Let $ f: X to S $ a finite morphism between affine schemas $ X = Spec (A), S = Spec (R) $. Note by $ phi: R to A $ the corresponding circular card.

I am looking for pure ** theoretical / algebraic ring** tools / criteria to define if $ f $ is a

**(in the topological sense). More concretely in the sense of what conditions the ring morphism $ phi $ and resp induced morphisms $ phi_p: R_p to A_p $ on the family of locations to $ p $ implies that $ f $ is open.**

*open the map*The bottom of my question is the following thread: The Finite and Locally Free map is open.

Here we have the situation $ f $ is a finite morphism and locally free and I want to infer that this already implies that $ f $ is open.

Obviously, the problem is local so we can work with the above setting and assume that X $, S $ affine and $ A = R ^ n $ as $ R $-module since $ phi $ is in the given context exactly the map $ R to R ^ n $.

The author observes that, due to local weakness, the stems of $ f _ * mathcal {O} _X $ are nonzero on an open subset of $ S $.

What does he mean? That at every point $ s in S $ there is a rod in $ ( phi _ * mathcal {O} _X) _s cong mathcal {O} ^ n_ {S, s} $ which can be extended to a section on an open subset $ U subset S $? Is it not regulated by definition stems as direct limit representatives?

Again, since the problem is local so wlog $ U = D (f) $ or $ f in R $. Why does this stem condition imply that $ f $ is open? Does this stem from a more general criterion of openness based on switched algebra methods?