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user438991
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Let $DM^{\text{eff}}(k)$$DM_{\text{gm}}$ be the category of Voevodsky´s effectivegeometric motives. Let $p,q\in \mathbb{Z}$ be integers with $p<0$.

Is it true that $$\text{Hom}_{DM^{\text{eff}}(k)}(M(X),\mathbb{Z}(p)[q])=0,$$$$\text{Hom}_{DM_{\text{gm}}}(M_{\text{gm}}(X),\mathbb{Z}(p)[q])=0,$$ where $M(X)$ is the motive of a smooth scheme $X$ over a field $k$ and $\mathbb{Z}(p)$ is the Tate motive?

Let $DM^{\text{eff}}(k)$ be the category of Voevodsky´s effective motives. Let $p,q\in \mathbb{Z}$ be integers with $p<0$.

Is it true that $$\text{Hom}_{DM^{\text{eff}}(k)}(M(X),\mathbb{Z}(p)[q])=0,$$ where $M(X)$ is the motive of a smooth scheme $X$ over a field $k$ and $\mathbb{Z}(p)$ is the Tate motive?

Let $DM_{\text{gm}}$ be the category of Voevodsky´s geometric motives. Let $p,q\in \mathbb{Z}$ be integers with $p<0$.

Is it true that $$\text{Hom}_{DM_{\text{gm}}}(M_{\text{gm}}(X),\mathbb{Z}(p)[q])=0,$$ where $M(X)$ is the motive of a smooth scheme $X$ over a field $k$ and $\mathbb{Z}(p)$ is the Tate motive?

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user438991
user438991
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A question about the vanishing of motivic cohomology in negative Tate twist

Let $DM^{\text{eff}}(k)$ be the category of Voevodsky´s effective motives. Let $p,q\in \mathbb{Z}$ be integers with $p<0$.

Is it true that $$\text{Hom}_{DM^{\text{eff}}(k)}(M(X),\mathbb{Z}(p)[q])=0,$$ where $M(X)$ is the motive of a smooth scheme $X$ over a field $k$ and $\mathbb{Z}(p)$ is the Tate motive?