[Merged by Bors] - feat(linear_algebra/invariant_basis_number): strong_rank_condition_iff_succ #9128
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Co-authored-by: Eric Wieser <wieser.eric@gmail.com>
Co-authored-by: Eric Wieser <wieser.eric@gmail.com>
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Co-authored-by: Eric Wieser <wieser.eric@gmail.com>
Co-authored-by: Eric Wieser <wieser.eric@gmail.com>
Co-authored-by: Eric Wieser <wieser.eric@gmail.com>
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…f_succ (#9128) We add `strong_rank_condition_iff_succ`: a ring satisfies the strong rank condition if and only if, for all `n : ℕ`, there are no injective linear maps `(fin (n + 1) → R) →ₗ[R] (fin n → R)`. This will be used to prove that any commutative ring satisfies the strong rank condition. The proof is simple and it uses the natural inclusion `R^n → R^m`, for `n ≤ m` (adding zeros at the end). We provide this in general as `extend_by_zero.linear_map : (ι → R) →ₗ[R] (η → R)` where `ι` and `η` are types endowed with a function `ι → η`.
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…f_succ (#9128) We add `strong_rank_condition_iff_succ`: a ring satisfies the strong rank condition if and only if, for all `n : ℕ`, there are no injective linear maps `(fin (n + 1) → R) →ₗ[R] (fin n → R)`. This will be used to prove that any commutative ring satisfies the strong rank condition. The proof is simple and it uses the natural inclusion `R^n → R^m`, for `n ≤ m` (adding zeros at the end). We provide this in general as `extend_by_zero.linear_map : (ι → R) →ₗ[R] (η → R)` where `ι` and `η` are types endowed with a function `ι → η`.
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…f_succ (#9128) We add `strong_rank_condition_iff_succ`: a ring satisfies the strong rank condition if and only if, for all `n : ℕ`, there are no injective linear maps `(fin (n + 1) → R) →ₗ[R] (fin n → R)`. This will be used to prove that any commutative ring satisfies the strong rank condition. The proof is simple and it uses the natural inclusion `R^n → R^m`, for `n ≤ m` (adding zeros at the end). We provide this in general as `extend_by_zero.linear_map : (ι → R) →ₗ[R] (η → R)` where `ι` and `η` are types endowed with a function `ι → η`.
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…f_succ (#9128) We add `strong_rank_condition_iff_succ`: a ring satisfies the strong rank condition if and only if, for all `n : ℕ`, there are no injective linear maps `(fin (n + 1) → R) →ₗ[R] (fin n → R)`. This will be used to prove that any commutative ring satisfies the strong rank condition. The proof is simple and it uses the natural inclusion `R^n → R^m`, for `n ≤ m` (adding zeros at the end). We provide this in general as `extend_by_zero.linear_map : (ι → R) →ₗ[R] (η → R)` where `ι` and `η` are types endowed with a function `ι → η`.
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We add
strong_rank_condition_iff_succ: a ring satisfies the strong rank condition if and only if, for alln : ℕ, there are noinjective linear maps
(fin (n + 1) → R) →ₗ[R] (fin n → R). This will be used to prove that any commutative ring satisfies the strong rank condition.The proof is simple and it uses the natural inclusion
R^n → R^m, forn ≤ m(adding zeros at the end). We provide this in general asextend_by_zero.linear_map : (ι → R) →ₗ[R] (η → R)whereιandηare types endowed with a functionι → η.