The basic geometry of Witt vectors, II: Spaces

This is an account of the algebraic geometry of Witt vectors and arithmetic jet spaces. The usual, "p-typical" Witt vectors of p-adic schemes of finite type are already reasonably well understood. The main point here is to generalize this theory in two ways. We allow not just p-typical Witt vectors but those taken with respect to any set of primes in any ring of integers in any global field, for example. This includes the "big" Witt vectors. We also allow not just p-adic schemes of finite type but arbitrary algebraic spaces over the ring of integers in the global field. We give similar generalizations of Buium's formal arithmetic jet functor, which is dual to the Witt functor. We also give concrete geometric descriptions of Witt spaces and arithmetic jet spaces and investigate whether a number of standard geometric properties are preserved by these functors.Comment: Final versio

The basic geometry of Witt vectors, I: The affine case

We give a concrete description of the category of etale algebras over the ring of Witt vectors of a given finite length with entries in an arbitrary ring. We do this not only for the classical p-typical and big Witt vector functors but also for variants of these functors which are in a certain sense their analogues over arbitrary local and global fields. The basic theory of these generalized Witt vectors is developed from the point of view of commuting Frobenius lifts and their universal properties, which is a new approach even for the classical Witt vectors. The larger purpose of this paper is to provide the affine foundations for the algebraic geometry of generalized Witt schemes and arithmetic jet spaces. So the basics here are developed somewhat fully, with an eye toward future applications.Comment: Final versio

Isocrystals associated to arithmetic jet spaces of abelian schemes

Using Buium's theory of arithmetic differential characters, we construct a filtered $F$-isocrystal ${\bf H}(A)_K$ associated to an abelian scheme $A$ over a $p$-adically complete discrete valuation ring with perfect residue field. As a filtered vector space, ${\bf H}(A)_K$ admits a natural map to the usual de Rham cohomology of $A$, but the Frobenius operator comes from arithmetic differential theory and is not the same as the usual crystalline one. When $A$ is an elliptic curve, we show that ${\bf H}(A)_K$ has a natural integral model ${\bf H}(A)$, which implies an integral refinement of a result of Buium's on arithmetic differential characters. The weak admissibility of ${\bf H}(A)_K$ depends on the invertibility of an arithmetic-differential modular parameter. Thus the Fontaine functor associates to suitably generic $A$ a local Galois representation of an apparently new kind.Comment: Final version, to appear in Advances in Mathematics. arXiv admin note: text overlap with arXiv:1703.0567

Plethystic algebra

The notion of a Z-algebra has a non-linear analogue, whose purpose it is to control operations on commutative rings rather than linear operations on abelian groups. These plethories can also be considered non-linear generalizations of cocommutative bialgebras. We establish a number of category-theoretic facts about plethories and their actions, including a Tannaka-Krein-style reconstruction theorem. We show that the classical ring of Witt vectors, with all its concomitant structure, can be understood in a formula-free way in terms of a plethystic version of an affine blow-up applied to the plethory generated by the Frobenius map. We also discuss the linear and infinitesimal structure of plethories and explain how this gives Bloch's Frobenius operator on the de Rham-Witt complex.Comment: 32 pages. To appear in Adv. Mat