Let $R$ be a Commutative Noetherian ring. For a chain complex $A_*$ of finitely generated $R$-modules which is homologically finite (i.e. only finitely many homologies $H_i(A_*)$ are non-zero) and each Homology has finite length as an $R$-module, define $\chi(A_*):=\sum_i (-1)^i l(H_i(A_*))$ , where $l(-)$ denotes length of a module.
So now let $0\to A_*\to B_*\to C_*\to 0$ be a short exact sequence of homologically finite chain complexes of finitely generated $R$-modules such that each of the complexes have finite length homologies.
Then, is it true that $\chi(B_*)=\chi(A_*)+\chi(C_*)$ ?
I think I have to use the induced Long exact sequence of the homologies of the complexes and use that length is alternating on a Long exact sequence, but I'm having trouble writing down a rigorous proof.
Please help