ROST S DEGREE FORMULA

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1 ROST S DEGREE FORMULA ALEXANDER MERKURJEV Some parts of algebraic quadratic form theory and theory of simple algebras with involutions) can be translated into the language of algebraic geometry. Example 0.1. Let V, q) be a quadratic form over a field F, Q the associated projective quadric in PV ). Then q is isotropic iff QF ). 1. Numbers n X and n L Notation: scheme = separated scheme of finite type over a field F. variety = integral scheme over F, F X) function field of X. Let X be a scheme over F. For a closed point x X we define degree of x: degx) = [F x) : F ] and the number n X = gcd degx), where the gcd is taken over all closed points of X. Example 1.1. For an anisotropic quadric Q, n Q = 2. Example 1.2. For a Severi-Brauer variety S = SBA), n S = inda). Proposition 1.3. i) If for a scheme X over F, XF ), then n X = 1; ii) Let X and Y be two schemes. If there is a morphism f : Y X, then n X divides n Y. Proof. ii) For a closed point y Y, the point x = fy) is closed and degx) divides degy). Let L/F be a field extension, v a valuation on L over F with residue field F v). We define degree of v: degv) = [F v) : F ] and the number n L = gcd degv), where the gcd is taken over all valuations of finite degree. Proposition 1.4. i) Let X be a complete scheme over F. If XL), then n X divides n L. ii) Let X be a variety, L = F X). If X is smooth, then n L divides n X. iii) If a variety X is complete and smooth, then n X = n L where L = F X). In particular, n X is a birational invariant of a complete smooth variety X. iv) If f : Y X is a rational morphism of complete varieties with Y smooth, then n X divides n Y. Date: June,

2 2 A. MERKURJEV Proof. i) Let O L be the valuation ring. Since X is complete, a point in XL) factors as Spec L Spec O f X. Let m be the closed point in Spec O, x = fm). Then F x) O/m = F v) and therefore, degx) divides degv). ii) Let x X. Since x is smooth, there is a valuation v on L with residue field F x), so that degv) = degx). iii) follows from i) and ii). iv) Let L = F Y ). Since XL), by i), n X divides n L. Finally, by ii), n L divides n Y. 2. Vector bundles and Chern classes A vector bundle over a schemes X is a morphism E X which is locally on X in Zariski topology) is isomorphic to the projection X A n X with linear transition functions. Let f : V W be a dominant morphism of integral schemes. We define degree of f by { [F V ) : F W )] if dim V = dim W, degf) = 0 otherwise. For a scheme X denote by CH n X) the Chow group of classes of cycles in X of dimension n. For a proper morphism f : Y X there is push-forward homomorphism f : CH n Y ) CH n X) defined by f [V ] ) = deg V fv ) ) [fv )]. The group CH 0 X) is generated by classes [x] of closed points x X. If X is complete, the push-forward homomorphism with respect to the structure morphism X Spec F is called the degree map deg : CH 0 X) Z, [x] degx) = [F x) : F ]. If X is a smooth scheme, the ring CH X) is a commutative ring with the identity 1 = [X]. For a vector bundle E X there are operational Chern classes c i E) : CH n X) CH n i X), α c i E) α. If X is a smooth variety, CH X) is a commutative ring and c i E) is the multiplication by c i E) [X]. The latter element we will also denote by c i E) Thus, c i E) CH d i X) where d = dim X. For a smooth X the Chern classes extend to a map c i : K 0 X) CH X). In particular, for a vector bundle E X, the classes c i E) CH d i X) are defined.

3 3. Definition of the invariant η p X) Let p be a prime number, F a field. We assume that charf ) p and µ p F. We fix a primitive p-th root of unity ξ. Let X be quasi-projective scheme over F. The group G = Z/pZ acts by cyclic permutations on the product X p = X X X. The factor scheme X p /G we denote by C p X. The image X of the diagonal X X p under the natural morphism X p C p X is a closed subscheme in C p X, isomorphic to X. In particular, n X = n X. Consider a G-action on the trivial linear bundle X p A 1 over X p by x 1, x 2,..., x p, t) x 2,..., x p, x 1, ξt). The projection X p \ X C p X \ X is unramified, hence the restriction of the factor vector bundle X p A 1 )/G to C p X \X is a linear bundle over C p X \X. Denote it by L X. Let d = dim X. Set l X = c 1 L X ) pd [C p X \ X] CH 0 C p X \ X). Remark 3.1. The element l X is known as the Euler class of the bundle L pd X, i.e. l X is the image of [X] under the composition CH pd C p X \ X) s CH pd L pd X ) p CH 0 C p X \ X), where s is the zero section of the vector bundle L pd X and p is the pull-back with respect to the flat) morphism p : L pd X Cp X \ X. If X is projective, we have degree homomorphism deg : CH 0 C p X \ X) Z/n X Z. Thus, the degree of l X defines an element η p X) Z/n X Z. Thus, for a projective scheme we have defined an invariant η p X) Z/n X Z. Note that X is projective but not necessarily smooth scheme over F. Lemma 3.2. p l X = 0 CH 0 C p X \ X). In particular, Proof. Let p η p X) = 0 Z/n X Z. f : X p \ X C p X \ X be the projection of degree p). Since the linear bundle f L X ) is trivial, by the projection formula, 0 = f c1 f L X ) pd [X p \ X] ) = c 1 L X ) pd f [X p \ X] = 3 p c 1 L X ) pd [C p X \ X] = p l X.

4 4 A. MERKURJEV 4. Degree formula We prove two degree formulas due to M. Rost. Theorem 4.1. Regular Degree Formula) Let f : Y X be a morphism of projective schemes of the same dimension d. Then n X divides n Y and η p Y ) = degf) η p X) Z/n X Z. Proof. Denote by Ỹ the inverse image of X under Cp f : C p Y C p X, so that Y Ỹ. In particular, n X = n X nỹ n Y = n Y. We have open embedding and proper morphism j : C p Y ) \ Ỹ Cp Y ) \ Y i : C p Y ) \ Ỹ Cp X) \ X of degree d p. Denote by L the restriction of the vector bundle L Y on C p Y \ Ỹ. Clearly, L is the inverse image of L X with respect to i, L = i L X ). We have by projection formula, i j l Y ) = i j c 1 L Y ) pd [C p Y \ Y ] ) = i c1 L ) pd [C p Y \ Ỹ ]) = i c1 i L X ) pd [C p Y \ Ỹ ]) = c 1 L X ) pd i [C p Y \ Ỹ ] = degi) c 1 L X ) pd [C p X \ X] ) = degf) p l X. Note, that by Lemma 3.2, Finally, taking degree, degf) p l X = degf) l X. η p Y ) = degl Y ) = deg i j l Y ) ) = degf) η p X) Z/n X Z. Theorem 4.2. Rational Degree Formula) Let f : Y X be a rational morphism of projective varieties of dimension d with Y smooth. Then n X divides n Y and η p Y ) = degf) η p X) Z/n X Z. Proof. By Proposition 1.4iv), n X divides n Y. Let Y 1 be closure of the graph of f in Y X. Applying Theorem 4.1 to the birational isomorphism Y 1 Y of degree 1), we get η p Y 1 ) = η p Y ) Z/n Y Z. On the other hand, applying Theorem 4.1 to the projection Y 1 Xof degree = degf)), we get η p Y 1 ) = degf) η p X) Z/n X Z.

5 Thus, η p Y ) = degf) η p X) Z/n X Z. Corollary 4.3. The class η p X) Z/n X Z is a birational invariant of a smooth projective variety X. Problem 4.4. How to define the invariant η p X) Z/n X Z out of the function field L = F X)? Note that n X = n L by Proposition 1.4.) Here is the main application: Theorem 4.5. Let X and Y be two projective varieties with Y smooth. Assume that X has a rational point over the field F Y ). Then n X divides n Y and if η p Y ) 0 mod n X ), the following holds: i) dimx) dimy ). ii) If dimx) = dimy ), then Y has a closed point over F X) of degree prime to p. Proof. By assumption, there exists a rational morphism f : Y X, hence n X divides n Y by Proposition 1.4iv). i) Assume that m = dimy ) dimx) > 0. Consider the composition g : Y f X X P m F. Clearly, degg) = 0 and n X P m F = n X. By Theorem 4.2, applied to g, η p Y ) 0 mod n X ), a contradiction. ii) By the degree formula, applied to f, and Lemma 3.2, the degree degf) is not divisible by p. Hence the generic point of Y determines a point over F X) of degree = degf). Remark 4.6. The first statement of Theorem 4.5 shows that a variety Y cannot be compressed to a variety X of smaller dimension if η p Y ) 0 Z/n X Z. 5. Computation of η 2 for a quadric Let Q = QV, q) be anisotropic smooth projective quadric of dimension d, so that n Q = 2. Proposition 5.1. η 2 Q) = { 1 + 2Z, if d = 2 k 1 for some k, 2Z, otherwise. Proof. Each pair of distinct points in Q determines a line in PV ), i.e. a plane in V. Thus we have a rational morphism α : C 2 Q Gr2, V ). 5

6 6 A. MERKURJEV Clearly, α is a birational isomorphism! Indeed, let U be the open subvariety in C 2 Q consisting of all pairs of points [v], [u]) such that [v] [u] and the restriction of the quadratic form q on the 2-dimensional subspace generated by v and u is nondegenerate. Then the restriction α U is an open immersion identifying U with the open subvariety U Gr2, V ) consisting of all planes W V such that the restriction q W is nondegenerate. The inverse rational morphism α 1 takes a plane W V to the intersection PW ) Q. If q W is nondegenerate, this intersection is an effective 0-cycle of degree 2, i.e. is a point of C 2 X. Remark 5.2. If dimq) = 1, i.e. if Q is a conic, α is an isomorphism between C 2 Q and Gr2, V ) = PV ). In the split case this isomorphism looks as follows: C 2 P 1 F P2 F. Let E be the tautological linear bundle over Gr2, V ) the second exterior power of rank 2 tautological vector bundle). Denote by L the restriction of the linear bundle L Q to the open subvariety U C 2 Q. Lemma 5.3. α U ) E) L. Proof. Let Ũ be the inverse image of U under the natural morphism Q2 C 2 Q. We have the following morphism of vector bundles: β : Ũ A1 F E, [v], [u], t) v, u, t v u bv, u) ) where b is the polar form of q. The action of G = Z/2Z on Ũ A1 F by [v], [u], t) [u], [v], t) commutes with the trivial action of G on E. Hence β induces an isomorphism L E over α U. At every point of bad loci C 2 Q\U and Gr2, V )\U the form q is isotropic. Hence the numbers n C 2 Q\U and n Gr2,V )\U are even. Lemma 5.3 implies that Let η 2 Q) deg c 1 E) ) 2d mod 2). i : Gr2, V ) P 2 V ) be the canonical closed embedding. The vector bundle E on Gr2, V ) is the inverse image under i of the tautological vector bundle on P 2 V ). The first Chern class of the tautological vector bundle is negative of the class of a hyperplane section. Thus, the degree degc 1 E) 2d is equal to 1) d times the degree of the subvariety igr2, V )) in the projective space P 2 V ). The later degree is known as the Catalan number 1 d + 1 ) 2d. d One easily checks that this number is odd iff d = 2 k 1 for some k.

7 Theorem 5.4. Let Q = QV, q) be a smooth projective quadric of dimension d 2 k 1 and let X be a variety over F such that n X is even and X has a point over F Q). Then i) dimx) 2 k 1. ii) If dimx) = 2 k 1, then Q has a point over F X). Proof. Let Q X be a rational morphism. Choose a subquadric Q Q of dimension 2 k 1 such that the composition f : Q Q X is defined. i) Since η 2 Q ) 0 Z/2Z and hence in Z/n X Z, by Theorem 4.51), dimx) dimq ) = 2 k 1. ii) By Theorem 4.52), Q has a point of odd degree over F X). By Springer s Theorem, Q and hence Q has a point over F X). Corollary 5.5. Hoffmann) Let Q 1 and Q 2 be two anisotropic quadrics. If dimq 1 ) 2 k 1 and Q 2 is isotropic over F Q 1 ), then dimq 2 ) 2 k 1. Corollary 5.6. Izhboldin) Let Q 1 and Q 2 be two anisotropic quadrics. If dimq 1 ) = dimq 2 ) = 2 k 1 and Q 2 is isotropic over F Q 1 ), then Q 1 is isotropic over F Q 2 ). 6. Computing η 2 for smooth varieties For a smooth variety X let T X be the tangent bundle over X. Theorem 6.1. Let X be a proper smooth variety of dimension d > 0. Then the degree of the 0-cycle c d T X ) is even and η 2 X) = deg c d T X ) 2 Z/n X Z. Proof. We would like first to compactify smoothly C 2 X \ X and extend the line bundle L X to the compactification. Note that C 2 X is smooth only if dimx) = 1. Let W be the blow-up of X 2 along the diagonal X X 2. The G-action on X 2 extends to one on W. The subvariety W G coincides with the exceptional divisor P ) T X T X )/T X = PTX ) : PT X ) W W \ PT X ) X X 2 X 2 \ X Since W G is of codimension 1 in W, W/G is smooth and therefore can be taken as a smooth compactification of C 2 X \ X. Now we would like to construct a canonical extension L of L X to the whole W/G. There is a canonical linear bundle L can over the blow-up W with the induced G-action. The group G acts by 1 on the restriction of L can on the exceptional divisor PT X ). The restriction of L can to the complement of the 7

8 8 A. MERKURJEV exceptional divisor W \ PT X ) X 2 \ X is the trivial vector bundle with the trivial G-action. Now we modify the G-action on L can by 1. The new G-action of the restriction of L can on the exceptional divisor PT X ) is trivial, hence L can descends to a linear vector bundle L on W/G which is a desired extension of L X on C 2 X \ X. Now we compute deg c 1 L ) 2d. At one hand, since L X is the restriction of L on C 2 X \ X, we have deg c 1 L ) 2d = deg c 1 L X ) 2d = η 2 X) Z/n X Z. On the other hand, consider morphisms PT X ) i W p W/G We have q X. p c 1 L ) = c 1 L can ) = [PT X )] = i 1) CH 1 W ). In particular, p c 1 L ) 2d = c 1 L can ) 2d and hence 2 deg c 1 L ) 2d = deg c 1 L can ) 2d. The restriction of L can on PT X ) is the canonical line bundle E can with the sheaf of sections O PTX )1). Hence, by the projection formula note that d > 0), hence The class c 1 L can ) 2d = i 1) c 1 L can ) 2d 1 = i c1 E can ) 2d 1) CH 0 W ), deg c 1 L can ) 2d = deg c 1 E can ) 2d 1 q c1 E can ) 2d 1) CH 0 X) is known as the Segre class of T X and coincides with c d T X ). In general, q c1 E can ) i+d 1) = c i T X ).) Finally, 2 deg c 1 L ) 2d = deg c 1 L can ) 2d = deg c 1 E can ) 2d 1 = deg c d T X ). Remark 6.2. The integer deg c d T X ) 2 does not change under field extensions and therefore can be computed over an algebraically closed field. Remark 6.3. If p > 2, the blow up W of the diagonal in X p does not have smooth orbit space W/G and the linear bundle L X cannot be extended to a linear bundle on W/G. Remark 6.4. The number deg c d T X ) does not change under field extensions and hence can be computed over algebraically closed fields.

9 Remark 6.5. The Theorem shows that the degree m of the cycle c d T X ) is always even. Hence this cycle defines a cycle on the symmetric square C 2 X of degree m 2. In the proof we construct this class canonically it is c 1L X ) 2d ). The degree formula follows from canonical nature of this class. One should study other cases of divisibility of characteristic numbers. 7. Computing η 2 for projective spaces Proposition 7.1. Let X = P d F. Then deg c d T X ) = 1) d 2d d Proof. It is known that [ T X ] = d+1)[e can ]+1 K 0 X). We can compute the total Chern class: 1 c T X ) = ce can ) = 1 d t). d+1 Hence ). ) d 1 2d deg c d T X ) = = 1) d d d Let v p be the p-adic discrete valuation and s p a) be the sum of digits of a written in base p. Corollary 7.2. Let Y be a Severi-Brauer variety of dimension d. Then v 2 deg cd T Y ) ) = s 2 d), η 2 Y ) = 2 s 2d) 1 + n Y Z Z/n Y Z. Corollary 7.3. Karpenko) Let A be a division algebra with orthogonal involution σ, Y the Severi-Brauer variety of A. Then the involution σ is anisotropic over F Y ). Proof. Let X = IA, σ) Y be the involution variety, dega) = 2 k. Then 2 k = n Y divides n X. If σ is isotropic over F Y ) then there is a rational morphism Y X. By Corollary 7.2, η 2 Y ) = 2 k 1 + n X Z is nontrivial in Z/n X Z. A contradiction by Theorem 4.51). Remark 7.4. In fact, we proved that the Severi-Brauer variety of a division algebra of degree 2 k cannot be compressed to a variety X of smaller dimension with IX) = 2 k Z. Remark 7.5. Let p be any prime integer. One can show that for a Severi- Brauer variety Y of dimension d divisible by p 1, η p X) = p s pd) p 1 1 Z/n Y Z. As in Corollary 7.3 one can prove that the Severi-Brauer variety of a division algebra of degree p k cannot be compressed to a variety X of smaller dimension with n X = p k Z. ). 9

10 10 A. MERKURJEV 8. Fundamental class of a smooth variety A partition α = α 1, α 2,..., α k ) is a sequence of integers possibly empty) α 1 α 2 α k > 0. The degree of α is the integer α = α 1 + α α k. Denote by πd) the set of all partitions α with α = d and by πd) the number of all partitions in πd). Consider the polynomial ring Z[b 1, b 2,... ] = Z[b] as a graded ring with deg b i = i. For every partition α set b α = b α1 b α2... b αk. The monomials b α form a basis of the polynomial ring over Z, and more precisely, the b α with α = d form a basis of Z[b] d. Example 8.1. Z[b] 0 = Z, Z[b] 1 = Zb 1, Z[b] 2 = Zb 2 Zb 2 1, Z[b] 3 = Zb 3 Zb 1 b 2 Zb 3 1. Consider another polynomial ring Z[c 1, c 2,... ] = Z[c] with similar grading deg c i = i. The elements of Z[c] we call characteristic classes and the c n - the Chern classes. For any partition α we define the smallest symmetric polynomial P α x 1, x 2,... ) = i 1,i 2,...,i k ) x α 1 i 1 x α 2 i 2... x α k i k = Q α σ 1, σ 2,... ), containing the monomial x α 1 1 x α x α k k where the σ i are the standard symmetric functions, and set c α = Q α c 1, c 2,... ). Example c n = c 1,1,...,1) n units). 2. The classes s d = c d) are called the additive characteristic classes. Note that Q d) is the Newton polynomial. The characteristic classes c α with α = d form a basis of Z[c] d. We consider a pairing between Z[b] d and Z[c] d such that the b α and c α form dual bases. In particular, the additive class s d corresponds to b d. For a smooth complete variety X of dimension d we define the fundamental class [X] = deg c α T X )b α Z[b] d. α =d Thus, [X] is a homogeneous polynomial in the b i of degree d. Note that [X] does not change under field extensions. The numbers deg c α T X ) are called characteristic numbers. They are well known in topology. The class T X plays the role of the normal bundle of an embedding of X into R n. Example [pt] = 1; 2. For a smooth projective curve X of genus g, [X] = 2g 1)b 1. In particular, [P 1 ] = 2b 1.

11 3. [P 2 ] = 3b 2 + 6b 2 1, [P 1 P 1 ] = 4b [P 3 ] = 4b b 1 b 2 20b 3 1, [Q 3 ] = 6b 3 10b 3 1. Exercise 8.4. Show that [X Y ] = [X] [Y ] for every smooth complete varieties X and Y. 9. Formal group laws Let R be a commutative ring. A one-dimensional, commutative) formal group law over R is a power series with r ij R such that 1) Φ Φx, y), z ) = Φ x, Φy, z) ), 2) Φx, y) = Φy, x). Φx, y) = x + y + i,j 1 r ij x i y j, Example 9.1. Φx, y) = x + y additive group law), Φx, y) = x + y + rxy, r R. Let f : R S be a commutative ring homomorphism, Φ is a formal group law over R. Then fφ) is a formal group law over S. A formal group law Φ univ over a ring L is called universal if for a any group law Φ over a ring R there is unique ring homomorphism f : L R such that Φ = fφ univ ). Proposition 9.2. A universal group law exists. Proof. Consider the polynomial ring Z[a ij ], i, j 1, and let L be the factor ring modulo the ideal generated by elements encoded in the definition of a formal group law. The ring L is unique up to canonical isomorphism) and it represents the functor R {formal group laws over R} from the category of commutative rings to the category of sets. The ring L is called the Lazard ring. Note that L is generated by the coefficients of Φ univ. We would like to find a convenient model for L. Consider the power series and its formal inverse exp t = t + b 1 t 2 + b 2 t 3 + Z[b][[t]] log t = t + m 1 t 2 + m 2 t 3 + Z[b][[t]]. Exercise 9.3. Prove that m d = [P d ]/d + 1). Consider the formal group law over Z[b]: Φ = explog x + log y) = x + y + i,j 1 r ij x i y j. By the universal property, there is unique ring homomorphism L Z[b] taking Φ univ to Φ. The following algebraic result is well known. 11

12 12 A. MERKURJEV Theorem 9.4. The homomorphism L Z[b] is injective and its image is generated by the coefficients of Φ. We shall identify the Lazard ring L with its image in Z[b]. Note that L is a graded subring in Z[b] generated by the coefficients a ij. Example 9.5. a 11 = 2b 1 = [P 1 ], a 12 = 3b 2 2b 2 1 = [P 2 ] 2[P 1 P 1 ], a 13 = 4b 3 8b 1 b 2 + 4b 3 1, a 22 = 6b 3 6b 1 b 2 + 2b 3 1,... Remark 9.6. The power series log is called the logarithm of the formal group law Φ. It turns out that the Lazard ring is a polynomial ring Z[M 1, M 2,... ] with deg M i = i, where the M i Z[b] can be chosen as follows: pb i + {decomposable terms}, if i = p k 1 M i = for some prime p and integer k > 0, b i + {decomposable terms}, otherwise. Example 9.7. Set M 1 = 2b 1 = [P 1 ], M 2 = 3b 2 2b 2 1 = [P 2 ], M 3 = 2b b 1 b 2 204b 3 1 = [P 3 ] + [Q 3 ], p, if i = p k 1 r i = for some prime p and integer k > 0, 1, otherwise. For any partition α of degree d set r α = r α1 r α2... r αk Z. Proposition 9.8. The factor group Z[b] d /L d is finite and isomorphic to α Z/r αz, where the coproduct is taken over all partitions α of degree d. In particular, L d is a free group of rank πd). Thus, Z[b] d /L d is a finite group of order r = r α = α =d p prime p [ d p 1 ]. This number coincides with the denominator of the Todd class td d.

13 13 Example 9.9. r 1 = 2, r 2 = 12, r 3 = 24, r 4 = Fundamental classes and Lazard ring Let L be the subgroup in Z[b] generated by the classes [X] for all smooth varieties X over F. Clearly, L is a graded subring in Z[b]. Theorem If the base field F is infinite, L = L. Remark In topology the fundamental class of a manifold takes values in L, the cobordism group of the point. We first prove that L L. The Lazard ring L is generated by the coefficients a ij of the universal group law Φ. Hence it is sufficient to prove that a ij L. Let H n,m be a smooth hypersurface of type 1, 1) in P n P m. Proposition [H n,m ] = i,j 0 a ij [P n i ][P m j ]. It follows from the formula by induction on n + m that a nm L. We need to show now that L L. The idea due to Stong) is to construct a fundamental class of a smooth complete variety in K-theory. We define the first Chern class of a linear bundle L over X by c K 1 L) = 1 [L ] K 0 X) and then extend the definition to get classes c K α E) K 0 X) for arbitrary partition α and vector bundle E over X. We define the fundamental class in K-theory for a smooth projective variety X of dimension d by [X] K = α f c K α T X ) ) b α Z[b], where f : K 0 X) K 0 pt) = Z is the push-forward homomorphism with respect to f : X pt. The class [X] K is a polynomial over Z of degree at most d. Example [P 1 ] K = 1 2b 1, [P 2 ] K = 1 3b 1 + 3b 2 + 6b 1 b 2 ). We compare two fundamental classes of X using the Chern character The Riemann-Roch theorem gives ch : K 0 X) CH X) Q

14 14 A. MERKURJEV Proposition For a smooth projective variety X, f c K α T X ) ) = chc K α )td 1) T X ) for all α. In particular, [X] = [X] K) d. Denote by L K d the subgroup of Z[b] generated by the classes [X]K for all projective smooth varieties X of dimension d. By Proposition 10.5, the restriction on L K d of the projection Z[b] Z[b] d is injective with image L d. Hence is a free abelian group of rank at most πd). L K d Proposition For every prime p and every integer d, there are varieties M p α for α πd) such that the classes [M p α] K in Z[b]/pZ[b] are linearly independent. It follows from Proposition that dimension of the image of L K d in Z[b]/pZ[b] is at least πd). Hence, L K d is a free abelian group of rank exactly πd) and hence L K d is generated by [M α] p K for all p and α. Also, the map L K d /pl K d Z[b]/pZ[b] is an isomorphism, hence L K d is a direct summand of Z[b] Hattori-Stong theorem). Note that [Mα] p generate L. Computing classes [P n ] K and [H n,m ] K in terms of the classes of generators [Mα] p K we get Corollary The ring L is generated by the classes [P n ] and [H n,m ] for all n and m. Finally one shows that the classes [P n ] and [H n,m ] can be expressed in terms of the coefficients a ij of the formal group law Φ, hence L L. Proposition Let X be a smooth projective variety of dimension d = p k 1 where p is prime. Then deg s d T X ) is divisible by p. 11. General degree formulas Let X be an algebraic proper smooth variety over F of dimension d. We define the graded) ideal MX) in L generated by the classes [Y ] L for all smooth proper varieties Y with dimy ) < d and such that there exists a morphism Y X. Thus, as an abelian group, MX) is generated by the classes [Y Z] of varieties of dimension d, where dim Z > 0 and there is a morphism Y X. Conjecture General Degree Formula) For any morphism f : Y X of proper smooth varieties of dimension d, [Y ] = degf) [X] L d /MX) d. Remark If f : Y X is étale morphism, then T Y = f T X ) and hence [Y ] = degf) [X] L d, i.e. the degree formula holds in a stronger form. In general, the difference [Y ] degf) [X] can be non-trivial and is caused by the ramification of f.

15 Remark In topology the statement of the conjecture is known as Quillen s theorem. The group L d is a subgroup in Z[b] d of finite index. Hence, the inclusion of dual groups Z[c] d = Z[b] d ) Ld ) = HomLd, Z) is an equality after tensoring with Q. Thus, the group L d ) can be identified with the group of rational characteristic classes c Z[c] d Q such that cl d ) Z, i.e. c T X ) Z for every smooth complete X of dimension d. We call such rational characteristic classes c special. Example By Riemann-Roch theorem, for a smooth complete variety X, deg tdt X ) = i 0 1) i dim H i X, O X ), thus, the inverse Todd class td 1 is special. Let c Z[c] d Q be a special characteristic class. We can consider c as a homomorphism c : L d Z. Applying c to both sides of the degree formula, we get a degree formula c[y ] = degf) c[x] Z/cMX) d ). Thus, to every special characteristic class c we have associated special degree formula. Example Let c = 1c 2 d c is the Chern class). The group MX) d is generated by the classes [Y Z] of varieties of dimension d, where k = dim Z > 0 and there is a morphism Y X. Let p and q be two projections of Y Z on Y and Z respectively. Then T Y Z = p T Y ) q T Z ). The highest Chern class the Euler class) is multiplicative, hence 1 2 deg c d T Y Z ) = deg c d k T Y ) 1 2 deg c k T Z ). Note that the second factor is integer by Theorem 6.1. Clearly, deg c k T Y ) is divisible by n Y. Since there is a morphism Y X, n X n Y by Proposition 1.4 and hence deg c k T Y ) is divisible by n X. Thus cmx) d ) = n X Z) and we get the degree formula 4.1) in the case of smooth varieties and p = 2. Example The additive integral class s d = c d) vanishes on decomposable elements in L d. Hence if d = p k 1, we get s d Y ) degf) s dx) mod IX). p p Recently, F. Morel and M. Levine prove the conjecture over fields of characteristic zero. They used heavily resolution of singularities. References 15