Consider the action of a group G on an ordinary commutative k-variety X = Spec(A). In this note we define the category of A–G-modules and their deformation theory. We then prove that this deformation theory is equivalent to the deformation theory of modules over the noncommutative k-algebra A[G] = A#G. The classification of orbits can then be studied over a commutative ring, and we give an example of this on surface cyclic singularities.
1. Eriksen, E. An introduction to noncommutative deformations of modules. Lect. Notes Pure Appl. Math., 2005, 243(2), 90–126.
doi:10.1201/9781420028102.ch5
2. Laudal, O. A. Matric Massey products and formal moduli I. In Algebra, Algebraic Topology and Their Interactions (Roos, J.-E., ed.). Lecture Notes in Math., 1183, 218–240. Springer Verlag, 1986.
3. Laudal, O. A. Noncommutative deformations of modules. Homology Homotopy Appl., 2002, 4(2), 357–396.
4. Laudal, O. A. Noncommutative algebraic geometry. Rev. Mat. Iberoamericana, 2003, 19(2), 509–580.
5. Siqveland, A. The method of computing formal moduli. J. Algebra, 2001, 241, 292–327.
doi:10.1006/jabr.2001.8757
6. Siqveland, A. Global Matric Massey products and the compactified Jacobian of the E6-singularity. J. Algebra, 2001, 241, 259–291.
doi:10.1006/jabr.2001.8758
7. Siqveland, A. A standard example in noncommutative deformation theory. J. Gen. Lie Theory Appl., 2008, 2(3), 251–255.
