Degree formula for the Euler characteristic

We give a proof of the degree formula for the Euler characteristic previously obtained by Kirill Zainoulline. The arguments used here are considerably simpler, and allow us to remove all restrictions on the characteristic of the base field.


Introduction
The degree formula for the Euler characteristic says that if f : Y X is a rational map, with X and Y projective connected smooth varieties of the same dimension d, then X possesses a zero-cycle of degree Here deg f is the degree of the function fields extension (zero when f is not dominant), χ the Euler characteristic, and τ d−1 the d − 1-st Todd number (the denominator in the d − 1-st Todd class, see (5)). This formula is useful to prove incompressibility properties of some varieties.
In the paper [Zai10], where this formula is introduced, two distinct proofs are given, based on different results: (a) the generalized degree formula for algebraic cobordism, (b) or the Rost's degree formula. Both techniques are quite sophisticated, and moreover require to make some assumptions on the characteristic of the base field. It is known that (a) (together with the universal property of algebraic cobordism) implies (b), but (b) has the advantage of being known for some fields of positive characteristic, while (a) requires to work over a field of characteristic zero.
Indeed (b) is not proved at the moment when no information on the characteristic of the base field is available, even under the assumption of resolution of singularities. In [Hau,Section 8] we showed that the p-primary part of the result of [Zai10] can be obtained in arbitrary characteristic when one disposes of the so-called p-resolution of singularities. This suggests that this result, the degree formula for the Euler characteristic, does not lie at the same depth as the classical degree formula (b).
The purpose of this article is to give a simpler proof of this result, over any field. In contrast to (a) or (b), the main ingredients here are the Grothendieck-Riemann-Roch theorem, and a small fraction of [Mer02].
Acknowledgements. The support of EPSRC Responsive Mode grant EP/G032556/1 is gratefully acknowledged. I thank Alexander Vishik for the useful discussions that we had on the subject of this paper.
1. Notations 1.1. Varieties. We fix a base field k. A variety is a finite type, separated, quasiprojective scheme over k. A morphism of varieties is a morphism of schemes over k. When X is a smooth variety, we denote by T X its tangent bundle.
1.2. Grothendieck groups of schemes. Let X be a noetherian scheme. We denote by K ′ 0 (X) (resp. K 0 (X)) the Grothendieck group (resp. ring) of coherent O X -modules (resp. locally-free O X -modules).
If f : Y → X is a flat morphism of noetherian schemes, then it induces a pullback f * : There is a natural map which is an isomorphism when X is regular (i.e. for every point x of X the local ring O X,x is regular).
1.4. Rank homomorphism. When X is connected, there is a ring homomorphism, sending a vector bundle to its rank rank : K 0 (X) → Z.
1.5. First Chern class. Let E be a vector bundle over a connected, noetherian scheme X. We denote its first Chern class by 1.6. Subgroup of generically trivial classes. Let X be an integral variety. We denote by K ′ 0 (X) (1) the subgroup of K ′ 0 (X) generated by the elements i * [O W ] with i : W ֒→ X a non-dominant closed embedding of varieties. We have an exact sequence ) → 0, where η : Spec(k(X)) → X is the generic point. 1.7. Euler characteristic. Let X be a projective variety, and x : X → Spec(k) its structural morphism. The Euler characteristic of a coherent O X -module E is For the last equality we have used the identification K ′ 0 (Spec(k)) = Z.

Degree formula for K-theory
Lemma 2.1. Let V be a vector bundle over a smooth connected variety X. Then there exists smooth varieties On the other hand, we have, by [Mer02, Theorem 9.8], with n i , f i , Z i as requested. Note that this element belongs to K 1 (X) = K 0 (X) · t −1 , therefore in view of (2) we can choose the varieties Z i so that dim Z i = dim X − 1. We obtain the result by applying to (3) and (4) the composite Proof. Since X is a regular variety, the map K 0 (X) → K ′ 0 (X) is an isomorphism. Under this identification, the subgroup K ′ 0 (X) (1) corresponds to the kernel of the rank homomorphism. Any element x of this kernel can be written as [E] − [F ], for some vector bundles E and F on X, having the same rank n. Then Finally we apply Lemma 2.1 above. Definition 2.3. Let f : Y → X be a projective morphism of varieties, with X integral. Consider the generic fiber Y × X k(X) as a variety over k(X), and define an integer being understood that χ(∅) = 0.
Lemma 2.4. Let f : Y → X be a projective morphism, with X integral. Then Proof. This follows from the sequence (1), and from the commutative diagram If X is a projective variety, we denote by n X the positive integer such that deg CH(X) = n X · Z.
This integer coincides with the greatest common divisor of the degrees of closed points of X.
Proposition 3.2. Let f : Y → X be a projective morphism. Assume that X is smooth, projective and connected. Then we have, using Definition 2.3, Proof. Project the formula of Theorem 2.5 to K ′ 0 (Spec(k)) = Z. This gives Note that n X |n Z i . Since every Z i is smooth and of dimension < dim X, Lemma 3.1 gives the result.
Let X, Y be projective integral varieties. A correspondence Y X is an element γ ∈ CH(Y × X). The multiplicity mult γ is the image of γ under the map The transpose t γ of γ is the correspondence X Y corresponding to the image of γ under the morphism exchanging factors.
When γ = [Γ] for some integral closed subvariety Γ of Y × X, then mult γ can be non-zero only if dim Γ = dim Y . In this case, mult γ coincides with the integer deg(Γ → Y ) of Definition 2.3. Lemma 3.3. Let γ : Y X be a correspondence between projective integral varieties, with Y smooth. Then n X | mult γ · n Y .
Proof. The map mult : CH(Y × X) → Z is linear and vanishes on cycles of dimension = dim Y , hence we can assume that γ = i * [Γ], where i : Γ ֒→ Y × X is an integral closed subvariety of dimension dim Y . We have a diagram where ∆ : Y ֒→ Y × Y is the diagonal embedding, and x : X → Spec(k), y : Y → Spec(k), γ : Γ → Spec(k) the structural morphisms. The square on the left is commutative because the push-forward along the projective morphism Γ → Y sends [Γ] to mult γ · [Y ]. Commutativity of the other squares is clear. We see that the bottom composite is mult γ · y * and factors through x * .
Theorem 3.4. Let γ : Y X be a correspondence between smooth, projective, connected varieties of the same dimension d. Then we have Proof. We can assume as above that γ is represented by an integral d-dimensional closed subvariety Γ of Y × X. The result then follows from the application of Proposition 3.2 to the projective morphisms Γ → X and Γ → Y .
Let f : Y X be a rational map of integral projective varieties. The closure of its graph defines a correspondence γ f of multiplicity one. We define the integer deg f as mult t γ f . This is compatible with Definition 2.3.
Corollary 3.5. Let f : Y X be a rational map of projective, smooth, connected varieties of the same dimension d. Then we have

4.2.
Resolution of singularities. When resolution of singularities is available, one can obtain Proposition 2.2 or Lemma 3.1, and therefore Proposition 3.2 for singular X. One can thus remove the smoothness assumptions in Theorem 3.4.
When the dimension of X is < p(p − 1), where p is the characteristic of the base field, one can also use [Hau11] to obtain the same result, over any field. 4.3. Perfect fields of positive characteristic. A statement of Rost ([Ros08, Corollary 1]) says that for any projective variety X over a perfect field of positive characteristic p, one has v p (n X ) ≤ v p (χ(O X )).
It follows that the p-primary content of Theorem 3.4 is empty when the base field is perfect. 4.4. Generalized degree formula for periodic multiplicative theories. By [LM07, Theorem 4.2.10], we know that K-theory is the universal oriented weak cohomology theory with periodic multiplicative formal group law. Therefore Theorem 2.5 implies Proposition 4.1. Let A be a oriented weak cohomology theory ([LM07, Definition 4.1.13]) over an arbitrary field k. Assume that the formal group law of A is F (x, y) = x + y − α · xy, for some invertible element α ∈ A(Spec(k)). Then for any projective morphism of smooth connected varieties f : Y → X, we have in for some smooth varieties Z i of dimension dim X−1, projective morphisms f i : Z i → X, and integers n i .
In particular we obtain that the subgroup of A(X) generated by projective push-forwards of fundamental classes of smooth varieties of arbitrary dimensions ("the image of cobordism") coincides with the Z[α]-submodule of A(X) generated by 1 X and the projective push-forwards of fundamental classes of smooth varieties of dimensions < dim X.
4.5. Generalized degree formula for connective K-theory. Connective Ktheory CK p,q has been introduced, over any field, in [Cai08]. It appears that the degree formula for this theory is equivalent to the degree formula for K-theory.
In order to make a precise statement, we use the notations of [Cai08]. In addition, for an integral variety X, we denote by [X] the element [O X ] considered as an element of CK dim X,− dim X (X). Thus the Bott element β ∈ CK 1,−1 (Spec(k)) is p * [P 1 ], where p : P 1 → Spec(k) is the projective line.
Proposition 4.2. Let f : Y → X be a projective morphism of integral varieties, with X smooth. Assume that c = dim X − dim Y is ≤ 0. Then we have in for some smooth varieties Z i of dimension dim X−1, projective morphisms f i : Z i → X, and integers n i .
Proof. It follows from the construction of CK-groups, and from the inequality dim Y ≥ dim X, that the natural map CK dim Y,− dim Y (X) → K ′ 0 (X) is an isomorphism. This maps sends the formula of the proposition to the formula of Theorem 2.5.