[Merged by Bors] - feat(RepresentationTheory/Homological/GroupHomology/LowDegree): Identify cycles A n with cyclesₙ A for n = 0, 1, 2#25888
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[Merged by Bors] - feat(RepresentationTheory/Homological/GroupHomology): long exact sequences
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cycles A n with nCycles A for n = 0, 1, 2cycles A n with cyclesₙ A for n = 0, 1, 2
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…ify `cycles A n` with `cyclesₙ A` for `n = 0, 1, 2` (#25888) Given a `k`-linear `G`-representation `A`, this PR provides isomorphisms between: - The abstract 0-cycles of `A` and `A` as a `k`-module - The abstract 0-opcycles of `A` and `A_G` - The abstract 1-cycles of `A` and the `cycles₁ A` defined as a submodule of `G →₀ A` - The abstract 2-cycles of `A` and the `cycles₂ A` defined as a submodule of `G × G →₀ A`. Co-authored-by: 101damnations <101damnations@github.com>
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cycles A n with cyclesₙ A for n = 0, 1, 2cycles A n with cyclesₙ A for n = 0, 1, 2
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…ify `cycles A n` with `cyclesₙ A` for `n = 0, 1, 2` (leanprover-community#25888) Given a `k`-linear `G`-representation `A`, this PR provides isomorphisms between: - The abstract 0-cycles of `A` and `A` as a `k`-module - The abstract 0-opcycles of `A` and `A_G` - The abstract 1-cycles of `A` and the `cycles₁ A` defined as a submodule of `G →₀ A` - The abstract 2-cycles of `A` and the `cycles₂ A` defined as a submodule of `G × G →₀ A`. Co-authored-by: 101damnations <101damnations@github.com>
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…ify `cycles A n` with `cyclesₙ A` for `n = 0, 1, 2` (leanprover-community#25888) Given a `k`-linear `G`-representation `A`, this PR provides isomorphisms between: - The abstract 0-cycles of `A` and `A` as a `k`-module - The abstract 0-opcycles of `A` and `A_G` - The abstract 1-cycles of `A` and the `cycles₁ A` defined as a submodule of `G →₀ A` - The abstract 2-cycles of `A` and the `cycles₂ A` defined as a submodule of `G × G →₀ A`. Co-authored-by: 101damnations <101damnations@github.com>
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Jul 28, 2025
…ify `cycles A n` with `cyclesₙ A` for `n = 0, 1, 2` (leanprover-community#25888) Given a `k`-linear `G`-representation `A`, this PR provides isomorphisms between: - The abstract 0-cycles of `A` and `A` as a `k`-module - The abstract 0-opcycles of `A` and `A_G` - The abstract 1-cycles of `A` and the `cycles₁ A` defined as a submodule of `G →₀ A` - The abstract 2-cycles of `A` and the `cycles₂ A` defined as a submodule of `G × G →₀ A`. Co-authored-by: 101damnations <101damnations@github.com>
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Given a
k-linearG-representationA, this PR provides isomorphisms between:AandAas ak-moduleAandA_GAand thecycles₁ Adefined as a submodule ofG →₀ AAand thecycles₂ Adefined as a submodule ofG × G →₀ A.Action.rhoaMonoidHominstead of a morphism inMonCat#21652Finsupp#21732Rep.diagonal k G (n + 1) ≅ Rep.free k G (Fin n → G)#21736IsCycle₁predicate and friends #25884This PR continues the work from #21759.
Original PR: #21759