# A curious construction of a chain complex and its homology

... curious to me, that is.

Suppose two module filtrations $$\cdots < A_3 < A_2 < A_1 < \cdots$$ and $$\cdots < B_3 < B_2 < B_1 < \cdots$$ are comparable in the sense that for all $j$, $B_{j+1} < A_{j} < B_{j-1}$; then there are natural complexes $$\cdots \to \frac{A_3}{B_4} \to \frac{A_2}{B_3} \to \frac{A_1}{B_2} \to \cdots$$ and $$\cdots \to \frac{B_3}{A_4} \to \frac{B_2}{A_3} \to \frac{B_1}{A_2} \to \cdots$$ both of which have homology groups $$\frac{A_i\cap B_i}{A_{i+1} + B_{i+1}} .$$

My question is in two parts:

1. this canonical isomorphism $H(A^+/B)\simeq H(B^+/A)$, has it got a name?

2. is it useful?

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$\cdots\to \frac{B_2}{B_4} \to \frac{B_1}{B_3}\to\frac{B_0}{B_2}\to\cdots$
$0\to \frac{A_2}{B_3}\to\frac{B_1}{B_3}\to\frac{B_1}{A_2}\to 0.$