Galois cohomology and invertible Morava motives

• Applicant: Dr. Pavel Sechin
• Subject Area: Mathematics
• Term: since 2025
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 575300506

Algebraic Morava K-theories are oriented cohomology theories of algebraic varieties, which are indexed by a prime number p and a natural number. They play an intermediate role between the algebraic K-theory of vector bundles modulo p and the Chow groups of algebraic cycles modulo p. Moreover, there is a category of motives with respect to the algebraic Morava K-theory (Morava motives), which also plays an intermediate role. In particular, the motives of smooth projective varieties contain more information as K-motives, but are structurally simpler as Chow motives. The goal of our proposal is the construction of invertible objects in the category of Morava motives that correspond to the elements of Galois cohomology with finite coefficients. These invertible objects can be seen as a generalization of Rost motives, which played a central role in the proof of the Bloch-Kato conjecture. The construction of invertible objects creates a foundation for the investigation of questions about Galois cohomology from the perspective of the algebraic geometry. We will study the properties of the invertible objects that we construct, especially in the relation to the norm functors. To this aim we will also define these norm functors on the level of the categories of Morava motives, using the Weil restriction functors.

• DFG Programme: Research Grants