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This is data I am thinking about after reading sections 1,2,3 of chapter 2 on schemes from Hartshorne's Algebraic Geometry.

Basically, I know very little and I'm very uncomfortable with schemes.

Let $X$ be a scheme.

We know that every point is in some open affine $U_i \cong \operatorname{Spec}(A_i)$. So, we can cover $X$ be open affines $U_i \cong \operatorname{Spec}(A_i)$.

Now, we can intersect any open subset of $X$ with the cover of open affines.

(1) Does this mean that any open subset of $X$ be can covered (abusing notation) by basic open subsets $D(f_{i_j}) \subset \operatorname{Spec}(A_i)$? Therefore, any point in $X$ is in some (abusing notation) $D(f_{i_j}) \cong \operatorname{Spec}(A_{i_{f_{i_j}}})$?

An exercise shows that any open subset is a scheme via the induced scheme structure.

(2) Does this mean that any cover of $X$ will give us a cover by open affines? For example, take any open subset $U$. Then $U$ is a scheme via the induced scheme structure. So, we can cover $U$ via open affines. Since $U$ is open, then these open affines are also open affines of $X$?

(3) If $p \in X$ is in some open affine $U \cong \operatorname{Spec}(A)$, can we also keep finding smaller and smaller open affines containing $p$? How do these smaller and smaller open affines relate to $U$ and $X$? How do the rings relate to each other?

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    $\begingroup$ (1): yes. (2): yes. Note that affine-ness for an open subset $U$ of the scheme $X$ doesn’t depend on $X$, apart from $X$ defining the structure sheaf on $U$. (3): theoretically, yes (if you allow for equalities). But beware, the topology on a scheme isn’t like a Euclidean topology – Zariski open subsets are relatively scarce. In important special cases (local rings, fields) it’s possible that $U$ is a minimal open subset containing $p$ (ie there is no smaller one). This being said, in many examples you still have enough cases to restrict the open subset further. $\endgroup$
    – Aphelli
    Oct 6, 2020 at 15:34
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    $\begingroup$ As an aside, I think Vakil does a much better job than Hartshorne at describing how to communicate between affine opens of a scheme. If you're still feeling a bit shaky on this, I heartily recommend reading section 5.3 of his book (p. 157). A key result not mentioned in Hartshorne is Nike's trick, which Vakil then uses to prove what he calls the Affine Communication Lemma. Both of these are incredibly useful for combining affine local data on several affine opens. $\endgroup$
    – Viktor Vaughn
    Oct 6, 2020 at 15:52
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    $\begingroup$ I've improved the formatting of your post by replacing $Spec$ ("Spec") with $\operatorname{Spec}$ ("\operatorname{Spec}"), which produces better spacing and is more readable. Please keep this in mind for the future. $\endgroup$
    – KReiser
    Oct 6, 2020 at 18:11
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    $\begingroup$ @Mindlack that looks like an answer to me - would you care to record it as such below? $\endgroup$
    – KReiser
    Oct 6, 2020 at 18:12
  • $\begingroup$ @RichardD.James I just skimmed some of that section. Thanks. So, (abusing notation) we can say that a basis for $\operatorname{Spec}(A)$ is $\{ \operatorname{Spec}(A_f) \}_{f \in A}$? $\endgroup$
    – user5826
    Oct 6, 2020 at 18:40

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(1): yes, exactly.

(2): yes. Note that affine-ness for an open subset $U$ of the scheme $X$ doesn’t depend on $X$ itself, only on $U$, apart from $X$ defining the structure sheaf on $U$.

(3): theoretically, yes (if you allow for equalities). The property is the following: if $U$ is any open subset of a scheme and $p \in U$, there exists an affine open subset $p \in W \subset U$.

But beware, the topology on a scheme isn’t like a Euclidean topology – Zariski open subsets are relatively scarce. In important special cases (local rings, fields) it’s possible that $U$ is a minimal open subset containing $p$, (that is, there is no smaller one). It’s also (that’s rather the opposite phenomenon) possible that any nonempty open subset $U$ contains $p$. This being said, in many examples, you still have enough cases to restrict the open subset further.

Note that it’s rarely useful to ask for a strict restriction – most early algebraic phenomena can be appropriately studied through Zariski open subsets.

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