TY - UNPB

T1 - Pairs of Lie-type and large orbits of group actions on filtered modules. (A characteristic-free approach to finite determinacy.)

AU - Boix, Alberto F.

AU - Greuel, Gert-Martin

AU - Kerner, Dmitry

PY - 2018/8/1

Y1 - 2018/8/1

N2 - Finite determinacy for mappings has been classically thoroughly studied
in numerous scenarios in the real- and complex-analytic category and in
the differentiable case. It means that the map-germ is determined, up to
a given equivalence relation, by a finite part of its Taylor expansion.
The equivalence relation is usually given by a group action and the
first step is always to reduce the determinacy question to an
"infinitesimal determinacy", i.e., to the tangent spaces at the orbits
of the group action. In this work we formulate a universal,
characteristic-free approach to finite determinacy, not necessarily over
a field, and for a large class of group actions. We do not restrict to
pro-algebraic or Lie groups, rather we introduce the notion of "pairs of
(weak) Lie type". These are groups together with a substitute for the
tangent space to the orbit such that the orbit is locally approximated
by its tangent space, in a precise sense. This construction may be
considered as a kind of replacement of the exponential resp. logarithmic
maps and is of independent interest. In this generality we establish the
"determinacy versus infinitesimal determinacy" criteria, a far reaching
generalization of numerous classical and recent results, together with
some new applications.

AB - Finite determinacy for mappings has been classically thoroughly studied
in numerous scenarios in the real- and complex-analytic category and in
the differentiable case. It means that the map-germ is determined, up to
a given equivalence relation, by a finite part of its Taylor expansion.
The equivalence relation is usually given by a group action and the
first step is always to reduce the determinacy question to an
"infinitesimal determinacy", i.e., to the tangent spaces at the orbits
of the group action. In this work we formulate a universal,
characteristic-free approach to finite determinacy, not necessarily over
a field, and for a large class of group actions. We do not restrict to
pro-algebraic or Lie groups, rather we introduce the notion of "pairs of
(weak) Lie type". These are groups together with a substitute for the
tangent space to the orbit such that the orbit is locally approximated
by its tangent space, in a precise sense. This construction may be
considered as a kind of replacement of the exponential resp. logarithmic
maps and is of independent interest. In this generality we establish the
"determinacy versus infinitesimal determinacy" criteria, a far reaching
generalization of numerous classical and recent results, together with
some new applications.

KW - Mathematics - Algebraic Geometry

M3 - ???researchoutput.researchoutputtypes.workingpaper.preprint???

T3 - arXiv180806185B

BT - Pairs of Lie-type and large orbits of group actions on filtered modules. (A characteristic-free approach to finite determinacy.)

ER -