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216fb31 | 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 | import OAI.Algebra.Finitistic.Dimension.PDMinimalData
import Mathlib.CategoryTheory.Abelian.Projective.Resolution
import Mathlib.Algebra.Homology.ShortComplex.ModuleCat
namespace OAI
noncomputable section
namespace LittleFinitistic.PDMinimal
open CategoryTheory CategoryTheory.Limits
universe u
variable {k A : Type u} [Field k] [Ring A] [Algebra k A] [FiniteDimensional k A]
variable (M : FiniteModule k A)
/-- The actual chain complex of the successive projective covers. -/
def complex : ChainComplex (ModuleCat A) ℕ := ChainComplex.of
(fun n => (term M n).cat)
(fun n => ModuleCat.ofHom (differential M n))
(fun n => ModuleCat.hom_ext (differential_comp M n))
@[simp] theorem complex_d (n : ℕ) :
(complex M).d (n+1) n = ModuleCat.ofHom (differential M n) := by
exact ChainComplex.of_d (fun n => (term M n).cat)
(fun n => ModuleCat.ofHom (differential M n)) n
instance term_projective (n : ℕ) : Projective ((complex M).X n) := by
exact (IsProjective.iff_projective _).mp (cover (syzygy M n)).projective
lemma complex_exactAt (n : ℕ) : (complex M).ExactAt (n+1) := by
rw [HomologicalComplex.exactAt_iff' _ (n+2) (n+1) n (by simp) (by simp)]
apply (ShortComplex.moduleCat_exact_iff_range_eq_ker _).mpr
change LinearMap.range ((complex M).d (n+2) (n+1)).hom =
LinearMap.ker ((complex M).d (n+1) n).hom
rw [complex_d, complex_d]
exact exact_differential M n
lemma complex_d_augment : (complex M).d 1 0 ≫ ModuleCat.ofHom (augment M 0) = 0 := by
rw [complex_d]
exact ModuleCat.hom_ext (augment_differential M 0)
lemma complex_exact_augment :
(ShortComplex.mk _ _ (complex_d_augment M)).Exact := by
apply (ShortComplex.moduleCat_exact_iff_range_eq_ker _).mpr
change LinearMap.range ((complex M).d 1 0).hom = LinearMap.ker (augment M 0)
rw [complex_d]
exact range_differential M 0
instance augment_epi : Epi (ModuleCat.ofHom (augment M 0)) :=
(ModuleCat.epi_iff_surjective _).mpr (cover _).surjective
/-- A projective resolution, including its augmentation quasi-isomorphism. -/
def resolution : ProjectiveResolution M.cat where
complex := complex M
π := (ChainComplex.toSingle₀Equiv _ _).symm ⟨ModuleCat.ofHom (augment M 0),
complex_d_augment M⟩
quasiIso := ⟨fun n => by
cases n with
| zero =>
rw [ChainComplex.quasiIsoAt₀_iff, ShortComplex.quasiIso_iff_of_zeros']
·
refine (ShortComplex.exact_and_epi_g_iff_of_iso ?_).2
⟨complex_exact_augment M, augment_epi M⟩
exact ShortComplex.isoMk (Iso.refl _) (Iso.refl _) (Iso.refl _)
(by erw [Category.id_comp])
(by erw [Category.id_comp])
all_goals rfl
| succ n =>
rw [quasiIsoAt_iff_exactAt']
· exact complex_exactAt M n
· apply ChainComplex.exactAt_succ_single_obj⟩
end LittleFinitistic.PDMinimal
end
end OAI
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