Seminar in Real and Complex Geometry

Thursday, December 28, 2017, 11:00-12:00, Schreiber building, room 209

Dmitry Kerner, Ben Gurion University

Group actions on modules, large orbits and finite determinacy


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 case-by-case, restricted mostly to the real/complex analytic setup. And the group was usually assumed to be Lie or pro-algebraic. 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, Lie-type groups and pro-algebraic 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 ready-to-use determinacy criteria in terms of the classical matrix invariants.