# Split short exact sequences and the associated graded algebra

Let $G$ be a group and $K$ be a field. Let $K(G)$ be the group ring and $I$ be its augmentation ideal. You are given that there exists a section of the projection $\pi_s:I^s\to I^s/I^{s+1}$ for all $s$, i.e. a module homomorphism $\alpha_s:I^s/I^{s+1}\to I^s$ such that $\pi_s\circ \alpha_s$ is the identity on $I^s/I^{s+1}$.

Since we are working over a field $K$, the split short exact sequences $I^{s+1}\to I^s\to I^s/I^{s+1}$ imply that $K(G) \cong I^{n+1} \oplus \bigoplus_{s=0}^n I^s/I^{s+1}$. If $\bigcap_{s=0}^\infty I^s = 0$, does this imply that $K(G) \cong\bigoplus_{s=0}^\infty I^s/I^{s+1}$ as $K$-algbera? How do I prove this?

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You surely have to assume that $\operatorname{char} K\nmid G$. Otherwise your exact sequences will not split in general. –  Julian Kuelshammer Jun 24 '13 at 20:14