The following is part of the Exercise 3, page 200 of the book: An Algebraic Introduction to Complex Projective Geometry by Peskine. It is easy to prove (i), but (ii) is hard for me. Could you give some help?
Let $R$ be a Noetherian normal domain.
(i) If $T$ be a torsion module we put $\mathrm{div}_W(T) = \sum_{\mathrm{ht}(P)= 1} \ell(T_P) [P]$. Show that if $0 \to T' \to T \to T'' \to 0$ is an exact sequence of finitely generated torsion module, then $\mathrm{div}_W(T) = \mathrm{div}_W(T') + \mathrm{div}_W(T'')$.
(ii) Assume there exists an exact sequence $$0 \to L \overset{f}{\to} L' \to T \to 0$$ where $L$ and $L'$ are free modules. Then $\mathrm{div}_W(T) = \mathrm{div}_W(R/\mathrm{det}(f))$.
For (ii), suffices to do this after localizing at a height one prime and so assume $R$ is a dvr. Now, the square matrix $f$ can be put in upper triangular form and then the answer should be clear.