Let $R$ be a commutative unitary ring and $I_1,..., I_n$ ideals in $R$. For each $p\in\{0,...,n-1\}$ consider the direct sums $\bigoplus_{i_0<...<i_p} I_{i_0}\cap...\cap I_{i_p}$ and define an $R$-homomorphism

$$\bigoplus_{i_0<...<i_p}I_{i_0}\cap...\cap I_{i_p}\xrightarrow{d^p}\bigoplus_{i_0<...<i_{p-1}}I_{i_0}\cap...\cap I_{i_{p-1}}$$

as follows: let $\pi_{j_0...j_{p-1}}:\oplus_{i_0<...<i_{p-1}}I_{i_0}\cap...\cap I_{i_{p-1}}\rightarrow I_{j_0}\cap...\cap I_{j_{p-1}}$ the canonical projection. Given $x\in I_{i_0}\cap...\cap I_{i_p}$ then $\pi_{j_0...j_{p-1}}\circ d^p(x)=0$ if $\{j_0,...,j_{p-1}\}\not\subseteq\{i_0,...,i_p\}$ and $\pi_{j_0...j_{p-1}}\circ d^p(x)=(-1)^jx$ if $\{j_0,...,j_{p-1}\}=\{i_0,...,\hat{i_j},...,i_p\}$ where $\hat{i_j}$ indicates that $i_j$ is ommited.

For $n=2$ and $n=3$ it is easy to prove that the sequence $\{\bigoplus_{i_0<...<i_p} I_{i_0}\cap...\cap I_{i_p}, d^p\}$ is exact. But I cannot prove it for $n\geq4$ (neither know if it is indeed true, but for me it seems ok).

Actually I was trying to prove that the sequence $\{Hom_R(\bigoplus_{i_0<...<i_p} I_{i_0}\cap...\cap I_{i_p}, E), Hom_R(d^p,E)\}$ (which can be obtained by applying $Hom_R(\_,E)$ in the sequence above), where $E$ is an injective $R$-module, is exact, but I think that the first one is already exact.

I will really appreciate any hint, suggestion or even proof to the exactness of any sequence above.