Finite type morphisms and localizations I am trying to get some geometric intuition for finite type and locally finite type morphisms of schemes. I have found similar questions but was not able to understand the answers sufficiently well to feel I have gained any real intuition.
For starters, finite type seems to be a strong form of compactness (i.e of pulling back compacts to compacts).
In this MO answer it is said that "finite type" is about finite dimensional fibers while this answer to the same question says "locally finite type" is about "finite dimensionality of small neighborhoods of the source of the map". I don't understand these intuitions at all and would appreciate examples to sharpen the "finite dimensionality of small neighborhoods".
The answers to this MSE question both dwell on localizations and their topological structure, seemingly hinting that localizations provide geometric intuition for finite type morphisms. Unfortunately, I don't understand at all: what is this geometric intuition?
I would especially like to understand the latter example of localization and its relation to finite type since it does not feel as vague as "finite dimensionality of neighborhoods".
 A: In regards to your question about 'finite dimensionality of small neighbourhoods', recall that if $f:X\rightarrow Y$ is locally of finite type, then for any affine open $U = \text{spec} A \subseteq Y$, we can cover $f^{-1}(U)$ by open affines $\text{spec} B_i \subset f^{-1}(U)$ such that $B_i$ is a finitely generated $A-$algebra. 
So if you fix any $x\in f^{-1}(U)$ (i.e. the source of the map), then if you take some small affine neighbourhood $\text{spec} B_i$ around $x$, then the corresponding morphism $\text{spec} B_i \rightarrow \text{spec} A$ has finite dimensional fibres. This is not the same as saying $f$ has finite dimensional fibres, since you can imagine that your preimage has infinitely many disjoint copies of open affines with increasing dimension. 
You are right that there is finite type gives some extra compactness condition, in fact finite type = locally finite type + quasi-compact (Hartshorne exercise 3.3(a)). In this case the problem we had above doesn't happen, since we can cover the preimage by finitely many open affines. 
