5 elaboration / references

I definitely agree with Peter's general intuitive description.

In response to some of the subsequent comments, here are some implications to keep in mind:

Finite ==> finite fibres (1971 EGA I 6.11.1) and projective (EGA II 6.1.11), hence proper (EGA II 5.5.3), but not conversely, contrary to popular belief ;)

Proper + locally finite presentation + finite fibres ==> finite (EGA IV (part 3) 8.11.1)

When reading about these, you'll need to know that "quasi-finite" means "finite type with finite fibres." Also be warned that in EGA , projectiveness is defined slightly more generaly than in Hartshorne (for exampleII.5.5.2) projective means $X$ is a closed subscheme of a "finite type projective bundle" $\mathbb{P}_Y(\mathcal{E})$, which gives a nice description via relative Proj, in EGA it turns out to be local on the target scheme).whereas "Hartshorne-projective" more restrictively means that $X$ is closed subscheme of "projective n-space" $\mathbb{P}^n_Y$.

When the target (or "base" scheme) is locally Noetherian, like pretty much anything that comes up in "geometry", a proper morphism is automatically of locally finite presentation, so in that case we do have

finite <==> proper + finite fibres

Regarding "locally finite type", its does not imply finite dimensionality of the fibres; rather, it's about finite dimensionality of small neighborhoods of the source of the map. For example, you can cover a scheme by some super-duper-uncountably-infinite disjoint union of copies of itself that is LFT but not FT, since it has gigantic fibres.

4 "1971"

I definitely agree with Peter's general intuitive description.

In response to some of the subsequent comments, here are some implications to keep in mind:

Finite ==> finite fibres (1971 EGA I 6.11.1) and projective (EGA II 6.1.11), hence proper (EGA II 5.5.3), but not conversely, contrary to popular belief ;)

Proper + locally finite presentation + finite fibres ==> finite (EGA IV (part 3) 8.11.1)

When reading about these, you'll need to know that "quasi-finite" means "finite type with finite fibres." Also be warned that in EGA, projectiveness is defined slightly more generaly than in Hartshorne (for example, in EGA it turns out to be local on the target scheme).

When the target (or "base" scheme) is locally Noetherian, like pretty much anything that comes up in "geometry", a proper morphism is automatically of locally finite presentation, so in that case we do have

finite <==> proper + finite fibres

Regarding "locally finite type", its does not imply finite dimensionality of the fibres; rather, it's about finite dimensionality of small neighborhoods of the source of the map. For example, you can cover a scheme by some super-duper-uncountably-infinite disjoint union of copies of itself that is LFT but not FT, since it has gigantic fibres.

3 clarified which scheme is loc. noeth.

I definitely agree with Peter's general intuitive description.

In response to some of the subsequent comments, here are some implications to keep in mind:

Finite ==> finite fibres (EGA I 6.11.1) and projective (EGA II 6.1.11), hence proper (EGA II 5.5.3), but not conversely, contrary to popular belief ;)

Proper + locally finite presentation + finite fibres ==> finite (EGA IV (part 3) 8.11.1)

When reading about these, you'll need to know that "quasi-finite" means "finite type with finite fibres." Also be warned that in EGA, projectiveness is defined slightly more generaly than in Hartshorne (for example, in EGA it turns out to be local on the target scheme).

For a

When the target (or "base" scheme) is locally Noetherianscheme, which like pretty much everything anything that comes up in "geometry" is, geometry", a proper morphisms are morphism is automatically of locally finite presentation, so in that case we do have

finite <==> proper + finite fibres

Regarding "locally finite type", its does not imply finite dimensionality of the fibres; rather, it's about finite dimensionality of small neighborhoods of the source of the map. For example, you can cover a scheme by some super-duper-uncountably-infinite disjoint union of copies of itself that is LFT but not FT, since it has gigantic fibres.

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