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Added two paragraph breaks to help clarify the question.
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edit approved Jun 28, 2017 at 3:39
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edit approved Jun 28, 2017 at 3:39
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Explicit Descriptiondescription of Verdier Quotientquotient of Effective Motiveseffective motives

Let $DM^{eff}_{gm}(k)$ be the triangulated category of effective geometric motives over some field $k$ (in the sense of Voevodsky), and let $DM^{eff}_{gm}(k)(1)$ be its full triangulated subcategory consisting of those effective motives Tate-twisted once. 

Consider $C:=DM^{eff}_{gm}(k)/DM^{eff}_{gm}(k)(1)$. 

Is there an explicit descritiondescription of this category? What I actually want to understand is $K_0(\cal{C})$. More precisely, I have a nonzero object in $C$, and I want to show that its class in $K_0(C)$ is nonzero by computing its image under some function $K_0(C)\rightarrow A$ into some abelian group $A$...probably coming from some realization functor.

Explicit Description of Verdier Quotient of Effective Motives

Let $DM^{eff}_{gm}(k)$ be the triangulated category of effective geometric motives over some field $k$ (in the sense of Voevodsky), and let $DM^{eff}_{gm}(k)(1)$ be its full triangulated subcategory consisting of those effective motives Tate-twisted once. Consider $C:=DM^{eff}_{gm}(k)/DM^{eff}_{gm}(k)(1)$. Is there an explicit descrition of this category? What I actually want to understand is $K_0(\cal{C})$. More precisely, I have a nonzero object in $C$, and I want to show that its class in $K_0(C)$ is nonzero by computing its image under some function $K_0(C)\rightarrow A$ into some abelian group $A$...probably coming from some realization functor.

Explicit description of Verdier quotient of effective motives

Let $DM^{eff}_{gm}(k)$ be the triangulated category of effective geometric motives over some field $k$ (in the sense of Voevodsky), and let $DM^{eff}_{gm}(k)(1)$ be its full triangulated subcategory consisting of those effective motives Tate-twisted once. 

Consider $C:=DM^{eff}_{gm}(k)/DM^{eff}_{gm}(k)(1)$. 

Is there an explicit description of this category? What I actually want to understand is $K_0(\cal{C})$. More precisely, I have a nonzero object in $C$, and I want to show that its class in $K_0(C)$ is nonzero by computing its image under some function $K_0(C)\rightarrow A$ into some abelian group $A$...probably coming from some realization functor.

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Explicit Description of Verdier Quotient of Effective Motives

Let $DM^{eff}_{gm}(k)$ be the triangulated category of effective geometric motives over some field $k$ (in the sense of Voevodsky), and let $DM^{eff}_{gm}(k)(1)$ be its full triangulated subcategory consisting of those effective motives Tate-twisted once. Consider $C:=DM^{eff}_{gm}(k)/DM^{eff}_{gm}(k)(1)$. Is there an explicit descrition of this category? What I actually want to understand is $K_0(\cal{C})$. More precisely, I have a nonzero object in $C$, and I want to show that its class in $K_0(C)$ is nonzero by computing its image under some function $K_0(C)\rightarrow A$ into some abelian group $A$...probably coming from some realization functor.