\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*{derived noncommutative geometry}

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

\hypertarget{higher_geometry}{}\paragraph*{{Higher geometry}}\label{higher_geometry}

[[!include higher geometry - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{some_history}{Some history}\dotfill \pageref*{some_history} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

In [[noncommutative algebraic geometry]] one represents a [[scheme]] by an [[abelian category]] of [[quasicoherent sheaf|quasicoherent sheaves]] on the [[scheme]], and looks at more general abelian categories as being categories of quasicoherent sheaves on a noncommutative space.

In \emph{derived (higher) noncommutative (algebraic) geometry} one instead considers the [[derived category]] of [[quasicoherent sheaf|quasicoherent sheaves]], or more precisely its [[enhanced triangulated category|dg-enhancement]] or [[A-infinity-category|A-infinity-enhancement]]; dg-enhancements for the derived categories of quasiprojective smooth varieties are essentially unique by the results of Lunts and Orlov. Taking the derived category instead of the abelian loses a bit of information but sometimes the information is sufficient.

In general one represents complex noncommutative spaces by [[pretriangulated dg-category|pretriangulated dg-categories]]. They may be viewed as models for stable $(\infty,1)$-categories. Note that accessible stable $(\infty,1)$-categories are quite close to Grothendieck $(\infty,1)$-topoi; more flexibility one gets from pretriangulated $A_\infty$-categories or, even better, certain class of [[spectral categories]].\newline This is well into [[homotopy theory]] area. Quillen [[model category]] structures and [[homotopy limit]]s in the context of dg-categories were studied by a number of people (including the impressive thesis by Tabuada). On the other hand, over a mixed characteristics, the meaning of such representations is less well understood.

Derived noncommutative geometry has been informally introduced by Kapranov-Bondal and later Orlov around 1990; full framework belongs to [[Kontsevich]], [[Valery Lunts|Lunts]], van den Bergh, Katzarkov, Kuznetsov, Kaledin. Some of the works of [[Bertrand Toen|Toen]], Vaquie, Keller, Cisinski, Tabuada are properly in this area as well.

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

In \hyperlink{KatzarkovKontsevichPantev}{Katzarkov-Kontsevich-Pantev} the following definition is given.

\begin{udefinition}
A \textbf{graded complex nc-space} is a $\mathbb{C}$-linear [[dg-category|differential graded category]] $C$ which is homotopy complete and cocomplete (has all [[homotopy limit]]s and colimits).

\end{udefinition}
The [[derived category|derived categories]] of [[quasicoherent sheaf|quasicoherent sheaves]] on a [[scheme]] over $Spec(\mathbb{C})$ is one of the examples; another example is the category of dg-modules over a fixed [[dg-algebra]] $A$, which are such that $A$ admits an exhaustive filtration such that the associated graded is a sum of shifts of $A$. Call that category $A$-$Mod$.

Kontsevich calls a complex differential $\mathbb{Z}/2$-graded algebra

\begin{itemize}%
\item \textbf{smooth} if $A$ is a perfect object in the category of $A$-$A$-dg-bimodules (perfect object here means that $Hom(A,-)$ preserves small homotopy colimits);


\item \textbf{compact} if the total complex dimension of its cohomology $H^\bullet(A,d_A)$ is finite



\end{itemize}
The category $A$-$Mod$ is a smooth (resp. compact) nc space if the underlying dg-algebras $A$ is; this notion depends on the category and not on the underlying dg-algebra.

The above definition implies that a category $C$ which represents a nc-space in the sense above is [[triangulated category|triangulated]] and [[Karoubi envelope|Karoubi closed]]. Sometimes this are requirements in another variant of the definition.

\begin{udefinition}
A \textbf{noncommutative space} $X$ is a small [[triangulated category]] $C_X$ which is [[Karoubi envelope|Karoubi closed]] (=every [[idempotent]] is a [[split idempotent]]) and appropriately [[enriched category|enriched]] over either

\begin{itemize}%
\item [[spectrum|spectra]] $Hom_{C_X}(E,F[i])=\pi_{-i}\mathbf{Hom}_{C_X}(E,F)$


\item [[chain complex|complexes]] of $k$-vector spaces (i.e. is a [[dg-category]]): $Hom(E,F[i])= H^i(\mathbf{Hom}_{C_X}(E,F))$. $C_X$ is $k$-linear over a field $k$ and one writes $X/k$. If instead $k$ is replaced by a ring $R$ then one enriches over complexes of $R$-modules which are [[model structure on chain complexes|cofibrant]].



\end{itemize}
\end{udefinition}
This receives good motivation from the fact that [[stable (infinity,1)-categries]] with a small set of generators are equivalently the [[categories of modules]] over an $A_\infty$-ring(oid). See \href{stable+(infinity%2C1}{there}-category\#AsCategoriesOfModules).

\hypertarget{some_history}{}\subsection*{{Some history}}\label{some_history}

In the early works of the Moscow school ([[Kapranov]], [[Alexei Bondal|Bondal]], [[Dmitri Orlov|Orlov]], [[Kontsevich]]) one replaces a [[variety]] by the [[derived category]] of [[coherent sheaves]] (or [[quasicoherent sheaves]] on that variety, or [[dg-category]] (or [[A-infinity category]]) [[enhanced triangulated category|enhancements]] thereof. There are also noncommutative deformations of such derived categories and analogues like the categories corresponding to the so-called [[Landau-Ginzburg model]]s. Therefore [[noncommutative derived algebraic geometry]] (and even noncommutative motives).

Notice that the [[derived category]] of coherent sheaves on a variety does \emph{not} remember all the structure of the original variety hence derived geometry loses often some information (sometimes not); thus derived algebraic geometry is sometimes easier than nonderived.

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

Survey includes

\begin{itemize}%
\item [[Maxim Kontsevich]], \emph{Geometry in triangulated categories}, talk at \emph{[[New Spaces for Mathematics and Physics]]}, IHP Paris, Oct-Sept 2015 (\href{https://www.youtube.com/watch?v=XnmO9RAbl50}{video recording})

\end{itemize}
Original articles include

\begin{itemize}%
\item L. Katzarkov, [[Maxim Kontsevich]], [[Tony Pantev]], \emph{Hodge theoretic aspects of mirror symmetry}, \href{http://arxiv.org/abs/0806.0107}{arxiv:0806.0107}

\end{itemize}
and a bit earlier this treatise on formal (infinitesimal in the sense of [[formal schemes]]) aspect as used in the [[deformation theory]] is in

\begin{itemize}%
\item [[Maxim Kontsevich]], [[Yan Soibelman]], \emph{Notes on A-infinity algebras, A-infinity categories and non-commutative geometry. I}, \href{http://arxiv.org/abs/math/0606241}{math.AG/0606241}


\item [[Dmitry Kaledin]], \emph{Homological methods in non-commutative geometry} (2008) (\href{http://imperium.lenin.ru/~kaledin/math/tokyo/final.pdf}{pdf})



\end{itemize}
The relations to tropical and symplectic geometry are in recent Kontsevich's talk at 2009 Arbeitstagung:

\begin{itemize}%
\item [[Maxim Kontsevich]], Mathematische Arbeitstagung 2009, \emph{Symplectic geometry of homological algebra}, preprint MPIM2009-40a, \href{http://www.mpim-bonn.mpg.de/preprints/send?bid=4024}{pdf}

\end{itemize}
Homological mirror symmetry is one of the main motivations and statements of the derived noncommutative algebraic geometry

\begin{itemize}%
\item [[Maxim Kontsevich]], \emph{Homological algebra of mirror symmetry}, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Z\"u{}rich, 1994), 120--139, Birkh\"a{}user, Basel, 1995.


\item [[Maxim Kontsevich]], [[Yan Soibelman]], \emph{Homological mirror symmetry and torus fibrations}, in Symplectic geometry and mirror symmetry (Seoul, 2000), 203--263, World Sci. Publ., River Edge, NJ, 2001.


\item [[Dmitri Orlov]], \emph{Smooth and proper noncommutative schemes and gluing of DG categories}, \href{http://arxiv.org/abs/1402.7364v1}{arXiv}.



\end{itemize}
Algebraic geometry over [[formal duals]] of [[E-n algebras]] is considered in

\begin{itemize}%
\item [[John Francis]], \emph{Derived algebraic geometry over $\mathcal{E}_n$-Rings} (\href{http://math.northwestern.edu/~jnkf/writ/thezrev.pdf}{pdf})


\item [[John Francis]], \emph{The tangent complex and Hochschild cohomology of $\mathcal{E}_n$-rings} (\href{http://math.northwestern.edu/~jnkf/writ/cotangentcomplex.pdf}{pdf})



\end{itemize}
Notice that for $n \geq 2$ the underlying ordinary rings (under $\pi_0$) are commutative. Therefore this has similarities with the [[formal geometry|formal]] [[noncommutative algebraic geometry]] perturbating around abelian schemes that is discussed in

\begin{itemize}%
\item [[Mikhail Kapranov]], \emph{Noncommutative geometry based on commutator expansions} (\href{http://arxiv.org/abs/math/9802041}{arXiv:9802041})

\end{itemize}
For more on this see at \emph{[[Kapranov's noncommutative geometry]]}

[[!redirects derived noncommutative algebraic geometry]] [[!redirects noncommutative derived algebraic geometry]]



\end{document}