The exercise 5.10 of Atiyah's Commutative Algebra gives the definition of going-up map:
A ring homomorphism $f:A\rightarrow B$ is said to have the going-up (resp. the going-down property) if the conclusion of going-up theorem (resp. the going-down theorem) holds for $B$ and its subring $f(A)$.
I think this definition is unnatural and the definition should be
....... if the conclusion of going-up theorem (resp. the going-down theorem) holds for $B$ and $A$.
It seems these two definitions are different. Why the book chooses the "unnatural" definition? Also one part of exercise is
i) $f$ has the going-down property
ii) For any prime ideal $q$ of $B$, if $p=q^{c}$, then $f^{*}: \mathrm{Spec}(B_{q})\to\mathrm{Spec}(A_{p})$ is surjective
Prove that i) $\Leftrightarrow$ ii).
It is trivial if using the definition which I think. But I do not know how to prove if using the book's definition. I am very confused now.