\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*{minimal dg-module}

[[!redirects KS-model]] [[!redirects minimal KS-extension]]

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

\hypertarget{rational_homotopy_theory}{}\paragraph*{{Rational homotopy theory}}\label{rational_homotopy_theory}

[[!include differential graded objects - 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{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak


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

A \emph{minimal dg-module} (\hyperlink{Roig92}{Roig 92}, \hyperlink{Roig94}{Roig 94, section 1}) is a [[minimal model]] in the context of the [[homotopy theory]] of [[dg-modules]].

Hence over [[dgc-algebras]] in non-positve degree, minimal dg-modules are the analogue of [[minimal Sullivan model]] as one passes from [[dg-algebras]] to (just) [[dg-modules]]. Minimal KS-extensions hence play the role of formal duals of [[minimal fibration]] in some applications of [[rational homotopy theory]].

(In \hyperlink{Halperin83}{Halperin 83} it has ``Koszul-Sullivan extensions'' for [[relative Sullivan algebras]], and ``KS'' in ``KS-models'' refers to that usage.)

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

Throughout, let $A$ be a [[dg-algebra]]. Eventually this is thought of as being a [[Sullivan model]] for the [[rationalization]] of the [[quotient]] of a [[topological space]] by a circle action.

We take all [[differentials]] to have degree +1. For $V$ a [[vector space]] and $n$ a [[natural number]], we write $V[n]$ for the [[chain complex]] concentrated on $V$ in degree $n$.

\begin{defn}
\label{HirschExtension}\hypertarget{HirschExtension}{}
\textbf{(Hirsch extension of dg-modules)}

Let $N$ be a [[dg-module]] over $A$ and let $n \in \mathbb{N}$ be a [[natural number]]. Then a degree $n$ \textbf{[[Hirsch extension]]} of $N$ is a [[monomorphisms]] of [[dg-modules]] of the form

\begin{displaymath}
N \hookrightarrow (N \oplus (A \otimes V[n]), d_{\phi})
\end{displaymath}
given by a choice of

\begin{enumerate}%
\item a [[vector space]] $V$


\item a [[linear map]] $d_V \;\colon\; V \to N_{n+1}$



\end{enumerate}
where the differential $d_{\phi}$ is $d_N$ on $N$, is $d_A$ on $A$, and is given on $V$ by $\phi$ followed by the [[action]] $\rho$ of $A$ on $V$:

\begin{displaymath}
d_\phi (a \otimes v) = (d_A a) \otimes v + (-1)^{\vert a \vert}  \rho(a,\phi(v))
  \,.
\end{displaymath}
\end{defn}
(\hyperlink{Roig94}{Roig 94, def. 1.8})

\begin{remark}
\label{}\hypertarget{}{}
It follows that for $N_1, N_2$ two $A$-[[dg-modules]] then homomorphisms $f$ out of a Hirsch extension of the former (def. \ref{HirschExtension})

\begin{displaymath}
f 
   \;\colon\;
  (N_1 \olpus (A \otimes V[n]), d_\phi)
   \longrightarrow
  N_2
\end{displaymath}
is equivalently

\begin{enumerate}%
\item a homomorphism of dg-modules $h \colon N_1 \to N_2$;


\item a [[linear map]] $g \colon V \to (N_2)_{n}$



\end{enumerate}
such that

\begin{itemize}%
\item $h \circ d_{\phi} = d_{N_2} \circ g$.

\end{itemize}
\end{remark}
The following defines a kind of [[minimal cofibrations]] of dg-modules.

\begin{defn}
\label{KSFactorization}\hypertarget{KSFactorization}{}
\textbf{(minimal KS-extension)}

For $N$ a [[dg-module]] over $A$, then a \textbf{[[minimal KS-extension]]} of $N$ is a certain [[transfinite composition]] of Hirsch extensions, namely a [[monomorphism]]

\begin{displaymath}
N \hookrightarrow \hat N
\end{displaymath}
equipped with an [[exhaustive filtration]] $\{\hat N(n,q)\}_{n,q \in \mathbb{N}}$ such that:

\begin{enumerate}%
\item $\hat N(0,0) \simeq N$;


\item the inclusions $\hat N(n,q) \hookrightarrow \hat N(n,q+1)$ are Hirsch extensions (def. \ref{HirschExtension}).


\item $N(n+1,0) = \underset{\longrightarrow}{\lim}_q \hat N(n,q)$ ([[colimit]] over the sequence of Hirsch extensions in the previous degree).



\end{enumerate}
Accordingly, a \textbf{minimal KS-factorization} of a morphism $N_1 \to N_2$ of $A$-dg-modules is a factorization as a minimal KS-extension followed by a [[quasi-isomorphism]]

\begin{displaymath}
\itexarray{
    N _1 && \longrightarrow && N_2
    \\
    & {}_{\mathllap{\text{minimal} \atop \text{KS-extension}}}\searrow && \nearrow_{\mathrlap{\text{quasi-iso}}}
    \\
    && N_1 \oplus (A \otimes V)
  }
  \,.
\end{displaymath}
Finally a \textbf{minimal KS-model} is a [[dg-module]] $N$ such that $0 \hookrightarrow N$ is a minimal KS-fibration.

\end{defn}
(\hyperlink{Roig94}{Roig 94, def. 1.9})

\begin{prop}
\label{}\hypertarget{}{}
Let the [[ground field]] be of [[characteristic zero]].

Let $f \colon N_1 \longrightarrow N_2$ be a morphism of $A$-[[dg-modules]] such that it induces a [[monomorphism]] in degree-0 [[cochain cohomology]], $H^0(f) \colon H^0(N_1) \hookrightarrow H^0(N_2)$ then it admits a minimal KS-factorization (def. \ref{KSFactorization}).

In particular, every [[dg-module]] has a minimal KS-model (def. \ref{KSFactorization}).

\end{prop}
\hyperlink{RoigSaralegiAranguren00}{Roig \& Saralegi-Aranguren 00, theorem 1.3.1}

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item \href{4-sphere#CircleAction}{4-sphere -- circle action}

\end{itemize}
\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[circle action]]

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

\begin{itemize}%
\item [[Agustí Roig]], \emph{Alguns punts d'\`a{}lgebra homot\`o{}pica}, Barcelona (1992)


\item [[Agustí Roig]], \emph{Minimal resolutions and other minimal models}, Publicacions Matem\`a{}tiques (1993) Volume: 37, Issue: 2, page 285-303 (\href{https://eudml.org/doc/41535}{web})


\item [[Agustí Roig]], \emph{Formalizability of dg modules and morphisms of cdg algebras}, Volume 38, Issue 3 (1994), 434-451 (\href{http://projecteuclid.org/euclid.ijm/1255986724}{euclid})


\item [[Igor Kriz]], [[Peter May]], section IV.3 of \emph{Operads, Algebras, Modules and Motives}, 1994 (\href{http://www.math.uchicago.edu/~may/PAPERS/kmbooklatex.pdf}{pdf})


\item [[Agustí Roig]], [[Martintxo Saralegi-Aranguren]], \emph{Minimal Models for Non-Free Circle Actions}, Illinois Journal of Mathematics, volume 44, number 4 (2000) (\href{https://arxiv.org/abs/math/0004141}{arXiv:math/0004141})



\end{itemize}
See also

\begin{itemize}%
\item [[Steve Halperin]], \emph{Lectures on minimal models}, Mem. Soc. Math. Franc. no 9/10 (1983) (\href{https://eudml.org/doc/94833}{web})


\item [[Flavio da Silveira]], \emph{Rational homotopy theory of fibrations}, Pacific Journal of Mathematics, Vol. 113, No. 1 (1984) (\href{http://msp.org/pjm/1984/113-1/pjm-v113-n1-p01-s.pdf}{pdf})



\end{itemize}
[[!redirects minimal dg-modules]]

[[!redirects Hirsch extension]] [[!redirects Hirsch extensions]]

[[!redirects minimal Hirsch cofibration]] [[!redirects minimal Hirsch cofibrations]]

[[!redirects minimal KS-extension]] [[!redirects minimal KS-extensions]]

[[!redirects minimal KS-model]] [[!redirects minimal KS-models]]



\end{document}