From e07833d40b77ef17a865f762180b1e76d4de4b7e Mon Sep 17 00:00:00 2001 From: Joshua Moerman Date: Thu, 10 Jul 2014 17:54:11 +0200 Subject: [PATCH] Some small fixups --- thesis/notes/Algebra.tex | 2 +- thesis/notes/CDGA_Basic_Examples.tex | 2 +- thesis/notes/CDGA_Of_Polynomials.tex | 4 ++-- thesis/notes/Free_CDGA.tex | 12 ++++++------ thesis/notes/Minimal_Models.tex | 10 +++++----- thesis/preamble.tex | 5 +---- thesis/style.tex | 3 +++ thesis/thesis.tex | 10 +++++++--- 8 files changed, 26 insertions(+), 22 deletions(-) diff --git a/thesis/notes/Algebra.tex b/thesis/notes/Algebra.tex index 8b76544..92a2d9c 100644 --- a/thesis/notes/Algebra.tex +++ b/thesis/notes/Algebra.tex @@ -2,7 +2,7 @@ \section{Differential Graded Algebra} \label{sec:algebra} -In this section $\k$ will be any commutative ring. We will recap some of the basic definitions of commutative algebra in a graded setting. By \emph{linear}, \emph{module}, \emph{tensor product}, etc \dots we always mean $\k$-linear, $\k$-module, tensor product over $\k$, etc \dots. +In this section $\k$ will be any commutative ring. We will recap some of the basic definitions of commutative algebra in a graded setting. By \emph{linear}, \emph{module}, \emph{tensor product}, etc\dots we always mean $\k$-linear, $\k$-module, tensor product over $\k$, etc\dots. \subsection{Graded algebra} diff --git a/thesis/notes/CDGA_Basic_Examples.tex b/thesis/notes/CDGA_Basic_Examples.tex index 3ba8cd9..d35c2ad 100644 --- a/thesis/notes/CDGA_Basic_Examples.tex +++ b/thesis/notes/CDGA_Basic_Examples.tex @@ -1,5 +1,5 @@ -\subsection{Cochain models for the $n$-disk and $n$-sphere} +\section{Cochain models for the $n$-disk and $n$-sphere} We will first define some basic cochain complexes which model the $n$-disk and $n$-sphere. $D(n)$ is the cochain complex generated by one element $b \in D(n)^n$ and its differential $c = d(b) \in D(n)^{n+1}$. $S(n)$ is the cochain complex generated by one element $a \in S(n)^n$ which differential vanishes (i.e. $da = 0$). In other words: $$ D(n) = ... \to 0 \to \k \to \k \to 0 \to ... $$ diff --git a/thesis/notes/CDGA_Of_Polynomials.tex b/thesis/notes/CDGA_Of_Polynomials.tex index bfb657a..a8d038b 100644 --- a/thesis/notes/CDGA_Of_Polynomials.tex +++ b/thesis/notes/CDGA_Of_Polynomials.tex @@ -1,5 +1,5 @@ -\subsection{CDGA of Polynomials} +\section{CDGA of Polynomials} \newcommand{\Apl}[0]{{A_{PL}}} @@ -7,7 +7,7 @@ We will now give a cdga model for the $n$-simplex $\Delta^n$. This then allows f \begin{definition} For all $n \in \N$ define the following cdga: - $$ (\Apl)_n = \frac{\Lambda(x_0, \ldots, x_n, dx_0, \ldots, dx_n)}{(\sum_{i=0}^n) x_i - 1, \sum_{i=0}^n dx_i)} $$ + $$ (\Apl)_n = \frac{\Lambda(x_0, \ldots, x_n, dx_0, \ldots, dx_n)}{(\sum_{i=0}^n x_i - 1, \sum_{i=0}^n dx_i)} $$ So it is the free cdga with $n+1$ generators and their differentials such that $\sum_{i=0}^n x_i = 1$ and in order to be well behaved $\sum_{i=0}^n dx_i = 0$. \end{definition} diff --git a/thesis/notes/Free_CDGA.tex b/thesis/notes/Free_CDGA.tex index 62701e0..b9ee7fc 100644 --- a/thesis/notes/Free_CDGA.tex +++ b/thesis/notes/Free_CDGA.tex @@ -21,10 +21,10 @@ Note that this construction is functorial and it is free in the following sense. \begin{corollary} Let $U$ be the forgetful functor from graded algebras to graded modules, then $T$ and $U$ form an adjoint pair: - $$ T: \grMod{\k} \leftadj \grAlg{\k} $$ + $$ T: \grMod{\k} \leftadj \grAlg{\k} :U $$ Moreover it extends and restricts to - $$ T: \dgMod{\k} \leftadj \dgAlg{\k} $$ - $$ T: \CoCh{\k} \leftadj \DGA{\k} $$ + $$ T: \dgMod{\k} \leftadj \dgAlg{\k} :U $$ + $$ T: \CoCh{\k} \leftadj \DGA{\k} :U $$ \end{corollary} As with the symmetric algebra and exterior algebra of a vector space, we can turn this graded tensor algebra in a commutative graded algebra. @@ -42,9 +42,9 @@ Again this extends to differential graded modules (i.e. the ideal is preserved b \begin{lemma} We have the following adjunctions: - $$ \Lambda: \grMod{\k} \leftadj \grAlg{\k}^{comm} $$ - $$ \Lambda: \dgMod{\k} \leftadj \dgAlg{\k}^{comm} $$ - $$ \Lambda: \CoCh{\k} \leftadj \CDGA_\k $$ + $$ \Lambda: \grMod{\k} \leftadj \grAlg{\k}^{comm} :U $$ + $$ \Lambda: \dgMod{\k} \leftadj \dgAlg{\k}^{comm} :U $$ + $$ \Lambda: \CoCh{\k} \leftadj \CDGA_\k :U $$ \end{lemma} We can now easily construct cdga's by specifying generators and their differentials. diff --git a/thesis/notes/Minimal_Models.tex b/thesis/notes/Minimal_Models.tex index e3778ca..be13b31 100644 --- a/thesis/notes/Minimal_Models.tex +++ b/thesis/notes/Minimal_Models.tex @@ -7,7 +7,7 @@ In this section we will discuss the so called minimal models. These are cdga's w An cdga $(A, d)$ is a \emph{Sullivan algebra} if \begin{itemize} \item $(A, d)$ is quasi-free (or semi-free), i.e. $A = \Lambda V$ is free as a cdga, and - \item $V$ has a filtration $V(0) \subset V(1) \subset \cdots \subset \bigcup{k \in \N} V(k) = V$ such that $d(V(k)) \subset \Lambda V(k-1)$. + \item $V$ has a filtration $V(0) \subset V(1) \subset \cdots \subset \bigcup_{k \in \N} V(k) = V$ such that $d(V(k)) \subset \Lambda V(k-1)$. \end{itemize} An cdga $(A, d)$ is a \emph{minimal (Sullivan) algebra} if in addition @@ -27,7 +27,7 @@ The requirement that there exists a filtration can be replaced by a stronger sta Let $(A, d)$ be a cdga which is $1$-reduced, quasi-free and with a decomposable differential. Then $(A, d)$ is a minimal algebra. \end{lemma} \begin{proof} - Take $V(n) = \bigoplus_{k=0}^n V^k$ (note that $V^0 = v^1 = 0$). Since $d$ is decomposable we see that for $v \in V^n$: $d(v) = x \cdot y$ for some $x, y \in A$. Assuming $dv$ to be non-zero we can compute the degrees: + Let $V$ generate $A$. Take $V(n) = \bigoplus_{k=0}^n V^k$ (note that $V^0 = V^1 = 0$). Since $d$ is decomposable we see that for $v \in V^n$: $d(v) = x \cdot y$ for some $x, y \in A$. Assuming $dv$ to be non-zero we can compute the degrees: $$ \deg{x} + \deg{y} = \deg{xy} = \deg{dv} = \deg{v} + 1 = n + 1. $$ As $A$ is $1$-reduced we have $\deg{x}, \deg{y} \geq 2$ and so by the above $\deg{x}, \deg{y} \leq n-1$. Conclude that $d(V(k)) \subset \Lambda(V(n-1))$. \end{proof} @@ -67,7 +67,7 @@ Before we state the uniqueness theorem we need some more properties of minimal m \cimage[scale=0.5]{Sullivan_Lifting} - By the left adjointness of $\Lambda$ we only have to specify a map $\phi: V \to X$ sucht that $p \circ \phi = g$. We will do this by induction. + By the left adjointness of $\Lambda$ we only have to specify a map $\phi: V \to X$ such that $p \circ \phi = g$. We will do this by induction. \begin{itemize} \item Suppose $\{v_\alpha\}$ is a basis for $V(0)$. Define $V(0) \to X$ by choosing preimages $x_\alpha$ such that $p(x_\alpha) = g(v_\alpha)$ ($p$ is surjective). Define $\phi(v_\alpha) = x_\alpha$. \item Suppose $\phi$ has been defined on $V(n)$. Write $V(n+1) = V(n) \oplus V'$ and let $\{v_\alpha\}$ be a basis for $V'$. Then $dv_\alpha \in \Lambda V(n)$, hence $\phi(dv_\alpha)$ is defined and @@ -101,7 +101,7 @@ Before we state the uniqueness theorem we need some more properties of minimal m Let $\phi: (M, d) \we (M', d')$ be a weak equivalence between minimal algebras. Then $\phi$ is an isomorphism. \end{lemma} \begin{proof} - Let $M$ and $M'$ be generated by $V$ and $V'$. Then $\phi$ induces a weak equivalence on the linear part $\phi_0: V \we V'$ \cite[Theorem 1.5.10]{loday}. Since the differentials are decomposable, their linear part vanishes. So we see that $\phi_0: (V, 0) \tot{\iso} (V', 0)$ is an isomorphism. + Let $M$ and $M'$ be generated by $V$ and $V'$. Then $\phi$ induces a weak equivalence on the linear part $\phi_0: V \we V'$ \cite[Theorem 1.5.2]{loday}. Since the differentials are decomposable, their linear part vanishes. So we see that $\phi_0: (V, 0) \tot{\iso} (V', 0)$ is an isomorphism. Conclude that $\phi = \Lambda \phi_0$ is an isomorphism. \end{proof} @@ -109,5 +109,5 @@ Before we state the uniqueness theorem we need some more properties of minimal m Let $m: (M, d) \we (A, d)$ and $m': (M', d') \we (A, d)$ be two minimal models. Then there is an isomorphism $\phi (M, d) \tot{\iso} (M', d')$ such that $m' \circ \phi \eq m$. \end{theorem} \begin{proof} - By the previous lemmas we have $[M', M] \iso [M', A]$. By going from right to elft we get a map $\phi: M' \to M$ such that $m' \circ \phi \eq m$. On homology we get $H(m') \circ H(\phi) = H(m)$, proving that (2-out-of-3) $\phi$ is a weak equivalence. The previous lemma states that $\phi$ is then an isomorphism. + By the previous lemmas we have $[M', M] \iso [M', A]$. By going from right to left we get a map $\phi: M' \to M$ such that $m' \circ \phi \eq m$. On homology we get $H(m') \circ H(\phi) = H(m)$, proving that (2-out-of-3) $\phi$ is a weak equivalence. The previous lemma states that $\phi$ is then an isomorphism. \end{proof} diff --git a/thesis/preamble.tex b/thesis/preamble.tex index d7f98fd..f9a4ad4 100644 --- a/thesis/preamble.tex +++ b/thesis/preamble.tex @@ -24,9 +24,6 @@ % Matrices have a upper bound for its size \setcounter{MaxMatrixCols}{20} -% Remove trailing `contents` after toc -\renewcommand{\contentsname}{} - % for the fib arrow \usepackage{amssymb} @@ -51,7 +48,7 @@ % Categories \newcommand{\Set}{\cat{Set}} % sets \newcommand{\Top}{\cat{Top}} % topological spaces -\newcommand{\DELTA}{\cat{\Delta}} % the simplicial cat +\newcommand{\DELTA}{\boldsymbol{\Delta}}% the simplicial cat \newcommand{\simplicial}[1]{\cat{s{#1}}}% simplicial objects \newcommand{\sSet}{\simplicial{\Set}} % simplicial sets \newcommand{\Mod}[1]{\cat{{#1}Mod}} % modules over a ring diff --git a/thesis/style.tex b/thesis/style.tex index a36543b..61fe496 100644 --- a/thesis/style.tex +++ b/thesis/style.tex @@ -7,3 +7,6 @@ % no indent, but vertical spacing \usepackage[parfill]{parskip} \setlength{\marginparwidth}{2cm} + +% skip subsections in toc +\setcounter{tocdepth}{1} diff --git a/thesis/thesis.tex b/thesis/thesis.tex index a2ee497..a1368f7 100644 --- a/thesis/thesis.tex +++ b/thesis/thesis.tex @@ -9,7 +9,13 @@ \begin{document} \maketitle +\begin{center} + {\bf \today} +\end{center} + +\vspace{2cm} \tableofcontents +\vspace{2cm} Some general notation: \todo{leave this out, or define somewhere else?} \begin{itemize} @@ -17,9 +23,7 @@ Some general notation: \todo{leave this out, or define somewhere else?} \item $\Hom_\cat{C}(A, B)$ will denote the set of maps from $A$ to $B$ in the category $\cat{C}$. \end{itemize} -\vspace{1cm} - -\newcommand{\myinput}[1]{\input{#1} \vspace{2cm}} +\newcommand{\myinput}[1]{\include{#1}} \myinput{notes/Algebra} \myinput{notes/Free_CDGA}