is smooth over k, this homomorphism can be identified with a ring homomorphism from the Chow ring to etale cohomology. Relation to K-theory An (algebraic) vector bundle E on a smooth scheme X over a field has Chern classes ci(E) in CHi(X), with the same formal properties as in topology. The Chern classes give a close connection between vector bundles and Chow groups. Namely, let K0(X) be the Grothendieck group of vector bundles on X. As part of the Grothendieck–Riemann–Roch theorem, Grothendieck showed that the Chern character gives an isomorphism This isomorphism shows the importance of rational equivalence, compared to any other adequate equivalence relation on algebraic cycles. Conjectures Some of the deepest conjectures in algebraic geometry and number theory are attempts to understand Chow groups. For example: The Mordell–Weil theorem implies that the divisor class group CHn-1(X) is finitely generated for any variety X of dimension n over a number field. It is an open problem whether all Chow groups are finitely generated for every variety over a number field. The Bloch–Kato conjecture on values of L-functions predicts that these groups are finitely generated. Moreover, the rank of the group of cycles modulo homological equivalence, and also of the group of cycles homologically equivalent to zero, should be equal to the order of vanishing of an L-function of the given variety at certain integer points. Finiteness of these ranks would also follow from the Bass conjecture in algebraic K-theory. For a smooth complex projective variety X, the Hodge conjecture predicts the image (tensored with the rationals Q) of the cycle map from the Chow groups to singular cohomology. For a smooth projective variety over a finitely generated field (such as a finite field or number field), the Tate conjecture predicts the image (tensored with Ql) of the cycle map from Chow groups to l-adic cohomology. For a smooth projective variety X over any field, the Bloch–Beilinson conjecture predicts a filtration on the Chow groups of X (tensored with the rationals) with strong properties. The conjecture would imply a tight connection between the singular or etale cohomology of X and the Chow groups of X. For example, let X be a smooth complex projective surface. The Chow group of zero-cycles on X maps onto the integers by the degree homomorphism; let K be the kernel. If the geometric genus h0(X, Ω2) is not zero, Mumford showed that K is "infinite-dimensional"