Formalizing the Ring of Witt Vectors

Abstract

The ring of Witt vectors W over a base ring is an important tool in algebraic number theory and lies at the foundations of modern -adic Hodge theory. W has the interesting property that it constructs a ring of characteristic 0 out of a ring of characteristic > 1, and it can be used more specifically to construct from a finite field containing Z/ Z the corresponding unramified field extension of the -adic numbers Q (which is unique up to isomorphism).We formalize the notion of a Witt vector in the Lean proof assistant, along with the corresponding ring operations and other algebraic structure. We prove in Lean that, for prime , the ring of Witt vectors over Z/ Z is isomorphic to the ring of -adic integers Z . We develop idioms to cleanly handle calculations of identities between operations on the ring of Witt vectors. These calculations are intractable with a naive approach, and require a proof technique that is usually skimmed over in the informal literature.