Abstract: To a triple of a ring $R$, an $R$-module $M$, and an ideal $I$ we can associate the local cohomology modules $H^n_I(M)$. One classical problem dating back to a question of Grothendieck is to identify when these vanish for all $n > i$ and all modules $M$, the smallest such $i$ we denote the cohomological dimension of $R$ with respect to $I$. Grothendieck showed that this is bounded by $d = \dim R$, while a very simple condition on $\widehat{R}$ and $I\widehat{R}$ controls the vanishing at $d$. A more subtle topological condition controls the vanishing at $d-1$ for regular local rings as posed by Hartshorne in the late 60s, and was proven in equicharacteristic in the early 70s. After 50 years it was shown to be true in unramified mixed characteristic, and the main result of this talk is the ramified mixed characteristic case.

We will discuss the history of the problem, as well as our solution in the ramified case. This proof can recover the vanishing in all characteristics, providing a unified proof, and requires only classical techniques making use of formal schemes.