tool for Chow groups is the localization sequence, as follows. For a scheme over a field and a closed subscheme of , there is an exact sequence where the first homomorphism is the pushforward associated to the proper morphism , and the second homomorphism is pullback with respect to the flat morphism . The localization sequence can be extended to the left using a generalization of Chow groups, (Borel–Moore) motivic homology groups, also known as higher Chow groups. For any morphism of smooth schemes over , there is a pullback homomorphism , which is in fact a ring homomorphism . Examples of flat pullbacks Note that non-examples can be constructed using blowups; for example, if we take the blowup of the origin in then the fiber over the origin is isomorphic to . Branched coverings of curves Consider the branched covering of curves Since the morphism ramifies whenever we get a factorization where one of the . This implies that the points have multiplicities respectively. The flat pullback of the point is then Flat family of varieties Consider a flat family of varieties and a subvariety . Then, using the cartesian square we see that the image of is a subvariety of . Therefore, we have Cycle maps There are several homomorphisms (known as cycle maps) from Chow groups to more computable theories. First, for a scheme X over the complex numbers, there is a homomorphism from Chow groups to Borel–Moore homology: The factor of 2 appears because an i-dimensional subvariety of X has real dimension 2i. When X is smooth over the complex numbers, this cycle map can be rewritten using Poincaré duality as a homomorphism In this case (X smooth over C), these homomorphisms form a ring homomorphism from the Chow ring to the cohomology ring. Intuitively, this is because the products in both the Chow ring and the cohomology ring describe the intersection of cycles. For a smooth complex projective variety, the cycle map from the Chow ring to ordinary cohomology factors through a richer theory, Deligne cohomology. This incorporates the Abel–Jacobi map from cycles homologically equivalent to zero to the intermediate Jacobian. The exponential sequence shows that CH1(X) maps isomorphically to Deligne cohomology, but that fails for CHj(X) with j > 1. For a scheme X over an arbitrary field k, there is an analogous cycle map from Chow groups to (Borel–Moore) etale homology. When X