Sorry, but can you give some simple examples?

I mean that maps $H_n(f)=H_n(g)$ for all $n$. The term "homological" is used in the book Y.A. Drozd, V.V Kirichenko "Finite-Dimensional Algebras".

Sorry, but if You have time, can You write the proof above without using the definition of (general?) linear bundle, but only the definition of divisors? I'm beginner in algebraic geometry, so I don't understand formally every step in the proof yet. My main book is Shafarevich's "Basic algebraic geometry", the book of Hartshorne is too comprehensive to me at this moment. Also all links to the used propositions, lemmas etc. will be very useful. Thanks.