  We study a question raised by Eisenbud, Mustata, and Stillman regarding the
injectivity of natural maps from Ext modules to local cohomology modules. We
obtain some positive answers to this question which extend earlier results of
Lyubeznik. In the process, we also prove a vanishing theorem for local
cohomology modules which connects theorems previously known in the case of
positive characteristic and in the case of monomial ideals.
