nLab Greenberg scheme

Idea

In the context of motivic integration, the term Greenberg scheme refers to one of multiple generalizations of arc spaces (here referred to as “arc schemes”). Greenberg schemes are built on schemes over a complete discrete valuation ring (DVR). In the locally Noetherian setting, if $R$ is a complete DVR then Greenberg schemes of formal schemes generalize this prior notion of Greenberg schemes. Greenberg schemes associated to such formal schemes are the setting for motivic integration.

Definitions

Let $R$ be a complete DVR, let $\mathfrak{m}$ be its unique maximal ideal, and let $k=R/\mathfrak{m}$ be its residue field.

The ring schemes $\mathscr{R}_n$

For every non-negative integer $n\ge0$, the functor $\mathscr{R}_n\colon\mathbf{Alg}_k\to\mathbf{Ring}$ from the category of $k$-algebras to the category of rings given by

$$A\mapsto A\otimes_k R_n = A\otimes_k R/\mathfrak{m}^{n+1}$$

is a representable functor if $R$ is equicharacteristic (that is, $R$ and $k$ have the same characteristic). Denote by $\mathscr{R}_\infty$ the limit of these functors, which is again representable. As schemes, these are identified with $\mathbb{A}_k^{n+1}$ and $\mathbb{A}_k^\infty$, respectively, but the ring scheme structures endowed on $\mathscr{R}_n$ and $\mathscr{R}_\infty$ are not the same as the usual ones on affine space.

In the mixed characteristic case, let $W(-)$ be the functor which assigns a $k$-algebra to its ring of Witt vectors. The functor

$$\mathscr{R}_n\colon A\mapsto W(A)\otimes_{W(k)} R_n$$

is not always representable, but its fpqc-sheafification (its sheafification over the fpqc site) is representable; in this case, we replace the functor $\mathscr{R}_n$ with its fpqc-sheafification.

Functorial description of $\mathrm{Gr}_n$

Given a $k$-scheme $(Y,\mathscr{O}_Y)$, define the locally ringed space $h_n(Y)=(Y,\mathscr{R}_n(\mathscr{O}_Y))$ whose structure sheaf is valued in $R_n$-algebras, given by

$$U\mapsto\mathrm{Hom}_k(U,\mathscr{R}_n)$$

which has the structure of an $R_n$-algebra given by the ring scheme structure on $\mathscr{R}_n$. $h_n(-)$ is isomorphic to the functor $(-)\times_{k} R_n$, and thus $h_n(Y)$ is an $R_n$-scheme.

With this construction, the Greenberg functor $\mathrm{Gr}_n$ can be defined abstractly as the right adjoint to the functor $h_n$; that is, to every $R$-scheme $X$, let

$$\mathrm{Gr}_n(X)(-)\coloneqq \mathrm{Hom}_{\mathbf{Sch}_R}(h_n(-), X).$$

This functor is representable; write the corresponding scheme as $\mathrm{Gr}_n(X)$.

The Greenberg schemes $\mathrm{Gr}_n(X)$ form an inverse system, where the morphisms $\mathrm{Gr}_m(X)\to\mathrm{Gr}_n(X)$ for $m\ge n$ come from the obvious natural transformations $h_n\Rightarrow h_m$. The limit of these functors

$$\mathrm{Gr}_\infty(X)(-)\coloneqq \lim_n\,\mathrm{Hom}_{\mathbf{Sch}_R}(h_n(-),X)$$

is also represented by a $k$-scheme.

Greenberg schemes of formal schemes

To any locally Noetherian $R$-adic formal scheme $\mathfrak{X}$, one can define the Greenberg scheme $\mathrm{Gr}_n(\mathfrak{X})$ by simply applying the already constructed Greenberg functor to $\mathfrak{X}_n=\mathfrak{X}\times_{R} R_n$. Similarly, the limit of these functors

$$\mathrm{Gr}_\infty(\mathfrak{X})\coloneqq \lim_n\,\mathrm{Gr}_n(\mathfrak{X})$$

again is representable, hence defines a $k$-scheme, called the Greenberg scheme of $\mathfrak{X}$.

This construction generalizes Greenberg schemes associated to locally Noetherian $R$-schemes, and therefore generalizes, in particular, the construction of arc schemes.

Related concepts

• arc space

• motivic integration

• p-adic integration

• motivic cohomology

References

Antoine Chambert-Loir, Johannes Nicaise, Julien Sebag, Motivic integration, Progress in Mathematics 325 Birkhaeuser 2018 (doi:10.1007/978-1-4939-7887-8)