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This is a statement made in Eisenbud Commutative Algebra, Chapter 2, Localization.

"Perhaps this is because interest was focused on finitely generated algebras on the one hand, and power series rings on the other, and neither of these classes of rings is closed under localization. Instead of passing to a localized ring, as we would now, people often used ideal quotients as a substitute."

My questions are in the following.

  1. Why localization fails to be closed for finitely generated algebra and power series? What is the example corresponding to this? I thought analytic functions agreeing on small neighborhood agree everywhere. So how come this will fail for analytic power series?

  2. Instead of passing to a localized ring, as we would now, people often used ideal quotients as a substitute. In the exercise, passing to ideal quotients implying passing to a localized ring. Why would one even consider passing to ideal quotients? I guess he means we do localization nowadays but not in the past.

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  1. Localization may take a finitely generated algebra to a non-finitely generated algebra. Let $k$ be a field and consider the example of $k[x]$, which is a finitely-generated $k$-algebra. When we localize $k[x]$ at the multiplicative system given by all non-zero-divisors, we get $k(x)$, which is not finitely generated as a $k$-algebra.

  2. I'm not sure why this was common, but given that it expresses set difference, it seems like a reasonable operation to use if you're not interested in much past set difference. I would guess that as algebraic geometry advanced in the mid 20th century, localization became more obviously the right choice. Localization is a more universal construction which can be used in many more contexts.