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namespace OAI
namespace PiExponent.NatToIntGrading
noncomputable section
open DirectSum
variable {M σM : Type*} [AddCommGroup M] [SetLike σM M] [AddSubgroupClass σM M]
(𝓜 : ℕ → σM) [DirectSum.Decomposition 𝓜]
def piece : ℤ → AddSubgroup M
| .ofNat n => AddSubgroup.ofClass (𝓜 n)
| .negSucc _ => ⊥
def component (m : M) : ∀ d, piece 𝓜 d
| .ofNat n => ⟨DirectSum.decompose 𝓜 m n, (DirectSum.decompose 𝓜 m n).property⟩
| .negSucc _ => 0
def support (m : M) : Finset ℤ := by
classical
exact (DirectSum.decompose 𝓜 m).support.image (fun n : ℕ => (n : ℤ))
theorem component_eq_zero_of_not_mem (m : M) (d : ℤ) (hd : d ∉ support 𝓜 m) :
component 𝓜 m d = 0 := by
classical
cases d with
| ofNat n =>
apply Subtype.ext
have hn : n ∉ (DirectSum.decompose 𝓜 m).support := by
intro hn
exact hd (Finset.mem_image.mpr ⟨n, hn, rfl⟩)
change (DirectSum.decompose 𝓜 m n : M) = 0
rw [DFinsupp.notMem_support_iff.mp hn]
rfl
| negSucc n => rfl
def decompositionMap : M →+ ⨁ d, piece 𝓜 d where
toFun m := DFinsupp.mk' (component 𝓜 m) (Trunc.mk
⟨(support 𝓜 m).val, fun d => by
classical
by_cases hd : d ∈ support 𝓜 m
· exact Or.inl hd
· exact Or.inr (component_eq_zero_of_not_mem 𝓜 m d hd)⟩)
map_zero' := by
apply DFinsupp.ext
intro d
apply Subtype.ext
cases d with
| ofNat n =>
change (DirectSum.decompose 𝓜 (0 : M) n : M) = 0
rw [DirectSum.decompose_zero]
rfl
| negSucc n => rfl
map_add' m n := by
apply DFinsupp.ext
intro d
apply Subtype.ext
cases d with
| ofNat k =>
change (DirectSum.decompose 𝓜 (m + n) k : M) =
(DirectSum.decompose 𝓜 m k : M) + (DirectSum.decompose 𝓜 n k : M)
rw [DirectSum.decompose_add]
rfl
| negSucc k => exact (zero_add (0 : M)).symm
@[simp] theorem decompositionMap_nat (m : M) (n : ℕ) :
(decompositionMap 𝓜 m (n : ℤ) : M) = (DirectSum.decompose 𝓜 m n : M) := rfl
theorem decompositionMap_injective : Function.Injective (decompositionMap 𝓜) := by
intro m n h
apply (DirectSum.decompose 𝓜).injective
ext k
exact congrArg (fun z : ⨁ d, piece 𝓜 d => (z (k : ℤ) : M)) h
theorem decompositionMap_homogeneous (d : ℤ) (m : piece 𝓜 d) :
decompositionMap 𝓜 m.val = DirectSum.of (fun d => piece 𝓜 d) d m := by
classical
cases d with
| negSucc n =>
have hm : m = 0 := Subtype.ext m.property
subst m
simp
| ofNat n =>
apply DFinsupp.ext
intro e
apply Subtype.ext
cases e with
| negSucc k =>
rw [DirectSum.of_eq_of_ne _ _ _ (show Int.negSucc k ≠ Int.ofNat n from by intro h; cases h)]
rfl
| ofNat k =>
by_cases hnk : n = k
· subst k
rw [DirectSum.of_eq_same]
exact DirectSum.decompose_of_mem_same 𝓜 m.property
· rw [DirectSum.of_eq_of_ne _ _ _ (show Int.ofNat k ≠ Int.ofNat n from
fun h => hnk (Int.ofNat.inj h).symm)]
exact DirectSum.decompose_of_mem_ne 𝓜 m.property hnk
instance decomposition : DirectSum.Decomposition (piece 𝓜) := by
have hr : (decompositionMap 𝓜).comp (DirectSum.coeAddMonoidHom (piece 𝓜)) = AddMonoidHom.id _ := by
apply DirectSum.addHom_ext
intro d m
simp only [AddMonoidHom.comp_apply, DirectSum.coeAddMonoidHom_of,
AddMonoidHom.id_apply, decompositionMap_homogeneous]
apply DirectSum.Decomposition.ofAddHom (piece 𝓜) (decompositionMap 𝓜) ?_ hr
apply AddMonoidHom.ext
intro m
apply decompositionMap_injective 𝓜
exact DFunLike.congr_fun hr (decompositionMap 𝓜 m)
@[simp] theorem decompose_nat (m : M) (n : ℕ) :
(DirectSum.decompose (piece 𝓜) m (n : ℤ) : M) = (DirectSum.decompose 𝓜 m n : M) := rfl
@[simp] theorem decompose_negSucc (m : M) (n : ℕ) :
(DirectSum.decompose (piece 𝓜) m (.negSucc n) : M) = 0 := rfl
section Scalars
variable {R σR : Type*} [Ring R] [Module R M] [SetLike σR R] [AddSubgroupClass σR R]
(𝒜 : ℕ → σR) [SetLike.GradedSMul 𝒜 𝓜]
instance gradedSMul : SetLike.GradedSMul (piece 𝒜) (piece 𝓜) where
smul_mem := by
intro i j a m ha hm
cases i with
| negSucc n =>
have ha' : a = 0 := ha
rw [ha', zero_smul]
exact zero_mem _
| ofNat n =>
cases j with
| negSucc k =>
have hm' : m = 0 := hm
rw [hm', smul_zero]
exact zero_mem _
| ofNat k =>
change a • m ∈ 𝓜 (n + k)
exact SetLike.GradedSMul.smul_mem (show a ∈ 𝒜 n from ha) (show m ∈ 𝓜 k from hm)
end Scalars
section Ring
variable {R σR : Type*} [Ring R] [SetLike σR R] [AddSubgroupClass σR R]
(𝒜 : ℕ → σR) [SetLike.GradedMonoid 𝒜]
instance gradedMonoid : SetLike.GradedMonoid (piece 𝒜) where
one_mem := by
change (1 : R) ∈ 𝒜 0
exact SetLike.GradedOne.one_mem
mul_mem := by
intro i j a b ha hb
cases i with
| negSucc n =>
have ha' : a = 0 := ha
rw [ha', zero_mul]
exact zero_mem _
| ofNat n =>
cases j with
| negSucc k =>
have hb' : b = 0 := hb
rw [hb', mul_zero]
exact zero_mem _
| ofNat k =>
change a * b ∈ 𝒜 (n + k)
exact SetLike.mul_mem_graded (show a ∈ 𝒜 n from ha) (show b ∈ 𝒜 k from hb)
end Ring
end
end PiExponent.NatToIntGrading
end OAI
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