... 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:

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

is it

*useful*?