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Galois Modules arising from Faltings's strict modules V. Abrashkin

Summary: Galois Modules arising from Faltings's strict modules
V. Abrashkin
Suppose O = F q [#] is a polynomial ring and R is a commutative unitary O­algebra.
The category of finite flat group schemes over R with a strict action of O was recently
introduced by Faltings and appears as an equal characteristic analogue of the classical
category of finite flat group schemes in the equal characteristic case. In this paper we
obtain a classification of these modules and apply it to prove analogues of properties
which were known earlier for classical group schemes.
0. Introduction
Throughout all this paper p is a fixed prime number. Suppose R is a commutative unitary ring and
GrR is the category of finite flat commutative group schemes over R. By definition its objects are
G = Spec A(G), where A(G) is a commutative flat R­algebra, which is a locally free R­module of
finite rank, with the comultiplication # : A(G) -#
A(G)# A(G), the counit e : A(G) -# R and the
coinversion i : A(G) -# A(G) satisfying well­known axioms. Denote by Gr # R the full subcategory
of GrR consisting of p­group schemes G, i.e. such that G is killed by some power of p id G .
If R = k is a perfect field of characteristic p then the objects G of Gr # k can be described in
terms of Dieudonne theory, i.e. in terms of finitely generated W (k)­modules M(G) with a #­linear
operator F and a # -1 ­linear operator V such that FV = V F = p id M(G) . (W (k) is the ring of Witt


Source: Abrashkin, Victor - Department of Mathematical Sciences, University of Durham


Collections: Mathematics