I'm currently reading through Hartshorne, and have come across a few things that have left me wondering.
(i) Somewhat pedantic, but also because I don't actually know the answer, (in Example 2.3.4) he looks at the affine plane over an algebraically closed field, defined as the scheme of $k[x,y]$, and discusses some of its properties. He says
Also, for each irreducible polynomial $f(x,y)$, there is a point $\eta$ whose closure consists of $\eta$ together with all closed points $(a,b)$ for which $f(a,b) = 0$. We say that $\eta$ is a generic point of the curve $f(x,y) = 0$.
Now, is $\eta$ unique? The way I read the first sentence, I feel there should be only one such point, but then he says $\eta$ is a generic point instead of the generic point, which really made me feel it was not unique when I read it.
(ii) The two definitions before Example 3.2.1 define certain morphisms to be locally of finite type, finite type or simply finite, but the very next example says a scheme (with no morphism between schemes in the example) is of finite type. What does this mean? Perhaps that the scheme itself admits a covering via affine subsets that are the Spec of rings?
(iii) Consider $R = \mathbb{C}[x]/(x^2)$. Then Spec $R$ has only one element, namely $(x)$. Hence Spec $R$ is irreducible. In a somewhat entertaining way however, Spec $R$ is not reduced. This follows quite easily from Example 3.0.1 in the book (in particular "$X$ is reduced if and only if nil $A = 0$", where nil $A$ is the nilradical of $A$, which in our case is $(x)$) but I'm fairly new to localization, so what hoping someone could make sure I'm not making a silly mistake in trying to show this directly. Is the localization of $R$ at $(x)$ simple $R$ again? (And hence the sheaf $\mathcal{O}((x)) = R$, which certainly has non-zero nilpotents.)
I should add, for the last one, I'm sure it was not Hartshorne that decided to use irreducible and reduced for distinct things, but I'm curious if anyone knows why the terminology came about this way? Perhaps a raw translation from french, where the words are slightly more distinct?
Thanks!