Abstract : We introduce a generalization of Temkin's reduction in an absolute setting. It takes the form of a category of graded log schemes, containing valuative spaces as a full subcategory, as well as more exotic objects such as the reduction mod $p^{n}$ of a p-adic rigid space. We will compare the log étale and log syntomic topologies on these objects, and we will show that the ramification filtrations of Abbes-Saito, Saito and Kato-Thatte measure precisely the lack of topological invariance of the corresponding log syntomic toposes. As a byproduct, we recover and generalize results of Deligne and Hattori on the ramification of extensions of truncated discrete valuation rings.

Keywords : Adic space; valuative space; ramification