Certain results have a variant for generic points, and a variant which works over a dense open. As an example let’s discuss “generically finite morphisms” of schemes.<\/p>\n
The first variant is Lemma Tag 02NW<\/a>: If f : X —> Y is of finite type and quasi-separated, \u03b7 is a generic point of an irreducible component of Y with f^{-1}(\u03b7) finite, then there exists an affine open V of Y containing \u03b7 such that f^{-1}(V) —> V is finite.<\/p>\n The second variant is Lemma Tag 03I1<\/a>: If f : X —> Y is a quasi-finite morphism, then there exists a dense open V of Y such that f^{-1}(V) —> V is finite.<\/p>\n Comments: (a) In the second variant it isn’t necessarily the case that every generic point of every irreducible component of Y is contained in the open V, although this would follow from the first variant if we assumed f quasi-separated. (b) The proof of the first variant in the stacks project is basically elementary; the proof of the second variant currently uses (a technical version of) Zariski’s main theorem.<\/p>\n The point I am trying to make (badly) is that you can often get around making any separation assumptions by trying to prove a variant “over a dense open”. Maybe the archetype is the following result (Lemma Tag 03J1<\/a>): Every quasi-compact scheme has a dense open subscheme which is separated.<\/p>\n","protected":false},"excerpt":{"rendered":" Certain results have a variant for generic points, and a variant which works over a dense open. As an example let’s discuss “generically finite morphisms” of schemes. The first variant is Lemma Tag 02NW: If f : X —> Y … Continue reading