We develop some of the foundations of affinoid pre-adic spaces without
Noetherian or finiteness hypotheses. We give some explicit examples of non-adic
affinoid pre-adic spaces (including a locally perfectoid one). On the positive
side, we also show that if every affinoid subspace of an affinoid pre-adic
space is uniform, then the structure presheaf is a sheaf; note in particular
that we assume no finiteness hypotheses on our rings here. One can use our
result to give a new proof that the spectrum of a perfectoid algebra is an adic
space.Comment: Version 2 of the manuscript -- the arguments are now presented for
general f-adic rings with a topologically nilpotent unit (the original proofs
still go through in this generality
