The Second Vanishing Theorem for Local Cohomology Modules

We prove the Second Vanishing Theorem for local cohomology modules of an unramified regular local ring in its full generality and provide a new proof of the Second Vanishing Theorem in prime characteristic $p$. As an application of our vanishing theorem for unramified regular local rings, we extend our topological characterization of the highest Lyubeznik number of an equi-characteristic local ring to the setting of mixed characteristic. An upper bound of local cohomological dimension in mixed characteristic is also obtained by partially extending Lyubeznik's vanishing theorem in prime characteristic $p$ to mixed characteristic.