Definition: finite type vs finitely generated

Question

The mathematical term "finite type" appears more and more in the modern articles nowadays. But it is still hard to be found in the standard textbooks. I learned the definition of it from Stacks Project http://stacks.math.columbia.edu/tag/00F2 , it is defining on the ring maps. What is the right point of view when we say "ring of finite type", "group of finite type", "module of finite type"? Would the definition in each case be exactly equivalent to the definition of "finitely generated"?

Answer 1

1. I think that "finite type" and "finitely generated" ring homomorphisms are really just synonyms. But "finitely generated" is also used for modules and in fact arbitrary algebraic structures (see below), so that one often prefers "finite type" in the setting of ring homomorphisms. In order to differentiate these notions even more, one says "finite" if the corresponding module is finitely generated. Similarly, for schemes, one can define finite morphisms and morphisms (locally) of finite type. See here for the relations between these two notions.

2. If $C$ is a variety in the sense of universal algebra, then an object $M \in C$ is called finitely generated if there are elements $a_1,\dotsc,a_n$ such that $M = \langle a_1,\dotsc,a_n \rangle$, where the right hand side is the smallest subobject of $M$ containing the $a_1,\dotsc,a_n$. This yields the usual notion when $C=\mathsf{Set},\, \mathsf{Grp},\,R\mathsf{-Mod},\, R\mathsf{-CAlg}$ etc.

3. Even more generally, an object $M$ of an arbitrary category $C$ is called finitely generated if for every directed diagram $\{N_i\}$ of objects whose transition maps are monomorphisms the canonical map $\varinjlim_i \hom(M,N_i) \to \hom(M,\varinjlim_i N_i)$ is bijective. This coincides with the definition above if $C$ is a variety (easy exercise).