5 problems

Let $\mathbb S$ denote the sphere spectrum over $\mathbb Z$. A spectrum $\mathbb E$ over $\mathbb Z$ is absolutely pure if, for every smoothable morphism $f:X\to S$ between regular…

Let $H\mathbb{Z}^{\mathbb{A}^1}$ denote the motivic spectrum representing motivic cohomology in the $\mathbb{A}^1$-homotopy setting. A motivic spectrum is absolutely pure when it s…

Let $B$ be a base scheme, let $\mathrm{SH}(B)$ be the motivic stable homotopy category over $B$, and let $\mathbf{1}\in\mathrm{SH}(B)$ denote its motivic sphere spectrum. Absolute…

Let $\operatorname{SH}$ denote the motivic homotopy category equipped with its six-functor formalism. Déglise's absolute purity conjecture. Absolute purity should hold for the moti…

Let $i:Y\to X$ be a closed immersion between regular noetherian schemes of pure codimension $c$, and let $n$ be an integer invertible on $X$. Grothendieck's absolute cohomological…