# Exactness of a certain sequence

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 $$\bigoplus_{i_0<... and define an $$R$$-homomorphism

$$\bigoplus_{i_0<...

as follows: let $$\pi_{j_0...j_{p-1}}:\oplus_{i_0<... the canonical projection. Given $$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 $$\{\bigoplus_{i_0<... is exact. But I cannot prove it for $$n\geq4$$ (neither know if it is indeed true, but for me it seems ok).

Actually I was trying to prove that the sequence $$\{Hom_R(\bigoplus_{i_0<... (which can be obtained by applying $$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.