Chow groups with coefficients.

*(English)*Zbl 0864.14002Summary: The paper considers generalities for localization complexes for varieties. Examples of these complexes are given by the Gersten resolutions in various contexts, in particular in \(K\)-theory and in étale cohomology. The paper gives a general notion of coefficient systems for such complexes, the so-called cycle modules. There are the corresponding “complexes of cycles with coefficients” and their homology groups, the “Chow groups with coefficients”. For these some general constructions are developed: proper push-forward, flat pull-back, spectral sequences for fibrations, homotopy invariance and intersection theory.

If one specializes the material to the case of Milnor’s \(K\)-theory as coefficient system, one obtains in particular an elementary development of intersections for the classical Chow groups. This treatment is somewhat different to former approaches. The main tool is still the deformation to the normal cone. The major different is that homotopy invariance is not established alone for the Chow groups, but for the “cycle complex with coefficients in Milnor’s \(K\)-theory”. This enables one to keep control in fibered situations. The proof of associativity of intersections is based on a doubled version of the deformation to the normal cone.

If one specializes the material to the case of Milnor’s \(K\)-theory as coefficient system, one obtains in particular an elementary development of intersections for the classical Chow groups. This treatment is somewhat different to former approaches. The main tool is still the deformation to the normal cone. The major different is that homotopy invariance is not established alone for the Chow groups, but for the “cycle complex with coefficients in Milnor’s \(K\)-theory”. This enables one to keep control in fibered situations. The proof of associativity of intersections is based on a doubled version of the deformation to the normal cone.

##### MSC:

14C05 | Parametrization (Chow and Hilbert schemes) |

14C15 | (Equivariant) Chow groups and rings; motives |

14C35 | Applications of methods of algebraic \(K\)-theory in algebraic geometry |

19D45 | Higher symbols, Milnor \(K\)-theory |