Error in Algebraic Curves by Fulton?

Question

The following lemma is from section 3.3 of Fulton's algebraic curves.

Notation: $\psi$ is a map from $k[X,Y]/I^n×k[X,Y]/I^m$ to $k[X,Y]/I^{m+n}$

I'm having some difficulty understanding the paragraph marked in yellow. Should it be $r<n$ or $s<m$ instead?

Otherwise, how does it follow that $r+m=s+n$ and $A_rF_m=-B_sG_n$? I realize that this must have something to do with the degree of $A_rF_m$ or $B_sG_n$ being less than $m+n$, but it seems to me as though this doesn't follow from the premise $r<m$ or $s<n$ (although replacing or with and seems to suffice ).

Answer 1

Yes, it should say "Suppose $r<n$ or $s<m$" just before your yellow markings. Assuming this, one of $A_r F_m$, $B_s G_n$, has degree less than $m+n$. Possibly there are other terms that are also nonzero (i.e. have degree less than $m+n$). Arguing by degrees of the terms, these two terms must have the same degree, the smallest degree term is $A_r F_m + B_s G_n$, and this must be $0$.