5 problems

Let $F$ be a field. The étale cohomology pointed set $H^1_{{\mathrm{et}}}(F,\mathrm{F}_4)$ parametrizes isomorphism classes of groups of type $\mathrm{F}_4$. Consider the map … ind…

Let $k$ be a field, let $p$ be a prime, and let $\operatorname{K}(n)$ be algebraic Morava K-theory at $p$. Assume that the invertible motives $L_{\alpha}$ exist for the indicated c…

Let $k$ be a field containing $p^r$-th roots of unity for all $r$. Let $n\ge 1$, and for $m\ge 0$ let $\operatorname{K}(m)$ be the $m$-th algebraic Morava K-theory at a prime $p$.…

Let $F$ be a field of cohomological dimension $2$. For such fields, $H^n_{\mathrm{et}}(F,M)=0$ for every $n\geq 3$ and every torsion module $M$, so there are no cohomological invar…

Let $F$ be a field of cohomological dimension $2$. Then $H^1_{\mathrm{et}}(F,\mathrm{E}_8)=1$, so every group of type $\mathrm{E}_8$ over $F$ is split. Serre conjecture II. If the…