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\begin{document}

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\section*{free abelian group}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{group_theory}{}\paragraph*{{Group Theory}}\label{group_theory}

[[!include group theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{Definition}{Definition}\dotfill \pageref*{Definition} \linebreak
\noindent\hyperlink{Properties}{Properties}\dotfill \pageref*{Properties} \linebreak
\noindent\hyperlink{in_terms_of_formal_linear_combinations}{In terms of formal linear combinations}\dotfill \pageref*{in_terms_of_formal_linear_combinations} \linebreak
\noindent\hyperlink{Subgroups}{Subgroups}\dotfill \pageref*{Subgroups} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The \emph{free abelian group} $\mathbb{Z}[S]$ on a [[set]] $S$ is the [[abelian group]] whose elements are \emph{formal $\mathbb{Z}$-[[linear combinations]]} of elements of $S$.

\hypertarget{Definition}{}\subsection*{{Definition}}\label{Definition}

\begin{defn}
\label{FreeAbelianGroup}\hypertarget{FreeAbelianGroup}{}
Let

\begin{displaymath}
U \colon Ab \longrightarrow Set
\end{displaymath}
be the [[forgetful functor]] from the category [[Ab]] of abelian groups, to the category [[Set]] of sets. This has a [[left adjoint]] [[free construction]]:

\begin{displaymath}
\mathbb{Z}[-] \colon Set \longrightarrow Ab
  \,.
\end{displaymath}
This is the \emph{free abelian group functor}. For $S \in$ [[Set]], the \textbf{free abelian group} $\mathbb{Z}[S] \in$ [[Ab]] is the [[free object]] on $S$ with respect to this [[free-forgetful adjunction]].

\end{defn}
Explicit descriptions of free abelian groups are discussed \hyperlink{Properties}{below}.

\hypertarget{Properties}{}\subsection*{{Properties}}\label{Properties}

\hypertarget{in_terms_of_formal_linear_combinations}{}\subsubsection*{{In terms of formal linear combinations}}\label{in_terms_of_formal_linear_combinations}

\begin{defn}
\label{FormalLinearCombination}\hypertarget{FormalLinearCombination}{}
A \textbf{formal linear combination} of elements of a set $S$ is a [[function]]

\begin{displaymath}
a : S \to \mathbb{Z}
\end{displaymath}
such that only finitely many of the values $a_s \in \mathbb{Z}$ are non-zero.

Identifying an element $s \in S$ with the function $S \to \mathbb{Z}$ which sends $s$ to $1 \in \mathbb{Z}$ and all other elements to 0, this is written as

\begin{displaymath}
a = \sum_{s \in S} a_s \cdot s
  \,.
\end{displaymath}
In this expression one calls $a_s \in \mathbb{Z}$ the [[coefficient]] of $s$ in the formal linear combination.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
Definition \ref{FormalLinearCombination} of formal linear combinations makes sense with [[coefficients]] in any [[abelian group]] $A$, not necessarily the integers.

\begin{displaymath}
A[S] \coloneqq \mathbb{Z}[S] \otimes A
  \,.
\end{displaymath}
\end{remark}
\begin{defn}
\label{GroupOfFormalLinearCombinations}\hypertarget{GroupOfFormalLinearCombinations}{}
For $S \in$ [[Set]], the \textbf{group of formal linear combinations} $\mathbb{Z}[S]$ is the [[group]] whose underlying [[set]] is that of formal linear combinations, def. \ref{FormalLinearCombination}, and whose group operation is the pointwise addition in $\mathbb{Z}$:

\begin{displaymath}
(\sum_{s \in S} a_s \cdot s)
  + 
  (\sum_{s \in S} b_s \cdot s)
  =
  \sum_{s \in S} (a_s + b_s) \cdot s
  \,.
\end{displaymath}
\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
The free abelian group on $S \in Set$ is, up to [[isomorphism]], the group of formal linear combinations, def. \ref{GroupOfFormalLinearCombinations}, on $S$.

\end{prop}
\begin{defn}
\label{}\hypertarget{}{}
For $S$ a set, the free abelian group $\mathbb{Z}[S]$ is the [[direct sum]] in [[Ab]] of ${|S|}$-copies of $\mathbb{Z}$ with itself:

\begin{displaymath}
\mathbb{Z}[S] \simeq \oplus_{s \in S} \mathbb{Z}
  \,.
\end{displaymath}
\end{defn}
\hypertarget{Subgroups}{}\subsubsection*{{Subgroups}}\label{Subgroups}

\begin{prop}
\label{SubgroupsOfFreeAbelianGroupsAreFree}\hypertarget{SubgroupsOfFreeAbelianGroupsAreFree}{}
Assuming the [[axiom of choice]], then every [[subgroup]] of a free abelian group (def. \ref{FreeAbelianGroup}) is itself a free abelian group.

\end{prop}
(e.g. \hyperlink{Lang02}{Lang 02, Appendix 2 \S{}2, page 880}) For a full \textbf{proof} see at \emph{[[principal ideal domain]]} \href{principal+ideal+domain#free}{this theorem}.

\begin{remrk}
\label{}\hypertarget{}{}
Prop. \ref{SubgroupsOfFreeAbelianGroupsAreFree} implies that (assuming AC) every abelian group admits a [[free resolution]] of length 2, hence with trivial [[syzygies]]. See \href{free+resolution#AbelianGroupHasFreeResolutionOfLength2}{there}.

\end{remrk}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item The free abelian group on the [[singular simplicial complex]] of a [[topological space]] $X$ consists of the [[singular chains]] on $X$.


\item For $R$ a ring and $S$ a set, the [[tensor product of abelian groups]] $\mathbb{Z}[S] \otimes R$ is the [[free module]] over $R$ on the [[basis]] $S$. If $R = k$ is a [[field]], then this is the [[vector space]] over $k$ with basis $S$.


\item For $R$ a ring, the [[tensor product of abelian groups]] $\mathbb{Z}[\mathbb{N}]\otimes R$ is the abelian group underlying the [[ring]] of [[polynomials]] over $R$.



\end{itemize}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[free group]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[Serge Lang]] \emph{Algebra}, Graduate Texts in Mathematics 211 (Revised third ed.), Springer 2002

\end{itemize}
[[!redirects free abelian group]] [[!redirects free abelian groups]]

[[!redirects formal linear combination]] [[!redirects formal linear combinations]]

[[!redirects formal sum]] [[!redirects formal sums]]



\end{document}