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\begin{document}

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\section*{mixed motive}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{motivic_cohomology}{}\paragraph*{{Motivic cohomology}}\label{motivic_cohomology}

[[!include motivic cohomology - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{voevodskys_mixed_motives}{Voevodsky's mixed motives}\dotfill \pageref*{voevodskys_mixed_motives} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Where the [[category of pure motives]] has [[smooth variety|smooth]] [[projective varieties]] as its [[objects]], the \emph{category of mixed motives} is supposed to be constructed from all [[smooth varieties]].

The category of mixed motives is supposed to be an [[abelian category|abelian]] [[tensor category]] which contains the [[pure motives]] as the [[full subcategory]] of [[semisimple objects]].

So far there is no realisation of such a category, but there are proposals by [[Vladimir Voevodsky]] and [[Marc Levine]] of [[triangulated categories]] that behave as its [[derived category]] is expected to.

\hypertarget{voevodskys_mixed_motives}{}\subsection*{{Voevodsky's mixed motives}}\label{voevodskys_mixed_motives}

Here we construct [[Voevodsky]]`s [[triangulated category]] of mixed motives following \hyperlink{CD}{Cisinski-Deglise}.

\begin{enumerate}%
\item Let $S$ be a [[regular scheme|regular]] and [[noetherian scheme|noetherian]] base [[scheme]]. Let $Sm_S$ be the category of [[schemes]] [[smooth]] and of [[finite type]] over $S$. Let $N_S^{tr}$ denote the [[closed symmetric monoidal category]] of [[Nisnevich sheaves with transfer]]. We will write $L_S[X]$ for the [[sheaf]] [[representable|represented]] by $X \in Sm_S$.


\item Let $G_S$ denote the set of representable sheaves and $H_S$ as the set of [[complexes]] in $Cpx(N_S^{tr})$ which are the [[cones]] of morphisms $L_S[Y] \to L_S[X]$ induced by [[hypercovers]] $Y\to X$. The pair $(G_S, H_S)$ then defines a weakly flat [[descent structure]] (in the sense of \hyperlink{CD}{Cisinski-Deglise}, see discussion at [[model structure on chain complexes]]) and therefore induces a [[symmetric monoidal model structure]] on $Cpx(N_S^{tr})$ where the [[weak equivalences]] are the [[quasi-isomorphisms]].


\item Let $\mathcal{T}_S$ denote the set of [[complexes]] $Cone(L_S[A^1_X] \to L_S[X])$ where $A^1_X = A^1_S \times_S X$ is the [[affine line]] over $X$. Call a [[complex]] $K$ \textbf{$A^1$-local} if $Hom_{D(A)}(T, K[n]) = 0$ for all $T \in \mathcal{T}$ and integers $n$. Equivalently, the [[Nisnevich]] [[hypercohomology sheaves]] are [[homotopy invariant]]. In particular, for $F$ [[fibrant]] with respect to the above [[model structure]], $F$ is $A^1$-local iff the morphism $F(X) \to F(A^1_X)$ induced by the projection is a [[quasi-isomorphism]] for all $X \in Sm_S$. This is again equivalent to the [[cohomology]] [[presheaves]] of $F$ being [[homotopy invariant]].


\item Consider the [[left Bousfield localization]] of the above [[model structure]] at the class of morphisms $0 \to T[n]$ for $T \in \mathcal{T}_S$ and $n \in \mathbf{Z}$. The [[fibrant objects]] are $G_S$-local and $A^1$-local complexes. One can prove that this [[model structure]] is still [[symmetric monoidal model structure||symmetric monoidal]].


\item The [[homotopy category]] of this [[model category]] is called the \textbf{triangulated category of effective motives} over $S$, denoted $DM^{eff}(S)$. It is canonically equivalent to the [[full subcategory]] of the [[derived category]] $D(N_S^{tr})$ spanned by Nisnevich fibrant and $A^1$-local complexes.


\item (introduce symmetric Tate spectra and DM(S))



\end{enumerate}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[motive]]

\begin{itemize}%
\item mixed motive,

\begin{itemize}%
\item [[mixed Tate motive]]

\end{itemize}

\item [[pure motive]],

\begin{itemize}%
\item [[Chow motive]], [[numerical motive]]

\end{itemize}

\item [[Voevodsky motive]]


\item [[noncommutative motive]]



\end{itemize}

\item [[motivic cohomology]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[Marc Levine]], \emph{Mixed Motives}, Handbook of K-theory (\href{http://www.uni-due.de/~bm0032/publ/MixMotKHB.pdf}{pdf})


\item [[Denis-Charles Cisinski]], [[Frédéric Déglise]], \emph{Local and stable homological algebra in Grothendieck abelian categories}, \href{http://arxiv.org/abs/0712.3296}{arXiv}.



\end{itemize}
Section 8.3 of

\begin{itemize}%
\item [[Alain Connes]], [[Matilde Marcolli]], \emph{[[Noncommutative Geometry, Quantum Fields and Motives]]} (\href{http://www.alainconnes.org/docs/bookwebfinal.pdf}{pdf})

\end{itemize}
[[!redirects mixed motives]]



\end{document}