From ed3eadf75ad5530288b0d08babf9de9d6e794d4b Mon Sep 17 00:00:00 2001 From: Joshua Moerman Date: Thu, 3 Jul 2014 17:27:51 +0200 Subject: [PATCH] Moves everything to notes, as there is no intended order yet. Adds some notes. --- .gitignore | 2 + thesis/{1_Algebra.tex => notes/Algebra.tex} | 8 +-- thesis/notes/CDGA_Basic_Examples.tex | 51 +++++++++++++ thesis/notes/CDGA_Of_Polynomials.tex | 71 +++++++++++++++++++ thesis/notes/Free_CDGA.tex | 50 +++++++++++++ .../Model_Categories.tex} | 0 .../Model_Of_CDGA.tex} | 0 thesis/notes/Polynomial_Forms.tex | 11 +++ thesis/preamble.tex | 19 ++++- thesis/references.bib | 14 ++++ thesis/thesis.tex | 10 ++- 11 files changed, 226 insertions(+), 10 deletions(-) rename thesis/{1_Algebra.tex => notes/Algebra.tex} (92%) create mode 100644 thesis/notes/CDGA_Basic_Examples.tex create mode 100644 thesis/notes/CDGA_Of_Polynomials.tex create mode 100644 thesis/notes/Free_CDGA.tex rename thesis/{2_Model_Cats.tex => notes/Model_Categories.tex} (100%) rename thesis/{CDGA_Model.tex => notes/Model_Of_CDGA.tex} (100%) create mode 100644 thesis/notes/Polynomial_Forms.tex diff --git a/.gitignore b/.gitignore index c090668..f46bf71 100644 --- a/.gitignore +++ b/.gitignore @@ -3,3 +3,5 @@ *.pdf build +*sublime* + diff --git a/thesis/1_Algebra.tex b/thesis/notes/Algebra.tex similarity index 92% rename from thesis/1_Algebra.tex rename to thesis/notes/Algebra.tex index f9463fa..86415a9 100644 --- a/thesis/1_Algebra.tex +++ b/thesis/notes/Algebra.tex @@ -33,7 +33,7 @@ Recall that the tensor product of modules distributes over direct sums. This def $$ (f \tensor g)(a \tensor x) = (-1)^{\deg{a}\deg{g}} \cdot f(a) \tensor g(x). $$ \end{definition} -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 +The sign is due to \emph{Koszul's 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. @@ -78,11 +78,11 @@ Finally we come to the definition of a differential graded algebra. This will be \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. +It is not hard to see that this definition precisely defines the monoidal objects in the category of differential graded modules. The category of dga's will be denoted by $\DGA_\k$, the category of commutative dga's (cdga's) 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}. -\todo{The notation $\CDGA$ seem to refer to cochain algebras in literature and not arbitrary CDGAs.} +\todo{The notation $\CDGA$ seem to refer to cochain algebras in literature and not arbitrary cdga's.} \subsection{Homology} @@ -110,7 +110,7 @@ For differential graded algebras we can consider the (co)homology by forgetting \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 Coalgebras and Hopf algebras? \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/notes/CDGA_Basic_Examples.tex b/thesis/notes/CDGA_Basic_Examples.tex new file mode 100644 index 0000000..3ba8cd9 --- /dev/null +++ b/thesis/notes/CDGA_Basic_Examples.tex @@ -0,0 +1,51 @@ + +\subsection{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: + +$$ D(n) = ... \to 0 \to \k \to \k \to 0 \to ... $$ +$$ S(n) = ... \to 0 \to \k \to 0 \to 0 \to ... $$ + +Note that $D(n)$ is acyclic for all $n$, or put in different words: $j_n : 0 \to D(n)$ is a quasi isomorphism. The sphere $S(n)$ has exactly one non-trivial cohomology group $H^n(S(n)) = \k \cdot [a]$. There is an injective function $i_n : S(n+1) \to D(n)$, sending $a$ to $c$. The maps $j_n$ and $i_n$ play the following important role in the model structure of cochain complexes: + +\begin{claim} + The set $I = \{i_n : S(n+1) \to D(n) \I n \in \N\}$ generates all cofibrations and the set $J = \{j_n : 0 \to D(n) \I n \in \N\}$ generates all trivial cofibrations. +\end{claim} + +The proof is omitted but can be found in different sources \todo{Cite sources}. In the next section we will prove a similar result for cdga's, so the reader can also refer to that proof. + +$S(n)$ plays a another special role: maps from $S(n)$ to some cochain complex $X$ correspond directly to elements in the kernel of $\restr{d}{X^n}$. Any such map is null-homotopic precisely when the corresponding elements in the kernel is a coboundary. So there is a natural isomorphism: $\Hom(S(n), X) / ~ \iso H^n(X)$. So the cohomology groups can be considered as honest homotopy groups. + +By using the free cdga functor we can turn these cochain complexes into cdga's $\Lambda(D(n))$ and $\Lambda(S(n))$. So $\Lambda(D(n))$ consists of linear combinations of $b^n$ and $c b^n$ when $n$ is even, and $c^n b$ and $c^n$ when $n$ is odd. In both cases we can compute the differentials using the Leibniz rule: +$$ d(b^n) = n \cdot c b^{n-1} $$ +$$ d(c b^n) = 0 $$ + +$$ d(c^n b) = c^{n+1} $$ +$$ d(c^n) = 0 $$ + +Those cocycles are in fact coboundaries (remember that $\k$ is a field of characteristic $0$): +$$ c b^n = \frac{1}{n} d(b^{n+1}) $$ +$$ c^n = d(b c^{n-1}) $$ + +There are no additional cocycles in $\Lambda(D(n))$ besides the constants and $c$. So we conclude that $\Lambda(D(n))$ is acyclic as an algebra. In other words $\Lambda(j_n): \k \to \Lambda D(n)$ is a quasi isomorphism. + +The situation for $\Lambda S(n)$ is easier: when $n$ is even it is given by polynomials in $a$, if $n$ is odd it is an exterior algebra (i.e. $a^2 = 0$). Again the sets $\Lambda(I) = \{ \Lambda(i_n) : \Lambda S(n+1) \to \Lambda D(n) \I n \in \N\}$ and $\Lambda(J) = \{ \Lambda(j_n) : \k \to \Lambda D(n) \I n \in \N\}$ play an important role. + +\begin{theorem} + The sets $\Lambda(I)$ and $\Lambda(J)$ generate a model structure on $\CDGA_\k$ where: + \begin{itemize} + \item weak equivalences are quasi isomorphisms, + \item fibrations are (degree wise) surjective maps and + \item cofibrations are maps with the left lifting property against trivial fibrations. + \end{itemize} +\end{theorem} + +We will prove this theorem in the next section. Note that the functors $\Lambda$ and $U$ thus form a Quillen pair with this model structure. + +\subsection{Why we need $\Char{\k} = 0$ for algebras} +The above Quillen pair $(\Lambda, U)$ fails to be a Quillen pair if $\Char{\k} = p \neq 0$. We will show this by proving that the maps $\Lambda(j_n)$ are not weak equivalences for even $n$. Consider $b^p \in D(n)$, then by the Leibniz rule: +$$ d(b^p) = p \cdot c b^{p-1} = 0. $$ +So $b^p$ is a cocycle. Now assume $b^p = dx$ for some $x$ of degree $pn - 1$, then $x$ contains a factor $c$ for degree reasons. By the calculations above we see that any element containing $c$ has a trivial differential or has a factor $c$ in its differential, contradicting $b^p = dx$. So this cocycle is not a coboundary and $\Lambda D(n)$ is not acyclic. + + + + diff --git a/thesis/notes/CDGA_Of_Polynomials.tex b/thesis/notes/CDGA_Of_Polynomials.tex new file mode 100644 index 0000000..bfb657a --- /dev/null +++ b/thesis/notes/CDGA_Of_Polynomials.tex @@ -0,0 +1,71 @@ + +\subsection{CDGA of Polynomials} + +\newcommand{\Apl}[0]{{A_{PL}}} + +We will now give a cdga model for the $n$-simplex $\Delta^n$. This then allows for simplicial methods. In the following definition one should be reminded of the topological $n$-simplex defined as convex span. + +\begin{definition} + For all $n \in \N$ define the following cdga: + $$ (\Apl)_n = \frac{\Lambda(x_0, \ldots, x_n, dx_0, \ldots, dx_n)}{(\sum_{i=0}^n) x_i - 1, \sum_{i=0}^n dx_i)} $$ + 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} + +Note that the inclusion $\Lambda(x_1, \ldots, x_n, dx_1, \ldots, dx_n) \to \Apl_n$ is an isomorphism of cdga's. So $\Apl_n$ is free and (algebra) maps from it are determined by their images on $x_i$ for $i = 1, \ldots, n$ (also note that this determines the images for $dx_i$). This fact will be used throughout. + +These cdga's will assemble into a simplicial cdga when we define the face and degeneracy maps as follows ($j = 1, \ldots, n$): + +$$ d_i(x_j) = \begin{cases} + x_{j-1}, &\text{ if } i < j \\ + 0, &\text{ if } i = j \\ + x_j, &\text{ if } i > j +\end{cases} \qquad d_i : \Apl_n \to \Apl_{n-1} $$ +$$ s_i(x_j) = \begin{cases} + x_{j+1}, &\text{ if } i < j \\ + x_j + x_{j+1}, &\text{ if } i = j \\ + x_j, &\text{ if } i > j +\end{cases} \qquad s_i : \Apl_n \to \Apl_{n+1} $$ + +One can check that $\Apl \in \simplicial{\CDGA_\k}$. We will denote the subspace of homogeneous elements of degree $k$ as $\Apl^k \in \simplicial{\Mod{\k}}$, this is indeed a simplicial $\k$-module as the maps $d_i$ and $s_i$ are graded maps of degree $0$. + +\begin{lemma} + $\Apl^k$ is contractible. +\end{lemma} +\begin{proof} + We will prove this by defining an extra degeneracy $s: \Apl_n \to \Apl_{n+1}$. Define for $i = 1, \ldots, n$: + \begin{align*} + s(1) &= (1-x_0)^2 \\ + s(x_i) &= (1-x_0) \cdot x_{i+1} + \end{align*} + Extend on the differentials and multiplicatively on $\Apl_n$. As $s(1) \neq 1$ this map is not an algebra map, however it well-defined as a map of cochain complexes. In particular when restricted to degree $k$ we get a linear map: + $$ s: \Apl^k_n \to \Apl^k_{n+1}. $$ + Proving the necessary properties of an extra degeneracy is fairly easy. For $n \geq 1$ we get (on generators): + \begin{align*} + d_0 s(1) &= d_0 (1 - x_0)^2 = (1 - 0) \cdot (1 - 0) = 1 \\ + d_0 s(x_i) &= d_0((1-x_0)x_{i+1}) = d_0(1-x_0) \cdot x_i \\ + &= (1-0) \cdot x_i = x_i + \end{align*} + So $d_0 s = \id$. + \begin{align*} + d_{i+1} s(1) &= d_{i+1} (1 - x_0)^2 = d_{i+1} (\sum_{j=1}^n x_j)^2 \\ + &= (\sum_{j=1}^{n-1} x_j)^2 = (1-x_0)^2 = s d_i(1) \\ + d_{i+1} s(x_j) &= d_{i+1}(1-x_0) d_{i+1}(x_j) = (1-x_0) d_i(x_{j+1}) = s d_i (x_j) + \end{align*} + So $d_{i+1} s = s d_i$. Similarly $s_{i+1} s = s s_i$. And finally for $n=0$ we have $d_1 s = 0$. + + So we have an extra degeneracy $s: \Apl^k \to \Apl^k$, and hence (see for example \cite{goerss}) we have that $\Apl^k$ is contractible. As a consequence $\Apl \to \ast$ is a weak equivalence. +\end{proof} + +\begin{lemma} + $\Apl_n^k$ is a Kan complex. +\end{lemma} +\begin{proof} + By the simple fact that $\Apl_n^k$ is a simplicial group, it is a Kan complex \cite{goerss}. +\end{proof} + +\begin{corollary} + $\Apl^k \to \ast$ is a trivial fibration in the standard model structure on $\sSet$. +\end{corollary} + + + diff --git a/thesis/notes/Free_CDGA.tex b/thesis/notes/Free_CDGA.tex new file mode 100644 index 0000000..62701e0 --- /dev/null +++ b/thesis/notes/Free_CDGA.tex @@ -0,0 +1,50 @@ + +\subsection{The free cdga} + +Just as in ordinary linear algebra we can form an algebra from any graded module. Furthermore we will see that a differential induces a derivation. + +\begin{definition} + The \emph{tensor algebra} of a graded module $M$ is defined as + $$ T(M) = \bigoplus_{n\in\N} M^{\tensor n}, $$ + where $M^{\tensor 0} = \k$. An element $m = m_1 \tensor \ldots \tensor m_n$ has a \emph{word length} of $n$ and its degree is $\deg{m} = \sum_{i=i}^n \deg{m_i}$. The multiplication is given by the tensor product (note that the bilinearity follows immediately). +\end{definition} + +Note that this construction is functorial and it is free in the following sense. + +\begin{lemma} + Let $M$ be a graded module and $A$ a graded algebra. + \begin{itemize} + \item A graded map $f: M \to A$ of degree $0$ extends uniquely to an algebra map $\overline{f} : TM \to A$. + \item A differential $d: M \to M$ extends uniquely to a derivation $d: TM \to TM$. + \end{itemize} +\end{lemma} + +\begin{corollary} + Let $U$ be the forgetful functor from graded algebras to graded modules, then $T$ and $U$ form an adjoint pair: + $$ T: \grMod{\k} \leftadj \grAlg{\k} $$ + Moreover it extends and restricts to + $$ T: \dgMod{\k} \leftadj \dgAlg{\k} $$ + $$ T: \CoCh{\k} \leftadj \DGA{\k} $$ +\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. + +\begin{definition} + Let $A$ be a graded algebra and define + $$ I = < ab - (-1)^{\deg{a}\deg{b}}ba \I a,b \in A >. $$ + Then $A / I$ is a commutative graded algebra. + + For a graded module $M$ we define the \emph{free commutative graded algebra} as + $$ \Lambda(M) = TM / I $$ +\end{definition} + +Again this extends to differential graded modules (i.e. the ideal is preserved by the derivative) and restricts to cochain complexes. + +\begin{lemma} + We have the following adjunctions: + $$ \Lambda: \grMod{\k} \leftadj \grAlg{\k}^{comm} $$ + $$ \Lambda: \dgMod{\k} \leftadj \dgAlg{\k}^{comm} $$ + $$ \Lambda: \CoCh{\k} \leftadj \CDGA_\k $$ +\end{lemma} + +We can now easily construct cdga's by specifying generators and their differentials. diff --git a/thesis/2_Model_Cats.tex b/thesis/notes/Model_Categories.tex similarity index 100% rename from thesis/2_Model_Cats.tex rename to thesis/notes/Model_Categories.tex diff --git a/thesis/CDGA_Model.tex b/thesis/notes/Model_Of_CDGA.tex similarity index 100% rename from thesis/CDGA_Model.tex rename to thesis/notes/Model_Of_CDGA.tex diff --git a/thesis/notes/Polynomial_Forms.tex b/thesis/notes/Polynomial_Forms.tex new file mode 100644 index 0000000..f88e0a6 --- /dev/null +++ b/thesis/notes/Polynomial_Forms.tex @@ -0,0 +1,11 @@ + +\subsection{Polynomial Forms} + +There is a general way to construct functors from $\sSet$ whenever we have some simplicial object. In our case we have the simplicial cdga $\Apl$ (which is nothing more than a functor $\opCat{\DELTA} \to \CDGA$) and we want to extend to a contravariant functor $\sSet \to \CDGA_\k$. This will be done via Kan extensions. + +Given a category $\cat{C}$ and a functor $F: \DELTA \to \cat{C}$, then define the following on objects: +\begin{align*} + F_!(X) &= \colim_{\Delta[n] \to X} F[n] &\quad X \in \sSet \\ + F^\ast(C)_n &= \Hom_{\cat{C}}(F[n], Y) &\quad C \in \cat{C} +\end{align*} +A simplicial map $X \to Y$ induces a map of the diagrams of which we take colimits. Applying $F$ on these diagrams, make it clear that $F_!$ is functorial. diff --git a/thesis/preamble.tex b/thesis/preamble.tex index e06aaaf..7af14ff 100644 --- a/thesis/preamble.tex +++ b/thesis/preamble.tex @@ -39,23 +39,34 @@ % Basic category stuff \newcommand{\cat}[1]{\mathbf{#1}} % the category of ... +\newcommand{\opCat}[1]{{#1}^{\text{op}}}% opposite category \newcommand{\Hom}{\mathbf{Hom}} \newcommand{\id}{\mathbf{id}} % Categories \newcommand{\Set}{\cat{Set}} % sets -\newcommand{\sSet}{\cat{sSet}} % simplicial sets \newcommand{\Top}{\cat{Top}} % topological spaces \newcommand{\DELTA}{\cat{\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 +\newcommand{\Alg}[1]{\cat{{#1}Alg}} % algebras over a ring \newcommand{\grMod}[1]{\cat{gr\mbox{-}{#1}Mod}} % graded modules over a ring \newcommand{\grAlg}[1]{\cat{gr\mbox{-}{#1}Alg}} % graded algebras over a ring -\newcommand{\DGA}{\cat{DGA}} % differential graded algebras -\newcommand{\CDGA}{\cat{CDGA}} % commutative dgas +\newcommand{\dgMod}[1]{\cat{dg\mbox{-}{#1}Mod}} % differential graded modules over a ring +\newcommand{\dgAlg}[1]{\cat{dg\mbox{-}{#1}Alg}} % differential graded algebras over a ring +\newcommand{\Ch}[1]{\cat{Ch_{n\geq0}({#1})}} % chain complexes +\newcommand{\CoCh}[1]{\cat{Ch^{n\geq0}({#1})}} % cochain complexes +\newcommand{\DGA}{\cat{DGA}} % cochain algebras +\newcommand{\CDGA}{\cat{CDGA}} % commutative cochain algebras \newcommand{\cof}{\hookrightarrow} % cofibration \newcommand{\fib}{\twoheadrightarrow} % fibration \newcommand{\we}{\tot{\simeq}} % weak equivalence +%\newcommand{\leftadj}{\ooalign{\hss\rightleftarrows\hss\cr\bot}} +\newcommand{\leftadj}{\rightleftarrows} + % Notation and operators \newcommand{\I}{\,\mid\,} % seperator in set notation \newcommand{\del}{\partial} % boundary @@ -68,6 +79,7 @@ \DeclareMathOperator*{\tensor}{\otimes} \DeclareMathOperator*{\bigtensor}{\bigotimes} \renewcommand{\deg}[1]{{|{#1}|}} +\newcommand{\Char}[1]{char({#1})} % restriction of a function \newcommand\restr[2]{{% we make the whole thing an ordinary symbol @@ -102,6 +114,7 @@ \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} +\newtheorem{claim}[theorem]{Claim} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} diff --git a/thesis/references.bib b/thesis/references.bib index 5f12847..a7e35e5 100644 --- a/thesis/references.bib +++ b/thesis/references.bib @@ -23,3 +23,17 @@ year={2007}, publisher={Providence, RI; American Mathematical Society; 1999} } + +@article{goerss, + title={Simplicial Homotopy Theory}, + author={Goerss, PG and Jardine, JF}, + publisher={Birkh{\"a}user}, + year={1999} +} + +@book{griffiths, + title={Rational homotopy theory and differential forms}, + author={Griffiths, Phillip A and Morgan, John W}, + year={2013}, + publisher={Birkh{\"a}user} +} diff --git a/thesis/thesis.tex b/thesis/thesis.tex index fea4d70..d186020 100644 --- a/thesis/thesis.tex +++ b/thesis/thesis.tex @@ -19,9 +19,13 @@ Some general notation: \todo{leave this out, or define somewhere else?} \vspace{1cm} -\input{1_Algebra} \vspace{2cm} -\input{2_Model_Cats} \vspace{2cm} -\input{CDGA_Model} \vspace{2cm} +\input{notes/Algebra} \vspace{2cm} +\input{notes/Free_CDGA} \vspace{2cm} +\input{notes/CDGA_Basic_Examples} \vspace{2cm} +\input{notes/Model_Categories} \vspace{2cm} +\input{notes/Model_Of_CDGA} \vspace{2cm} +\input{notes/CDGA_Of_Polynomials} \vspace{2cm} +\input{notes/Polynomial_Forms} \vspace{2cm} % \listoftodos