We give an algorithm to compute a standard basis of the tangent space to the orbit of an algebraic group action. Using this standard basis we can compute the codimension of the tangent space, an important invariant in the classification of map germs. The task is not completely trivial since the tangent space is usually described as the sum of two infinite dimensional vector spaces given by two modules over different rings. We also explain how the standard basis can be computed using modular methods.

Key Words: standard basis, tangent space at orbit, right-left equivalence, map germs.

2010 Mathematics Subject Classification: Primary 14B05. Secondary 14H20, 14J17.