De Rham Cohomology of Local Cohomology Modules

Abstract

Let K be a field of characteristic zero and let \(\mathcal {O}_n\) be the ring \(K[[X_1,\ldots ,X_n]]\). Let \(\mathcal {D}_n = \mathcal {O}_n[\partial _1,\ldots ,\partial _n]\) be the ring of K-linear differential operators on \(\mathcal {O}_n\). Let M be a holonomic \(\mathcal {D}_n\)-module. In this paper we prove \(H^i({\partial }, M) = 0\) for \(i \) be the nth Weyl algebra over K. By a result due to Lyubeznik the local cohomology modules \(H^i_I(R)\) are holonomic \(A_n(K)\)-modules for each \(i \ge 0\). In this article we also compute the de Rham cohomology modules \(H^*(\partial _1,\ldots ,\partial _n ; H^*_I(R))\) for certain classes of ideals.