One says that "finite determinacy" holds in a given situation if the object is determined (up to
some equivalence or
group action) by some finite part of its Taylor expansion.
Finite determinacy is the minimalistic notion of stability. It often holds for the germs of
functions, maps, matrices, modules etc.
The works of Mather, Tougeron and many other have reduced (in many particular cases) the
verification of finite determinacy to the level of tangent spaces to the orbit and to the
ambient space. Yet, the approach was often casebycase, restricted mostly to the real/complex
analytic setup. And the group was usually assumed to be Lie or
proalgebraic.
We give a vast generalization of this to the case of arbitrary ring k (of any characteristic),
filtered module M
over k, filtered group action of G on M, under the assumption: G posseses a "well behaved
tangent space at the unit
element". (In particular, Lietype groups and proalgebraic groups are of this type.)
This extends both the classical criteria of Mather/Tougeron (and many others) and some recent
results (in
zero/positive characteristics) to a broad class of rings, modules and group actions.
Such a ``linearization", i.e. reduction of determinacy questions to the tangent spaces/modules,
is not yet a complete
solution. In many cases, e.g. for group actions on matrices, the tangent module can be
complicated. We address
some most important actions and formulate the readytouse determinacy criteria in terms of the
classical matrix
invariants.
