D-modules in positive characteristic and Frobenius descent

Let R = k[x1, . . . , xd] be the ring of polynomials in a finite number of variables over a field k and let DR|k be the corresponding ring of k-linear differential operators. The theory of DR|k-modules has been successfully applied in Commutative Algebra in order to study local cohomology modules due to the fact that, despite not being finitely generated as R-modules, they are so when considered as modules over DR|k. When k is a field of characteristic zero, DR|k is the ring extension of R generated by the partial derivatives {∂i := d dxi | i = 1, . . . , d}. In this setting, G. Lyubeznik [3] proved some finiteness properties of local cohomology modules using the fact that they are holonomic. This is a nice class of DR|k-modules satisfying some good properties, in particular they have finite length. When k is a field of characteristic p > 0, DR|k is the ring extension of R generated by the set of differential operators {∂ i := 1 t! dt dxi | t ∈ N , i = 1, . . . , d} so it is no longer a Noetherian ring. Therefore, the theory of DR|k-modules in positive characteristic do not behave as in the case of characteristic zero as it was pointed out in [1]. The aim of this talk is to give a better understanding of DR|k-modules in positive characteristic. In particular, we are interested in the notion of holonomic modules and its comparison with the category of F-finite F-modules introduced by G. Lyubeznik [4]. The main ingredients we are going to use are the rings of differential operators of level e given by P. Berthelot [2] and the so-called Frobenius descent.