Cristian D. Gonzalez-Aviles

The Greenberg functor revisited

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

Flasque resolutions of reductive group schemes

We generalize Colliot-Thélène’s construction of flasque resolutions of reductive group schemes over a field to a broad class of base schemes.

The algebraic fundamental group of a reductive group scheme over an arbitrary base scheme

We define the algebraic fundamental group π1(G) of a reductive group scheme G over an arbitrary non-empty base scheme and show that the resulting functor G↦ π1(G) is exact.

Finite modules over non-semisimple group rings

LetG be an abelian group of ordern and letR be a commutative ring which admits a homomorphism ℤ[ζn] →R, where ζn is a (complex) primitiven-th root of unity. Given a finiteR[G]-moduleM, we derive a formula relating the order ofM to the product of the orders of the various isotypic componentsMx ofM, where ξ ranges over the group ofR-valued characters ofG. ForG cyclic, we give conditions under which the order ofM is exactly equal to the product of the orders of theMx. To derive these conditions, we build on work of Aljadeff and Ginosar and obtain, in particular, a new criterion for cohomological triviality which improves upon the well-known critetion of T. Nakayama. We also give applications to abelian varieties and to ideal class groups of number fields, obtaining in particular some new class number relations. In an Appendix to the paper, we use étale cohomology to obtain some additional class number relations. Our results also have applications to “non-semisimple” Iwasawa theory, but we do not develop these here. In general, the results of this paper could be used to strengthen a variety of known results involving finiteR[G]-modules whose hypotheses include (an equivalent forn of) the following assumption: “the order ofG is invertible inR”.

Galois sets of connected components and Weil restriction

Let k be a field, A a finite k-algebra and X a smooth A-scheme. We describe the Galois set of connected components of the Weil restriction

A generalization of the Cassels-Tate dual exact sequence

\mathfrak {R}_{A/ k}(X)

Abelian class groups of reductive group schemes

RA/k(X) in terms of the sets of connected components of the geometric fibers of X.

On the Hasse principle for finite group schemes over global function fields

We obtain finiteness theorems for algebraic cycles of small codimension on quadric fibrations over curves over perfect fields. For example, if k is finitely generated over ℚ and X → C is a quadric fibration of odd relative dimension at least 11, then CHi(X) is finitely generated for i ≤ 4.