On affine local properties In learning the third section First Properties of Schemes of the second chapter of Hartshorne, I heard someone mentioned Nike's lemma (I don't know if I spell the name right): if a scheme $X$ has a property $P$ affine locally (i.e. there is an affine open cover $ \{ \operatorname{Spec}A_i \}$ of $X$, such that each $\operatorname{Spec}A_i$ has property $P$), then for every affine open cover $\{\operatorname{Spec}B_j\}$, each $\operatorname{Spec}B_j$ has property $P$. Is the statement correct? Is it right? Will someone be kind enough to give me some referrence on this lemma or say something on this (the proof, the applications, etc)? Thank you very much!
 A: I believe that Nike's Lemma is Prop. 6.3.1 of Ravi Vakil's notes, namely the statement that the intersection of two affine opens can be covered by open sets that are simultaneously distinguished in both of the original affine opens --- see e.g. this post on Ravi's blog.  The actual statement that  you give in your post is not this result, but rather is an application of this result --- it is the affine communication lemma mentioned by Akhil Mathew in his comment above, which is Lemma 6.3.2 of Ravi's notes.  This lemma has many applications, which you can find by looking through Ravi's notes.

Regarding the origin of the name:
Nike is Vinayak "Nike" Vatsal, and the name the Nike trick for what is Ravi's Lemma 6.3.1 was coined by some of his fellow grad students while he was a grad student at Princeton (I think after he pointed it out during their joint study group on Hartshorne's book).  It was used (for example) under this name in Brian Conrad's algebraic geometry course at Harvard in the late '90s, and thus gained some currency among the current generation of American algebraic geometers (including Ravi Vakil).  
