From d83c87339f0d90ea05b6e165cf288e98a11202d2 Mon Sep 17 00:00:00 2001 From: Joshua Moerman Date: Fri, 16 May 2014 14:25:20 +0100 Subject: [PATCH] Adds differential on tensor. Koszul sign conv. Adds todos. --- thesis/1_Algebra.tex | 29 +++++++++++++++++++++++------ thesis/2_Model_Cats.tex | 16 +++++++--------- thesis/CDGA_Model.tex | 6 ++++-- thesis/preamble.tex | 17 ++++++++++++++++- thesis/thesis.tex | 11 +++++++---- 5 files changed, 57 insertions(+), 22 deletions(-) diff --git a/thesis/1_Algebra.tex b/thesis/1_Algebra.tex index efcc7ed..f9463fa 100644 --- a/thesis/1_Algebra.tex +++ b/thesis/1_Algebra.tex @@ -29,9 +29,14 @@ Recall that the tensor product of modules distributes over direct sums. This def \begin{definition} The graded tensor product is defined as: $$ (M \tensor N)_n = \bigoplus_{i + j = n} M_i \tensor N_j. $$ + The tensor product extends to graded maps. Let $f: A \to B$ and $g:X \to Y$ be two graded maps, then their tensor product $f \tensor g: A \tensor B \to X \tensor Y$ is defined as: + $$ (f \tensor g)(a \tensor x) = (-1)^{\deg{a}\deg{g}} \cdot f(a) \tensor g(x). $$ \end{definition} -The graded modules together with graded maps of degree $0$ form the category $\grMod{\k}$ of graded modules. From now on we will simply refer to maps instead of graded maps. Together with the tensor product and the ground ring, $(\grMod{\k}, \tensor, \k)$ is a monoidal category. This now dictates the definition of a graded algebra. +The sign is due to \emph{Koszuls sign convention}: whenever two elements next to each other are swapped (in this case $g$ and $a$) a minus sign appears if both elements are of odd degree. More formally we can define a swap map +$$ \tau : A \tensor B \to B \tensor A : a \tensor b \mapsto (-1)^{\deg{a}\deg{b}} b \tensor a. $$ + +The graded modules together with graded maps of degree $0$ form the category $\grMod{\k}$ of graded modules. From now on we will simply refer to maps instead of graded maps. Together with the tensor product and the ground ring, $(\grMod{\k}, \tensor, \k)$ is a symmetric monoidal category (with the symmetry given by $\tau$). This now dictates the definition of a graded algebra. \begin{definition} A \emph{graded algebra} consists of a graded module $A$ together with two maps of degree $0$: @@ -48,8 +53,6 @@ Again these objects and maps form a category, denoted as $\grAlg{\k}$. We will d $$ xy = (-1)^{\deg{x}\deg{y}} yx. $$ \end{definition} -\todo{Add a remark about the signs somewhere} - \subsection{Differential graded algebra} @@ -59,15 +62,22 @@ Again these objects and maps form a category, denoted as $\grAlg{\k}$. We will d A differential graded module $(M, d)$ with $M_i = 0$ for all $i < 0$ is a \emph{chain complex}. A differential graded module $(M, d)$ with $M_i = 0$ for all $i > 0$ is a \emph{cochain complex}. It will be convenient to define $M^i = M_{-i}$ in the latter case, so that $M = \bigoplus_{n \in \N} M^i$ and $d$ is a map of \emph{upper degree} $+1$. -The tensor product of two differential graded modules is again a differential graded module if we define the differential as follows. \todo{Define this} +\begin{definition} + Let $(M, d_M)$ and $(N, d_N)$ be two differential graded modules, their tensor product $M \tensor N$ is a differential graded module with the differential given by: + $$ d_{M \tensor N} = d_M \tensor \id_N + \id_M \tensor d_N. $$ +\end{definition} + +\todo{Prove that this is in fact a differential?} Finally we come to the definition of a differential graded algebra. This will be a graded algebra with a differential. Of course we want this to be compatible with the algebra structure, or stated differently: we want $\mu$ and $\eta$ to be chain maps. \begin{definition} - A \emph{differential graded algebra (DGA)} is a graded algebra $A$ together with an differential $d$ such that in addition: + A \emph{differential graded algebra (DGA)} is a graded algebra $A$ together with an differential $d$ such that in addition the \emph{Leibniz rule} holds: $$ d(xy) = d(x) y + (-1)^{\deg{x}} x d(y) \quad\text{ for all } x, y \in A. $$ \end{definition} +\todo{Define the notion of derivation?} + It is not hard to see that this definition precisely defines the monoidal objects in the category of differential graded modules. The category of DGAs will be denoted by $\DGA_\k$, the category of commutative DGAs (CDGAs) will be denoted by $\CDGA_\k$. If no confusion can arise, the ground ring $\k$ will be suppressed in this notation. Let $M$ be a DGA, just as before $M$ is called a \emph{chain algebras} if $M_i = 0$ for $i < 0$. Similarly if $M^i = 0$ for all $i < 0$, then $M$ is a \emph{cochain algebra}. @@ -94,6 +104,13 @@ For differential graded algebras we can consider the (co)homology by forgetting is a graded algebra. \end{lemma} \begin{proof} - \todo{} + \todo{Maybe just state this?} \end{proof} +\TODO{Discuss: +\titem The Künneth theorem (especially in the case of fields) +\titem The tensor algebra $T : Ch^\ast(\Q) \to \DGA_\Q$ and free cdga $\Lambda : Ch^\ast(\Q) \to \CDGA_\Q$ +\titem Coalgebras and Hopfalgebras? +\titem Define reduced/connected differential graded things +\titem Singular (co)homology as a quick example? +} \ No newline at end of file diff --git a/thesis/2_Model_Cats.tex b/thesis/2_Model_Cats.tex index 4d129d7..4493333 100644 --- a/thesis/2_Model_Cats.tex +++ b/thesis/2_Model_Cats.tex @@ -86,18 +86,16 @@ Note that axiom [MC5a] allows us to replace any object $X$ with a weakly equival \end{tikzpicture} \end{center} -\todo{Maybe some basic propositions: -> Over/under category (or simply pointed objects) \\ -> If a map has LLP/RLP wrt fib/cof, it is a cof/fib \\ -> Fibs are preserved under pullbacks/limits \\ +\TODO{Maybe some basic propositions (refer to Dwyer \& Spalinski): +\titem Over/under category (or simply pointed objects) +\titem If a map has LLP/RLP wrt fib/cof, it is a cof/fib +\titem Fibs are preserved under pullbacks/limits +\titem Cofibrantly generated mod. cats. +\titem Small object argument } \todo{Define homotopy category} -\todo{Cofibrantly generated mod cats?} - -\todo{Small obj. argument?} - \subsection{Quillen pairs} -In order to relate model categories and their associated homotopy categories we need a notion of maps between them. +In order to relate model categories and their associated homotopy categories we need a notion of maps between them. We want the maps such that they induce maps on the homotopy categories. \todo{Definition etc} diff --git a/thesis/CDGA_Model.tex b/thesis/CDGA_Model.tex index 210f2d6..f6ca0cc 100644 --- a/thesis/CDGA_Model.tex +++ b/thesis/CDGA_Model.tex @@ -1,7 +1,9 @@ \section{Model structure on $\CDGA_\k$} -In this section we will define a model structure on CDGAs over a field $\k$ \todo{Can $\k$ be a c. ring here?}, where the weak equivalences are quasi isomorphisms and fibrations are surjective maps. The cofibrations are defined to be the maps with a left lifting property with respect to trivial fibrations. +\TODO{First discuss the model structure on (co)chain complexes. Then discuss that we want the adjunction $(\Lambda, U)$ to be a Quillen pair. Then state that (co)chain complexes are cofib. generated, so we can cofib. generate CDGAs.} + +In this section we will define a model structure on CDGAs over a field $\k$ of characteristic zero\todo{Can $\k$ be a c. ring here?}, where the weak equivalences are quasi isomorphisms and fibrations are surjective maps. The cofibrations are defined to be the maps with a left lifting property with respect to trivial fibrations. \begin{proposition} There is a model structure on $\CDGA_\k$ where $f: A \to B$ is @@ -51,7 +53,7 @@ Next we will prove the factorization property [MC5]. We will do this by Quillen' The maps $i_n$ are trivial cofibrations and the maps $j_n$ are cofibrations. \end{lemma} \begin{proof} - Since $H(T(n)) = \k$ we see that indeed $H(i_n)$ is an isomorphism. For the lifting property of $i_n$ and $j_n$ simply use surjectivity of the fibrations. \todo{give a bit more detail} + Since $H(T(n)) = \k$ \todo{Note that this only hold when characteristic = 0} we see that indeed $H(i_n)$ is an isomorphism. For the lifting property of $i_n$ and $j_n$ simply use surjectivity of the fibrations. \todo{give a bit more detail} \end{proof} \begin{lemma} diff --git a/thesis/preamble.tex b/thesis/preamble.tex index b5f7b9d..e06aaaf 100644 --- a/thesis/preamble.tex +++ b/thesis/preamble.tex @@ -34,6 +34,7 @@ \newcommand{\Np}{{\mathbb{N}^{>0}}} % positive numbers \newcommand{\Z}{\mathbb{Z}} % integers \newcommand{\R}{\mathbb{R}} % reals +\newcommand{\Q}{\mathbb{Q}} % rationals \renewcommand{\k}{\mathbbm{k}} % default ground ring % Basic category stuff @@ -76,11 +77,25 @@ \right|_{#2} % this is the delimiter }} -% todos +% Todos in the margin \newcommand{\todo}[1]{ \addcontentsline{tdo}{todo}{\protect{#1}} $\ast$ \marginpar{\tiny $\ast$ #1} } +% Big todos in text +\newcommand{\TODO}[1]{ + \addcontentsline{tdo}{todo}{\protect{#1}} + {\tiny $\ast$ #1} +} +% TODO item, as itemize does not work +\newcommand{\titem}[0]{\\-\qquad} +% List of todos +\makeatletter + \newcommand \listoftodos{\section*{Todo list} \@starttoc{tdo}} + \newcommand\l@todo[2]{ + \par\noindent \textit{#2}, \parbox{10cm}{#1}\par + } +\makeatother \theoremstyle{plain} \newtheorem{theorem}{Theorem}[section] diff --git a/thesis/thesis.tex b/thesis/thesis.tex index 4b17612..fea4d70 100644 --- a/thesis/thesis.tex +++ b/thesis/thesis.tex @@ -16,11 +16,14 @@ Some general notation: \todo{leave this out, or define somewhere else?} \item $\cat{0}$ (resp. $\cat{1}$) will denote the initial (resp. final) objects in a category. \item $\Hom_\cat{C}(A, B)$ will denote the set of maps from $A$ to $B$ in the category $\cat{C}$. \end{itemize} -\newpage -\input{1_Algebra} \newpage -\input{2_Model_Cats} \newpage -\input{CDGA_Model} \newpage +\vspace{1cm} + +\input{1_Algebra} \vspace{2cm} +\input{2_Model_Cats} \vspace{2cm} +\input{CDGA_Model} \vspace{2cm} + +% \listoftodos \nocite{*} \bibliographystyle{alpha}