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What are stalks of (structure sheaf of) affine scheme Spec $\mathbb{Z}/(60)=\{(2),(3),(5)\}$? What are its global sections?

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    $\begingroup$ Schemes don't have stalks; sheaves have. $\endgroup$ – Georges Elencwajg Nov 16 '12 at 8:53
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    $\begingroup$ ?? I mean the structure sheaf of the affine scheme. $\endgroup$ – Tom Nov 16 '12 at 9:08
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    $\begingroup$ and what have you tried? $\endgroup$ – user27126 Nov 16 '12 at 9:34
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Hint: By the Chinese Remainder Theorem, we have an isomorphism $\mathbb{Z}/60 \cong \mathbb{Z}/4 \times \mathbb{Z}/3 \times \mathbb{Z}/5$. The prime ideals are generated by $(2,1,1),(1,0,1)$ and $(1,1,0)$, so localization is easy.

I leave to you how to localize the leftmost factor (hint: since the prime ideal annihilates some the ring, the localization map should not be injective).

To find the stalk at $(2)$, that is, the local ring $(\mathbb{Z}/4 \times \mathbb{Z}/3 \times \mathbb{Z}/5)_{(2,1,1)}$, we are allowed to divide by anything of the form $(1,a,b)$ and $(3,a,b)$. This annihilates the second and third factor and leaves the first factor unchanged. So we are left with $\mathbb{Z}/4$. (the two last factors perish because we are allowed to divide by zero inside them)

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  • $\begingroup$ Thank you. In Vakil's FOAG notes (5.4.11,p131,line18), they are $Z/4,Z/3,Z/5$. It is wrong, isn't it? $\endgroup$ – Tom Nov 16 '12 at 13:30
  • $\begingroup$ Second question(though it is minor): The prime ideals should be generated by $(2,1,1), (1,0,1),(1,1,0)$, shouldn't they? $\endgroup$ – Tom Nov 16 '12 at 14:01
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    $\begingroup$ @Tom: Thanks, I was of course mistaken. I've edited my post. No, Vakil's right, see my edited answer. $\endgroup$ – Fredrik Meyer Nov 16 '12 at 14:20
  • $\begingroup$ Thank you, I have understood! And the global sections are $\mathbb{Z}/(60)$, aren't they? $\endgroup$ – Tom Nov 17 '12 at 12:08
  • $\begingroup$ @Tom: Yes, the global sections are $\mathbb{Z}/60$. Think of them as the product of the stalks. I've edited my answer again: it's clearer and more correct. $\endgroup$ – Fredrik Meyer Nov 17 '12 at 13:46

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