Re 2: I think there is a certain amount of "inbetweener" material in algebra that is too advanced for, say, a college linear algebra class, but is already taken for granted in graduate algebra. After getting my undergraduate degree at Harvard, I'm pretty sure I didn't know what a tensor product was and I definitely didn't know what an exact sequence was. When I started grad school at the same place a year later, I felt a bit of a hillbilly for not knowing these definitions.
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