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In Vakil's notes on the foundations of algebraic geometry, he defines the structure sheaf $ \mathscr{O}_{\text{Spec}A}(D(f))$ for distinguished open sets $D(f)$ by $\mathscr{O}_{\text{Spec}A}(D(f)) := A_f$. He then defines the restriction map for $D(g) \subseteq D(f)$ by

$$\rho^{D(f)}_{D(g)} : \mathscr{O}_{\text{Spec}A}(D(f)) \rightarrow \mathscr{O}_{\text{Spec}A}(D(g))$$

"in the obvious way". I assume this means

$$\frac{a}{f^{n}} \mapsto \frac{b^{m}a}{g^{mn}}?$$

He then goes on and proves that this is indeed a sheaf of rings on the distinguished open sets. Now, he claims that this then indeed gives a sheaf of rings on the topological space Spec $A$. I know that every open set $U$ of Spec $A$ may be written as the union of distinguished open sets such that

$$U = \bigcup_{i \in I} D(f_{i}).$$

But how do I get the sheaf $\mathscr{O}_{\text{Spec}A}(U)$? I.e. is this the localization at the sum $\sum_{i \in I} f_i$ or the product of the $A_{f_{i}}$?

KReiser
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    Neither. It will be the ring of matching sections, which is a subring of the product. – Zhen Lin Oct 29 '21 at 14:33
  • Is it not important in practice to have an explicit description of the sheaf on an arbitrary open set? I kept reading in Vakil's notes, but so far he never states it. –  Oct 30 '21 at 12:15

1 Answers1

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It is not important to know what the sections are on a generic open set, as you can always work with basic open sets, but you can find a description of the sections of $\text{Spec} A$ in Hartshorne's Algebraic geometry, page 70

Aitor Iribar Lopez
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  • In the case you do need to know the sections on an arbitrary open, another approach which is often more tractable than the citation from Hartshorne is the description of the sections of the sheaf as an equalizer: see here. This is also an exercise in the early parts of Vakil. – KReiser Nov 02 '21 at 20:42