3 problems

Let $K/k$ be a purely inseparable extension of degree $p$, let $X$ be a connected scheme proper over $k$, and let $x_K\in X_K(K)$ lie over $x\in X(k)$. Let $\mathcal{C}_X$ and…

Let $K$ be a complete discrete valued field of characteristic $p>0$ with residue field $k$, and let $A$ be a $p$-algebra over $K$, meaning a central simple algebra whose center has…

Let $K'/K$ be a geometric, purely inseparable, untwisted extension of geometric discrete valuation fields with finite exponent. Write $e$ for its ramification index, $f$ for its du…