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.