In this post we give a definition of spectral spaces, as both and are spectral.
Recall that a topological space is irreducible if it cannot be expressed as the union of two proper closed subsets. Equivalently, every non-empty open subset is dense. This is also equivalent to any two non-empty open subsets have non-empty intersection. A subset of a topological space is irreducible, if is irreducible for the topology induced from . Every subset of the form is irreducible for .
We say that is a generic point if . If then we call a generization of (or a specialization of ) if .
Definition 9 A topological space is sober if every irreducible closed subset of has a unique generic point.
Definition 10 A topological space is spectral if it is quasi-compact, sober and has a basis consisting of quasi-compact open subsets which is closed under finite intersections.
We remark that for a ring the space is spectral when endowed with the usual topology. A basis can be taken to be , where .
Theorem 11 For a topological space the following assertions are equivalent
- is spectral.
- is homeomorphic to for some ring .
- is homeomorphic to the underlying topological space of a quasi-compact quasi-separated scheme.
- is the projective limit of finite -spaces.
Proof: This is a result of Hochster. See Theorem 3.15 in [Wed].
Recall that a topological space is called quasi-separated if the intersection of any two quasi-compact open subsets is again quasi-compact. We remark that a spectral space is quasi-separated. Recall that a subset of is constructible if it is in the Boolean algebra of subsets of generated by all open quasi-compact subsets (i.e. the smallest set of subsets of that contains all open quasi-compact subsets, is stable under finite intersections and under taking complements, hence also stable under finite unions).
Proposition 12 Let be a ring. Then the valuation spectrum is a spectral space. The sets of the form for are open and quasi-compact. The Boolean algebra generated by them is the set of constructible sets.
Proof: This is Proposition 4.7 in [Wed] or Proposition 2.6 in [Hub1].
Building on this result, one can prove (see Theorem 7.10 in [Wed] or Corollary 3.2 in [Hub1]) that is spectral. The main result of this section is
Theorem 13 Let be an affinoid ring. Then is a spectral space and the rational subsets of form a basis of quasi-compact open subsets of which is stable under finite intersections.
Proof: The statement follows from writing as an intersection of and certain spaces for , which we know to be spectral. See Theorem 7.35 in [Wed] or Theorem 3.5 in [Hub1].
[Hub1] R. Huber, “`Continuous valuations”‘
[Wed] T. Wedhorn, “`Adic spaces”‘, unpublished notes