In algebraic geometry, an irreducible scheme has a point called "the generic point." The justification for this terminology is that under reasonable finiteness hypotheses, a property that is true at the generic point is actually generically true (i.e. is true on a dense open subset).
For example, there is a result called "generic flatness" (EGA IV (2), Theorem 6.9.1). Suppose Y is locally noetherian and integral, with f:X→Y a morphism of finite type and F is a coherent OX-module. If F is flat over the generic point of Y (a condition which is always satisfied, since anything is flat over a point), then there is a dense open subscheme U⊆Y such that F is flat over U.
I'm sure that there are lots of instances of this "yoga of generic points", but whenever I try to come up with one, it's kind of lame (in my example above, the condition at the generic point is vacuous). What are the main examples of the yoga of generic points?
Best Answer
Greg's example from Hartshorne is actually a special case of a more general situation. Under any dominant map of affine varieties, the inverse image of the generic point is the scheme associated to a finitely generated domain over the function field of the target. Hence by Noether's normalization, this inverse image scheme is a finite cover of an affine space over that function field. It follows that over an open set U of the target, the map factors as a finite cover of the projection of U x k^n --> U, where n is the transcendence degree of the field extension defined by the original map. In Hartshorne's exercise of course n = 0. This is the argument for the structure of a dominant morphism in Mumford's red book, I.8, proof of theorem 3.
In my experience the word yoga has been explained to mean "yoke" or "union", from the Sanscrit, and refers to any practice meant to help achieve oneness, or perhaps understanding, as the unknown touched upon above. But my impression is that practicing yoga is more spiritual than intellectual.