nLab Greenberg scheme

Contents

Contents

Idea

In the context of motivic integration, the term Greenberg scheme refers to one of multiple generalizations of arc spaces (here referred to as β€œarc schemes”). Greenberg schemes are built on schemes over a complete discrete valuation ring (DVR). In the locally Noetherian setting, if RR 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 RR be a complete DVR, let π”ͺ\mathfrak{m} be its unique maximal ideal, and let k=R/π”ͺk=R/\mathfrak{m} be its residue field.

The ring schemes β„› n\mathscr{R}_n

For every non-negative integer nβ‰₯0n\ge0, the functor β„› n:Alg kβ†’Ring\mathscr{R}_n\colon\mathbf{Alg}_k\to\mathbf{Ring} from the category of kk-algebras to the category of rings given by

A↦AβŠ— kR n=AβŠ— kR/π”ͺ n+1 A\mapsto A\otimes_k R_n = A\otimes_k R/\mathfrak{m}^{n+1}

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

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

β„› n:A↦W(A)βŠ— W(k)R n \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 β„› n\mathscr{R}_n with its fpqc-sheafification.

Functorial description of Gr n\mathrm{Gr}_n

Given a kk-scheme (Y,π’ͺ Y)(Y,\mathscr{O}_Y), define the locally ringed space h n(Y)=(Y,β„› n(π’ͺ Y))h_n(Y)=(Y,\mathscr{R}_n(\mathscr{O}_Y)) whose structure sheaf is valued in R nR_n-algebras, given by

U↦Hom k(U,β„› n) U\mapsto\mathrm{Hom}_k(U,\mathscr{R}_n)

which has the structure of an R nR_n-algebra given by the ring scheme structure on β„› n\mathscr{R}_n. h n(βˆ’)h_n(-) is isomorphic to the functor (βˆ’)Γ— kR n(-)\times_{k} R_n, and thus h n(Y)h_n(Y) is an R nR_n-scheme.

With this construction, the Greenberg functor Gr n\mathrm{Gr}_n can be defined abstractly as the right adjoint to the functor h nh_n; that is, to every RR-scheme XX, let

Gr n(X)(βˆ’)≔Hom Sch R(h n(βˆ’),X). \mathrm{Gr}_n(X)(-)\coloneqq \mathrm{Hom}_{\mathbf{Sch}_R}(h_n(-), X).

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

The Greenberg schemes Gr n(X)\mathrm{Gr}_n(X) form an inverse system, where the morphisms Gr m(X)β†’Gr n(X)\mathrm{Gr}_m(X)\to\mathrm{Gr}_n(X) for mβ‰₯nm\ge n come from the obvious natural transformations h nβ‡’h mh_n\Rightarrow h_m. The limit of these functors

Gr ∞(X)(βˆ’)≔lim nHom Sch R(h n(βˆ’),X) \mathrm{Gr}_\infty(X)(-)\coloneqq \lim_n\,\mathrm{Hom}_{\mathbf{Sch}_R}(h_n(-),X)

is also represented by a kk-scheme.

Greenberg schemes of formal schemes

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

Gr ∞(𝔛)≔lim nGr n(𝔛) \mathrm{Gr}_\infty(\mathfrak{X})\coloneqq \lim_n\,\mathrm{Gr}_n(\mathfrak{X})

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

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

References

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

Last revised on July 3, 2023 at 06:59:22. See the history of this page for a list of all contributions to it.