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I am (slowly) reading Eisenbud and Harris and trying to get my head around (affine) schemes.

Let $R$ be a ring, and $X=\text{spec}(R)$, as usual with the Zariski topology. We have a basis of open sets; for $f \in R$, $X_f=\{ p \subset R | f \notin p \}$ where $p$ is a prime ideal. Then the structure sheaf is $\mathcal{O}(X_f) = R_f$, the localization of the ring $R$ with respect to the multiplicative subset $\{ 1,f,f^2,\ldots \}$.

Exercise I-20 in E-H is to calculate the points and sheaf of functions for some schemes


1) $X_1 = \text{Spec } \mathbb{C}[x]/(x^2)$

This shouldn't be to hard - there is exactly one (closed) point corresponding to the maximal ideal $(x)$ in which case $X_1 = \{(x)\}$. Thus the only open sets are $\emptyset \subset X_1$ and then $\mathcal{O}(\emptyset) = 0$ and $\mathcal{O}(X_1) = \mathbb C[x]/(x^2)$

Is this correct?

2) $X_2 = \text{Spec } \mathbb{C}[x](x^2-x)$ Here we should have exactly two (closed) points: $(x),(x-1)$. Call these $\{ a,b \}$. The topology should then be $\{\emptyset,\{a \}, \{ b \}, \{a, b\} \}$ (the discrete topology). Again we have $\mathcal{O}(\emptyset) =0 $ and $\mathcal{O}(\{ a,b \}) = \mathbb C[x]/(x^2-x)$.

Now $ \begin{align} \mathcal{O}(\{ a \}) &= [\mathbb C[x]/(x^2-x)]_{(x)} \\ &\simeq [\mathbb C[x]/(x(x-1)]_{(x)} \end{align} $

Am I now localizaing with respect to the multiplicative set $R - \mathfrak{p}$ where $\mathfrak{p}=(x)$ and $R = \mathbb C[x]/(x^2-x)$? And then is this just: $ \begin{align} \mathcal{O}(\{ a \}) &\simeq [\mathbb C[x]/(x(x-1)]_{(x)} \\ &\simeq [\mathbb C[x]/(x-1)]_{(x)} \\ &\simeq \mathbb{C} \end{align} $ ?

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    @Keenan - thank you! (And I agree)2012-06-04

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Your arguement are correct. I am not sure what are your question? But what you write is correct. In fact, as you already know, as for the first example, the underlying topology is easy which is clearly irreducible, but note that the ring of global sections is not reduced; as for the second example, please note that the underlying toplogy is not irreducible, and the ring of global sections is isomorphic to $\mathbb{C}\times \mathbb{C}$, which is reduced.

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    You are welcome. I also learnt it by myself. So go ahead bravely!2012-06-04