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…


Hodge-Newton paradise

I just posted a new version of my preprint on local shtukas and Harris’s conjecture.  To be clear, the goal of this paper is to make good on the optimism I expressed in this previous post.  This project has been one of the most intense mathematical experiences of my life, and I hope to write a proper blog post about it soon.

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!