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In chapter 2 of this note, the author defined variety as separated scheme of finite type over Spec$(k)$ for any field. But in Hartshorn there is still an integral.

I presume that he wants the variety to be able to be non-irreducible, i.e., the alegraic set. But integral is equivalent to irreducible and reduced. Does it hurt anything if we removed the reducedness?

Also, what does separated refer to? If we say separated scheme, then it is separated over Spec$(\mathbb{Z})$. However, i have seen another definition saying that it is separted over Spec$(k)$. Are they equivalent?

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Depends on what you want to do. If you remove the reducedness you will in general have nilpotency in your structure sheaf. The notion of a variety varies a lot in the literature depending on what the author wants to do.

A morphism $f \colon X \rightarrow Y$ between two schemes is separated if the diagonal $\Delta_X \subset X \times_Y X$ is closed. If you take $Y = \text{Spec}(\mathbb{Z})$, then the fibre product will be the regular product and you will get your definition of a separated scheme back. Now you can also take $Y =\text{Spec}(k)$ and think about that situation.

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