On localization rings

From: Jose Capco (nospam_at_nospam.org)
Date: 09/29/04 

Date: Wed, 29 Sep 2004 10:44:10 +0200

Dear NG,

I'm a bit confused on working with localization rings. I am trying to
understand some of the proofs on Schemes by Hartshorne, and whenever he
tries to prove injectiveness of some mapping between a ring and a
localized ring I get a bit confused.

Let for instance P be a prime ideal of a ring A (here Hartshorne always
considers the ring itself not to be a prime ideal, thus P is a strict
subset). Then A_P being the localized ring (I think this will be
A/(A\P), but correct me if I am wrong!) with respect to P. Then whenever
he gets a/f=b/g in A_P (a,b in A and f,g not in P) then there is an h
not in P such that h(ga-fb)=0 ... um, I might be a bit stupid or have
misunderstood localized rings.. but can't one just automatically argue
that ga-fb=0? ... where and how did that "h" come from?

Sincerely,
Jose Capco