**Definition.** Let $k$ be a commutative ring. Let $V$ be a $k$-module. We turn the symmetric algebra $\mathrm{S}\left(V\right)$ of $V$ into a graded Hopf algebra by defining the comultiplication

$$\Delta : \mathrm{S}(V) \to \mathrm{S}(V) \otimes \mathrm{S}(V)$$

by

\begin{align}
& \Delta(v_1v_2\ldots v_n) \\ = {} & \sum\limits_{i=0}^n \sum\limits_{\sigma\in\operatorname{Sh}(i,n-i)} \left(v_{\sigma(1)}v_{\sigma(2)}\ldots v_{\sigma(i)}\right) \otimes (v_{\sigma(i+1)}v_{\sigma(i+2)}\ldots v_{\sigma(n)}),
\end{align}

where $\operatorname{Sh}(i,n-i)$ denotes the set $\left\lbrace \sigma\in S_n \mid \sigma(1) < \sigma(2) < \cdots < \sigma\left(i\right) \text{ and } \sigma(i+1) < \sigma(i+2) < \cdots < \sigma\left(n\right) \right\rbrace$ of all $(i,n-i)$-shuffles. The counit of this Hopf algebra is simply the projection from $S(V)$ onto $k$.

**Definition.** Let $k$ be a commutative ring, and $C$ be a $k$-coalgebra. A *coderivation* of $C$ means a $k$-linear map $c:C\to C$ such that $\Delta \circ c = \left(c\otimes\mathrm{id} + \mathrm{id}\otimes c\right)\circ \Delta$.

*Remark.* Coderivations behave, in some sense, dually to derivations (which is not surprising since the condition $\Delta \circ c = \left(c\otimes\mathrm{id} + \mathrm{id}\otimes c\right)\circ \Delta$ is a kind of dual to the Leibniz identity, when the latter is written in pointfree notation): First of all, if $c:C\to C$ is a coderivation, then $c^{\ast} : C^{\ast}\to C^{\ast}$ is a derivation. The converse holds at least when $C$ is finite-dimensional. As an exercise in reversing arrows, the reader can prove that $\varepsilon\circ c=0$ for every coderivation $c$ of a coalgebra (in analogy to the equality $d\left(1\right)=0$ which holds for every derivation $d$ of an algebra).

**Theorem.** Let $k$ be a field of characteristic $0$. Let $V$ be a $k$-vector space. Then, the maps

$\mathrm{Hom}\left(S\left(V\right),V\right) \to \mathrm{Coder}\left(S\left(V\right)\right)$,

$X\mapsto \mu \circ \left(\mathrm{id}\otimes X\right) \circ \Delta$  (where $\mu$ is the multiplication morphism $S\left(V\right)\otimes S\left(V\right)\to S\left(V\right)$)

and

$\mathrm{Coder}\left(S\left(V\right)\right) \to \mathrm{Hom}\left(S\left(V\right),V\right)$,

$c\mapsto \pi_1\circ c$ (where $\pi_1$ is the projection from $\mathrm{S}\left(V\right)=\bigoplus\limits_{i\in\mathbb N}\mathrm{S}^i\left(V\right)$ onto the addend $\mathrm{S}^1\left(V\right)=V$)

are mutually inverse isomorphisms.

This is how I understand Chapter 5, Theorem 4.20 in Eckhard Meinrenken, [Clifford algebras and Lie groups (revision 12/2011)][1]. (Unfortunately, the statement of the theorem in Meinrenken's text is obscured by the fact that one direction of the isomorphism - the easy one - is not written out explicitly.) The proof given in this text uses the characteristic-$0$ assumption: first, by assuming "WLOG" that a generic element of $\mathrm{S}\left(V\right)$ has the form $e^v$ for some $v\in V$ (this is made formal by going over to formal power series, but stripped of this formality, this is exactly what is known as umbral calculus for over a century), and second, by using the fact that the primitives of $\mathrm{S}\left(V\right)$ all come from $V$.

**Question.** Does the above theorem hold in arbitrary characteristic?

I am sure this is intimately related to the question whether $\mathrm{S}\left(V\right)$ is the cofree graded cocommutative coalgebra over $V$ (or at least whether it is "cogenerated" in degree $1$, whatever this means precisely!). Unfortunately, the only case when I am sure of this is the characteristic-$0$ case, so this is of no help to me. [Loday-Valette][2] does not seem to care for positive characteristic too much, either, and it is difficult for me to find any other source.


  [1]: https://web.archive.org/web/20120303204145/http://www.math.toronto.edu:80/mein/teaching/clif_main.pdf
  [2]: https://www.math.univ-paris13.fr/~vallette/Operads.pdf