2017-03-10 16:51:24 +00:00
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/-
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Author: Jeremy Avigad
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-/
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import homotopy.chain_complex ..algebra.module ..algebra.is_short_exact ..move_to_lib
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open eq pointed sigma fiber equiv is_equiv sigma.ops is_trunc nat trunc
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open algebra function
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open chain_complex
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open succ_str
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2017-03-30 19:43:54 +00:00
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open left_module
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2017-03-10 16:51:24 +00:00
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structure module_chain_complex (R : Ring) (N : succ_str) : Type :=
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(mod : N → LeftModule R)
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(hom : Π (n : N), left_module.homomorphism (mod (S n)) (mod n))
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(is_chain_complex :
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Π (n : N) (x : mod (S (S n))), hom n (hom (S n) x) = 0)
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namespace module_chain_complex
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variables {R : Ring} {N : succ_str}
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definition mcc_mod [unfold 2] [coercion] (C : module_chain_complex R N) (n : N) :
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LeftModule R :=
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module_chain_complex.mod C n
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definition mcc_carr [unfold 2] [coercion] (C : module_chain_complex R N) (n : N) :
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Type :=
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C n
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definition mcc_pcarr [unfold 2] [coercion] (C : module_chain_complex R N) (n : N) :
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Set* :=
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mcc_mod C n
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definition mcc_hom (C : module_chain_complex R N) {n : N} : C (S n) →lm C n :=
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module_chain_complex.hom C n
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definition mcc_is_chain_complex (C : module_chain_complex R N) (n : N) (x : C (S (S n))) :
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mcc_hom C (mcc_hom C x) = 0 :=
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module_chain_complex.is_chain_complex C n x
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protected definition to_chain_complex [coercion] (C : module_chain_complex R N) :
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chain_complex N :=
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chain_complex.mk
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(λ n, mcc_pcarr C n)
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(λ n, pmap_of_homomorphism (@mcc_hom R N C n))
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(mcc_is_chain_complex C)
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-- maybe we don't even need this?
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definition is_exact_at_m (C : module_chain_complex R N) (n : N) : Type :=
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is_exact_at C n
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end module_chain_complex
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namespace left_module
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variable {R : Ring}
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variables {A₀ B₀ C₀ : LeftModule R}
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variables (f₀ : A₀ →lm B₀) (g₀ : B₀ →lm C₀)
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definition is_short_exact := @_root_.is_short_exact _ _ C₀ f₀ g₀
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end left_module
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