[Merged by Bors] - feat(Mathlib.RingTheory.TensorProduct.MvPolynomial) : tensor product of a (multivariate) polynomial ring #12293
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Change i to iota Co-authored-by: Junyan Xu <junyanxu.math@gmail.com>
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…of a (multivariate) polynomial ring (#12293) Let `Semiring R`, `Algebra R S` and `Module R N`. * `MvPolynomial.rTensor` gives the linear equivalence `MvPolynomial σ S ⊗[R] N ≃ₗ[R] (σ →₀ ℕ) →₀ (S ⊗[R] N)` characterized, for `p : MvPolynomial σ S`, `n : N` and `d : σ →₀ ℕ`, by `rTensor (p ⊗ₜ[R] n) d = (coeff d p) ⊗ₜ[R] n` * `MvPolynomial.scalarRTensor` gives the linear equivalence `MvPolynomial σ R ⊗[R] N ≃ₗ[R] (σ →₀ ℕ) →₀ N` such that `MvPolynomial.scalarRTensor (p ⊗ₜ[R] n) d = coeff d p • n` for `p : MvPolynomial σ R`, `n : N` and `d : σ →₀ ℕ`, by * `MvPolynomial.rTensorAlgHom`, the algebra morphism from the tensor product of a polynomial algebra by an algebra to a polynomial algebra * `MvPolynomial.rTensorAlgEquiv`, `MvPolynomial.scalarRTensorAlgEquiv`, the tensor product of a polynomial algebra by an algebra is algebraically equivalent to a polynomial algebra Co-authored-by: Oliver Nash <github@olivernash.org> Co-authored-by: Antoine Chambert-Loir <antoine.chambert-loir@math.univ-paris-diderot.fr>
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…of a (multivariate) polynomial ring (#12293) Let `Semiring R`, `Algebra R S` and `Module R N`. * `MvPolynomial.rTensor` gives the linear equivalence `MvPolynomial σ S ⊗[R] N ≃ₗ[R] (σ →₀ ℕ) →₀ (S ⊗[R] N)` characterized, for `p : MvPolynomial σ S`, `n : N` and `d : σ →₀ ℕ`, by `rTensor (p ⊗ₜ[R] n) d = (coeff d p) ⊗ₜ[R] n` * `MvPolynomial.scalarRTensor` gives the linear equivalence `MvPolynomial σ R ⊗[R] N ≃ₗ[R] (σ →₀ ℕ) →₀ N` such that `MvPolynomial.scalarRTensor (p ⊗ₜ[R] n) d = coeff d p • n` for `p : MvPolynomial σ R`, `n : N` and `d : σ →₀ ℕ`, by * `MvPolynomial.rTensorAlgHom`, the algebra morphism from the tensor product of a polynomial algebra by an algebra to a polynomial algebra * `MvPolynomial.rTensorAlgEquiv`, `MvPolynomial.scalarRTensorAlgEquiv`, the tensor product of a polynomial algebra by an algebra is algebraically equivalent to a polynomial algebra Co-authored-by: Oliver Nash <github@olivernash.org> Co-authored-by: Antoine Chambert-Loir <antoine.chambert-loir@math.univ-paris-diderot.fr>
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…of a (multivariate) polynomial ring (#12293) Let `Semiring R`, `Algebra R S` and `Module R N`. * `MvPolynomial.rTensor` gives the linear equivalence `MvPolynomial σ S ⊗[R] N ≃ₗ[R] (σ →₀ ℕ) →₀ (S ⊗[R] N)` characterized, for `p : MvPolynomial σ S`, `n : N` and `d : σ →₀ ℕ`, by `rTensor (p ⊗ₜ[R] n) d = (coeff d p) ⊗ₜ[R] n` * `MvPolynomial.scalarRTensor` gives the linear equivalence `MvPolynomial σ R ⊗[R] N ≃ₗ[R] (σ →₀ ℕ) →₀ N` such that `MvPolynomial.scalarRTensor (p ⊗ₜ[R] n) d = coeff d p • n` for `p : MvPolynomial σ R`, `n : N` and `d : σ →₀ ℕ`, by * `MvPolynomial.rTensorAlgHom`, the algebra morphism from the tensor product of a polynomial algebra by an algebra to a polynomial algebra * `MvPolynomial.rTensorAlgEquiv`, `MvPolynomial.scalarRTensorAlgEquiv`, the tensor product of a polynomial algebra by an algebra is algebraically equivalent to a polynomial algebra Co-authored-by: Oliver Nash <github@olivernash.org> Co-authored-by: Antoine Chambert-Loir <antoine.chambert-loir@math.univ-paris-diderot.fr>
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Let
Semiring R,Algebra R SandModule R N.MvPolynomial.rTensorgives the linear equivalenceMvPolynomial σ S ⊗[R] N ≃ₗ[R] (σ →₀ ℕ) →₀ (S ⊗[R] N)characterized,for
p : MvPolynomial σ S,n : Nandd : σ →₀ ℕ, byrTensor (p ⊗ₜ[R] n) d = (coeff d p) ⊗ₜ[R] nMvPolynomial.scalarRTensorgives the linear equivalenceMvPolynomial σ R ⊗[R] N ≃ₗ[R] (σ →₀ ℕ) →₀ Nsuch that
MvPolynomial.scalarRTensor (p ⊗ₜ[R] n) d = coeff d p • nfor
p : MvPolynomial σ R,n : Nandd : σ →₀ ℕ, byMvPolynomial.rTensorAlgHom, the algebra morphism from the tensor productof a polynomial algebra by an algebra to a polynomial algebra
MvPolynomial.rTensorAlgEquiv,MvPolynomial.scalarRTensorAlgEquiv,the tensor product of a polynomial algebra by an algebra
is algebraically equivalent to a polynomial algebra