Regular readers of this blog probably know that I’m obsessed with diamonds. They can thus imagine my happiness when Peter posted an official foundational reference for diamonds a few weeks ago.
I want to use this occasion to make a remark aimed at graduate students etc. who might be wondering whether they should bother learning this stuff: in my opinion, spending time with difficult* manuscripts like the one above usually pays off in the long run. Of course, this only works if you invest a reasonable amount of time, and there’s some initial period where you’re completely befuddled, but after some months the befuddlement metamorphoses into understanding, and then you have a new set of tools in your toolkit! This shouldn’t be so surprising, though; after all, papers like this are difficult precisely because they are so rich in new ideas and tools.
Really, I’ve had this experience many times now – with the paper linked above and its precedent, with the Kedlaya-Liu “Relative p-adic Hodge theory” series, with Kato’s paper on p-adic Hodge theory and zeta functions of modular forms, etc. – and it was the same every time: for some period of months (or years) I would just read the thing for its own sake, but then at some point something in it would congeal with the rest of the swirling fragments in my head and stimulate me to an idea which never would’ve occurred to me otherwise. It’s the most fun thing in the world. Try it yourself.
*Here by “difficult” I don’t mean anything negative, but rather some combination of dense/forbidding/technical – something with a learning curve. Of course, there are plenty of papers which are difficult for bad reasons, e.g. because they’re poorly written. Don’t read them.