By William Fulton

Intersection thought has performed a principal function in arithmetic, from the traditional origins of algebraic geometry within the strategies of polynomial equations to the triumphs of algebraic geometry over the past centuries. This publication develops the rules of the speculation and exhibits the variety of classical and glossy purposes. The hardcover variation obtained the celebrated Steele Prize in 1996 for top exposition.

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B) Either f is flat in a neighborhood of Z, or Z is generically transverse to f , in the sense that f −1 (Z) is generically smooth. r ∗ Under these assumptions, if [f −1 (Z)] = i=1 mi Wi , then f (cl(Z)) = r m cl(W ) (note that if Z is generically transverse to f , then mi = 1 i i i=1 for all i). (A8) (Case of a point) If x = Spec(k), then cl(x) = 1 and Trx (1) = 1. A basic example of a Weil cohomology theory is given by singular cohomology in the case k = C, when we may take K = Q. In the last section we will discuss an example of a Weil cohomology theory when char(k) = p > 0, the -adic cohomology (with K = Q , for some = p).

If α = ni [Vi ] is an r-cycle that is rationally equivalent to zero, then i ni cl(Vi ) = 0 in A2(n−r) (X). Proof. We may assume that there is an irreducible (r + 1)-dimensional subvariety W of X, and a nonzero rational function φ on W such that α = divW (φ). We have a rational map φ : W P1 defined by φ. Let π : W → W be a projective, generically finite morphism, with W an integral scheme, such that φ ◦ π is a morphism ψ. After possibly replacing W by a nonsingular alteration (see [deJ2]), we may assume that W is nonsingular, connected, and projective.

A2) (K¨ unneth property) If X and Y are nonsingular, connected, projective varieties, and if pX : X × Y → X and pY : X × Y → Y are the canonical 1Recall that graded commutative means that αβ = (−1)deg(α) deg(β) βα for every homogeneous elements α and β. 25 26 4. WEIL COHOMOLOGY THEORIES AND THE WEIL CONJECTURES projections, then the K-algebra homomorphism H ∗ (X) ⊗K H ∗ (Y ) → H ∗ (X × Y ), α ⊗ β → p∗X (α) ∪ p∗Y (β) is an isomorphsim. (A3) (Poincar´e duality) For every nonsingular, connected, projective variety X, the trace map Tr : H 2 dim(X) (X) → K is an isomorphism, and for every i with 0 ≤ i ≤ 2 dim(X), the bilinear map H i (X) ⊗K H 2 dim(X)−i (X) → K, α ⊗ β → TrX (α ∪ β) is a perfect pairing.

### Intersection Theory by William Fulton

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