Master thesis on Rational Homotopy Theory
https://github.com/Jaxan/Rational-Homotopy-Theory
You can not select more than 25 topics
Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
50 lines
2.1 KiB
50 lines
2.1 KiB
|
|
\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:
|
|
$$ T: \grMod{\k} \leftadj \grAlg{\k} :U $$
|
|
Moreover it extends and restricts to
|
|
$$ T: \dgMod{\k} \leftadj \dgAlg{\k} :U $$
|
|
$$ T: \CoCh{\k} \leftadj \DGA{\k} :U $$
|
|
\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.
|
|
|
|
\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:
|
|
$$ \Lambda: \grMod{\k} \leftadj \grAlg{\k}^{comm} :U $$
|
|
$$ \Lambda: \dgMod{\k} \leftadj \dgAlg{\k}^{comm} :U $$
|
|
$$ \Lambda: \CoCh{\k} \leftadj \CDGA_\k :U $$
|
|
\end{lemma}
|
|
|
|
We can now easily construct cdga's by specifying generators and their differentials.
|
|
|