Diversion

The following excerpt from Wikipedia made me laugh out loud:

The large power output of the Sun is mainly due to the huge size and density of its core (compared to Earth and objects on Earth), with only a fairly small amount of power being generated per cubic metre. Theoretical models of the Sun’s interior indicate a maximum power density, or energy production, of approximately 276.5 watts per cubic metre at the center of the core,[76] which is about the same rate of power production as takes place in reptile metabolism or a compost pile.

Takeaway: If the sun were a giant ball of lizards, nothing would change.

spectral spaces; snark

(Update 12/24: Unfortunately the argument below doesn’t work as written. I overlooked the following detail in my “proof” that X^{wl} \to X is open: if S \to S' is a surjective continuous map of finite T_0 spaces, it’s not necessarily true that S^Z \to S'^Z is surjective. For instance, if S'= \{ \eta \rightsquigarrow s \} is the spectrum of a DVR and S=\{ \eta, s \} is S' equipped with the discrete topology, then S'^Z has three points but S=S^Z has two.

For the argument below to work, it would be enough to know that for any open subset V \subset X^{wl}, its image in X contains an open subset. Is this true?

In any case, the Corollary is still true, although by a totally different argument.)

*****

I spent about six hours yesterday and today proving the following thing.

Lemma. Let X be a spectral space, and let Z be a closed generalizing nowhere-dense subset of X. Then Z is nowhere-dense for the constructible topology on X, i.e. Z doesn’t contain any nonempty constructible subset of X.

This has the following concrete consequence, which is what I really needed.

Corollary. Let (A,A^+) be some Tate-Huber pair with adic spectrum X=\mathrm{Spa}(A,A^+), and let Z \subset X be a Zariski-closed nowhere-dense subset. Suppose U_1 and U_2 are quasicompact open subsets of X such that U_1 \cap (X - Z) = U_2 \cap (X - Z). Then U_1 = U_2.

Proof of Corollary. We need to check that V= (U_1 \cup U_2 ) - (U_1 \cap U_2) is empty. But V is a constructible subset of X contained in Z, so this is immediate from the lemma.

Amusingly, even though this corollary is pretty down-to-earth, I only managed to prove it by proving the lemma, and I only managed to prove the lemma by exploiting the structure of the w-localization X^{wl} of X. Is there a more direct approach? Am I missing something obvious?

(Sketch of actual argument: the profinite set of closed points X^{wl}_{c} maps homeomorphically onto X equipped with the constructible topology, so if V \subset X is constructible it is clopen when viewed as a subset of X^{wl}_{c}. The key point is then to check that Z is nowhere-dense when viewed as a subset of X^{wl}_c. This can be done, using that the natural surjection t: X^{wl} \to X is open and that t^{-1}(Z) (which is then closed, generalizing and nowhere-dense in X^{wl}, the last point by openness of t) is the preimage of its image in \pi_0(X^{wl}) \cong X^{wl}_c.

The openness of t doesn’t seem to be stated in the literature, but it can be deduced from the proof of Lemma 2.1.10 in Bhatt-Scholze, using the fact that it’s obviously true for finite T_0 spaces.)


You may have noticed that RIMS is hosting a series of four workshops next year under the umbrella of a “RIMS Research Project” entitled Expanding Horizons of Inter-universal Teichmuller Theory.  The first of the workshops looks pretty reasonable, the other three not so much.  In case you’re wondering (as I did) how much money RIMS is ponying up for this, it seems to be capped at 5 million yen, or about $41k (according to e.g. this document). This doesn’t seem like very much money to support four workshops; I guess some funding is also coming from that infamous EPSRC grant.

Anyway, when you’re inside a black hole, your horizons might seem quite expansive indeed, but I doubt you’ll have much luck convincing others to join you.