The triangulated category of \(\mathrm K\)-motives \(DK_{-}^{\mathrm{eff}}(k)\) (Q2877707)

For any perfect field \(k\), \textit{V. Voevodsky} [Ann. Math. Stud. 143, 188--238 (2000; Zbl 1019.14009)] constructed a triangulated category of motives \(DM= DM^{\text{eff}}_-(k)\) in which there exists a motivic complex \(\mathbb{Z}(i)\) such that the associated motivic cohomology \(H^\ast_{{\mathcal M}}\) is given by NEWLINE\[NEWLINEH^j_{{\mathcal M}}(X,\mathbb{Z}(i)\simeq \Hom_{DM}(M(X),\mathbb{Z}(i)[j]).NEWLINE\]NEWLINE For every smooth variety \(X\) there is a spectral sequence NEWLINE\[NEWLINEE^{p,q}_2= H^{p-q}_{{\mathcal M}}(X, \mathbb{Z}(-g))\Rightarrow K_{p-q}(X),NEWLINE\]NEWLINE where \(K_*\) is Quillen's \(K\)-Theory.NEWLINENEWLINE The aim of this paper is to construct a triangulated category of \(K\)-motives \(DK= DK^{\text{eff}}_(k)\) such that, to each smooth variety \(X\) over \(k\) the associated motive \(M_{{\mathbf K}}(X)\in DK^{\text{eff}}_-(k)\) satisfies NEWLINE\[NEWLINEK_n(X)- \Hom_{DK}(M_{{\mathbf K}}(X)[n],\, M_{{\mathbf K}}(pt))NEWLINE\]NEWLINE for \(n\in\mathbb{Z}\), where \(pt=\text{Spec\,}k\) and \(K_*\) is Quillen's K-Theory.NEWLINENEWLINE In order to construct the triangulated category \(DK^{\text{eff}}_-(k)\) the authors introduce a spectral category \({\mathcal O}_{\text{naive}}\) whose objects are those of \(Sm/k\) and morphism spectra are defined as NEWLINE\[NEWLINE{\mathcal O}_{\text{naive}}(X,Y)_p= \Hom_{Sm/k}(X,Y)_+\wedge S^pNEWLINE\]NEWLINE for all \(p\geq 0\) and \(X,Y\in Sm/k\). A spectral category over \(Sm/k\) is a pair \(({\mathcal O},\sigma)\), where \({\mathcal O}\) is a spectral category whose objects are those of \(Sm/k\) and \(\sigma:{\mathcal O}_{\text{naive}}\to{\mathcal O}\) is a spectral functor which is identity on objects. A spectral category \(({\mathcal O},\sigma)\) is a \({\mathcal V}\)-spectral category if it is connective, Nisnevich excisive and satisfies some other conditions. Then, starting from a spectral category \(({\mathcal O},\sigma)\), one defines a triangulated category category \(D{\mathcal O}^{\text{eff}}_-(k)\) which maybe viewed as an analog of Voevodsky's triangulated category. In fact \(D{\mathcal O}^{\text{eff}}_-(k)\) is equivalent to \(DM^{\text{eff}}_-(k)\), if \({\mathcal O}={\mathcal O}_{cor}\). Here \({\mathcal O}_{cor}\) is the Eilenberg-Mac Lane spectral category associated to the ringoid of correspondences Cor. The \({\mathcal V}\)-spectral category \({\mathbf K}\) is obtained by taking K-theory symmetric spectra \(K({\mathcal A}(U, X))\) of certain additive categories \({\mathcal A}(U, X)\), with \(U,X\in Sm/k\). There is a spectral functor \(\sigma:{\mathcal O}_{\text{naive}}\to{\mathbf K}\) between spectral categories, such that the pair \(({\mathbf K},\sigma)\) is a spectral category over \(Sm/k\). For any perfect field \(k\) the triangulated category of \(K\)-motives \(DK^{\text{eft}}_-(k)\) is the triangulated category \(D{\mathcal O}^{\text{eff}}_-\) associated to the \({\mathcal V}\)-spectral category \({\mathbf K}\).