Classification of one-dimensional isocrystals

Robert Y. Lewis, Heather Macbeth — Sat, 06 Aug 2022 04:30:08 GMT

Last year, there was a big mathlib refactor to replace linear maps throughout the library with semilinear maps, a more abstract concept which, importantly, unifies linear and conjugate-linear maps.

But this is not the full extent of the generalization! Our number theorist friends here in mathlib told us that we should make sure we chose this full generality of semilinear maps, maps $f:M \to N$ such that $f(ax)=\sigma(a)f(x)$ for some ring homomorphism $\sigma$ between the scalar rings of the modules $M$ and $N$. So we and our coauthor Frédéric Dupuis implemented this full generality, and then asked them for an example to illustrate their need for this more abstract definition. This blog post tells the story of our little adventure in number theory.

It turns out that the standard use of semilinear maps in number theory is for Frobenius-semilinearity, semilinearity with respect to the ring homomorphism of the fraction field of the $p$-typical Witt vectors over a perfect characteristic-$p$ integral domain which is induced by the Frobenius automorphism of that domain. Let's backtrack to catch everyone up...

Lean community blog (Posts by Heather Macbeth)

Classification of one-dimensional isocrystals