formal/lean/mathlib/algebra/direct_sum/algebra.lean

/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import algebra.algebra.basic
import algebra.direct_sum.module
import algebra.direct_sum.ring

/-! # Additively-graded algebra structures on `⨁ i, A i`

This file provides `R`-algebra structures on external direct sums of `R`-modules.

Recall that if `A i` are a family of `add_comm_monoid`s indexed by an `add_monoid`, then an instance
of `direct_sum.gmonoid A` is a multiplication `A i → A j → A (i + j)` giving `⨁ i, A i` the
structure of a semiring. In this file, we introduce the `direct_sum.galgebra R A` class for the case
where all `A i` are `R`-modules. This is the extra structure needed to promote `⨁ i, A i` to an
`R`-algebra.

## Main definitions

* `direct_sum.galgebra R A`, the typeclass.
* `direct_sum.galgebra.of_submodules`, for creating the above instance from a collection of
  submodules.
* `direct_sum.to_algebra` extends `direct_sum.to_semiring` to produce an `alg_hom`.

-/

universes uι uR uA uB

variables {ι : Type uι}

namespace direct_sum
open_locale direct_sum

variables (R : Type uR) (A : ι → Type uA) {B : Type uB} [decidable_eq ι]

variables [comm_semiring R] [Π i, add_comm_monoid (A i)] [Π i, module R (A i)]
variables [add_monoid ι] [gsemiring A]

section

/-- A graded version of `algebra`. An instance of `direct_sum.galgebra R A` endows `(⨁ i, A i)`
with an `R`-algebra structure. -/
class galgebra :=
(to_fun : R →+ A 0)
(map_one : to_fun 1 = graded_monoid.ghas_one.one)
(map_mul : ∀ r s,
  graded_monoid.mk _ (to_fun (r * s)) = ⟨_, graded_monoid.ghas_mul.mul (to_fun r) (to_fun s)⟩)
(commutes : ∀ r x, graded_monoid.mk _ (to_fun r) * x = x * ⟨_, to_fun r⟩)
(smul_def : ∀ r (x : graded_monoid A), graded_monoid.mk x.1 (r • x.2) = ⟨_, to_fun (r)⟩ * x)

end

variables [semiring B] [galgebra R A] [algebra R B]

instance : algebra R (⨁ i, A i) :=
{ to_fun := (direct_sum.of A 0).comp galgebra.to_fun,
  map_zero' := add_monoid_hom.map_zero _,
  map_add' := add_monoid_hom.map_add _,
  map_one' := (direct_sum.of A 0).congr_arg galgebra.map_one,
  map_mul' := λ a b, begin
    simp only [add_monoid_hom.comp_apply],
    rw of_mul_of,
    apply dfinsupp.single_eq_of_sigma_eq (galgebra.map_mul a b),
  end,
  commutes' := λ r x, begin
    change add_monoid_hom.mul (direct_sum.of _ _ _) x =
      add_monoid_hom.mul.flip (direct_sum.of _ _ _) x,
    apply add_monoid_hom.congr_fun _ x,
    ext i xi : 2,
    dsimp only [add_monoid_hom.comp_apply, add_monoid_hom.mul_apply, add_monoid_hom.flip_apply],
    rw [of_mul_of, of_mul_of],
    apply dfinsupp.single_eq_of_sigma_eq (galgebra.commutes r ⟨i, xi⟩),
  end,
  smul_def' := λ r x, begin
    change distrib_mul_action.to_add_monoid_hom _ r x = add_monoid_hom.mul (direct_sum.of _ _ _) x,
    apply add_monoid_hom.congr_fun _ x,
    ext i xi : 2,
    dsimp only [add_monoid_hom.comp_apply, distrib_mul_action.to_add_monoid_hom_apply,
      add_monoid_hom.mul_apply],
    rw [direct_sum.of_mul_of, ←of_smul],
    apply dfinsupp.single_eq_of_sigma_eq (galgebra.smul_def r ⟨i, xi⟩),
  end }

lemma algebra_map_apply (r : R) :
  algebra_map R (⨁ i, A i) r = direct_sum.of A 0 (galgebra.to_fun r) := rfl

lemma algebra_map_to_add_monoid_hom :
  ↑(algebra_map R (⨁ i, A i)) = (direct_sum.of A 0).comp (galgebra.to_fun : R →+ A 0) := rfl

/-- A family of `linear_map`s preserving `direct_sum.ghas_one.one` and `direct_sum.ghas_mul.mul`
describes an `alg_hom` on `⨁ i, A i`. This is a stronger version of `direct_sum.to_semiring`.

Of particular interest is the case when `A i` are bundled subojects, `f` is the family of
coercions such as `submodule.subtype (A i)`, and the `[gmonoid A]` structure originates from
`direct_sum.gmonoid.of_add_submodules`, in which case the proofs about `ghas_one` and `ghas_mul`
can be discharged by `rfl`. -/
@[simps]
def to_algebra
  (f : Π i, A i →ₗ[R] B) (hone : f _ (graded_monoid.ghas_one.one) = 1)
  (hmul : ∀ {i j} (ai : A i) (aj : A j), f _ (graded_monoid.ghas_mul.mul ai aj) = f _ ai * f _ aj)
  (hcommutes : ∀ r, (f 0) (galgebra.to_fun r) = (algebra_map R B) r) :
  (⨁ i, A i) →ₐ[R] B :=
{ to_fun := to_semiring (λ i, (f i).to_add_monoid_hom) hone @hmul,
  commutes' := λ r, (direct_sum.to_semiring_of _ _ _ _ _).trans (hcommutes r),
  .. to_semiring (λ i, (f i).to_add_monoid_hom) hone @hmul}

/-- Two `alg_hom`s out of a direct sum are equal if they agree on the generators.

See note [partially-applied ext lemmas]. -/
@[ext]
lemma alg_hom_ext' ⦃f g : (⨁ i, A i) →ₐ[R] B⦄
  (h : ∀ i, f.to_linear_map.comp (lof _ _ A i) = g.to_linear_map.comp (lof _ _ A i)) : f = g :=
alg_hom.to_linear_map_injective $ direct_sum.linear_map_ext _ h

lemma alg_hom_ext ⦃f g : (⨁ i, A i) →ₐ[R] B⦄ (h : ∀ i x, f (of A i x) = g (of A i x)) : f = g :=
alg_hom_ext' R A $ λ i, linear_map.ext $ h i

end direct_sum

/-! ### Concrete instances -/

/-- A direct sum of copies of a `algebra` inherits the algebra structure.

-/
@[simps]
instance algebra.direct_sum_galgebra {R A : Type*} [decidable_eq ι]
  [add_monoid ι] [comm_semiring R] [semiring A] [algebra R A] :
  direct_sum.galgebra R (λ i : ι, A) :=
{ to_fun := (algebra_map R A).to_add_monoid_hom,
  map_one := (algebra_map R A).map_one,
  map_mul := λ a b, sigma.ext (zero_add _).symm (heq_of_eq $ (algebra_map R A).map_mul a b),
  commutes := λ r ⟨ai, a⟩, sigma.ext ((zero_add _).trans (add_zero _).symm)
                                    (heq_of_eq $ algebra.commutes _ _),
  smul_def := λ r ⟨ai, a⟩, sigma.ext (zero_add _).symm (heq_of_eq $ algebra.smul_def _ _) }

namespace submodule

variables {R A : Type*} [comm_semiring R]

end submodule