To prepare for my oral exams in algebraic geometry (covering Hartshorne's book "Algebraic Geometry" Chapter II and III) I sketched an overview diagram of morphism properties in the category of noetherian schemes. Maybe this is a good cheat sheet to keep with you while reading the book for the first or second time (ok, and I dropped a "Nisnevich" for no good reason, you can ignore it).

You can get a PDF version of the image or click on it to get a readable version.

I'm still in the process of writing down examples and counter-examples to these properties, maybe that list will be online some day (another kind of "counterexamples in algebraic geometry").

As always, I'm happy to hear any comments (did I miss an important arrow, did I get anything wrong) -- but I should stress that the diagram works in Hartshorne-world, not in EGA-terms (this kind of confusion cost me almost one entire day trying to prove wrong statements..)

UPDATE (2011-11-18): improved diagram (more information, less colour) and higher quality PNG file.