Theorems · Definition · category theory
CategoryTheory.ShortComplex.moduleCatToCycles
{R : Type u} →
[inst : Ring R] → (S : CategoryTheory.ShortComplex (ModuleCat R)) → ↑S.X₁ →ₗ[R] ↥(ModuleCat.Hom.hom S.g).kerThe canonical linear map S.X₁ →ₗ[R] LinearMap.ker S.g induced by S.f.
- Cited by
- 20 results in Mathlib
- Foundations
- Depth 51 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.
Cites16
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- RingHom.idstatement · cited by 18,349
- LinearMapstatement · cited by 10,215
- Ringstatement and proof · cited by 7,463
- Submodulestatement · cited by 7,192
- 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
- CategoryTheory.ShortComplex.X₁statement · cited by 889
- CategoryTheory.ShortComplex.X₃statement · cited by 876
- LinearMap.kerstatement and proof · cited by 848
- CategoryTheory.ShortComplex.gstatement and proof · cited by 658
Cited by22
Results whose statement or proof uses this declaration.
- CategoryTheory.ShortComplex.moduleCatLeftHomologyDataproof · cited by 106
- groupHomology.H1π_eq_zero_iffproof · cited by 4
- groupHomology.H1ToTensorOfIsTrivialproof · cited by 3
- CategoryTheory.ShortComplex.moduleCatLeftHomologyData_π_homstatement · cited by 3
- groupCohomology.H1π_eq_zero_iffproof · cited by 2
- groupHomology.H1ToTensorOfIsTrivial_H1π_singleproof · cited by 1
- groupCohomology.H2π_eq_zero_iffproof · cited by 1
- groupHomology.H2π_eq_zero_iffproof · cited by 1
- groupHomology.π_comp_H1Iso_hom_applystatement · cited by 1
- groupCohomology.toCocycles_comp_isoCocycles₁_hom_applystatement · cited by 0
- groupCohomology.toCocycles_comp_isoCocycles₂_hom_applystatement · cited by 0
- groupHomology.toCycles_comp_isoCycles₁_hom_applystatement · cited by 0