Sorry for the silence here, the last six months have been pretty busy. Anyway, here are some small and miscellaneous thoughts.
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.
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.)
Recently I had need of the following result:
Lemma. Let be the spectrum of a valuation ring, with generic point . Let be a separated and finite type map of schemes, and let be a section over the generic point of , with scheme-theoretic image . Then the induced map is an open immersion.
Note that if is proper, then 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 is a domain, then any flat finite type -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…