# fundamental theorem of coalgebras

Let $(C,\Delta,\varepsilon)$ be a coalgebra over a field $k$ and $x\in C$. Then there exists subcoalgebra $D\subseteq C$ such that $x\in D$ and $\mathrm{dim}_{k}\,D<\infty$.

Proof. Let

 $\Delta(x)=\sum_{i}b_{i}\otimes c_{i}.$

Consider the element

 $\Delta_{2}(x)=\sum_{i}\Delta(b_{i})\otimes c_{i}=\sum_{i,j}a_{j}\otimes b_{ij}% \otimes c_{i}.$

Note that we may assume that $(a_{j})$ are linearly independent  and so are $(c_{i})$. Let $D$ be a subspace  spanned by $(b_{ij})$. Of course $\mathrm{dim}_{k}\,D<\infty$. Furthermore $x\in D$, because

 $x=\sum_{i,j}\varepsilon(a_{j})\varepsilon(c_{i})b_{ij}.$

We will show that $D$ is a subcoalgebra, i.e. $\Delta(D)\subseteq D\otimes D$. Indeed, note that

 $\sum_{i,j}\Delta(a_{j})\otimes b_{ij}\otimes c_{i}=\sum_{i,j}a_{j}\otimes% \Delta(b_{ij})\otimes c_{i}$

and since $c_{i}$ are linearly independent we obtain that

 $\sum_{j}\Delta(a_{j})\otimes b_{ij}=\sum_{j}a_{j}\otimes\Delta(b_{ij})$

for all $i$. Thus

 $\sum_{j}a_{j}\otimes\Delta(b_{ij})\in C\otimes C\otimes D$

and since $a_{j}$ are linearly independent, we obtain that $\Delta(b_{ij})\in C\otimes D$ for all $i,j$. Analogously we show that $\Delta(b_{ij})\in D\otimes C$, thus

 $\Delta(b_{ij})\in C\otimes D\cap D\otimes C=D\otimes D,$

(please, see this entry (http://planetmath.org/TensorProductOfSubspacesOfVectorSpaces) for last equality) which completes   the proof. $\square$

Remark. The category of finite dimensional coalgebras is dual to the category of finite dimensional algebras   (via dual space  functor), so one could think that generally they are similar  . Unfortunetly Fundamental Theorem of Coalgebras is major diffrence between algebras and coalgebras. For example consider a field $k$ and its polynomial algebra $k[X]$. Then whenever $f\in k[X]$ is such that $\mathrm{deg}\,(f)>0$, then a subalgebra generated by $f$ is always infinite dimensional (if $\mathrm{deg}\,(f)=0$ then subalgebra generated by $f$ is $k$). This can never occur in coalgebras.

Title fundamental theorem of coalgebras FundamentalTheoremOfCoalgebras 2013-03-22 18:49:22 2013-03-22 18:49:22 joking (16130) joking (16130) 6 joking (16130) Theorem msc 16W30