I.3: Spectral spaces

In this post we give a definition of spectral spaces, as both {\mathrm{Spa}(A,A^+)} and {\mathrm{Spv} A} 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 {Z} of a topological space {X} is irreducible, if {Z} is irreducible for the topology induced from {X}. Every subset of the form {\overline{\{x\}}} is irreducible for {x \in X}.

We say that {x \in X} is a generic point if {\overline{\{x\}} = X}. If {x,y \in X} then we call {x} a generization of {y} (or {y} a specialization of {x}) if {y \in \overline{\{x\}}}.

Definition 9 A topological space {X} is sober if every irreducible closed subset of {X} 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 {A} the space {\mathrm{Spec} A} is spectral when endowed with the usual topology. A basis can be taken to be {(D(f))_{f\in A}}, where {D(f) = \{\mathfrak{p} \in \mathrm{Spec} A | f\notin \mathfrak{p}\}}.

Theorem 11 For a topological space {X} the following assertions are equivalent

  1. {X} is spectral.
  2. {X} is homeomorphic to {\mathrm{Spec} A} for some ring {A}.
  3. {X} is homeomorphic to the underlying topological space of a quasi-compact quasi-separated scheme.
  4. {X} is the projective limit of finite {T_0}-spaces.

Proof: This is a result of Hochster. See Theorem 3.15 in [Wed]. \Box

Recall that a topological space {X} 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 {Z} of {X} is constructible if it is in the Boolean algebra {\mathcal{C}} of subsets of {X} generated by all open quasi-compact subsets (i.e. the smallest set of subsets of {X} 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 {A} be a ring. Then the valuation spectrum {\mathrm{Spv} A} is a spectral space. The sets of the form {\mathrm{Spv}(A) (\frac{f}{s})} for {f,s \in A} 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]. \Box

Building on this result, one can prove (see Theorem 7.10 in [Wed] or Corollary 3.2 in [Hub1]) that {\mathrm{Cont} A} is spectral. The main result of this section is

Theorem 13 Let {(A,A^+)} be an affinoid ring. Then {\mathrm{Spa} (A,A^+)} is a spectral space and the rational subsets of {\mathrm{Spa} (A,A^+)} form a basis of quasi-compact open subsets of {\mathrm{Spa} (A,A^+)} which is stable under finite intersections.

Proof: The statement follows from writing {\mathrm{Spa}(A,A^+)} as an intersection of {\mathrm{Cont} A} and certain spaces {\mathrm{Spv} (A,I)(\frac{a}{1})} for {a \in A^+}, which we know to be spectral. See Theorem 7.35 in [Wed] or Theorem 3.5 in [Hub1]. \Box

References:

[Hub1] R. Huber, “`Continuous valuations”‘

[Wed] T. Wedhorn, “`Adic spaces”‘, unpublished notes

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