Chow groups of zero- and higher zero-cycles
Final Report Abstract
In algebraic geometry one studies the sets of solutions of equations with coefficients in for example the complex, rational or p-adic numbers or the integers. Endowed with more additional structure, these sets are called varieties. In order to study varieties one method is to analyse their subspaces, or subvarieties, up to a chosen notion of equivalence. In the theory of Chow groups, which is a classical theory of invariants, subspaces are classified up to rational equivalence. Two subspaces are rationally equivalent if one can be deformed into the other. The theory of Chow groups does not just give information on the geometry of the object which is studied but also on the coefficients over which these objects are defined. Understanding both of these aspects better is a central goal of arithmetic geometry. Of special interest is the Chow group of zero-cycles, i.e. points up to rational equivalence, since they can often be calculated by a reduction to curves. Of a different, and more general or conceptual, interest is the generalisation of Chow groups to so called higher Chow groups. These are a model for motivic cohomology, a universal theory of invariants. In our project, we first proved a local to global principle for higher zero-cycles on varieties over the rational numbers with Johann Haas. Following this we studied Chow groups over the p-adic numbers. A central tool for this is the Gersten conjecture for Milnor K-theory which we proved in this context with Matthew Morrow. The Gersten conjecture relates local properties of a space to its global properties. As an application we were able to prove important properties of different models of motivic cohomology and to deduce the contravariant functoriality of Chow groups in some cases. Here we used and developed a new technique, the so called left Kan extension from smooth algebras. Following this project we studied cohomological Chow groups and the Gersten conjecture for more general spaces such as singular, i.e. non-smooth, ones. We showed that these Chow groups can in some cases be calculated using smooth varieties, that the Gersten conjecture holds in some cases for semi-stable schemes and that certain subspaces of non-smooth schemes can be moved into good position up to rational equivalence.
Publications
- A local to global principle for higher zero-cycles, Journal of Number Theory, Volume 220, pp. 235-265, 2021
Johann Haas, Morten Lüders
(See online at https://doi.org/10.1016/j.jnt.2020.06.011) - Milnor K-theory of p-adic rings
Morten Lüders, Matthew Morrow
- Bloch-Ogus theory for smooth and semi-stable schemes in mixed characteristic
Morten Lüders
- On the relative Gersten conjecture for Milnor K-theory in the smooth case
Morten Lüders
