5 problems

Let $A$ be a discrete valuation ring with local parameter $\pi$, perfect residue field $k$ of characteristic $p>0$, and fraction field $K$ of characteristic zero. Let…

Let $S$ be a henselian discrete valuation ring with maximal ideal $\mathfrak m_S=(\pi)$. Let $A$ be a separated essentially of finite type $S$-algebra such that $(A,I=(\pi)A)$ is a…

Let $X$ be a regular scheme of finite type over $S$, with $S$ pure of characteristic $p>0$. Let $Sh_t(X/S)$ denote tame sheaves, let $i:Z\hookrightarrow X$ be the inclusion of a re…

Let $X$ be a scheme over a base scheme $S$, and let $pr:\mathbb A^1_X\to X$ be the natural projection. For a torsion sheaf $F\in \operatorname{Sh}_t(X/S)$, write $H^q_t(X/S,F)$ for…

Fix an algebraically closed field $k$ over which resolution of singularities holds, and let $X$ be a scheme of finite type over $k$. For integers $m\geq 1$ and $n\geq 0$, let … be…