I've used fulton's "algebraic curves" (here is the link http://xahlee.info/math/i/algebraic_geometry_by_william_fulton_73219.pdf) to prepare my exam and i had just 2 issues with proofs, 2 details i didn't understand:
1)in the lemma a) at page 39 (when proving the property 5) of intesection numbers) it proves the existence of an integer N for which $I^{2N} ⊂ (F,G)O_P(A_2)$ but why does this imply that $A_{ij}$ belongs to $(F,G)O_P(A_2)$? I cannot figure out why BF' belongs to $(F,G)O_P(A_2)$.enter image description here
2) in the third step of the proof of the bezout theorem (page 58), when proving the independence of the $a_i$, I cannot figure out how it gets $$z^r Σλ_i A_i = z^s B^*F+z^t C^*G$$ Using the proposition 5 in chapter 2 as suggested by the book i should have every $A_i$ multiplied by different powers of z as i do not know whether the $A_i$ are all divided by the same powers of z or not. Are all the $A_i$ actually divided by the same powers of z and why? Or is there another reason why the $A_i$ are all multiplied by the same factor $z^r$ ?enter image description here
