[Merged by Bors] - feat: generalize Module.Finite.trans
#9380
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Mathlib/Algebra/Algebra/Tower.lean
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| theorem span_algebraMap_image_of_tower {S T : Type*} [CommSemiring S] [Semiring T] [Module R S] | ||
| [IsScalarTower R S S] [Algebra R T] [Algebra S T] [IsScalarTower R S T] (a : Set S) : | ||
| [Algebra R T] [Algebra S T] [IsScalarTower R S T] (a : Set S) : |
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This can't be generalized yet with what we include in this PR; this is an application of the CompatibleSMul class but there's some unresolved controversy with @eric-wieser.
(restrictScalars consumes CompatibleSMul which was derived from LinearMap.IsScalarTower.compatibleSMul, but in the absence of IsScalarTower R S S it can be directly supplied by
variable {R S : Type*} [Semiring S] [SMul R M] [Module S M] [SMul R M₂] [Module S M₂]
instance IsScalarTower.compatibleSMul' [SMul R S] [IsScalarTower R S M] :
CompatibleSMul S M R S
in #9046.)
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Add `span_eq_closure` and `closure_induction` which say that `Submodule.span R s` is generated by `R • s` as an AddSubmonoid. I feel that the existing `span_induction` should be replaced by `closure_induction` as the latter is stronger, and allow us to remove the commutativity condition in `span_smul_of_span_eq_top` in Algebra/Tower and generalize `Module.Finite.trans` to allow a non-commutative base ring. Co-authored-by: Junyan Xu <junyanxu.math@gmail.com>
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Generalize the conditions of the tower law [FiniteDimensional.finrank_mul_finrank'](https://leanprover-community.github.io/mathlib4_docs/Mathlib/FieldTheory/Tower.html#FiniteDimensional.finrank_mul_finrank') in FieldTheory/Tower from `[CommRing F] [Algebra F K]` to `[Ring F] [Module F K]`, and remove the `[Module.Finite F K]` and `[Module.Finite K A]` conditions. The generalized version applies to situations when we have a tower C/B/A where the A-module structure on C is induced from the B-module structure via a RingHom from A to B, and the A-module structure on B is induced by the same RingHom. In particular, it applies when the A-module structure on B and the B-module structure on C come from two RingHoms, and the A-module structure on C comes from the composition of them, regardless of whether A and B are commutative or not. As prerequisites, I also generalized lemmas originally introduced by @kckennylau in [mathlib3#3355](https://github.com/leanprover-community/mathlib/pull/3355/files) to prove the tower law. They were split into three PRs: + LinearAlgebra/Span #9380: add `span_eq_closure` and `closure_induction` which say that `Submodule.span R s` is generated by `R • s` as an AddSubmonoid. I feel that the existing `span_induction` should be replaced by `closure_induction` as the latter is stronger, and allow us to remove the commutativity condition in `span_smul_of_span_eq_top` in Algebra/Tower. + Algebra/Tower #9382: switching from CommSemiring/Algebra to Semiring/Module here requires proving the curious lemma `IsScalarTower.isLinearMap` which states that for a tower of modules A/S/R, any S-linear map from S to A is also R-linear. If the map is injective, we can deduce that S/S/R also form a tower. (By `ringHomEquivModuleIsScalarTower` in #9381, there is therefore a canonical RingHom from R to S.) + Lemmas for free modules over rings including `finrank_mul_finrank'` are moved from FieldTheory/Tower to LinearAlgebra/Dimension/Free [Zulip](https://leanprover.zulipchat.com/#narrow/stream/116395-maths/topic/Puzzle.3A.20noncommutative.20RingHom.20as.20typeclasses/near/407347711) Co-authored-by: Junyan Xu <junyanxu.math@gmail.com> Co-authored-by: Eric Wieser <wieser.eric@gmail.com>
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Add
span_eq_closureandclosure_inductionwhich say thatSubmodule.span R sis generated byR • sas an AddSubmonoid. I feel that the existingspan_inductionshould be replaced byclosure_inductionas the latter is stronger, and allow us to remove the commutativity condition inspan_smul_of_span_eq_topin Algebra/Tower and generalizeModule.Finite.transto allow a non-commutative base ring.This is one of the uncontroversial parts of #9046.