Scheme theoretic 'class inclusions'

Question

For lack of a better name*, I call the following two things class inclusions: $$1) \quad\textbf{Magma}\supset \textbf{Semigroup}\supset\textbf{Monoid}\supset \textbf{Group}\supset \textbf{Abelian Group}$$

$$2)\quad \textbf{Commutative Rings}\supset \textbf{IDs}\supset \textbf{GCD domains}\supset \textbf{UFDs}\supset\textbf{PIDs}\supset \textbf{EDs}\supset\textbf{Fields}$$

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Looking in the index of Hartshorne - Algebraic geometry, we see the following words before Scheme: 'Affine', 'Connected', 'Formal', 'geometrically integral', 'geometrically irreducible', 'geometrically reduced', 'integral', 'irreducible', 'locally factorial', 'locally noetherian', 'noetherian', 'nonprojective', 'nonseparated', 'nonsingular in codimension one', 'normal', 'of finite type over a field', 'reduced', 'regular' and 'separate.

Is there some notion of a class inclusion diagram here? Where can I find one if so?

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*And the latter is called this on Wikipedia.

Answer 1

Konrad Voelkel licensed his nice diagram under the same CC BY-SA 3.0 license as this site, so we can luckily embed it here:

I would also like to add that Bjorn Poonen has nice tables of permanence properties for various types of scheme morphisms in the appendix to Rational Points on Varieties, although that isn't exactly what you are looking for.

Edit. I also realize that Appendices C and D in Görtz–Wedhorn are also very useful.

Answer 2

There are a lot of inclusions. Here are some of them (the list is not complete) :

$\bullet$ Geometrically integral $\subset$ geometrically irreducible $\subset$ irreducible

$\bullet$ Geometrically integral $\subset$ geometrically reduced $\subset$ reduced

$\bullet$ Geometrically integral $\subset$ integral $\subset$ irreducible $\subset$ connected

$\bullet$ Regular $\subset$ locally factorial $\subset$ normal (provided that you call a scheme "normal" if all its local rings are integrally closed, but do not require the scheme to be integral) $\subset$ reduced

$\bullet$ Projective over $k$ $\subset$ of finite type over $k$ $\subset$ Noetherian $\subset$ locally Noetherian

$\bullet$ Affine $\subset$ separated $\subset$ quasi-separated

All those inclusions can be found in classical books about schemes (Hartshorne, EGA, Liu, Görtz-Wedhorn, etc), but you probably have to look for them one by one.