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Algebraic Cycles Algebraic Cycles Anand Sawant Abstract Algebraic cycles arose in the study of intersection theory for algebraic varieties. This note, based on a a lecture in the Mathematics Students’ Seminar at TIFR on September 7, 2012 is meant to give an intorduction to algebraic cycles and various adequate equivalence relations on them. We then state the Standard Conjecture D and state some of its consequences including the conjectural theory of motives. 1 Intersection theory and algebraic cycles Intersection theory deals with the problem of studying intersection of two closed subschemes of a scheme. Perhaps the first result in intersection theory was Bezout’s theorem, which says that given 2 two plane curves of degree m and n in PC having no component in common, their intersection consists of at most mn points (exactly mn points if counted with multiplicity). A systematic study of intersection theory involves the study of the group of algebraic cycles on a scheme X. In this section, we give a brief account of basic definitions and properties of algebraic cycles; for details, the reader is referred to Fulton’s book [1]. For simplicity, all the schemes considered in this note will be of finite type over a field F. Definition 1.1. An algebraic cycle on a scheme X is an element of the free abelian group on the set of all the closed integral subschemes of X. We shall denote the group of algebraic cycles by Z(X). Since closed integral subschemes of X are determined uniquely by their generic points, we have Z(X) = ⊕ Z: ζ2X We usually grade this group by dimension, by letting Zp(X) denote the algberaic cycles of dimension p. If X(p) denotes the set of points of X of dimension p, then Zp(X):= ⊕ Z. A point x 2 X is x2X(p) said to have codimension p, if the local ring OX;x has Krull dimension p. We may grade Z(X) by codimension as well: writing X(p) for the set of points of X of codimension p, we set Z p(X):= ⊕ Z. If X is equidimensional of dimension n, we have x2X(p) p Zn−p(X) = Z (X): If Z ,! X is a closed integral subscheme, we denote its class in Z(X) by [Z], and call it a prime cycle. 1 Definition 1.2. An algebraic cycle associated to a closed subscheme Y ,! X is defined to be [Y]:= ⊕ `(OY;Yi )[Yi]; i where the sum is taken over all the irreducible components Yi of Y and `(OY;Yi ) denotes the length of OY;Yi as module over itself. Note that the length is finite, OY;Yi being an artinian ring. For a well-working theory of algebraic cycles, we need to have a notion of intersection product of the cycles associated to two subvarieties Y and Z of a scheme X of dimension n, which will closely reflect the geometry of their intersection Y \ Z. Just taking the set-theoretic intersection of subvarieties in question will not work, because subvarieties may occur in bad position - for instance, two parallel lines in a plane or the case of intersecting a subvariety with itself. In case the cycles intersect properly, that is, if dim Y + dim Z − dim Y \ Z = n, we can define [Y] · [Z] = [Y \ Z]: In order to take care of the bad cases, we must have an adequate equivalence relation on Z(X) which is broad enough to ensure that given any two cycles on X we can find equivalent cycles intersecting properly such that their intersection is equivalent to the original intersection. The most important such equivalence relation which is essential for the purposes of intersection theory on smooth varieties is rational equivalence. Definition 1.3 (Rational equivalence). An i-dimensional cycle α on a scheme X is said to be ratio- nally equivalent to zero and we write α ∼rat 0 if there exist (i+1)-dimensional subvarieties V1;::: Vr 1 1 of X × P , such that the projection maps π j : V j ! P are dominant and we have Xr α = [V j(0)] − [V j(1)]; j=1 −1 1 1 where [V j(P)] = p∗[π j (P)] for a point P 2 P . Here p denotes the projection X × P ! X. The group of i-dimensional cycles that are rationally equivalent to zero is denoted by Rati(X). We use the notation Rati(X), if the cycles are graded by codimension. Thus, in other words, an i-dimensional cycle α on a scheme X is rationally equivalent to zero if there is an (i + 1)-dimensional cycle β on X × P1 such that α = [β(0)] − [β(1)]: Definition 1.4. The Chow group of dimension i-cycles on a scheme X is defined to be CHi(X):= Zi(X)=Rati(X): We will henceforth assume that all the schemes considered here are equidimensional and will i write CH (X):= CHn−i(X), where n is the dimension of X. Theorem 1.5. If X is a smooth scheme of dimension n over a field F, then the intersection product n makes CH∗(X) = ⊕ CHi(X) into a commutative ring. i=1 2 This theorem was first proved in the Chevalley seminar for smooth quasiprojective varieties using the moving lemma and was later generalized by Fulton by replacing the moving lemma by using the techniques of reduction to the diagonal and of deformation to the normal cone. We summarize the imporatant properties of Chow groups in the next proposition: Proposition 1.6. (1) If f : Y ! X is proper, there is a pushforward map f∗ : CHp(Y) ! CHp(X); for all p. (2) If g : Y ! X is flat of relative dimension d, then there is a pullback map ∗ g : CHp(Y) ! CHp+d(X); for all p. (3) Let i : Z ,! X be a closed subscheme and let U = X − Z. Let j : U ,! X be the inclusion of U into X. Then the sequence ∗ i∗ j CHp(Z) −! CHp(X) −! CHp(U) ! 0 is exact for all p. (4) If π : E ! X is an affine bundle of rank d, then the flat pullback ∗ π : CHp(X) ! CHp+d(E); is an isomorphism for all p. 2 Examples of adequate equivalence relations We now see examples of some other adequate equivalence relations on algebraic cycles on smooth projective varieties over an algebraically closed field k. We shall denote the category of smooth projectivve varieties by SmProj=k. Definition 2.1 (Algebraic equivalence). We say that α 2 Zi(X) is algebraically equivalent to zero and write α ∼alg 0 if there exists a smooth curve C, points c1; c2 2 C and a cycle β 2 Zi+1(X × C) such that α = [β(c1)] − [β(c2)]: The subgroup of Zi(X) generated by cycles algebraically equivalent to zero is denoted by Algi(X). We use the notation Algi(X), if the cycles are graded by codimension. Clearly, it can be seen that Rati(X) ⊆ Algi(X). The inclusion is proper in general. We next introduce homological equivalence of algebraic cycles. This depends on the notion of a Weil cohomology theory, which we define next. Definition 2.2. A Weil cohomology theory is a contravariant functor H∗ : SmProj=k ! Graded K-algebras; ∗ i for a field K such that if X 2 SmProj=k is of dimension n, then H (X) = ⊕iH (X) satisfies the following properties: 3 1. Hi(X) is a finite dimensional K-vector space for each i. 2. Hi(X) = 0 for i < 0 and for i > 2n. 3. H2n ' K. 4. (Poincare´ duality) There exists a nondegenerate pairing Hi(X) × H2n−i(X) ! H2n(X) ' K; for each i. 5. (Kunneth¨ isomorphism) There is a canonical isomorphism ∼ H∗(X) ⊗ H∗(Y) −! H∗(X × Y); induced by the projection maps. 6. (Cycle map) There exists a group homomorphism ∗ γX : Z(X) ! H (X) that is functorial with proper pushforwards and flat pullbacks, respects products and direct sums and such that it coincides with the canonical map Z ! K in the case X = Spec k. 7. (Weak Lefschetz property) For any smooth hyperplane section j : W ,! X (that is, W = X\H for some hyperplane H), j∗ : Hi(X) ! Hi(W) is an isomorphism for i ≤ n − 2 and a monomorphism for i ≤ n − 1. 8. (Hard Lefschetz property) For any smooth hyperplane section j : W ,! X, let w = γX([W]) 2 H2(X). Let L : Hi(X) ! Hi+2(X) be the Lefschetz operator given by x 7! x [ w. Then Li : Hn−i(X) ! Hn+1(X) is an isomorphism for i = 1;:::; n. The classical examples of Weil cohomology theories include the singular (Betti) cohomology, de Rham cohomology, l-adic cohomology, crystalline cohomology and Hodge cohomology. We are now set ot define the adequate equivalence relation of homological equivalence. Definition 2.3 (Homological equivalence). Fix a Weil cohomology theory H∗ on SmProj=k. A cycle α 2 Zi(X) is said to be homologically equivalent to zero and we write α ∼hom 0 if γX(α) = 0. The subgroup of Zi(X) generated by cycles homologiically equivalent to zero is denoted by Homi(X). We use the notation Homi(X), if the cycles are graded by codimension. It is a fact that Algi(X) ⊆ Homi(X) for each i. The inclusion can be proper, as was shown first by Griffiths and later by Clemens and others. (Griffiths showed that Hom2(X)= Alg2(X) is nonzero 4 and actually has an element of infinite order, if X is a generic hypersurface of degree 5 in PC.) We now define the coarsest of the adequate equivalence relations.
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