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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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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