On the Witt Vector Frobenius

We study the kernel and cokernel of the Frobenius map on the p-typical Witt vectors of a commutative ring, not necessarily of characteristic p. We give many equivalent conditions to surjectivity of the Frobenus map on both finite and infinite length Witt vectors. In particular, surjectivity on finite Witt vectors turns out to be stable under certain integral extensions; this provides a clean formulation of a strong generalization of Faltings’s almost purity theorem from p-adic Hodge theory, incorporating recent improvements by Kedlaya–Liu and by Scholze.