Proposition 18.24.1424A6AE6F169

Proof. Pullback sends every base competitor to an upstairs competitor: it preserves positivity, the obstacle inequality, the ordinary order and, by the Lelong formula and the curve valuative criterion, the all-weight ideal order. Conversely, if $h$ is an upstairs competitor, its fiber maximum $m_ h$ is a base competitor by 5.43.15, since $h\leq F\circ r$ implies $m_ h\leq F$. Also $h\leq m_ h\circ r$. Thus the two unregularized suprema are equal after pullback. A finite nonconstant holomorphic map of curves is open; upper regularization commutes with its pullback, as is seen in the coordinates $t=z^ e$. This proves both identities. Take $F=0$. An ordinary model is fixed by its ordinary envelope, so its pullback is a model; surjectivity of $r$ gives injectivity. Applying the ideal identity to a model proves preservation of ideal projection. $\square$