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Suppose we have the following situation: $I = \oplus_{i=1}^{p-1}I_1$ is a decomposition of an $R$-ideal in invertible modules of rank $1$, $I_iI_j \subset I_{i+j \pmod{p-1}}$ and $I_1^i = I_{i \pmod{p-1}}$.

Then it is clear that $\text{Sym}_RI_1$ surjects onto $R$, but why is the kernel of the natural map $\text{Sym}_RI_1 \to R$ generated by $(a-1)\otimes I_1^{\otimes p}$, where $a \in I_1^{\otimes 1-p} = \text{Hom}(I_1^{\otimes p}, I_1)$ is the multiplication map $I_1^{\otimes p} \to I_1$?

It is somewhat clear, but I need a stringent argument.

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