ScalingStacks

Let $\ell\neq p$ be a prime. The categories of equivariant mixed Tate motives $\operatorname{DTM}_{G}(k)$ can be regarded as graded versions of the categories of equivariant sheaves $\operatorname{D}_{G}(\operatorname{pt},\mathbb{Q}_{\ell})$ defined by Bernstein–Lunts [BL94]. Under the realisation functor $\operatorname{Real}_{\ell},$ see (3.1), the Tate motive $\mathbb{Q}(1)$ gets mapped to the Tate module

$$\operatorname{Real}_{\ell}(\mathbb{Q}(1))=\mathbb{Q}_{\ell}(1)=\varprojlim\mu_{\ell^{n}}\otimes_{\mathbb{Z}_{\ell}}\mathbb{Q}_{\ell}$$

which can be identified with $\mathbb{Q}_{\ell}$ by choosing a compatible system of $\ell^{n}$-th roots of unity in $k=\overline{\mathbb{F}}_{p}.$ This induces a natural equivalence of functors

(3.9)

$\displaystyle\operatorname{Real}_{\ell}\circ(1)\to\operatorname{Real}_{\ell}.$

Hence, intuitively, the Tate twist $(1)$ can be regarded as a shift of grading and $\operatorname{Real}_{\ell}$ as a functor forgetting the grading. Restricted to mixed Tate motives, the functor $\operatorname{Real}_{\ell}$ becomes a degrading functor in the sense of [BGS96, Section 4.3].