Let $k$ be a field of characteristic $p \neq 2$, and $V = \oplus V_{n}$ be a graded vector space over $k$.
Then, can one compute the graded (counital) cofree cocommutative coalgebra $C(V)$ cogenerated by $V$?
The cases in which I am interested are:
(i) the case $V$ is concentrated in degree $1$.
I think $C(V)$ is just the exterior coalgebra $\Lambda^{\bullet}(V_{1})$, but I do not know how to prove it.
(ii) the case $V$ is concentrated in degree $2$.
Is it $\Gamma ^{\bullet /2}(V_{2})$? (where $\Gamma ^{\bullet}$ denotes the divided power Hopf $k$-algebra generated by $V_{2}$).
Please give me any advice or reference.
