The Stacks project

Definition 10.54.1. Let $R \to S$ be a ring map.

  1. We say that $R \to S$ is essentially of finite type if $S$ is the localization of an $R$-algebra of finite type.

  2. We say that $R \to S$ is essentially of finite presentation if $S$ is the localization of an $R$-algebra of finite presentation.

Comments (4)

Comment #2605 by Oliver on

Does this notion have an "official" scheme theoretic analogue? To me there are at least two choices. One would be to require the induced morphisms of stalks to be essentially of finite type as maps of rings, the other would be via open affine covers.

Comment #2606 by on

Do you think it would be useful to have such a notion?

Comment #6988 by on

In the definitions, it might be better to change the article: a localization of ..., since the multiplicative subset is unspecific (this article is used, say, in Weibel's book for the definition of essentially of finite type).

Comment #7221 by on

Well, I can argue that to say "the localization of an -algebra of finite type" means "the localization of an -algebra of finite type at some multiplicative subset".

There are also:

  • 5 comment(s) on Section 10.54: Homomorphisms essentially of finite type

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