# Reference for the fact that a coderivation of the (non reduced) tensor coalgebra is determined by its corestrictions

If $V$ is a vector space, let us consider the tensor coalgebra $TV=\bigoplus\limits_{k=0}^\infty V^{\otimes^k}$ with coproduct given by $$\Delta (x_1\otimes \dots \otimes x_n):= \sum\limits_{i=0}^{n}(x_1\otimes\dots\otimes x_i)\bigotimes (x_{i+1}\otimes\dots\otimes x_{n}) .$$

Its reduced version is $\overline{TV}=\bigoplus\limits_{k=1}^\infty V^{\otimes^k}$ with coproduct $\bar{\Delta}$ given by replacing $\sum\limits_{i=0}^{n}$ by $\sum\limits_{i=1}^{n-1}$ in the previous formula. In other words: $$\Delta (x)=\bar{\Delta}(x)+1\otimes x + x\otimes 1 .$$

It is well known that a coderivation of $\overline{TV}$, i.e. $Q\in L(\overline{TV},\overline{TV})$ satisfying $(Q\otimes Id + Id\otimes Q)\circ \Delta=\Delta \circ Q$, is determined by its corestrictions, i.e. by $\Pi_V\circ Q$, where $\Pi_V\colon \overline{TV}\rightarrow V$ is the canonical projection on $V$.

What I am looking for is a reference for a proof of the same statement for $TV$, the non reduced tensor coalgebra.

-
arxiv.org/abs/math/0310337 – Fernando Muro Dec 28 '11 at 21:30
Thanks for the reference, but I think it does not address my question since in his Lemme 1.1.2.2 he only deals about coderivations of the reduced tensor coalgebra. – yael fregier Dec 28 '11 at 23:07

## 1 Answer

This is Proposition 1.2.9. in Loday and Vallette, "Algebraic Operads".

-
It might be helpful to say that Loday-Valette require the derivation to satisfy $d\left(1\right)=0$, though. (But this requirement seems unsubstantial: The definition of a derivation yields that $d\left(1\right)$ is primitive, hence lies in the degree-$1$ component $V$ of $TV$, and thus it is "visible" after projection.) – darij grinberg Mar 1 '14 at 2:24
Actually, it is easy to reduce the $TV$ case to the $\overline{TV}$ case: First, the argument sketched above shows that $d\left(1\right)\in V$, and thus WLOG $d\left(1\right) = 0$; furthermore, it is an easy consequence of the definition of a coderivation that $\varepsilon\circ d=0$ (the dual result to the known fact that as every derivation sends $1$ to $0$), and thus $d$ can WLOG be regarded as a coderivation of $\overline{TV}$. – darij grinberg Mar 1 '14 at 2:28