⃝Springer Science+Business Media New York (2016)
RAMIFIED GALOIS COVERS VIA
Freie University of Berlin
Kaiserswerther Str. 16–18
14195 Berlin, Germany
Abstract. We interpret Galois covers in terms of particular monoidal functors, extending
the correspondence between torsors and ﬁber functors. As applications we characterize
tame G-covers between normal varieties for ﬁnite and ´etale group schemes and we prove
that, if G is a ﬁnite, ﬂat and ﬁnitely presented nonabelian and linearly reductive group
scheme over a ring, then the moduli stack of G-covers is reducible.
Let R be a base commutative ring and G be a ﬂat, ﬁnite and ﬁnitely presented
group scheme over R. In [Ton13a] I introduced the notion of a ramiﬁed Galois
cover with group G, brieﬂy a G-cover, and the stack G-Cov of such objects (see
1.2 for details). This stack is algebraic and of ﬁnite type over R and contains B
the stack of G-torsors, as an open substack. If G is diagonalizable, its nice rep-
resentation theory makes it possible to study G-covers in terms of simpliﬁed data
(collections of invertible sheaves and morphisms between them) and to investigate
the geometry of the moduli G-Cov (see [Ton13a]).
The general case is much harder, even when G is a constant group over an
algebraically closed ﬁeld of characteristic zero: a direct approach as in the di-
agonalizable case fails because of the complexity of the representation theory of
G. Thus in order to handle general G-covers one needs a diﬀerent perspective
and Tannaka’s duality comes into play. The G-torsors are very special G-covers
and the solution of Tannaka’s reconstruction problem asserts that they can be
described in terms of particular strong monoidal functors with domain Loc
the category of G-comodules over R which are projective and ﬁnitely generated
as R-modules. If X is an algebraic stack, denote by Loc X (resp. QCoh X ) the
category of locally free of ﬁnite rank (resp. quasi-coherent) sheaves on X , so that
G ≃ Loc
R. When X = Spec A we simply write Loc A and QCoh A. The
result about G-torsors can be stated as follows.
Theorem ([DM82, Thm. 3.2], [Sch13, Thm. 1.3.2]). Let SMon
be the stack over
R whose ﬁber over an R-scheme T is the category of R-linear, exact (on short
Received July 28, 2015. Accepted March 1, 2016.
Corresponding Author: F. Tonini, e-mail: email@example.com.
-016-939 -5 4
Vol. 22, No.
, 2017, pp.