The Greenberg functor revisited

Abstract

We extend Greenberg's original construction to arbitrary (in particular, non-reduced) schemes over (certain types of) local artinian rings. We then establish a number of basic properties of the extended functor and determine, for example, its behavior under Weil restriction. We also discuss a formal analog of the functor.

On the Inner Products of Some Deligne–Lusztig-type Representations

“…In [Sta09], this Greenberg functor technique was used to generalise the constructions and results in [Lus04], from the positive characteristic case to the general case. Detailed modern treatments of the Greenberg functors can be found in [Sta12] and [BDA16].…”

The norm map and the capitulation kernel

“…where the right-hand group above is connected by Lemma 3.8. Now, by[5, Proposition 20.2], R(1) B/K (G K ) is smooth, connected and unipotent over K. The rest of the proof is similar to the last part of the proof of Lemma 3.8.Proposition 3.10. Let S ′ → S be a finite and faithfully flat morphism of locally noetherian schemes and let G be a smooth, commutative and quasiprojective S-group scheme with connected fibers.…”

The Greenberg Functor is Site Cocontinuous

“…We begin by recalling the full-level Greenberg Transform for a complete field extension K/ Q p ; cf. [1] for a modern account and [12] and [13] for the introduction of the functors. Begin by assuming that R/ Z p is an absolutely unramified extension ring of Z p with residue field k/ F p and fraction field K/ Q p .…”

“…If the algebraic field extension F/ Q p has nontrivial ramification, say as in the quadratic extension Q p ( √ p), then there is an analogous story to the one told above that may be used to construct the Greenberg Transform and its left adjoint h. It essentially the same construction, save for now we take products of the W n O X sheaves to build up the Eisenstein equation of ramification termwise, and then build up our Witt Vector structure with these ramifications in mind. Explicit details may be found in [1], but will not be particularly relevant for this paper, save for in providing intuition for the explicit case to which we wish to apply our results.…”

“…While the Greenberg functor and the Greenberg Transform have given myriad tools with which to study schemes and their arithmetic properties, especially by relating the mixed-characteristic case of formal schemes over Spec R to schemes over the residue field Spec k, the Greenberg Transform was written in a pre-Grothendieck language, which made applying the rich theory around it more difficult. It was this issue that lead Bertapelle and González-Avilés to revist and recast the Greenberg Transform into scheme (and formal scheme)-theoretic language of modern algebraic geometry in [1]. This significantly helped in the study of the Greenberg Transform, as it established many site-theoretic and geometric properties of the functor itself, and studied the Greenberg Transform in great detail.…”