# 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