Saito criterion and its avatars (Q6623824)

The main result consists of a simple algebraic proof of a criterion to detect when a module of tangent derivations is free, a new avatar of Saito's criterion, as the authors described. From here, they applied this type of argument to some classes of varieties: square-free polynomials, non-reduced divisors and varieties defined by several algebraically independent polynomials.\N\NThe proof comes from a description of the sufficient and necessary conditions for a reflexive module to be free, by analyzing free submodules of the highest possible rank and the embedding of this submodule inside the free closure of the reflexive module. To show the effectiveness of this new approach, they provided numerous examples showing how to calculate this condition in the affine and projective settings. Furthermore, the authors generalized it to the context of non-reduced divisors by studying their multiderivations and provide an analogous result to the square-free case.\N\NTo conclude, they extend the criterion to ideals defined by more than one algebraically independent polynomial. This is achieved by reducing to their initial situation after considering the Jacobian scheme of these derivations. The paper finishes with some examples illustrating the applications to both affine and projective cases.