arXiv subjects

Kees Kok

Publications and source records attributed to Kees Kok.

3 recordsLinked to original sources

arXiv2024

On the functoriality of refined unramified cohomology

In this paper, we generalise the construction of the functorial pullback of refined unramified cohomology between smooth schemes, by following the ideas of Fulton's intersection theory and Rost's cycle modules. We also define standard actions of algebraic cycles on the refined unramified cohomology groups of smooth proper schemes avoiding Chow's moving lemma, which coincide with Schreieder's constructions for smooth projective schemes. As applications, we prove the projective bundle and blow-up formulas for refined unramified cohomology groups and we reduce the Rost nilpotence principle in characteristic zero to a statement concerning certain refined unramified cohomology groups. Moreover, we compute the refined unramified cohomology for smooth proper linear varieties and show that Rost's nilpotence principle holds for these varieties in characteristic zero.

math.AG↗

arXiv2023

On the failure of the integral Hodge/Tate conjecture for products with projective hypersurfaces

In this paper we show the failure of the integral Hodge/Tate conjecture for the product of an Enriques surface with a smooth odd-dimensional projective hypersurface. To do this, we use a specialization argument of Colliot-Th\'el\`ene applied to Schreieder's refined unramified cohomology. The results obtained in this way give an interpretation of Shen's result in terms of refined unramified cohomology. Moreover, using this interpretation, we avoid the need to work over the complex numbers so that we may conclude that Shen's result also holds over general algebraically closed fields of characteristic not 2.

math.AG↗

arXiv2023

Higher Chow groups with finite coefficients and refined unramified cohomology

In this paper we show that Bloch's higher cycle class map with finite coefficients for quasi-projective equi-dimensional schemes over a field fits naturally in a long exact sequence involving Schreieder's refined unramified cohomology. We also show that the refined unramified cohomology satisfies the localization sequence. Using this we conjecture in the end that refined unramified cohomology is a motivic homology theory and explain how this is related to the aforementioned results.

math.AG↗