ScalingStacks

Morphisms in $\operatorname{DM}(X,\mathbb{Q})$ are best understood in terms of higher Chow groups, as defined by Bloch [Blo86]. For $X$ smooth, there is a natural isomorphism

	$\displaystyle\operatorname{Hom}_{\operatorname{DM}(X,\mathbb{Q})}(\mathbb{Q}_{X},\mathbb{Q}_{X}(m)[n])$	$\displaystyle\cong\operatorname{Hom}_{\operatorname{DM}(k,\mathbb{Q})}(\operatorname{M}(X),\mathbb{Q}(m)[n])$
(3.2)			$\displaystyle\cong\operatorname{CH}^{m}(X,2m-n)_{\mathbb{Q}},$

where $\mathbb{Q}_{X}$ denotes the tensor unit and $\operatorname{CH}$ a higher Chow group. In particular, for $n=2m$ one obtains the usual Chow group of codimension-$m$ cycles $\operatorname{CH}^{m}(X,0)_{\mathbb{Q}}=\operatorname{CH}^{m}(X)_{\mathbb{Q}}.$ In this case the realisation functor $\operatorname{Real}_{\ell}$ yields the cycle class map to $\ell$-adic cohomology

$$\operatorname{CH}^{m}(X)_{\mathbb{Q}}\to H^{2m}_{\text{\'{e}t}}(X,\mathbb{Q}_{\ell}(m)).$$

For $X=\operatorname{pt}$ and $k=\mathbb{F}_{q}$ or $k=\overline{\mathbb{F}}_{p}$ these hom-groups are particularly simple: