes the proof of the theorem. MGl Remark: Let s denote the direct limit of by. Using the (q, a i /i q t 1) same techniques as in the proof of 2.1 one can show that (2.2.11) H, MGl ( (q, a i /i q k 1, k = 1, 2, ), Z/q) = A A (Q 0,, Q t, ) and (2.2.12) H, MGl ( (q, a i /i q k 1, k = 1, 2,, n), Z/q) = A A (Q 0,, Q n ) Therefore these two spectra could be taken as a model for the Brown-Peterson spectrum BP and BP n, respectively. We have now all the information needed to construct a tower for k (t). Consider first the following diagram in which we denote q t 1 = d and a q t 1 with v t for brevity: Φ 1 f 1 (2.2.13) Σ 2d,d k (t) v t k (t) v2 C 0 v t t Σ 4d,2d k (t)

16 S. BORGHESI 14 the multiplication by v t is the one by the MGl-module structure of k (t): (P 1 ) d k (t) vt id MGl k (t) m k (t) and both the sequences are exact triangles. From the cohomology long exact sequence associated to the exact triangle Σ 4d,2d k (t) v2 t k (t) Φ 1 H, (Φ 1, Z/q) = A A Q t x we can get that A A Q t y 1 as left A -module, with x H 0,0 (Φ 1, Z/q) being the class hitting the bottom class of k (t), and y 1 H4d+1,2d (Φ 1, Z/q) is the image of the bottom class of Σ 4d+1,2d k (t) 1 ; this follows since v i t all induce the zero map in cohomology, being the cohomology an A -cyclic module. Turning now to the cone C 1 of f 1, by observing the long exact sequence (2.2.14) H, (Φ 1, Z/q) A H, (C 1, Z/q) we can see that there will defenitely be a bottom class α H 2d+1,d (C 1, Z/q) going to hit Q t ι (ι is the bottom class of A ) which generates an A -cyclic module A isomorphic to and then another class γ H 2d+2,d (C A 1, Z/q), image of y Q 1; t no other generators over A are present. There is also the relation Q t α = γ which can be obtained by realizing the exact triangle in topology. The fact that there are no other relations follows immediately from Lemma 2.5. The Margolis homology group HM(A, Q t ) is zero. 1 y 1 4d+1,2d A generates a cyclic module isomorphic to Σ A just because Q t A H, (Σ 4d+1,2d k (t), Z/q) = Σ 4d+1,2d A. Q t

17 S. BORGHESI 15 Proof. Let I = (i 1, i 2, ) and aq I (r 1, ) be a generic element expressed in the Milnor basis. We have Q t (aq I (r 1, )) = Q t A aqi (r 1, ) ± aq t Q I (r 1, ). Therefore left multiplication by Q t is the differential of a double complex. Its homology is HM(H,, Q t ) Z/q HM(A top, Q t ) which must be zero because it is well known that HM(A top, Q t ) = 0. All this combined together yields H, (C 1, Z/q) = Σ 2d+1,d A. In the same way we can argue on the diagram Φ 2 f 2 (2.2.15) Σ 4d,2d k (t) v 2 t v3 k (t) Φ 1 v t t Σ 6d,4d k (t) and conclude with an exact triangle Φ 2 f 2 Φ 1 C 1 with H, (C 1, Z/q) = Σ 2d+1,d A. The tower Φ 2 f 2 Φ 1 f 1 C0 is the tower we will need for k (t). Define now the following exact triangle Φ 1 H Z/q have the commutative diagram Q t Σ 2d+1,d H Z/q so that we

18 S. BORGHESI 16 (2.2.16) Φ 1 H Z/q Σ 2d+1,d H Z/q i 1 1 Σ 2d+1,d 1 Φ 1 C 0 f 0 C 1 and i 1 is the fill in map; it will induce a cohomology isomorphism because the other two maps do. We can let y 1 = i y 1 y 1 and define a new exact triangle Φ 2 Φ 1 Σ 4d+1,2d H Z/q ; it can be nested in a diagram similar to the previous one which yields a map i 2 : Φ 2 Φ 2 inducing isomorphism in motivic cohomology, so that y 2 can be defined likewise and continue in this fashion. The colimit of this new tower will be denoted by k(t) and called the algebraic Morava K-theory for the positive integer t, at the prime q. What we will actually use is a slightly different version of this tower for k(t): infact we want to have a series of exact triangles of the kind Σ 2d,d Φ r 1 Φ r H Z/q. In the next diagram define F i to be the F i 1 Φi 1 Φ i : (2.2.17) p 2 Φ 2 p 1 Φ 1 p 0 H Z/q u F 2 F 1 Spec k and the map u is the composition Spec k Spec k + = S 0 H Z/q. Notice that F i is non canonically isomorphic to Σ 2d,d Φ i 1 for all i because the cone of p 0 is Σ 2d+1,d H Z/q, so F 1 is the homotopy fiber of Spec k Σ 2d+1,d H Z/q i.e. canonically

19 S. BORGHESI 17 (up to homotopy) Σ 2d,d H Z/q ; then by induction on the stage of the tower we get what stated.

20 CHAPTER 3 The theorem in the general form Fix an odd prime q, an embedding of the base field k C, smooth projective varieties X and Y of dimension dr with d = q t 1 and a morphism f : Y X. In this chapter, [X] will stand for the algebraic cobordism class of X in MGl,. Let e : MGl Φ r any map to the r-th stage of the H Z/q -resolution of the Morava K-theory at the prime q. Using the fibration Σ 2d,d φ ψ Φ r 1 Φr HZ/q we get the following commutative diagram: (3.3.1) π 2dr,dr (MGl Y + ) π 2dr,dr (Σ 2d,d Φ r 1 Y + ) e φ π 2dr,dr (Φ r Y + ) ψ π 2dr,dr (H Z/q Y + ) = CH dr (Y ) Z/q = Z/q[Y ] HZ/q f π 2dr,dr (Σ 2d,d Φ r 1 X + ) φ f π 2dr,dr (Φ r X + ) ψ deg f π 2dr,dr (H Z/q X + ) = CH dr (X) Z/q = Z/q[X] HZ/q p X π 2d(r 1),d(r 1) (Φ r 1 ) φ = p X π 2dr,dr (Φ r ) = Z/q where p X is the map induced in homotopy by the projection on the first factor Φ r X + Φ r. Proposition 3.1. For each smooth, projective scheme W of dimension n we have: 18

21 S. BORGHESI 19 (a) a class [W ] MGl MGl 2n,n (W + ) whose topological realization is the fundamental MU-homology class [W (C)] MU ; (b) a class [W ] MGl 2n,n (S 0 ) whose topological realization is the cobordism class [W (C)]; (c) the map MGl, (W + ) p 1 MGl, (S 0 ) induced by W + S 0 sends [W ] MGl to [W ]. Remark 3.1. Using this proposition we can define a fundamental class [W ] E for certain spectra E. Let E be a ring spectrum equipped with a Thom class, i.e. a stable map τ : MGl E such that u (τ) = 1 E 0,0 (S 0 ) where u : S 0 MGl is the unit. Then define [W ] E = τ ([W ] MGl ). Notice that this definition is compatible with the topological envinronment. Proof. As for the second and third part of the Proposition, once we have [W ] MGl, we can define [W ] to be the image of [W ] MGl under p 1. Infact, it suffices to prove the first part for W = P N for every positive integer N: since X is projective, there is a closed embedding W i P N for some N. Let ν W be the normal bundle of this closed embedding; then in [5] it is shown that P N /(P N i(w )) = T h(ν W ), hence there is a projection p : P N T h(ν W ). This map induces MGl, (P N +) p MGl, (T h(ν W ) + )

22 S. BORGHESI 20 and we call α N = p ([P N ] MGl ). There is a well known pairing called cap product with α N : (3.3.2) MGl 2N,N (T h(ν W ) + ) MGl 2(N n),n n (T h(ν W ) + ) α N MGl2n,n (W + ). we will define [W ] MGl to be α N τ νw, where τ νw : T h(ν W ) T h(η N n ) is the Thom class of ν W i.e. the MGl cohomology class induced by the classifying map ν W η N n (3.3.3) W G N n η N n being the universal N n-plane bundle. To prove the first statement of the Proposition for P N, we need to use the following Lemma 3.1. Let E be a ring spectra with multiplication m. If E, (P N + ) is a free and finitely generated E, (S 0 )-module with basis b 0, b 1, b N in bidegree (2i, i), then E P N + splits as b i E (P 1 ) i. Proof. There is an exact triangle P N 1 P N (P 1 ) N. We will denote with φ i the composition (3.3.4) E P N 1 b i E E (P 1 ) i m 1 E (P 1 ) i. Let also φ N be the wedge of the φ i for all the i = 0, 1,, N. Then we have the following commutative diagram

23 S. BORGHESI 21 E P N 1 E P N E (P 1 ) N (3.3.5) = φ N 1 N 1 i=0 E (P1 ) i φ N N i=0 E (P1 ) i E (P 1 ) N in which the left isomorphism holds by induction, and the assumption on the freeness of the E-cohomology of P N implies the commutativity of the right square. The fact that MGl satisfies the assumptions of the Lemma is proved in [4], therefore we can conclude that MGl P N + = N i=0 MGl (P1 ) i. It follows that a map (P 1 ) N MGl P N + is an element of a direct sum of the kind N i=0 π 2N,N(MGl (P 1 ) i ) = N i=0 π 2(N i),(n i)(mgl). Recall that the canonical formal group law of MGl gives a canonical embedding of the Lazard ring L, π, (MGl), therefore letting [P N ] MGl = N i=0 u i where the homotopy classes u i L 2i,i are described by [P N (C)] MU = N i=0 u i gives a definition compatible with the topological one. Remark 3.1 implies at once the following Corollary 3.1. Let e r : MGl Φ r be defined inductively as a lifting of e r 1 : (3.3.6) Φ r Φ r 1 Σ 2dr+1,dr H Z/q e r e r 1 0 MGl

24 S. BORGHESI 22 and e 0 : MGl H Z/q is the Thom class. Then (ψ (e r ) )[X] MGl = [X] HZ/q, and the same is true for Y. An immediate, fundamental consequence is the following congruence: we let I r (X) to be the image of p X φ and t r (W ) = p W ((e r ) ([W ] MGl )) π 2dr,dr (φ r ) = Z/q in 3.3.1; (3.3.7) t r (Y ) deg(f)t r (X) mod I r (X) The purpose of the remaining part of this chapter is to find sufficient conditions for t r (Y ) and t r (X) to be nonzero. We have the commutative diagram: π 2dr,dr (MGl Y + ) (er) π 2dr,dr (Φ r Y + ) (3.3.8) f π 2dr,dr (MGl X + ) f π 2dr,dr (Φ r X + ) p X π 2dr,dr (MGl) p X (e r) π 2dr,dr (Φ r ) which suggests us that if we had an MGl-orientation e r of Φ r such that (e r ) [Y ] is nonzero in π 2dr,dr (Φ r ) ([Y ] here stands for the algebraic cobordism class of Y ), then automatically p X (f (α Y )) 0 because of the part (c) of 3.1.

25 S. BORGHESI 23 Definition 3.1. Let v t denote any homotopy class in π 2d,d (MGl) such that there exists a lifting e 1 of a map representing the Thom class in τ H 0,0 (MGl, Z/q) with the property that (e 1 ) v t is a non zero homotopy class in π 2d,d (Φ 1 ). For example, we can take v t to be the algebraic cobordism class represented by a v t -variety (see next chapter for the definition), as shown by Voevodsky in [9], Lemma 3.6. Proposition 3.2. Any morphism MGl er Φ r such that the composition MGl er Φ r Φ r 1 Φ 1 H Z/q is nonzero, has the property that (e r ) (v r t ) 0. Proof. The topological realization of e r, which we will denote by e top r, is going to be nonzero when composed down to H Z/q in the tower. By the previous definition of v t, v r t will map to its topological counterpart so it is easy to see that we only have to prove the Proposition in the topological case. The base of the induction amounts to proving that for any lifting e of the Thom class, e v t is nonzero. The difference between two liftings e and e factors through Σ 2d H Z/q : (3.3.9) Σ 2d H Z/q Φ 1 e S 2d v t τ Q t MU H Z/q Σ 2d+1 H Z/q Since the image of v t by the Hurewicz map is zero, v t is zero in cohomology, hence the composition S 2d vt MU Σ 2d H Z/q is null homotopic. Let now e r be as in the

26 S. BORGHESI 24 statement of the Proposition; we have the commutative diagram (3.3.10) Σ 2d Φ r 1 Φ r H Z/q 0 f e r Σ 2d v t MU MU where f (vt r 1 ) is nonzero by inductive assumption and it must go to a nonzero homotopy class of Φ r which, by commutativity, is the same as (e r ) ((v t ) (v r 1 t )) = (e r ) (v r t ).

27 CHAPTER 4 The higher degree formula In this section we are going to specialize the formula to the case of smooth, projective algebraic varieties with the space of C-rational points having the same cobordism class of a product of smooth, irreducible, projective (v t, q)-varieties X i of dimension d = q t 1. As before [X] will denote the algebraic cobordism class of X, [X(C)] the topological cobordism class of the C-rational points of X and [X] E will be the E-fundamental class of X for a ring spectrum E. Let me recall the following (see [9]): Definition 4.1. A complex algebraic variety X is called a (v t, q)-variety if (a) dim(x) = q t 1; (b) all characteristic numbers of X(C) are divisible by q, and (c) s q t 1, [X(C)] MU 0 mod q 2. As a reminder, the classes s n are defined as the cohomology classes in H 2n (BG m, Z), the cohomology of the classifying space for the infinite Grassmanian of m-planes in C with m n, corresponding to t n 1 + tn tn m, if we identify the Chern classes c i of the universal m-plane bundle with σ i (t 1,, t m ), the elementary symmetric polynomials in the variables {t 1, t m }. This is usually being done by noticing that 25

28 S. BORGHESI 26 the map P P BG m that classifies the bundle µ = µ 1 µ m, with µ i being the universal plane bundle of the i-th copy of P, induces an injection in cohomology sending the total Chern class of the universal m-plane bundle to the total Chern class of µ which is m 1=i (1 + t i), and t i is the first Chern class of µ i. Voevodsky has shown in [9] that each of such X i can be taken as a model for a class representing v t π 2d,d (MGl) as defined in 3.1; therefore if X is cobordant to the product of r copies of X i, it is a model for v r t π 2dr,dr (MGl). Proposition 3.2 implies that there exists an MGl-orientation of Φ r 1 which we called e r, that sends [X] to a nonzero homotopy class of Φ r or, using the same notation as in 3.3.7, p X (f (α X )) 0 π 2dr,dr (Φ r ) = Z/q. Therefore we have the following theorem, which is a consequence of and which shows how assumptions on cobordism classes of the C-rational points of the varieties involved can affect the degree of a map between them and vice versa: Theorem 4.1. Let X and Y be smooth, projective algebraic varieties of dimension r(q t 1) for an odd prime number q and positive integers r and t; let also f : Y X be any map between them. If f is not dominant, then (e r ) ([Y ]) I r (X). If we assume that I r (X) = 0 and f dominant, then, (a) deg(f) mod q is a (topological) cobordism invariant in the source Y ; (b) if [Y (C)] = vt r, we have deg(f) 0 mod q and [X] n[y ] mod ker(e r) with n Z and n deg(f) mod q in π 2r(q t 1),r(q t 1)(MGl); in particular [X(C)] is not divisible by q in π 2r(q t 1)(MU);

29 S. BORGHESI 27 (c) if [Y (C)] = [X(C)] = v r t, then deg(f) 1 mod q; (d) if [X(C)] = v r t and deg(f) 0 mod q then [Y ] m[x] mod ker(e r ) with m Z and m deg(f) mod q; in particular [Y (C)] is not divisible by q. Remark 4.1. In the previous theorem, the condition [W (C)] = v r t with W = Y, X, can be replaced by [W (C)] = v r t mod ker(e r ) throughout, without changing the conclusions. The next task is to give conditions on X so that I r (X) = 0. Recall the definition of I r (X) from 3.3.1: it is the image of either of the two compositions in the commutative square (4.4.1) π 2dr,dr (Σ 2d,d Φ r 1 X) p X π 2dr,dr (Σ 2d,d Φ r 1 ) = π 0,0 (Φ r 1 ) φ π 2dr,dr (Φ r X) p X = π2dr,dr (Φ r ) = Z/q Remark: For r = 1, we have (4.4.2) π 2d,d (Σ 2d,d H Z/q X) = CH 0 (X) Z/q p X π 2d,d (Σ 2d,d H Z/q ) = CH 0 (Spec k) Z/q = Z/q and the map p X can be seen as g 1, where g : CH 0 (X) CH 0 (Spec k) = Z is the map induced by the canonical map X Spec k : it takes a closed point Spec L X to the positive integer [L; k].

30 S. BORGHESI 28 Notation: Denote with J(X) the ideal in Z generated by the image of g. More in general, if f : W Spec k is a proper scheme, and W E f Spec E is the base changed scheme by Spec E Spec k, E being a field extension of k, J(W E ) will be the ideal in Z generated by the direct image homomorphism f : CH 0 (W E ) CH 0 (Spec E) = Z. We can now formulate the following result: Proposition 4.1. Let X 1,, X r be smooth, irreducible, projective varieties of dimension d, X = X 1 X r ; assume that J((X i ) K(X1 X i 1 )) qz for every i = 1,, r. 1 Then I r (X) = 0. Proof. The first step is to show that (p X1 ) (p X2 ) = 0 in the diagram below, that is I 2 (X 1 X 2 ) = 0 (4.4.3) π 0,0 (H Z/q X 1 X 2 ) φ π 2d,d (Φ 1 X 1 X 2 ) ψ π 2d,d (H Z/q X 1 X 2 ) = CH d (X 1 X 2 ) Z/q (p X2 ) π 0,0 (H Z/q X 1 ) φ (p X2 ) π 2d,d (Φ 1 X 1 ) ψ (p X2 ) π 2d,d (H Z/q X 1 ) = CH d (X 1 ) Z/q = Z/q[X 1 ] (p X1 ) CH 0 (Spec k) Z/q = (p X1 ) π 2d,d (Φ 1 ) = Z/q By diagram chasing, it is enough to prove that, (a) (p X2 ) : CH d (X 1 X 2 ) Z/q CH d (X 1 ) Z/q is the zero map and also (b) (p X1 ) : CH 0 (X 1 ) Z/q CH 0 (Spec k) Z/q is the zero map. 1 here K(X 1 X i 1 ) denotes the function field of X 1 X i 1 and (X i ) K(X1 X i 1) is the base changed scheme by the canonical map Spec K(X 1 X i 1 ) Spec k.

31 S. BORGHESI 29 The second follows immediately by the assumption J(X 1 ) qz. To prove the first, we fix a cycle α of dimension d in X 1 X 2 ; (p X2 ) α is of the kind mx 1, where m is [K(α); K(X 1 )]. Let p X2 be the base change of p X2 over the generic point Spec K(X 1 ) of X 1 (4.4.4) (X 1 X 2 ) K(X1 ) can (X 2 ) K(X1 ) dp X2 X 1 X 2 p X2 X 2 Spec K(X 1 ) X 1 Spec k p X2 induces a map between the Chow groups CH 0 ((X 1 X 2 ) K(X1 )) and CH 0 (K(X 1 )). The ideal generated by the image of CH 0 ((X 1 X 2 ) K(X1 )) ( dp X 2 ) CH 0 (K(X 1 )) is canonically contained in J((X 2 ) K(X1 )), because of the factorization of the diagram, and the latter is in qz by assumption; therefore m must be divisible by q because mk(x 1 ) is the image through ( p X2 ) of the 0-cycle α K(X1 ) CH 0 ((X 1 X 2 ) K(X1 )). 2 For the last case, the diagram becomes (4.4.5) π 2d(r 2),d(r 2) (Φ r 2 X r ) φ π 2d(r 1),d(r 1) (Φ r 1 X r ) ψ CH d(r 1) (X r ) Z/q (p Xr ) π 2d(r 2),d(r 2) (Φ r 2 X r 1 ) φ (p Xr ) π 2d(r 1),d(r 1) (Φ r 1 X r ) ψ (p Xr ) CH d(r 1) (X r 1 ) Z/q (p Xr 1 ) π 2d(r 2),d(r 2) (Φ r 2 ) = I r 1 (X r 1 ) = (p Xr 1 ) π 2d(r 1),d(r 1) (Φ r 1 ) = Z/q 2 here it is used that m is [K(α); K(X 1 )].

32 S. BORGHESI 30 where we have used the abbreviation X r for X 1 X r and X r 1 for X 1 X r 1 We have to check that the following maps are zero: (a) p Xr : CH d(r 1) (X r ) Z/q CH d(r 1) (X r 1 ) Z/q (b) p Xr 1 : π 2d(r 2),d(r 2) (Φ r 2 X r 1 ) π 2d(r 2),d(r 2) (Φ r 2 ). The second map is zero by induction. Take β to be a d(r 1)-cycle in X r ; its direct image p Xr (β) will be mw with W a d(r 1)- cycle in X r 1 and m, as before, is [K(β); K(W )]. From β K(W ) β K(Xr 1 ) β (4.4.6) p K(W ) p K(Xr 1 ) p Xr Spec K(W ) Spec K(X r 1 ) W and the fact that p K(W ) is finite of degree m, we conclude that also p K(Xr 1 ) must be finite of the same degree and the direct image p K(Xr 1 ) (β K(Xr 1 )) = mk(x r 1 ) CH 0 (K(X r 1 )). And now, as for the case r = 2, we have the diagram (4.4.7) (X r ) K(Xr 1 ) can (X r ) K(Xr 1 ) p K(Xr 1 ) X r p Xr Spec K(X r 1 ) X r 1 with β K(Xr 1 ) that is a 0-cycle in (X r ) K(Xr 1 ) on which we have no assumption; but the crucial fact is, once again, that p K(Xr 1 ) factors through (X r ) K(Xr 1 ), hence

33 S. BORGHESI 31 p K(Xr 1 ) (β K(Xr 1 )) J((X r ) K(Xr 1 )) qz so q divides m and this finishes the proof of the proposition.

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35 S. BORGHESI 33 [10] Triangulated categories of motives over a field. Cycles, Transfers and Motivic Homology Theories., by Vladimir Voevodsky, Eric. M. Friedlander and Andrei Suslin. [11] V.Voevodsky. A 1 -Homotopy theory. Documenta Mathematica. Extra Volume ICM I