Open Collections

Description

Let $p$ be a prime number, and let $S$ be a scheme of characteristic $p>0$. For any integer $n>1$, one can build the scheme of Witt vectors of length $n$ of $S$, denoted by $W_n(S)$. It is a natural thickening of characteristic $p^n$ of $S$. Let us say "$W_n$-bundle" (or Witt vector bundle if $n$ is understood) for "vector bundle over $W_n(S)$". In this talk, we consider the following Question. Let $V$ be a vector bundle over $S$. $Q(n,V)$: Does $V$ extend to a $W_n$-bundle I will first give precise (hopefully elementary) definitions, and basic properties of Witt vector bundles. I will then show that the answer to question $Q$ is positive, when $V$ is the tautological vector bundle of the projective space of a vector bundle, defined over an affine base. I will discuss counterexamples in the general setting -- in a Galois-theoretic way. If time permits, I plan to discuss the main motivation for raising question $Q$: lifting Galois representations. This is joint work, with Charles De Clercq and Giancarlo Lucchini Arteche.