We discuss the capitulation kernel associated to a degree $n$ covering using Čech cohomology and the Kummer sequence. The main result is a five-term exact sequence that relates the capitulation kernel to the Čech cohomology of the $n$-th roots of unity and a certain subquotient of the group of units modulo $n$-th powers.
"Čech cohomology and the capitulation kernel." Funct. Approx. Comment. Math. 61 (1) 95 - 108, September 2019. https://doi.org/10.7169/facm/1758