4 problems

Let $k$ be the base field, let ${\sf DM}_{\mathrm{eff}}(k)$ be the category of effective motivic complexes, and let ${\sf DM}_{\leq n}(k)$ be its subcategory of $n$-motivic complex…

Assume that the characteristic exponent $p$ is inverted and take the étale topology. For every smooth $k$-variety $X$, write $h_0^{\text{\scriptsize{\rm {\'e}t}}}(X)$ for its zerot…

Let $k$ be the base field, let $n\rg e 2$, and let ${\sf HI}_{tr}^{\text{\scriptsize{\rm {\'e}t}}}(k)$ be the category of homotopy-invariant étale sheaves with transfers, with…

Let $k$ be a field of characteristic $0$ equipped with a complex embedding, and let $\operatorname{DA}^{\mathrm{\acute{e}t}}(-,\mathbb{Q})$ and…