[Merged by Bors] - feat: definition of linear topologies#14990
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AntoineChambert-Loir wants to merge 19 commits intomasterfrom
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[Merged by Bors] - feat: definition of linear topologies#14990AntoineChambert-Loir wants to merge 19 commits intomasterfrom
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This was referenced Jul 22, 2024
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(Note: we talked about this PR IRL, so I'm marking it as awaiting-author) |
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Well, I believe I am not ready for the moment to implement this big change, and I would like to go forward with the stuff that is already done and comes after this. So I propose that we leave this interesting refactoring to later work. |
ADedecker
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jcommelin
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I've just had a last look, to me this is ready to go.
urkud
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Dec 24, 2024
| section Ring | ||
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| /-- A topology on a ring is linear if its topology is defined by a family of ideals. -/ | ||
| class _root_.IsLinearTopology (R : Type*) [Ring R] [TopologicalSpace R] where |
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I know nothing about this, but Wikipedia defines linear topology in a different context, and it doesn't agree with this definition (it uses left submodules, not two sided ideals). I do not suggest that you change the definition (see start of this comment), but I think that existence of different definitions and some relations between them should be discussed in the module docstring.
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@urkud Thanks for your comment. Indeed what we were defining is a special case about linearly topologized modules. I adapted the file to talk about modules. The only subtlety is that, to check this does indeed give the same thing for non-commutative rings, one needs to start from a basis of left ideals and one of right ideals to build one made of two-sided ideals, which needs a slight trick. I can move this part to a later PR if needed, but I think it also fits here. |
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Thanks 🎉 bors merge |
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A topology on a module is linear if it is invariant by translation and if there is a basis of neighborhoods consisting of submodules. We are most interested in the case of rings: a topology on a ring is linear if it is linear for both the left- and right-module structures on R over itself. This is equivalent to being invariant by translation and admitting a basis of neighborhoods consisting of two-sided ideals. This will be used in a subsequent PR to evaluate multivariate power series. We will also show that the natural topology on `MvPowerSeries S R` is a linear topology when `S` has a linear topology (e.g the discrete topology). Co-authored-by: @mariainesdff and @ADedecker Co-authored-by: mariainesdff <mariaines.dff@gmail.com> Co-authored-by: ADedecker <anatolededecker@gmail.com>
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Pull request successfully merged into master. Build succeeded: |
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* polynomial-sequences: (149 commits) Aha, here's how to get Lean to stop showing S.elems' in the infoview. Try satisfying the linter gods again. Probably enough initial tidying to send the PR. Kill more temporary names. Touch more natDegree. Does protected satisfy the docstring linter? Bit shorter. More Quiet linters. Remove redundant imports. Copyright header and more twiddling. Rename lemma to 'degree_smul_of_leadingCoeff_rightRegular' and split out feat(Polynomial): polynomial sequences are bases for R[X] chore(Dynamics/PeriodicPts): don't import `MonoidWithZero` (#20765) chore(Associated): split out `Ring` results (#20737) feat(AlgebraicGeometry): flat morphisms of schemes (#19790) feat(AlgebraicGeometry): scheme-theoretic fibre (#19427) chore: split Mathlib.Analysis.Asymptotics.Asymptotics (#20785) doc: typo (#20829) feat(CategoryTheory): condition for an induced functor between comma categories to be final (#20139) feat(1000.yaml): allow statements of theorems also (#20637) feat(Algebra/Homology/Embedding): homology of truncGE' (#19570) chore: cleanup many porting notes in Combinatorics (#20823) chore: eliminate porting notes about `deriving Fintype` (#20820) feat(Algebra/Lie): a Lie algebra is solvable iff it is solvable after faithfully flat base change (#20808) feat: define bases of root pairings (#20667) feat(Tactic): basic ConcreteCategory support for elementwise (#20811) feat(CategoryTheory): define unbundled `ConcreteCategory` class (#20810) chore(CategoryTheory): rename `ConcreteCategory` to `HasForget` (#20809) feat: `CommSemiring (NonemptyInterval ℚ≥0)` (#20783) chore(yaml_check.py): re-format (#20807) feat: elementary estimate for Real.log (#20766) feat: definition of linear topologies (#14990) feat(RingTheory): flatness over a semiring (#19115) feat(Algebra/Homology/Embedding): the canonical truncation truncLE (#19550) feat(Algebra/Homology/Embedding): API for the homology of an extension of homological complex (#19203) feat(Algebra/Ring): `RingEquiv.piUnique` (#20794) feat(RingTheory/LocalRing): add instance `Unique (MaximalSpectrum R)` for a local ring `R` (#20801) chore(GroupExtension/Defs): define `Section` and redefine `Splitting` (#20802) chore: restore `def` to `adicCompletion` (#20796) refactor: rename `UniqueContinuousFunctionalCalculus` to `ContinuousMap.UniqueHom` (#20643) feat(Algebra/Homology/Embedding): the morphism from a complex to its `truncGE` truncation (#19544) chore(Mathlib/Computability/TuringMachine): split file (#20790) feat(Data/Finset/Card): add `InjOn` and `SurjOn` versions of finset cardinality lemmas (#20753) feat(Order/WellFoundedSet): add convenience constructors for IsWF and IsPWO for WellFoundedLT types (#20752) feat(Set/Finite): a set is finite if its image and fibers are finite (#20751) chore: cleanup .gitignore files (#20795) feat(Topology/Group/Profinite): Profinite group is limit of finite group (#16992) feat(Combinatorics/SimpleGraph): vertices in cycles (#20602) CI: merge `bot_fix_style` actions (#20789) ...
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A topology on a module is linear if it is invariant by translation and if there
is a basis of neighborhoods consisting of submodules.
We are most interested in the case of rings: a topology on a ring is linear
if it is linear for both the left- and right-module structures on R over itself.
This is equivalent to being invariant by translation and admitting a
basis of neighborhoods consisting of two-sided ideals.
This will be used in a subsequent PR to evaluate multivariate power series.
We will also show that the natural topology on
MvPowerSeries S Ris a linear topology whenShas a linear topology (e.g the discrete topology).Co-authored-by: @mariainesdff and @ADedecker