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fix(algebraic_geometry/projective_spectrum/scheme) : fix module doc s…
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…tring (#15633)

After renaming definitions/lemmas in the body of the `src/algebraic_geometry/projective_spectrum/scheme.lean`, the module doc string is left unchanged. This pr fix the doc string
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jjaassoonn committed Jul 25, 2022
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Showing 1 changed file with 8 additions and 9 deletions.
17 changes: 8 additions & 9 deletions src/algebraic_geometry/projective_spectrum/scheme.lean
Expand Up @@ -32,24 +32,23 @@ open sets in `Proj`, more specifically:
1. We prove that `Proj` can be covered by basic open sets at homogeneous element of positive degree.
2. We prove that for any `f : A`, `Proj.T | (pbo f)` is homeomorphic to `Spec.T A⁰_f`:
- forward direction :
- forward direction `to_Spec`:
for any `x : pbo f`, i.e. a relevant homogeneous prime ideal `x`, send it to
`x ∩ span {g / 1 | g ∈ A}` (see `Top_component.forward.carrier`). This ideal is prime, the proof
is in `Top_component.forward.to_fun`. The fact that this function is continuous is found in
`Top_component.forward`
- backward direction : TBC
`x ∩ span {g / 1 | g ∈ A}` (see `Proj_iso_Spec_Top_component.to_Spec.carrier`). This ideal is
prime, the proof is in `Proj_iso_Spec_Top_component.to_Spec.to_fun`. The fact that this function
is continuous is found in `Proj_iso_Spec_Top_component.to_Spec`
- backward direction `from_Spec`: TBC
## Main Definitions and Statements
* `degree_zero_part`: the degree zero part of the localized ring `Aₓ` where `x` is a homogeneous
element of degree `n` is the subring of elements of the form `a/f^m` where `a` has degree `mn`.
For a homogeneous element `f` of degree `n`
* `Top_component.forward`: `forward f` is the
* `Proj_iso_Spec_Top_component.to_Spec`: `forward f` is the
continuous map between `Proj.T| pbo f` and `Spec.T A⁰_f`
* `Top_component.forward.preimage_eq`: for any `a: A`, if `a/f^m` has degree zero, then the preimage
of `sbo a/f^m` under `forward f` is `pbo f ∩ pbo a`.
* `Proj_iso_Spec_Top_component.to_Spec.preimage_eq`: for any `a: A`, if `a/f^m` has degree zero,
then the preimage of `sbo a/f^m` under `to_Spec f` is `pbo f ∩ pbo a`.
* [Robin Hartshorne, *Algebraic Geometry*][Har77]: Chapter II.2 Proposition 2.5
-/
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