\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{noncommutative 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{Definition}{Definition}\dotfill \pageref*{Definition} \linebreak
\noindent\hyperlink{AsUniversalAditiveInvariant}{As the universal additive/localizing invariant}\dotfill \pageref*{AsUniversalAditiveInvariant} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{RelationToAlgebraicKTheory}{Relation to algebraic K-theory}\dotfill \pageref*{RelationToAlgebraicKTheory} \linebreak
\noindent\hyperlink{RelationToCorrespondences}{Relation to correspondences equipped with cocycles}\dotfill \pageref*{RelationToCorrespondences} \linebreak
\noindent\hyperlink{RelationToChowMotives}{Relation to Chow motives}\dotfill \pageref*{RelationToChowMotives} \linebreak
\noindent\hyperlink{relation_to_kktheory}{Relation to KK-theory}\dotfill \pageref*{relation_to_kktheory} \linebreak
\noindent\hyperlink{applications}{Applications}\dotfill \pageref*{applications} \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}

The notion of [[motives]] in algebraic geometry can be adapted to derived [[noncommutative geometry]]. The idea and the first version has been developed by [[Maxim Kontsevich]]. There is a remarkable observation that the [[category of Chow motives]] (after localizing at the [[Lefschetz motive]]) can be embedded into the category of Kontsevich's noncommutative motives. More recently this direction has been systematically studied by Cisinski and Tabuada.

A second approach is due to [[Bertrand Toën]], [[Michel Vaquié]], [[Gabriele Vezzosi]]. They construct a [[motivic stable homotopy theory]] for [[noncommutative geometry|noncommutative spaces]] (in the sense of [[Kontsevich]]).

There is another approach by [[Arne Ostvaer]].

In [[noncommutative geometry]] \`a{} la [[Alain Connes]], Connes and [[Matilde Marcolli]] have also introduced some motivic ideas. Marcolli also has a recent collaboration with Tabuada on the algebraic side, see her [[Matilde Marcolli|webpage]].

\hypertarget{Definition}{}\subsection*{{Definition}}\label{Definition}

\hypertarget{AsUniversalAditiveInvariant}{}\subsubsection*{{As the universal additive/localizing invariant}}\label{AsUniversalAditiveInvariant}

The definition in (\hyperlink{BlumbergGepnerTabuada10}{Blumberg-Gepner-Tabuada 10}) is the following.

\begin{defn}
\label{Catperf}\hypertarget{Catperf}{}
Write

\begin{itemize}%
\item (stable) $Cat_\infty^{stab}$ for the [[(∞,1)-category]] of [[stable (∞,1)-categories]];


\item (small with linear maps) $Cat_\infty^{ex}$ for the $(\infty,1)$-category of \emph{[[small (infinity,1)-category|small]]} stable $\infty$-categories with [[finite (infinity,1)-limit]]/colimit-preserving functors between them;


\item (idem-complete) $Cat_\infty^{perf} \hookrightarrow Cat_\infty^{ex}$ for the further [[full sub-(infinity,1)-category]] of the small stable $\infty$-categories on the [[idempotent complete (infinity,1)-categories]].



\end{itemize}
\end{defn}
(\hyperlink{BlumbergGepnerTabuada10}{Blumberg-Gepner-Tabuada 10, def. 2.12 and above def 2.14})

\begin{prop}
\label{IdempotentCompletion}\hypertarget{IdempotentCompletion}{}
The inclusion from def. \ref{Catperf} is a [[reflective sub-(infinity,1)-category]]

\begin{displaymath}
Cat^{perf} \stackrel{\overset{Idem}{\longleftarrow}}{\hookrightarrow} Cat^{ex}
\end{displaymath}
the reflector $Idem$ being [[idempotent completion]].

\end{prop}
\begin{remark}
\label{}\hypertarget{}{}
Passing to [[ind-objects]] yields an [[equivalence of (infinity,1)-categories]]

\begin{displaymath}
Ind \colon 
  Cat_\infty^{per}
  \stackrel{\simeq}{\longrightarrow}
  Pr_{St,cg}^{L}
  \hookrightarrow
  Pr_{St}^{L}
\end{displaymath}
with the [[stable (infinity,1)-category|stable]] [[compactly generated (∞,1)-categories]].

\end{remark}
([[Higher Topos Theory|HTT, 5.5.7]])

\begin{defn}
\label{MotitaEquivalence}\hypertarget{MotitaEquivalence}{}
Say that a morphism in $Cat_\inty^{ex}$ is a [[Morita equivalence]] if it is an $Idem$-equivalence, hence if it becomes an [[equivalence of (∞,1)-categories]] under idempotent completion, prop. \ref{IdempotentCompletion}.

\end{defn}
\begin{defn}
\label{ExactSequences}\hypertarget{ExactSequences}{}
Say that a sequence in $Cat_\infty^{ex}$ is (split-)exact if it is an exact sequence (\ldots{}see section 5\ldots{}) under idempotent completion, prop. \ref{IdempotentCompletion}

\end{defn}
\begin{defn}
\label{additiveinvariant}\hypertarget{additiveinvariant}{}
A functor $Cat_\infty^{ex} \to \mathcal{D}$ to a stable [[presentable (∞,1)-category]] is called a \textbf{localizing invariant} (\textbf{additive invariant}) if it

\begin{enumerate}%
\item inverts Morita equivalences, def. \ref{MotitaEquivalence};


\item preserves [[filtered (∞,1)-colimits]];


\item sends (split-) exact sequences, def. \ref{ExactSequences}, to (split) cofiber sequences (\ldots{}see section 5\ldots{}).



\end{enumerate}
\end{defn}
\begin{defn}
\label{BGTCharacterization}\hypertarget{BGTCharacterization}{}
The $(\infty,1)$-category $Mot_{add}$ or $Mot_{loc}$, respectively, of \textbf{noncommutative motives} is the [[universal construction|universal]] localizing/additive invariant, def. \ref{additiveinvariant}

\begin{displaymath}
\mathcal{U}_{loc} \colon Cat_\infty^{ex} \to Mot_{loc}
  \,.
\end{displaymath}
\begin{displaymath}
\mathcal{U}_{add} \colon Cat_\infty^{ex} \to Mot_{add}
  \,.
\end{displaymath}
\end{defn}
(\hyperlink{BlumbergGepnerTabuada10}{Blumberg-Gepner-Tabuada 10, theorem 1.1, section 8})

\begin{remark}
\label{}\hypertarget{}{}
The localization property here (be additive, invert Morita, preserve split sequences) is of the same form as that which defines the localization of [[C\emph{-algebras]] to [[KK-theory]] in [[noncommutative stable homotopy theory]]. See at \emph{\href{KK-theory#UniversalCharacterization}{KK-theory -- Universal characterization}}. See also (\hyperlink{BlumbergGepnerTabuada10}{Blumberg-Gepner-Tabuada 10, paragraph 1.5}).}

\end{remark}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{RelationToAlgebraicKTheory}{}\subsubsection*{{Relation to algebraic K-theory}}\label{RelationToAlgebraicKTheory}

\begin{defn}
\label{}\hypertarget{}{}
For $\mathcal{A}, \mathcal{B} \in Cat_\infty^{stab}$ with $\mathcal{B}$ smooth and proper, hence a [[compact object in an (infinity,1)-category|compact object]], then the hom-[[spectrum]] in $Mot_{loc}$ between $\mathcal{A}$ and $\mathcal{B}$ is given by the [[non-connective algebraic K-theory]] $\mathbb{K}$ of the tensor product in that there is a [[natural equivalence]]

\begin{displaymath}
Hom_{Mot_{loc}}(\mathcal{U}_{loc}(\mathcal{B}), \mathcal{U}_{loc}(\mathcal{A}))
  \simeq
  \mathbb{K}(\mathcal{B}^{op}\widehat \otimes \mathcal{A})
  \,.
\end{displaymath}
\end{defn}
(\hyperlink{BlumbergGepnerTabuada10}{Blumberg-Gepner-Tabuada 10}, theorem 9.36)

\hypertarget{RelationToCorrespondences}{}\subsubsection*{{Relation to correspondences equipped with cocycles}}\label{RelationToCorrespondences}

By (\hyperlink{BlumbergGepnerTabuada10}{Blumberg-Gepner-Tabuada 10, theorem 9.36}), the morphisms of noncommutative motives from $\mathcal{A}$ to $\mathcal{B}$ for $\mathcal{B}$ suitably dualizable/compact are given by

\begin{displaymath}
Maps(\mathcal{U}_{loc}(\mathcal{B}), \mathcal{U}_{loc}(\mathcal{A}))
  \simeq
   \mathbb{K}(\mathcal{B}^{op}\otimes\mathcal{A})
  \,,
\end{displaymath}
hence by the [[non-connective algebraic K-theory]] of the [[Deligne tensor product]] of the two categories.

Thinking of these as categories of [[quasicoherent sheaves]] on some spaces (by definition in [[noncommutative algebraic geometry]]), this are $\mathbb{K}$-cocycles on the product [[correspondence]] space.

(\ldots{})

\hypertarget{RelationToChowMotives}{}\subsubsection*{{Relation to Chow motives}}\label{RelationToChowMotives}

The category of ordinary [[Chow motives]], after factoizing out the action of the [[Tate motive]] essentially sits inside that of noncommutative Chow motives. This is recalled as (\hyperlink{Tabuada11}{Tabuada 11, theorem 4.6}). For more see (\hyperlink{Tabuada11Chow}{Tabuada 11 ChowNCG}).

This relation is best understood as being exhibited by [[K-motives]], see there.

\hypertarget{relation_to_kktheory}{}\subsubsection*{{Relation to KK-theory}}\label{relation_to_kktheory}

Noncommutative motives receive a universal functor from [[KK-theory]]

\begin{displaymath}
KK \longrightarrow NCC_{dg}
\end{displaymath}
which is given by sending a [[C\emph{-algebra]] to the [[dg-category]] of [[perfect complexes]] over (the [[unitalization]] of) its underlying [[associative algebra]] (\hyperlink{Mahanta13}{Mahanta 13}).}

\hypertarget{applications}{}\subsection*{{Applications}}\label{applications}

\begin{itemize}%
\item [[Tabuada]] has used noncommutative motives to compute the [[cyclic homology]] of [[twisted projective homogeneous varieties]]. Also, he showed that the noncommutative motive of such a variety is trivial if and only if the [[Brauer class|Brauer classes]] of the associated [[central simple algebras]] are trivial. See (\hyperlink{Tabuada13Twisted}{Tabuada 13}).


\item Bernardara and Tabuada have used noncommutative motives to compute the rational [[Chow groups]] of certain complete [[intersections]] of [[curves]]. See (\hyperlink{BerTab13}{Bernardara-Tabuada 13}).



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

\begin{itemize}%
\item [[bivariant cohomology]], [[bivariant algebraic K-theory]]


\item [[motivic symplectic category]]


\item [[motive]]

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

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

\end{itemize}

\item [[mixed motive]]



\end{itemize}

\item [[motivic cohomology]]



\end{itemize}
[[!include noncommutative motives - table]]

\hypertarget{References}{}\subsection*{{References}}\label{References}

A survey is in

\begin{itemize}%
\item [[Goncalo Tabuada]], \emph{A guided tour through the garden of noncommutative motives}, in Guillermo Cortinas, \emph{Topics in Noncommutative Geometry} Clay Mathematics Proceedings Vol 16, 2012 (\href{http://arxiv.org/abs/1108.3787}{arxiv1108.3787});

\end{itemize}
Discussion of [[Maxim Kontsevich]]`s definition of noncommutative motives include

\begin{itemize}%
\item [[Maxim Kontsevich]], \emph{Noncommutative motives}, talk at the conference on Pierre Deligne's 61st birthday (2005) ( of part of the talk, notes by [[Zoran Skoda]])


\item [[Maxim Kontsevich]], \emph{Geometry and Arithmetic - Non-commutative motives}, talk at Institute for Advanced Study October 20, 2005 (\href{http://video.ias.edu/Geometry-and-Arithmetic-Kontsevich}{video})



\end{itemize}
The following article has the treatment of $A_\infty$-categories representing smooth, proper, separated etc. noncommutative varieties, notions which are used in Kontsevich's approach to motives in the above talks.

\begin{itemize}%
\item [[Ludmil Katzarkov]], [[Maxim Kontsevich]], [[Tony Pantev]], \emph{Hodge theoretic aspects of mirror symmetry}, in [[Ron Donagi]] and [[Katrin Wendland]] (eds.) \emph{From Hodge theory to integrability and TQFT: $tt^\ast$-geometry}, Proceedings of Symposia in Pure Mathematics vol. 78 (2008), 87-174 (\href{http://arxiv.org/abs/0806.0107}{arXiv:0806.0107})

\end{itemize}
An abstract characterization of noncommutative motives in [[dg-category]] theory and higher [[algebraic K-theory]] is in

\begin{itemize}%
\item [[Denis-Charles Cisinski]], [[Gonçalo Tabuada]], \emph{Non connective K-theory via universal invariants}. Compositio Mathematica 147 (2011), 1281--1320 (\href{http://arxiv.org/abs/0903.3717}{arXiv:0903.3717})


\item [[Denis-Charles Cisinski]], [[Gonçalo Tabuada]], \emph{Symmetric monoidal structure on Non-commutative motives}, (\href{http://arxiv.org/abs/1001.0228}{arxiv/1001.0228})


\item [[Gonçalo Tabuada]], \emph{K-theory via universal invariants}, Duke Math. J. 145 (2008), no.1, 121--206.



\end{itemize}
and a further lift of this to [[(∞,1)-category theory]] is in

\begin{itemize}%
\item [[Andrew Blumberg]], [[David Gepner]], [[Gonçalo Tabuada]], \emph{A universal characterization of higher algebraic K-theory}, Geometry and Topology 17 (2013) 733--838 (\href{http://arxiv.org/abs/1001.2282}{arXiv:1001.2282})

\end{itemize}
with discussion of the corresponding [[cyclotomic trace]] in

\begin{itemize}%
\item [[Andrew Blumberg]], [[David Gepner]], [[Gonçalo Tabuada]], \emph{Uniqueness of the multiplicative cyclotomic trace}, Advances in Mathematics (\href{http://arxiv.org/abs/1103.3923}{arXiv:1103.3923})

\end{itemize}
See also

\begin{itemize}%
\item [[Goncalo Tabuada]],

\emph{Bivariant cyclic cohomology and Connes' bilinear pairings in Non-commutative motives}, \href{http://arxiv.org/abs/1005.2336}{arxiv/1005.2336};

\emph{Products, multiplicative Chern characters, and finite coefficients via Non-commutative motives}, \href{http://arxiv.org/abs/1101.0731}{arxiv/1101.0731};


\item [[Goncalo Tabuada]], \emph{Chow motives versus non-commutative motives} (\href{http://arxiv.org/abs/1103.0200}{arxiv/1103.0200}


\item [[Goncalo Tabuada]], \emph{Galois descent of additive invariants}, \href{http://arxiv.org/abs/1301.1928}{arxiv/1301.1928}


\item [[Matilde Marcolli]], [[Goncalo Tabuada]], \emph{Kontsevich's noncommutative numerical motives}, \href{http://arxiv.org/abs/1108.3785}{arxiv/1108.3785}; \emph{Noncommutative motives, numerical equivalence, and semi-simplicity}, \href{http://arxiv.org/abs/1105.2950}{arxiv/1105.2950}; \emph{Noncommutative numerical motives, Tannakian structures, and motivic Galois groups}, \href{http://arxiv.org/abs/1110.2438}{arxiv/1110.2438}


\item [[Ivo Dell'Ambrogio]], [[Gonçalo Tabuada]], \emph{Tensor triangular geometry of non-commutative motives}, \href{http://arxiv.org/abs/1104.2761}{arxiv/1104.2761}



\end{itemize}
For the approach of [[Bertrand Toën]]-[[Michel Vaquié]]-[[Gabriele Vezzosi]], see

\begin{itemize}%
\item [[Michel Vaquié]], \emph{A new approach on non commutative motives}, lecture at \href{http://www.mathematics.jhu.edu/new/jami2011/noncommgeo.htm}{JAMI 2011}. \href{http://streams1.nts.jhu.edu/mathematics/jami2011/}{video}, \href{http://streams1.nts.jhu.edu/mathematics/jami2011/jami2011pdf/Vaquie.pdf}{notes}

\end{itemize}
and the Ph.D. thesis of [[Marco Robalo]], under the supervision of [[Bertrand Toën]]:

\begin{itemize}%
\item [[Marco Robalo]], \emph{Noncommutative Motives I: A Universal Characterization of the Motivic Stable Homotopy Theory of Schemes}, June 2012 (\href{http://arxiv.org/abs/1206.3645}{arxiv:1206.3645})


\item [[Marco Robalo]], \emph{Noncommutative Motives II: K-Theory and Noncommutative Motives}, June 2013, (\href{http://arxiv.org/abs/1306.3795}{arxiv:1306.3795})



\end{itemize}
Also the lectures notes:

\begin{itemize}%
\item [[Marco Robalo]], \emph{Noncommutative motives and K-theory}, talk at \href{http://www.mat.univie.ac.at/~favero/Workshops/Higher.html}{Higher Categories and Topological Quantum Field Theories, Vienna, 2013}, \href{http://www.mat.univie.ac.at/~favero/Workshops/here/NotesRobalo.pdf}{notes}

\end{itemize}
Another survey article is

\begin{itemize}%
\item [[Snigdhayan Mahanta]], Noncommutative geometry in the framework of differential graded categories, \href{http://arxiv.org/abs/0805.1628}{arXiv:0805.1628}.

\end{itemize}
Discussion of how the [[triangulated categories of sheaves|derived category]] of a [[scheme]] determines its commutative and [[noncommutative motive|noncommutative]] [[Chow motive]] is in

\begin{itemize}%
\item [[Adeel Khan]], \emph{On derived categories and noncommutative motives of varieties}, \href{http://arxiv.org/abs/1401.7222}{arXiv}.

\end{itemize}
In

\begin{itemize}%
\item [[Snigdhayan Mahanta]], \emph{Higher nonunital Quillen $K'$-theory, KK-dualities, and applications to topological T-duality}, (\href{http://wwwmath.uni-muenster.de/u/snigdhayan.mahanta/papers/KQ.pdf}{pdf}, \href{http://wwwmath.uni-muenster.de/u/snigdhayan.mahanta/papers/KKTD.pdf}{talk notes})

\end{itemize}
it is shown that there is a universal functor $KK \longrightarrow NCC_{dg}$ from [[KK-theory]] to the category of [[noncommutative motives]], which is the category of [[dg-categories]] and dg-[[profunctors]] up to homotopy between them. This is given by sending a [[C\emph{-algebra]] to the [[dg-category]] of [[perfect complexes]] of (the unitalization of) its underlying [[associative algebra]].}

\begin{itemize}%
\item [[Goncalo Tabuada]], \emph{Additive invariants of toric and twisted projective homogeneous varieties via noncommutative motives}, (\href{http://arxiv.org/abs/1310.4063}{arXiv}).


\item [[Marcello Bernardara]], [[Goncalo Tabuada]]. \emph{Chow groups of intersections of quadrics via homological projective duality and (Jacobians of) noncommutative motives.} (\href{http://arxiv.org/abs/1310.6020}{arXiv})



\end{itemize}
category: algebraic geometry, noncommutative geometry

[[!redirects noncommutative motif]] [[!redirects noncommutative motive]] [[!redirects noncommutative motives]] [[!redirects noncommutative motives]]

[[!redirects noncommutative Chow motive]] [[!redirects noncommutative Chow motives]]

[[!redirects (infinity,1)-category of non-commutative motives]] [[!redirects (∞,1)-category of non-commutative motives]]

[[!redirects (infinity,1)-category of noncommutative motives]] [[!redirects (∞,1)-category of noncommutative motives]]

[[!redirects (infinity,1)-categories of non-commutative motives]] [[!redirects (∞,1)-categories of non-commutative motives]]

[[!redirects (infinity,1)-categories of noncommutative motives]] [[!redirects (∞,1)-categories of noncommutative motives]]

[[!redirects non-commutative motive]] [[!redirects non-commutative motives]]



\end{document}