Theorems · Theorem · category theory
CategoryTheory.ShortComplex.ShortExact.moduleCat_exact_iff_function_exact
∀ {R : Type u} [inst : Ring R] (S : CategoryTheory.ShortComplex (ModuleCat R)),
S.Exact ↔ Function.Exact ⇑(CategoryTheory.ConcreteCategory.hom S.f) ⇑(CategoryTheory.ConcreteCategory.hom S.g)- Cited by
- 11 results in Mathlib
- Foundations
- Depth 102 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- Ring
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites20
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- RingHom.idstatement · cited by 18,349
- LinearMapstatement · cited by 10,215
- Ringstatement and proof · cited by 7,463
- CategoryTheory.ConcreteCategory.homstatement and proof · cited by 4,022
- CategoryTheory.ShortComplexstatement and proof · cited by 1,850
- ModuleCatstatement and proof · cited by 1,429
- CategoryTheory.ShortComplex.X₂statement · cited by 1,115
- ModuleCat.carrierstatement · cited by 997
- LinearMap.rangeproof · cited by 893
- CategoryTheory.ShortComplex.X₁statement · cited by 889
- CategoryTheory.ShortComplex.X₃statement · cited by 876
Cited by11
Results whose statement or proof uses this declaration.
- LinearMap.shortExact_shortComplexKerproof · cited by 3
- ModuleCat.localizedModule_hasInjectiveDimensionLEproof · cited by 2
- ModuleCat.shortComplex_exactproof · cited by 2
- Module.Flat.iff_lTensor_preserves_shortComplex_exactproof · cited by 1
- Module.Flat.rTensor_shortComplex_exactproof · cited by 1
- ModuleCat.shortComplex_shortExactproof · cited by 1
- Module.Flat.lTensor_shortComplex_exactproof · cited by 1
- Module.Flat.iff_rTensor_preserves_shortComplex_exactproof · cited by 0
- ModuleCat.restrictScalars_map_exactproof · cited by 0
- ModuleCat.localizedModuleFunctor_map_exactproof · cited by 0
- ModuleCat.uliftFunctor_map_exactproof · cited by 0