8 problems

Let $F$ be a global field of characteristic $p\geq 0$, let $l\neq p$ be a prime, and for each place $v$ let $F_v$ be the completion of $F$ at $v$. Let $X$ be the scheme under consi…

Kato's residue conjecture. The map $\Delta_j$ is an isomorphism for all $r\geq 1$ and $j\geq 0$. This conjecture concerns comparison of Kato homology on a regular proper flat model…

Global localization conjecture. For $j>0$, there is an isomorphism

Residue invariance conjecture. If the model $\mathscr{X}$ is proper over $\mathcal{O}_F$ and regular, then

Finite-field computation. There is an isomorphism

Let $R$ be an excellent henselian discrete valuation ring with residue field $k$, and let $B=\operatorname{Spec}R$. Let $\mathcal Y$ be a regular scheme, flat and proper over $B$,…

Let $X$ be smooth and projective over a finite field, and let $H_i^K(X,\mathbb Q/\mathbb Z(1))$ denote Kato homology in weight $1$. Write…

Kato's conjecture. The displayed isomorphism holds. This is a cohomological Hasse-principle prediction for Kato homology of smooth projective varieties over the field $k$; the supp…