Let $f,g: K \to X$ two morphisms between preschemes. In order to "compare" these two morphisms in David Mumford's "Red Book of Varieties and Schemes" there is suggested (see page 118) for a $x \in K$ to say
$f(x)\equiv g(x)$ iff
(1) $f(x)= g(x)$ (as sets) and
(2) the induced maps of residue fields $f^*_x, g^*_x: \kappa(f(x)) \to \kappa(x) $
Here the excerpt:
My question is why do we need the condition (2)? The main obstruction is what happens with torsion in the induced stalk maps $f^*_x, g^*_x: \mathcal{O}_{X,f(x)} \to \mathcal{O}_{K,x}$.
But when we consider the induced maps $f^*_x, g^*_x: \kappa(f(x)) \to \kappa(x) $ of residue fields then this torsion is killed since by definition $\kappa(x)= \mathcal{O}_{K,x}/m_x$.
So these two maps $f^*_x, g^*_x$ are automatically the canonic inclusions. So they should coinside,right?
Or where is the error in my reasonings making the condition (2) neccessary?