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algebra.homology.augment

Augmentation and truncation of -indexed (co)chain complexes. #

The truncation of a -indexed chain complex, deleting the object at 0 and shifting everything else down.

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There is a canonical chain map from the truncation of a chain map C to the "single object" chain complex consisting of the truncated object C.X 0 in degree 0. The components of this chain map are C.d 1 0 in degree 0, and zero otherwise.

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We can "augment" a chain complex by inserting an arbitrary object in degree zero (shifting everything else up), along with a suitable differential.

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  • C.augment f w = {X := λ (i : ), chain_complex.augment._match_1 C i, d := λ (i j : ), chain_complex.augment._match_2 C f i j, shape' := _, d_comp_d' := _}
  • chain_complex.augment._match_1 C (i + 1) = C.X i
  • chain_complex.augment._match_1 C 0 = X
  • chain_complex.augment._match_2 C f (n.succ + 1) (j + 1) = C.d n.succ j
  • chain_complex.augment._match_2 C f n.succ.succ 0 = 0
  • chain_complex.augment._match_2 C f (0 + 1) (j + 1) = C.d 0 j
  • chain_complex.augment._match_2 C f 1 0 = f
  • chain_complex.augment._match_2 C f 0 _x = 0
@[simp]
theorem chain_complex.augment_X_zero {V : Type u} [category_theory.category V] [category_theory.limits.has_zero_morphisms V] (C : chain_complex V ) {X : V} (f : C.X 0 X) (w : C.d 1 0 f = 0) :
(C.augment f w).X 0 = X
@[simp]
theorem chain_complex.augment_X_succ {V : Type u} [category_theory.category V] [category_theory.limits.has_zero_morphisms V] (C : chain_complex V ) {X : V} (f : C.X 0 X) (w : C.d 1 0 f = 0) (i : ) :
(C.augment f w).X (i + 1) = C.X i
@[simp]
theorem chain_complex.augment_d_one_zero {V : Type u} [category_theory.category V] [category_theory.limits.has_zero_morphisms V] (C : chain_complex V ) {X : V} (f : C.X 0 X) (w : C.d 1 0 f = 0) :
(C.augment f w).d 1 0 = f
@[simp]
theorem chain_complex.augment_d_succ_succ {V : Type u} [category_theory.category V] [category_theory.limits.has_zero_morphisms V] (C : chain_complex V ) {X : V} (f : C.X 0 X) (w : C.d 1 0 f = 0) (i j : ) :
(C.augment f w).d (i + 1) (j + 1) = C.d i j

Truncating an augmented chain complex is isomorphic (with components the identity) to the original complex.

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@[simp]
theorem chain_complex.truncate_augment_hom_f {V : Type u} [category_theory.category V] [category_theory.limits.has_zero_morphisms V] (C : chain_complex V ) {X : V} (f : C.X 0 X) (w : C.d 1 0 f = 0) (i : ) :
(C.truncate_augment f w).hom.f i = 𝟙 (C.X i)
@[simp]

Augmenting a truncated complex with the original object and morphism is isomorphic (with components the identity) to the original complex.

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A chain map from a chain complex to a single object chain complex in degree zero can be reinterpreted as a chain complex.

Ths is the inverse construction of truncate_to.

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The truncation of a -indexed cochain complex, deleting the object at 0 and shifting everything else down.

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There is a canonical chain map from the truncation of a cochain complex C to the "single object" cochain complex consisting of the truncated object C.X 0 in degree 0. The components of this chain map are C.d 0 1 in degree 0, and zero otherwise.

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We can "augment" a cochain complex by inserting an arbitrary object in degree zero (shifting everything else up), along with a suitable differential.

Equations
  • C.augment f w = {X := λ (i : ), cochain_complex.augment._match_1 C i, d := λ (i j : ), cochain_complex.augment._match_2 C f i j, shape' := _, d_comp_d' := _}
  • cochain_complex.augment._match_1 C (i + 1) = C.X i
  • cochain_complex.augment._match_1 C 0 = X
  • cochain_complex.augment._match_2 C f (i + 1) (j + 1) = C.d i j
  • cochain_complex.augment._match_2 C f n.succ 0 = 0
  • cochain_complex.augment._match_2 C f 0 n.succ.succ = 0
  • cochain_complex.augment._match_2 C f 0 1 = f
  • cochain_complex.augment._match_2 C f 0 0 = 0
@[simp]
theorem cochain_complex.augment_X_zero {V : Type u} [category_theory.category V] [category_theory.limits.has_zero_morphisms V] (C : cochain_complex V ) {X : V} (f : X C.X 0) (w : f C.d 0 1 = 0) :
(C.augment f w).X 0 = X
@[simp]
theorem cochain_complex.augment_X_succ {V : Type u} [category_theory.category V] [category_theory.limits.has_zero_morphisms V] (C : cochain_complex V ) {X : V} (f : X C.X 0) (w : f C.d 0 1 = 0) (i : ) :
(C.augment f w).X (i + 1) = C.X i
@[simp]
theorem cochain_complex.augment_d_zero_one {V : Type u} [category_theory.category V] [category_theory.limits.has_zero_morphisms V] (C : cochain_complex V ) {X : V} (f : X C.X 0) (w : f C.d 0 1 = 0) :
(C.augment f w).d 0 1 = f
@[simp]
theorem cochain_complex.augment_d_succ_succ {V : Type u} [category_theory.category V] [category_theory.limits.has_zero_morphisms V] (C : cochain_complex V ) {X : V} (f : X C.X 0) (w : f C.d 0 1 = 0) (i j : ) :
(C.augment f w).d (i + 1) (j + 1) = C.d i j

Truncating an augmented cochain complex is isomorphic (with components the identity) to the original complex.

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@[simp]
theorem cochain_complex.truncate_augment_hom_f {V : Type u} [category_theory.category V] [category_theory.limits.has_zero_morphisms V] (C : cochain_complex V ) {X : V} (f : X C.X 0) (w : f C.d 0 1 = 0) (i : ) :
(C.truncate_augment f w).hom.f i = 𝟙 (C.X i)

Augmenting a truncated complex with the original object and morphism is isomorphic (with components the identity) to the original complex.

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A chain map from a single object cochain complex in degree zero to a cochain complex can be reinterpreted as a cochain complex.

Ths is the inverse construction of to_truncate.

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