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\begin{document}

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\section*{pure 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{construction}{Construction}\dotfill \pageref*{construction} \linebreak
\noindent\hyperlink{category_of_correspondences}{Category of correspondences}\dotfill \pageref*{category_of_correspondences} \linebreak
\noindent\hyperlink{category_of_effective_pure_motives}{Category of effective pure motives}\dotfill \pageref*{category_of_effective_pure_motives} \linebreak
\noindent\hyperlink{category_of_pure_motives}{Category of pure motives}\dotfill \pageref*{category_of_pure_motives} \linebreak
\noindent\hyperlink{category_of_pure_chow_motives}{Category of pure Chow motives}\dotfill \pageref*{category_of_pure_chow_motives} \linebreak
\noindent\hyperlink{category_of_pure_numerical_motives}{Category of pure numerical motives}\dotfill \pageref*{category_of_pure_numerical_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}

[[Grothendieck]] conjectured that every [[Weil cohomology theory]] factors uniquely through some [[category]], which he called the category of \emph{[[motives]]}. For [[smooth variety|smooth]] [[projective varieties]] (over some [[field]] $k$) such a category was given by Grothendieck himself, called the category of \emph{pure [[Chow motives]]}. For general smooth varieties the category is still conjectural, see at \emph{[[mixed motives]]}.

\hypertarget{construction}{}\subsection*{{Construction}}\label{construction}

Fix some [[adequate equivalence relation]] $\sim$ (e.g. [[rational equivalence]]). Let $Z^i(X)$ denote the group of $i$-codimensional [[algebraic cycles]] and let $A^i_\sim(X)$ denote the quotient $Z^i(X)/\sim$.

\hypertarget{category_of_correspondences}{}\subsubsection*{{Category of correspondences}}\label{category_of_correspondences}

Let $Corr_\sim(k)$, the category of [[correspondences]], be the [[category]] whose [[objects]] are [[smooth variety|smooth]] [[projective varieties]] and whose [[hom-sets]] are the [[direct sum]]

\begin{displaymath}
Corr_\sim(h(X),h(Y)) = \bigoplus_i A^{n_i}_\sim(X_i \times Y)
  \,,
\end{displaymath}
where $(X_i)$ are the [[irreducible components]] of $X$ and $n_i$ are their respective [[dimensions]]. The [[composition]] of two [[morphisms]] $\alpha \in Corr(X,Y)$ and $\beta \in Corr(Y,Z)$ is given by

\begin{displaymath}
p_{XZ,*} (p_{XY}^*(\alpha) . p_{YZ}^*(\beta))
\end{displaymath}
where $p_{XY}$ denotes the projection $X \times Y \times Z \to X \times Y$ and so on, and $.$ denotes the [[intersection product]] in $X \times Y \times Z$.

There is a canonical [[contravariant functor]]

\begin{displaymath}
h \colon SmProj(k) \to Corr_\sim(k)
\end{displaymath}
from the category of smooth projective varieties over $k$ given by mapping $X \mapsto X$ and a morphism $f : X \to Y$ to its [[graph of a function|graph]], the [[image]] of its [[graph morphism]] $\Gamma_f : X \to X \times Y$.

The category of correspondences is [[symmetric monoidal category|symmetric monoidal]] with $h(X) \otimes h(Y) \coloneqq h(X \times Y)$.

We also define a category $Corr_\sim(k, A)$ of correspondences with [[coefficients]] in some [[commutative ring]] $A$, by [[tensor product|tensoring]] the morphisms with $A$; this is an $A$-[[linear category]] [[additive category|additive]] [[symmetric monoidal category|symmetric monoidal]] category.

\hypertarget{category_of_effective_pure_motives}{}\subsubsection*{{Category of effective pure motives}}\label{category_of_effective_pure_motives}

The [[Karoubi envelope]] (pseudo-abelianisation) of $Corr_\sim(k, A)$ is called the category of \textbf{effective pure motives} (with coefficients in $A$ and with respect to the equivalence relation $\sim$), denoted $Mot^eff_\sim(k, A)$.

Explicitly its objects are pairs $(h(X), p)$ with $X$ a smooth projective variety and $p \in Corr(h(X), h(X))$ an [[idempotent]], and morphisms from $(h(X), p)$ to $(h(Y), q)$ are morphisms $h(X) \to h(Y)$ in $Corr_\sim$ of the form $q \circ \alpha \circ p$ with $\alpha \in Corr_{\sim}(h(X), h(Y))$.

This is still a [[symmetric monoidal category|symmetric monoidal]] category with $(h(X), p) \otimes (h(Y), q) = (h(X \times Y), p \times q)$. Further it is [[Karoubian category|Karoubian]], $A$-[[linear category|linear]] and [[additive category|additive]].

The image of $X \in SmProj(k)$ under the above functor

\begin{displaymath}
h \colon SmProj(k) \to Corr_\sim(k,A) \to Mot^{eff}_\sim(k,A)
\end{displaymath}
is the \textbf{the motive of $X$}.

\hypertarget{category_of_pure_motives}{}\subsubsection*{{Category of pure motives}}\label{category_of_pure_motives}

There exists a motive $\mathbf{L}$, called the \textbf{[[Lefschetz motive]]}, such that the motive of the [[projective line]] decomposes as

\begin{displaymath}
h(\mathbf{P}^1_k) = h(\Spec(k)) \oplus \mathbf{L}
\end{displaymath}
To get a [[rigid category]] we formally invert the Lefschetz motive and get a category

\begin{displaymath}
Mot_\sim(k, A)
  \coloneqq
  Mot^{eff}_\sim(k,A)[\mathbf{L}^{-1}]
  \,,
\end{displaymath}
the \textbf{category of pure motives} (with coefficients in $A$ and with respect to $\sim$).

This is a [[rigid category|rigid]], [[Karoubian category|Karoubian]], [[symmetric monoidal category]]. Its objects are triples $(h(X), p, n)$ with $n \in \mathbf{Z}$.

\hypertarget{category_of_pure_chow_motives}{}\subsubsection*{{Category of pure Chow motives}}\label{category_of_pure_chow_motives}

When the relation $\sim$ is [[rational equivalence]] then $A^*_\sim$ are the [[Chow groups]], and $Mot_\sim(k) = Mot_{rat}(k)$ is called the category of \textbf{pure [[Chow motives]]}.

\hypertarget{category_of_pure_numerical_motives}{}\subsubsection*{{Category of pure numerical motives}}\label{category_of_pure_numerical_motives}

When the relation $\sim$ is numerical equivalence, then one obtains \emph{[[numerical motives]]}.

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

\begin{itemize}%
\item [[sheaf with transfer]]


\item [[motive]]


\item [[mixed motive]]


\item [[Voevodsky motive]]


\item [[noncommutative motive]]



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

\begin{itemize}%
\item [[Yuri Manin]], \emph{Correspondences, motifs and monoidal transformations} , Math. USSR Sb. 6 439, 1968(\href{http://resources.agssp2012.torsor.org/documents/manin.pdf}{pdf}, \href{http://iopscience.iop.org/0025-5734/6/4/A01}{web})


\item [[Tony Scholl]], \emph{Classical motives}, in Motives, Seattle 1991. Proc Symp. Pure Math 55 (1994), part 1, 163-187 (\href{https://www.dpmms.cam.ac.uk/~ajs1005/preprints/classical_motives.pdf}{pdf})


\item [[James Milne]], \emph{Motives -- Grothendieck's Dream} (\href{http://www.jmilne.org/math/xnotes/MOT.pdf}{pdf})


\item [[Minhyong Kim]], \emph{Classical Motives: Motivic $L$-functions} (\href{http://www.ucl.ac.uk/~ucahmki/ihes3.pdf}{pdf})


\item [[Bruno Kahn]], \href{http://www.aimath.org/WWN/motivesdessins/PaloAlto1.pdf}{pdf slides} on pure motives


\item R. Sujatha, \emph{Motives from a categorical point of view}, Lecture notes (2008) (\href{http://www.math.tifr.res.in/~sujatha/ihes.pdf}{pdf})



\end{itemize}
Section 8.2 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}
category: algebraic geometry

[[!redirects pure motives]] [[!redirects category of pure motives]]



\end{document}