[Merged by Bors] - feat(algebraic_geometry): Being quasi-compact is a local property. #15995
Conversation
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
Member
erdOne
commented
Aug 11, 2022
erdOne
commented
erdOne
added
awaiting-review
jcommelin
reviewed
Aug 11, 2022
Comment on lines
172
to
196
| lemma quasi_compact.affine_open_cover_tfae {X Y : Scheme.{u}} (f : X ⟶ Y) : | ||
| tfae [quasi_compact f, | ||
| ∃ (𝒰 : Scheme.open_cover.{u} Y) [∀ i, is_affine (𝒰.obj i)], | ||
| ∀ (i : 𝒰.J), compact_space (pullback f (𝒰.map i)).carrier, | ||
| ∀ (𝒰 : Scheme.open_cover.{u} Y) [∀ i, is_affine (𝒰.obj i)] (i : 𝒰.J), | ||
| compact_space (pullback f (𝒰.map i)).carrier, | ||
| ∀ {U : Scheme} (g : U ⟶ Y) [is_affine U] [is_open_immersion g], | ||
| compact_space (pullback f g).carrier, | ||
| ∃ {ι : Type u} (U : ι → opens Y.carrier) (hU : supr U = ⊤) (hU' : ∀ i, is_affine_open (U i)), | ||
| ∀ i, compact_space (f.1.base ⁻¹' (U i).1)] := | ||
| quasi_compact_eq_affine_property.symm ▸ | ||
| quasi_compact_affine_property_is_local.affine_open_cover_tfae f | ||
|
|
||
| lemma quasi_compact.open_cover_tfae {X Y : Scheme.{u}} (f : X ⟶ Y) : | ||
| tfae [quasi_compact f, | ||
| ∃ (𝒰 : Scheme.open_cover.{u} Y), ∀ (i : 𝒰.J), | ||
| quasi_compact (pullback.snd : (𝒰.pullback_cover f).obj i ⟶ 𝒰.obj i), | ||
| ∀ (𝒰 : Scheme.open_cover.{u} Y) (i : 𝒰.J), | ||
| quasi_compact (pullback.snd : (𝒰.pullback_cover f).obj i ⟶ 𝒰.obj i), | ||
| ∀ (U : opens Y.carrier), quasi_compact (f ∣_ U), | ||
| ∀ {U : Scheme} (g : U ⟶ Y) [is_open_immersion g], | ||
| quasi_compact (pullback.snd : pullback f g ⟶ _), | ||
| ∃ {ι : Type u} (U : ι → opens Y.carrier) (hU : supr U = ⊤), ∀ i, quasi_compact (f ∣_ (U i))] := | ||
| quasi_compact_eq_affine_property.symm ▸ | ||
| quasi_compact_affine_property_is_local.target_affine_locally_is_local.open_cover_tfae f |
There was a problem hiding this comment.
Should these two be merged? Probably already at the general level of local properties?
There was a problem hiding this comment.
I don't think it is necessary. The use case of these lemmas are usually to and from quasi_compact, and their proofs are separate as well.
Co-authored-by: Johan Commelin <johan@commelin.net>
leanprover-community-bot-assistant
added
ready-to-merge
bors bot
pushed a commit
that referenced
this pull request
Aug 15, 2022
…15995) Co-authored-by: Andrew Yang <36414270+erdOne@users.noreply.github.com>
|
Build failed (retrying...): |
bors bot
pushed a commit
that referenced
this pull request
Aug 15, 2022
…15995) Co-authored-by: Andrew Yang <36414270+erdOne@users.noreply.github.com>
|
bors merge p=2 |
bors bot
pushed a commit
that referenced
this pull request
Aug 16, 2022
…15995) Co-authored-by: Andrew Yang <36414270+erdOne@users.noreply.github.com>
|
Pull request successfully merged into master. Build succeeded: |
bors
bot
changed the title
Sign up for free
to join this conversation on GitHub.
Already have an account?
Sign in to comment
Add this suggestion to a batch that can be applied as a single commit.
This suggestion is invalid because no changes were made to the code.
Suggestions cannot be applied while the pull request is closed.
Suggestions cannot be applied while viewing a subset of changes.
Only one suggestion per line can be applied in a batch.
Add this suggestion to a batch that can be applied as a single commit.
Applying suggestions on deleted lines is not supported.
You must change the existing code in this line in order to create a valid suggestion.
Outdated suggestions cannot be applied.
This suggestion has been applied or marked resolved.
Suggestions cannot be applied from pending reviews.
Suggestions cannot be applied on multi-line comments.
Suggestions cannot be applied while the pull request is queued to merge.
Suggestion cannot be applied right now. Please check back later.