## Some disconnected thoughts

Sorry for the silence here, the last six months have been pretty busy. Anyway, here are some small and miscellaneous thoughts.

$\bullet$ Dan Abramovich is a really excellent and entertaining writer.  I especially recommend this BAMS review of books by Cutkosky and Kollar, and this survey article for the 2018 ICM; after reading these, you’ll feel like you really understand what “resolution of singularities” looks like as a working field.  His mathscinet review of Fesenko’s IUTT survey is also deeply funny – let us hope that the phrase “resounding partial success” enters common usage.

$\bullet$ Speaking of the 2018 ICM, Scholze’s article is also now available here. I was amused to see a reference to “prismatic cohomology” in the last paragraph – when I visited Bhargav in October, this object had a truly horrible and useless provisional name, so it’s great to see that name replaced by such a perfect and suggestive one.  (This is a new cohomology theory which is closely related to crystalline cohomology, and is also morally related to diamonds – cf. these notes for the actual definition.)

$\bullet$ Recently I had need of the following result:
Lemma. Let $S=\mathrm{Spec}\,A$ be the spectrum of a valuation ring, with generic point $\eta \in S$. Let $X \to S$ be a separated and finite type map of schemes, and let $x: \eta \to X_{\eta}$ be a section over the generic point of $S$, with scheme-theoretic image $Z \subset X$. Then the induced map $Z \to S$ is an open immersion.

Note that if $X \to S$ is proper, then $Z \to S$ is an isomorphism. Anyway, I ended up working out two proofs: a short argument relying on Nagata compactification, and a longer argument which avoided this result but used some other pretty tricky stuff, including Zariski’s main theorem and Raynaud-Gruson’s magic theorem that if $R$ is a domain, then any flat finite type $R$-algebra is finitely presented. If someone knows a truly elementary proof of this lemma, I’d be interested to see it.

Next time I’ll say something about what I needed this for…

Anyway, the paper should basically be stable at this point, with the exception that $\S4.3$ will probably be rewritten to some degree once Peter’s six-functors book is done.  The only real difference from the the first version is that the material around the “pointwise criterion” in $\S2.2$ has been streamlined and clarified a bit.  All comments, questions or corrections are very welcome!