The limit of a directed inverse system of quasi-compact spaces need not be quasi-compact. Danger Will Robinson!
Nice exercise: what happens with an inverse limit of spectral spaces with spectral maps? A spectral space is a topological space which is sober, has a basis of quasi-compact opens, and is such that the intersection of any two quasi-compact opens is quasi-compact; actually Hochster showed these are always homeomorphic to spectra of rings.
As usual: don’t answer if you know the answer…
I think I got it. That is unexpected.
This is the first time I am hearing of Hochster’s characterization of spectral spaces. It seems to me that every quasi-compact open subset of a spectral space is again a spectral space. So, if I understand properly what you are saying, every quasi-compact open subset of Spec R is isomorphic to Spec S for some ring S (which, a priori, has nothing to do with R). Is this correct?
Yep!
Hochster actually went a little further: he found canonical constructions of rings inducing a given spectral space. More precisely, he found several subcategories of (spectral spaces) where one can find a left-inverse to Spec:(Rings) —–> (spectral spaces). For example, one can do so on the subcategory where all maps are required to be surjective. Another example, relevant to Jason’s comment, is that one can find such a left-inverse on the subcategory consisting of a given spectral space and all its spectral subobjects.