Given an ideal \(I\) in a regular local ring \(A\), the cohomological dimension of \(I\) in \(A\) is the index of the highest non-vanishing local cohomology of \(A\) supported at \(I\). Determining effective upper bounds on the cohomological dimension in terms of topological invariants of \(\textrm{Spec}(A/I)\) is a central problem in commutative algebra: foundational results include the Hartshorne--Lichtenbaum Vanishing Theorem and the Second Vanishing Theorem.

In equal characteristic, Faltings established in 1980 a general bound on the cohomological dimension of an ideal in terms of its “big height”. In this talk, we extend Faltings’ results to the unramified mixed characteristic setting and show that the resulting bound is sharp.