The Gersten conjecture and algebraic cycles in mixed characteristic

• Applicant: Dr. Morten Lüders
• Subject Area: Mathematics
• Term: since 2025
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 557768455

The aim of our project is to study the local and global structure of algebraic varieties, i.e. spaces considered in algebraic and arithmetic geometry, using the following three methods and to develop these methods further at the same time: (1) The Gersten resolution allows the local analysis of invariants such as Milnor K-theory by their values on subspaces which locally span the space. Such a resolution exists for many invariants on smooth varieties in equal characteristics. We would like to show the existence of such resolutions in further cases, for example for the K-theory of valuation rings, relative Milnor K-theory and Milnor-Witt K-theory in the mixed characteristic. Working in mixed characteristic i.e. roughly speaking simultaneously working at several different primes, is of central importance in number theory. (2) Chow groups are suitable objects for the global analysis of a variety. These groups classify the subspaces of a variety up to deformation. The generalization to so-called higher Chow groups can be identified with motivic cohomology, the motif behind each cohomological invariant, also called cohomology theory. In particular, there usually exist maps, called cycle class maps, from Chow groups to other cohomology theories. In our project, we mainly aim to study Chow groups and Chow groups with orientation in various degrees in families of varieties over discrete valuation rings. The restriction of Chow groups to a special fiber of such a family often requires the Gersten resolution, which we develop first. Working in families makes it for example possible to prove strong structural statements for Chow groups over local fields. Local fields are completions of the integers at a prime, in analogy to the real numbers, are of mixed characteristic and essential for understanding the integers in arithmetic geometry. Another aim of the project is to further develop the connection between the Gersten conjecture and Chow groups using deformation theory, recently initiated by Schreieder. (3) The cycle class map from Chow groups into syntomic cohomology is another method to study Chow groups. There has been great progress in the development of syntomic cohomology in recent years because of a fundamental reformulation by Bhatt-Morrow-Scholze. We would like to prove properties such as purity and duality, which hold for older variants of syntomic cohomology, in the greatest generality for the above reformulation, thus enabling the definition and study of cycle class maps for a larger class of spaces than before. From a similar point of view, we intend to study the p-adic tame Tate twists we have recently defined.

• DFG Programme: Research Grants