Let $A \to B$ be a $G$-Galois extension of rings, or more generally of $\mathbb{E}_\infty$-ring spectra in the sense of Rognes. A basic question in algebraic $K$-theory asks how close the map $K(A) \to K(B)^{hG}$ is to being an equivalence, i.e., how close algebraic $K$-theory is to satisfying Galois descent. An elementary argument with the transfer shows that this equivalence is true rationally in most cases of interest. Motivated by the classical descent theorem of Thomason, one also expects such a result after periodic localization. We formulate and prove a general result which enables one to promote rational descent statements as above into descent statements after periodic localization. This reduces the localized descent problem to establishing an elementary condition on $K_0(-)\otimes \mathbb{Q}$. As applications, we prove various descent results in the periodic localized $K$-theory, $TC$, $THH$, etc. of structured ring spectra, and verify several cases of a conjecture of Ausoni and Rognes.

翻译：让美元到B$是G$G$-Galois的环形延伸,或者更广义地说,是Rognes 意义上的“Gathbb{E ⁇ infty$-ring 光谱 ” 。 代数 $K-理论的一个基本问题问,地图 $K(A) $to K(B) ⁇ hG} 如何接近KK(B) $to K(hG) 是一个等值,也就是说, 接近代数 $K(-) 美元- 理论是满足 Galois 的下降。 一项基本的转移论证表明, 在大多数情况下,这种等值确实合理。 托马松的古典世系理论激发了这种结果, 在定期的本地化之后,我们也期待这样的结果。 我们制定并证明一个总的结果, 使一个人能够将上面的合理的血统声明在定期本地化之后, 将 合理的本地化 。 这可以减少本地化的问题, 以 $0(-) otime)\ mathbblates a case.