Adic spaces II

Abstract

This chapter defines adic spaces. A scheme is a ringed space which locally looks like the spectrum of a ring. An adic space will be something similar. The chapter then identifies the adic version of “locally ringed space.” Briefly, it is a topologically ringed topological space equipped with valuations. The chapter also reflects on the role of A+ in the definition of adic spaces. The subring A+ in a Huber pair may seem unnecessary at first: why not just consider all continuous valuations on A? Specifying A+ keeps track of which inequalities have been enforced among the continuous valuations. Finally, the chapter differentiates between sheafy and non-sheafy Huber pairs.