Specialization of graded modules and generic freeness of local cohomology

Often a good tactic to approach a challenging problem is to go all the way up to a generic case and then find sufficient conditions for the specialization to keep some of the main features of the former. The procedure depends on taking a dramatic number of variables to allow modifying the given data into a generic shape, and usually receives the name of specialization. This classical method is seemingly due to Kronecker and Krull.

We introduce and study a new notion of specialization that generalizes the classical approach. As applications, we also consider the specialization of rational maps and symmetric and Rees powers of a module.

Our main technical tool is to study the generic freeness of local cohomology modules in a graded setting. Our approach works in a quite unrestrictive setting by only assuming that the coefficient ring is Noetherian, and under additional assumptions (e.g., the coefficient ring is reduced or an integral domain) the results are considerably improved. This is joint work with Marc Chardin and Aron Simis.