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:
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}
\begin{definition}
A \emph{graded algebra} consists of a graded module $A$ together with two maps of degree $0$:
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
$$ xy =(-1)^{\deg{x}\deg{y}} yx. $$
$$ xy =(-1)^{\deg{x}\deg{y}} yx. $$
\end{definition}
\end{definition}
\todo{Add a remark about the signs somewhere}
\subsection{Differential graded algebra}
\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}$+1$.
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:
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.
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}
\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:
$$ d(xy)= d(x) y +(-1)^{\deg{x}} x d(y)\quad\text{ for all } x, y \in A. $$
$$ d(xy)= d(x) y +(-1)^{\deg{x}} x d(y)\quad\text{ for all } x, y \in A. $$
\end{definition}
\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.
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}.
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.
is a graded algebra.
\end{lemma}
\end{lemma}
\begin{proof}
\begin{proof}
\todo{}
\todo{Maybe just state this?}
\end{proof}
\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
> If a map has LLP/RLP wrt fib/cof, it is a cof/fib \\
\titem If a map has LLP/RLP wrt fib/cof, it is a cof/fib
> Fibs are preserved under pullbacks/limits \\
\titem Fibs are preserved under pullbacks/limits
\titem Cofibrantly generated mod. cats.
\titem Small object argument
}
}
\todo{Define homotopy category}
\todo{Define homotopy category}
\todo{Cofibrantly generated mod cats?}
\todo{Small obj. argument?}
\subsection{Quillen pairs}
\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.
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}
\begin{proposition}
There is a model structure on $\CDGA_\k$ where $f: A \to B$ is
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.
The maps $i_n$ are trivial cofibrations and the maps $j_n$ are cofibrations.
\end{lemma}
\end{lemma}
\begin{proof}
\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}
Joshua Moerman
10 years ago