[Merged by Bors] - feat(LinearAlgebra/BilinearForm/TensorProduct): base change of bilinear forms #6306
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…asic-TensorProduct.congr
…asic-TensorProduct.congr
…asic-TensorProduct.congr
…ra instance This also moves some lemmas about `IsUnit (_ : Module.End R M)` to an earlier file as they are nothing to do with `Algebra`.
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| variable [SMulCommClass R A M₁] [SMulCommClass A R M₁] [IsScalarTower R A M₁] | ||
| variable [Module R M₂] | ||
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| /-- The tensor product of two bilinear forms injects into bilinear forms on tensor products. |
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| /-- The tensor product of two bilinear forms injects into bilinear forms on tensor products. | |
| variable (R A) in | |
| /-- The tensor product of two bilinear forms injects into bilinear forms on tensor products. |
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| Note this is heterobasic; the bilinear form on the left can take values in a larger ring than | ||
| the one on the right. -/ |
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| Note this is heterobasic; the bilinear form on the left can take values in a larger ring than | |
| the one on the right. -/ | |
| Note this is heterobasic; the bilinear form on the left can take values in an algebra over | |
| the ring in which the right bilinear form is valued. -/ |
Mathlib/LinearAlgebra/Dual.lean
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| @@ -1608,6 +1609,25 @@ theorem dualDistrib_apply (f : Dual R M) (g : Dual R N) (m : M) (n : N) : | |||
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| end | |||
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| namespace AlgebraTensorModule | |||
| set_option autoImplicit false | |||
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| set_option autoImplicit false |
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…ar forms (#6306) This generalizes the existing `BilinForm.tensorDistrib` to be heterogenous in the rings it uses, such that a base change, ```lean protected def baseChange (B : BilinForm R M₂) : BilinForm A (A ⊗[R] M₂) := ``` falls out as a special case. I do not attempt to generalize `BilinForm.tensorDistribEquiv`. Unfortunately, this changes the defeq as ```diff -(B₁.tmul B₂) (m₁ ⊗ₜ m₂) (m₁' ⊗ₜ m₂') = B₁ m₁ m₁' * B₂ m₂ m₂' +(B₁.tmul B₂) (m₁ ⊗ₜ m₂) (m₁' ⊗ₜ m₂') = B₂ m₂ m₂' • B₁ m₁ m₁' ``` We could fix this by using the right action instead, but that's a lot of work for a very minorly annoying defeq. This also breaks the defeq of `tensorDistribEquiv B = tensorDistrib B`; though the reason is more complicated than the scalar action issue above. It would be fixed if we defined all the homogenous operations on tensor products as special cases of the heterogenous ones, but that's also a lot of work for a very small win. This is a port of work from pygae/lean-ga#31, and almost at the end of the path to a base change of quadratic forms and clifford algebras. This was independently developed at the Leiden workshop as [`BilinForm.baseChange`](https://github.com/alexjbest/ant-lorentz/blob/1f97add294b2d50f99537c15583666d78b0d7e24/AntLorentz/BaseChange.lean#L75-L85), though the results there are not sorry-free.
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…ar forms (#6306) This generalizes the existing `BilinForm.tensorDistrib` to be heterogenous in the rings it uses, such that a base change, ```lean protected def baseChange (B : BilinForm R M₂) : BilinForm A (A ⊗[R] M₂) := ``` falls out as a special case. I do not attempt to generalize `BilinForm.tensorDistribEquiv`. Unfortunately, this changes the defeq as ```diff -(B₁.tmul B₂) (m₁ ⊗ₜ m₂) (m₁' ⊗ₜ m₂') = B₁ m₁ m₁' * B₂ m₂ m₂' +(B₁.tmul B₂) (m₁ ⊗ₜ m₂) (m₁' ⊗ₜ m₂') = B₂ m₂ m₂' • B₁ m₁ m₁' ``` We could fix this by using the right action instead, but that's a lot of work for a very minorly annoying defeq. This also breaks the defeq of `tensorDistribEquiv B = tensorDistrib B`; though the reason is more complicated than the scalar action issue above. It would be fixed if we defined all the homogenous operations on tensor products as special cases of the heterogenous ones, but that's also a lot of work for a very small win. This is a port of work from pygae/lean-ga#31, and almost at the end of the path to a base change of quadratic forms and clifford algebras. This was independently developed at the Leiden workshop as [`BilinForm.baseChange`](https://github.com/alexjbest/ant-lorentz/blob/1f97add294b2d50f99537c15583666d78b0d7e24/AntLorentz/BaseChange.lean#L75-L85), though the results there are not sorry-free.
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This generalizes the existing
BilinForm.tensorDistribto be heterogenous in the rings it uses, such that a base change,falls out as a special case. I do not attempt to generalize
BilinForm.tensorDistribEquiv.Unfortunately, this changes the defeq as
We could fix this by using the right action instead, but that's a lot of work for a very minorly annoying defeq.
This also breaks the defeq of
tensorDistribEquiv B = tensorDistrib B; though the reason is more complicated than the scalar action issue above. It would be fixed if we defined all the homogenous operations on tensor products as special cases of the heterogenous ones, but that's also a lot of work for a very small win.This is a port of work from pygae/lean-ga#31, and almost at the end of the path to a base change of quadratic forms and clifford algebras.
This was independently developed at the Leiden workshop as
BilinForm.baseChange, though the results there are not sorry-free.