Hyperdescent and étale K-theory

We study the étale sheafification of algebraic K -theory, called étale K -theory. Our main results show that étale K -theory is very close to a noncommutative invariant called Selmer K -theory, which is defined at the level of categories. Consequently, we show that étale K -theory has surprisingly well-behaved properties, integrally and without finiteness assumptions. A key theoretical ingredient is the distinction, which we investigate in detail, between sheaves and hypersheaves of spectra on étale sites.