u/cjustinc

I'm posting this in response to a question that was posted here about an hour ago and deleted before I could answer it. Hopefully the OP will see this, but if not, maybe it will save others the same confusion in the future.

The question was about Danilov's book Algebraic Varieties and Schemes. In it, the structure sheaf on an affine scheme Spec R is defined by assigning to a Zariski open U the localization of R at the set of elements which don't vanish on U. Why, went the question, don't other authors define it this way? It seems simpler than taking the inverse limit over principal opens or whatever.

The reason is that this definition is incorrect! See this MSE question for some counterexamples:

https://math.stackexchange.com/questions/81858/what-is-an-example-of-mathscr-o-spec-ru-neq-s-1r-for-some-s-consisti