This question was first asked on MathSE but nobody answered.

In his proof of Lemma A2.5 in his book *Commutative Algebra with a View towards Algebraic Geometry*, Prof. Eisenbud writes something like this:

Let R be a commutative ring, $M$ an $R$-module, $S(M)$ the symmetric algebra (the quotient of the tensor algebra by the ideal of commuting relations) on $M$ considered as a coalgebra with $\Delta$ being the comultiplication, defined here as the algebra homomorphism induced by the diagonal map from $M$ to $M\oplus M$.

To prove that for all $x\in M$, there exists an element $y\in S(M)^{\otimes d}$ such that $$\Delta^d x\ =\ \Sigma_{\sigma \in G}\ \sigma(y)$$ where $G$ is the symmetric group on $d$ letters acting on $S(M)^{\otimes d}$ by the action coming from its action by permutation of factors on the $R$-module $M\oplus \cdots\oplus M$ ($d$ times), one can do it on elements that are product of elements in $M$, then use induction on the number of factors of the product. For the first step of the induction, he writes it is quite easy to see that if $x\in M$ and $y=x\otimes 1\otimes 1\cdots \otimes 1$, the symmetrization of $y$ is $\Delta^d x$.

I cannot easily see that since for me $(d-1)!\Delta^d x = \Sigma_{\sigma \in G}\sigma(y)$. Am I misunderstanding something obvious ? And if not how to do this first step ?

**Edit**: This lemma is used in the proof that the dual of the symmetric algebra has a system of divided powers the following way: to show that for $u\in S_n(M)^*$, $u^d$ is divisible by $d!$, one look at the action of $u^d$ on an element $x\in S(M)$, which is exactly the action of $u^{\otimes d}$ on $\Delta^d(x)\in S(M)^{\otimes d}$. But $u^{\otimes d}$ acts nontrivially only on tensor products of $d$ elements **all** of degree $n$ (which implies that the degree of $x$ is $nd$).

For this, there is a weak version of the lemma that works fine and can be proven by induction on $n$ which is the following: if $x\in S_{nd}(M)$, then $\Delta^dx$ is a sum of a symmetric terms like $\Sigma_{\sigma \in G}\ \sigma(y)$ where $y\in S_n(M)^{\otimes d}$ and other terms not in $S_n(M)^{\otimes d}$.

To conclude my too long question, I believe the lemma "Symmetry of Diagonalization" in not true on a general commutative ring, but the weak version is sufficient for the use of the Lemma in the book of Prof. Eisenbud. Can anybody give me an opinion here ?

**Edit 2**: When Prof. Eisenbud writes $\Delta^d$ it is actually $\Delta^{d-1}$. I sticked to his convention.