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 ≅ Spec ⁡ ( A i ). So, we can cover X be open affines U i ≅ 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 ) ⊂ Spec ⁡ ( A i )? Therefore, any point in X is in some (abusing notation) D ( f i j ) ≅ 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 ∈ X is in some open affine U ≅ 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&amp;

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This is data I am thinking about after reading sections 1,2,3 of chapter 2 on schemes from Hartshorne&#039;s Algebraic Geometry.Basically, I know very little and I&#039;m very uncomfortable with schemes.Let X be a scheme.We know that every point is in some open affine       U    i    ≅  Spec  ⁡  (      A    i    ). So, we can cover X be open affines       U    i    ≅  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              )  ⊂  Spec  ⁡  (      A    i    )? Therefore, any point in X is in some (abusing notation)   D  (      f                  i        j              )  ≅  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  ∈  X is in some open affine   U  ≅  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&amp;amp;

Jaylee Dodson · Accepted Answer

(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  ∈  U, there exists an affine open subset   p  ∈  W  ⊂  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.

This is data I am thinking about after reading sections 1,2,3 of chapter 2 on schemes from Hartshorn

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