[Paper Review] The basic geometry of Witt vectors, II: Spaces

James Borger|arXiv (Cornell University)|Jun 1, 2010

Algebraic Geometry and Number Theory13 references4 citations

TL;DR

This paper extends the theory of Witt vectors and arithmetic jet spaces to arbitrary algebraic spaces over rings of integers in global fields, generalizing p-typical and big Witt functors. It establishes that the Witt and jet functors preserve key geometric properties like étale maps and flatness, proving that the co-ghost map is affine and an isomorphism away from the set of primes E, under E-smoothness assumptions.

ABSTRACT

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.

Motivation & Objective

• To generalize p-typical and big Witt functors to E-typical Witt functors over rings of integers in global fields, including arbitrary algebraic spaces.
• To extend the Witt and arithmetic jet functors—dual to each other—from affine schemes to all algebraic spaces over a global base.
• To investigate whether standard geometric properties such as étale maps, flatness, and affineness are preserved under these functors.
• To provide concrete geometric descriptions of Witt spaces and arithmetic jet spaces in the generalized setting.
• To determine conditions under which the co-ghost map is affine or an isomorphism, particularly away from the set E of primes.

Proposed method

• Generalizes Witt functors $W_{R,E,n}$ to arbitrary algebraic spaces using the étale sheaf-theoretic extension of functors from affine schemes.
• Applies Grothendieck–Verdier’s general method to extend $W_n$ and its left adjoint $Λ_n ⊙ -$ to the category of algebraic spaces via the functors $W_{n*}$ and $W_n^*$.
• Uses van der Kallen’s theorem to show that $W_n$ preserves étale maps of $R$-algebras, ensuring the extended functors preserve étale topology.
• Employs colimit constructions $W_n^*(X) = ∙∙∙∙∙\operatorname{colim}_{U\to X} W_n(U)$ to define the left adjoint on algebraic spaces.
• Applies the terminal object property in categories of $Y_U$-spaces to construct maps $W_{n+1*}(X) \to Z_X$, proving key isomorphism results.
• Reduces the general case to the single-prime case via decomposition $E = E' \amalg E''$, leveraging functorial isomorphisms from (10.6.2) and (10.4.9).

Experimental results

Research questions

• RQ1 Does the Witt functor $W_{n*}$ preserve geometric properties such as étale maps and flatness when extended to algebraic spaces?
• RQ2 Under what conditions is the co-ghost map $W_{n*}(X) \to X^{[0,n]}$ affine or an isomorphism?
• RQ3 Can the adjunction between $W_n$ and $\Lambda_n \odot -$ be extended from affine schemes to arbitrary algebraic spaces over $R$?
• RQ4 Is the $W_{n*}$-image of an $E$-smooth algebraic space $X$ still $E$-smooth and ${\mathfrak{m}}$-flat?
• RQ5 How does the geometry of Witt spaces and arithmetic jet spaces behave when $E$ consists of multiple primes or a single prime?

Key findings

• The extended Witt functor $W_{n*}$ preserves étale maps and covers, ensuring it defines a well-behaved endofunctor on the category of algebraic spaces.
• The left adjoint $W_n^*$ extends the affine construction $\operatorname{Spec}A \mapsto \operatorname{Spec}W_n(A)$ to arbitrary algebraic spaces via colimits over étale covers.
• The co-ghost map $W_{n*}(X) \to X^{[0,n]}$ is affine and an isomorphism away from the set $E$ of primes when $X$ is $E$-smooth.
• The $W_{n*}$-image of an $E$-smooth algebraic space is $E$-flat and ${\mathfrak{m}}$-flat, ensuring good geometric behavior.
• The proof relies on reducing to the case where $E$ consists of a single maximal ideal, using functorial decompositions and terminal object arguments.
• A counterexample shows that the ${\mathfrak{m}}$-flatness assumption is necessary: without it, the image of $W_{n*}(X)$ may fail to be dense outside $\operatorname{Spec}{\mathcal{O}}_S/{\mathfrak{m}}$.