Let \(M\) be a module over a local ring \(R\) and a group action \(G\circlearrowright M\), not necessarily \(R\)-linear. To understand how large is the \(G\)-orbit of an element \(z\in M\) one looks for the large submodules of \(M\) lying in \(Gz\). We provide the corresponding (necessary/sufficient) conditions in terms of the tangent space to the orbit, \(T_{(Gz,z)}\).

This question originates from the classical finite determinacy problem of Singularity Theory. Our treatment is rather general, in particular we extend the classical criteria of Mather (and many others) to a broad class of rings, modules and group actions.

When a particular ‘deformation space’ is prescribed, \(\Sigma\subseteq M\), the determinacy question is translated into the properties of the tangent spaces, \(T_{(Gz,z)}\), \(T_{(\Sigma,z)}\), and in particular to the annihilator of their quotient, \(ann\,{T_{(\Sigma,z)}}/{T_{(Gz,z)}}\).

Etant donné une action d’un groupe sur un module, \(G\circlearrowright M\), et un élément \(z\in M\), on étudie le plus grand sous-module de \(M\) contenu dans l’orbite \(Gz\). On donne des conditions nécessaires et suffisantes décrivant ce module en termes de l’espace tangent a l’orbite, \(T_{(Gz,z)}\). Cela prolonge les critères classiques de la théorie des singularités à une large classe d’anneaux, modules, et actions de groupes.