In the context of motivic integration, the term Greenberg scheme refers to one of multiple generalizations of arc spaces (here referred to as βarc schemesβ). Greenberg schemes are built on schemes over a complete discrete valuation ring (DVR). In the locally Noetherian setting, if 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.
Let be a complete DVR, let be its unique maximal ideal, and let be its residue field.
For every non-negative integer , the functor from the category of -algebras to the category of rings given by
is a representable functor if is equicharacteristic (that is, and have the same characteristic). Denote by the limit of these functors, which is again representable. As schemes, these are identified with and , respectively, but the ring scheme structures endowed on and are not the same as the usual ones on affine space.
In the mixed characteristic case, let be the functor which assigns a -algebra to its ring of Witt vectors. The functor
is not always representable, but its fpqc-sheafification (its sheafification over the fpqc site) is representable; in this case, we replace the functor with its fpqc-sheafification.
Given a -scheme , define the locally ringed space whose structure sheaf is valued in -algebras, given by
which has the structure of an -algebra given by the ring scheme structure on . is isomorphic to the functor , and thus is an -scheme.
With this construction, the Greenberg functor can be defined abstractly as the right adjoint to the functor ; that is, to every -scheme , let
This functor is representable; write the corresponding scheme as .
The Greenberg schemes form an inverse system, where the morphisms for come from the obvious natural transformations . The limit of these functors
is also represented by a -scheme.
To any locally Noetherian -adic formal scheme , one can define the Greenberg scheme by simply applying the already constructed Greenberg functor to . Similarly, the limit of these functors
again is representable, hence defines a -scheme, called the Greenberg scheme of .
This construction generalizes Greenberg schemes associated to locally Noetherian -schemes, and therefore generalizes, in particular, the construction of arc schemes.
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.