4 problems

Let $K$ be a field, let $p$ be a prime distinct from the characteristic of $K$, assume that $K$ contains a primitive $p$th root of unity $\zeta$, and let $r\ge0$. For $w\in H^r(K)$…

Let $k$ be a field and let $q,r,q',r'$ be nonnegative integers. The condition $(N_q^r,k)$ asserts that, for every $k'\in\operatorname{\mathcal E}(k)$ and every discrete torsion abe…

Let $\mathcal{O}$ be a $\mathbb{Z}[1/\ell]$-algebra, where $\ell$ is prime, and let $n\geq 1$. Write $\operatorname{H}_{\textup{\'et}}^*(\mathcal{O},\mathbb{Z}/\ell^n(*))$ for étal…

Let $k$ be a perfect field. Suppose $X/k$ is a finite type $k$-scheme of pure dimension $n$, and let … be a saturated flag of points. Here $E^{\operatorname*{Yek}}$ and…