feat(Topology): A spectral map between quasi-separated, prespectral sober spaces has compact fibers#39663
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Raph-DG wants to merge 75 commits into
Open
feat(Topology): A spectral map between quasi-separated, prespectral sober spaces has compact fibers#39663Raph-DG wants to merge 75 commits into
Raph-DG wants to merge 75 commits into
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This reverts commit bb5278f.
…matic from topological assumptions
Co-authored-by: Christian Merten <christian@merten.dev>
Co-authored-by: Christian Merten <christian@merten.dev>
Co-authored-by: Christian Merten <christian@merten.dev>
Co-authored-by: Christian Merten <christian@merten.dev>
Co-authored-by: Christian Merten <christian@merten.dev>
Co-authored-by: Christian Merten <christian@merten.dev>
…into Raph-DG-AlgebraicCycle
Co-authored-by: Christian Merten <christian@merten.dev>
PR summary b6865bbcda
|
| File | Base Count | Head Count | Change |
|---|---|---|---|
| Mathlib.Topology.Spectral.ConstructibleTopology | 890 | 893 | +3 (+0.34%) |
Import changes for all files
| Files | Import difference |
|---|---|
Mathlib.Topology.Spectral.ConstructibleTopology |
3 |
Declarations diff
+ HasCompactFibers
+ IsOpenEmbedding.isSpectralMap_of_compactSpace
+ IsProperMap.hasCompactFibers
+ IsSpectralMap.hasCompactFibers
+ WithConstructibleTopology.continuous_equiv
+ WithConstructibleTopology.equiv
+ WithConstructibleTopology.map
+ WithConstructibleTopology.map_comp
+ WithConstructibleTopology.map_continuous
+ WithConstructibleTopology.map_id
+ instance [PrespectralSpace X] [T0Space X] [QuasiSober X] :
+ isOpen_ofConstructibleTopology_preimage_iff
+ toConstructibleTopology
You can run this locally as follows
## from your `mathlib4` directory:
git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci
## summary with just the declaration names:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh <optional_commit>
## more verbose report:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh long <optional_commit>The doc-module for scripts/pr_summary/declarations_diff.sh in the mathlib-ci repository contains some details about this script.
No changes to strong technical debt.
No changes to weak technical debt.
3 tasks
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In this PR, we develop some API around the constructible topology, culminating in the fact that a spectral map between quasi-separated, prespectral sober spaces has compact fibers. To see why this might be of interest, note that the analogous theorem in algebraic geometry (that a quasiseparated map between schemes has quasicompact fibers) does not require any global separatedness assumptions, and the proof of this is very algebraic. So we have the somewhat mysterious situation that it seems as though there are nontrvial topological restrictions on the kinds of spectral maps which can be the underlying maps of morphisms of schemes.
This PR was originally part of #26304, a PR on pushforwards of algebraic cycles. This is where the notion of compactness of fibers becomes relevant, as this guarantees each coefficient of the pushforward of a cycle is computed by a finite sum.