We study a question raised by Eisenbud, Mustaţa, 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.
"Local cohomology and pure morphisms." Illinois J. Math. 51 (1) 287 - 298, Spring 2007. https://doi.org/10.1215/ijm/1258735336