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d7f9a33 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 | import OAI.Algebra.AuslanderReiten.Complete.CompleteResolution
import OAI.Algebra.AuslanderReiten.Basic
import Mathlib.RingTheory.Finiteness.Finsupp
namespace OAI
noncomputable section
namespace ArExplicit
universe u v w x y
variable {R : Type u} [Ring R]
variable {K : Type v} [AddCommGroup K] [Module R K]
variable {L : Type w} [AddCommGroup L] [Module R L]
variable {M : Type x} [AddCommGroup M] [Module R M]
variable {Q : Type y} [AddCommGroup Q] [Module R Q]
theorem HomExactAt.arbitrary_projective [Module.Finite R L] [Module.Projective R Q]
{e : K →ₗ[R] L} {d : L →ₗ[R] M} (h : HomExactAt e d)
(f : L →ₗ[R] Q) (hf : f.comp e = 0) :
∃ g : M →ₗ[R] Q, g.comp d = f := by
classical
obtain ⟨s,hs⟩ := Module.projective_def'.mp (inferInstance : Module.Projective R Q)
obtain ⟨c,hc⟩ := (LinearMap.finsuppLinearMap_bijective_of_moduleFinite
R L R Q ℤ).surjective (s.comp f)
have he (q : Q) : (c q).comp e = 0 := by
ext z
have hh := congrArg (fun F : L →ₗ[R] (Q →₀ R) => F (e z) q) hc
change (c q) (e z) = (s (f (e z))) q at hh
rw [show f (e z) = 0 from DFunLike.congr_fun hf z, map_zero] at hh
exact hh
choose g hg using fun q => h (c q) (he q)
let G : M →ₗ[R] (Q →₀ R) := ∑ q ∈ c.support, (Finsupp.lsingle q).comp (g q)
have hG : G.comp d = s.comp f := by
ext z q
have hh := congrArg (fun F : L →ₗ[R] (Q →₀ R) => F z q) hc
change (c q) z = (s (f z)) q at hh
change G (d z) q = (s (f z)) q
rw [← hh]
change (∑ j ∈ c.support, (Finsupp.lsingle j).comp (g j)) (d z) q = (c q) z
simp only [LinearMap.sum_apply, LinearMap.comp_apply, Finsupp.finsetSum_apply]
have hg' (j : Q) : g j (d z) = (c j) z := DFunLike.congr_fun (hg j) z
simp only [hg', Finsupp.lsingle_apply]
rw [Finset.sum_eq_single q]
· simp
· intro b _ hb
simp [Ne.symm hb]
· intro hq
have hcz : c q = 0 := Finsupp.notMem_support_iff.mp hq
simp [hcz]
refine ⟨(Finsupp.linearCombination R id).comp G, ?_⟩
rw [LinearMap.comp_assoc, hG, ← LinearMap.comp_assoc, hs, LinearMap.id_comp]
def GorensteinProjectiveAllTargets (R : Type u) [Ring R]
(M : Type v) [AddCommGroup M] [Module R M] : Prop :=
∃ P : ChainComplex (ModuleCat.{max u v} R) ℤ,
(∀ n : ℤ, Module.Finite R (P.X n) ∧ Module.Projective R (P.X n)) ∧
(∀ n : ℤ, LinearMap.range (P.d (n+1) n).hom = LinearMap.ker (P.d n (n-1)).hom) ∧
(∀ (Q : Type y) [AddCommGroup Q] [Module R Q] [Module.Projective R Q]
(n : ℤ) (f : P.X n →ₗ[R] Q), f.comp (P.d (n+1) n).hom = 0 →
∃ g : P.X (n-1) →ₗ[R] Q, f = g.comp (P.d n (n-1)).hom) ∧
Nonempty (M ≃ₗ[R] (P.X 0 ⧸ LinearMap.range (P.d 1 0).hom))
theorem GorensteinProjective.all_projective_targets
{M : Type v} [AddCommGroup M] [Module R M]
(h : GorensteinProjective R M) : GorensteinProjectiveAllTargets.{u,v,y} R M := by
obtain ⟨P,hP,he,hd,hi⟩ := h
refine ⟨P,hP,he,?_,hi⟩
intro Q _ _ _ n f hf
let := (hP n).1
have hn : HomExactAt (P.d (n+1) n).hom (P.d n (n-1)).hom := by
intro f hf
obtain ⟨g,hg⟩ := hd n f hf
exact ⟨g,hg.symm⟩
obtain ⟨g,hg⟩ := hn.arbitrary_projective f hf
exact ⟨g,hg.symm⟩
theorem gorensteinProjective_iff_all_projective_targets
{M : Type v} [AddCommGroup M] [Module R M] :
GorensteinProjective R M ↔ GorensteinProjectiveAllTargets.{u,v,u} R M := by
refine ⟨GorensteinProjective.all_projective_targets, ?_⟩
rintro ⟨P,hP,he,hd,hi⟩
exact ⟨P,hP,he,hd R,hi⟩
end ArExplicit
end
end OAI
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