cofree comodules and embedding

Question

For an $R$-coalgebra C, is it possible for every C-comodule M to be embeded into a C-comodule of the form $\underset{i \in I}{\bigoplus} C$?

Answer 1

Let $C=R[t]$ with the usual coalgebra structure $(\varepsilon(t)=1$, $\Delta(t)=t \otimes t$). Then a $C$-comodule is the same as an $\mathbb{N}$-graded $R$-module. In fact, given a coaction $\alpha : M \to M[t]$, let $M_n = \{m \in M : \alpha(m)=m \cdot t^n\}$ and show $M = \oplus_n M_n$. Thus, in this case your question is: Does every graded $R$-module $M$ embed into a graded $R$-module whose components are free $R$-modules? Of course not, since for example the components of $M$ may have torsion!

If $R$ is a field, chances are better. If $M$ is a $C$-comodule, with underlying $R$-module $|M|$, the coaction can be seen as a morphism of $C$-comodules $M \to |M| \otimes_R C$. The underlying $R$-module homomorphism is split (by the counit of $C$), hence this morphism is a monomorphism. If $R$ is a field, $|M|$ is a direct sum of copies of $R$, so that $M$ embeds into a direct sum of copies of $C$.