Homology object of a chain complex

Depends on

• Cycle and boundary subobjects of a complex
• The boundary subobject factors through the cycle subobject
• Kernels and cokernels in a category with zero morphisms as equalizers and coequalizers

Used by

• Gaussian cancellation preserves homotopy type and abelian-category homology Corollary
• One Element Koszul Homology Corollary
• A universal-coefficient splitting cannot in general be chosen naturally Counterexample
• Sum and product totalisations on an infinite diagonal Counterexample
• The isolated differential 2 on the integers cannot be cancelled Counterexample
• Augmentation and reduced simplicial homology Definition
• Derived schemes and the cotangent complex of a morphism Definition
• Exact couple Definition
• Hochschild chains and Hochschild homology with coefficients Definition
• Homological spectral sequence Definition
• Koszul Regular And H One Regular Sequences Definition
• Left derived objects relative to supplied projective resolution data Definition
• Morse homology of a Morse--Smale pair Definition
• Simplicial cycles, boundaries, and homology Definition
• Simplicial objects, simplicial commutative rings and homotopy groups Definition
• The connecting morphism in homology Definition
• The evaluation map from cohomology to Hom of homology Definition
• The homology cross product for tensor complexes Definition
• The preconnecting arrow on cycles Definition
• The relative homology of a chain map Definition
• The singular chain complex and singular homology Definition
• A two-term complex has kernel and cokernel homology Example
• A unit pivot forces the minus Schur sign Example
• A zero-differential complex has homology equal to each term Example
• Universal-coefficient homology with cyclic coefficients Example
• H0 of the cotangent complex and the polynomial case Lemma
• Independence of the cotangent complex from the chosen simplicial resolution Lemma
• The cycle-boundary short exact sequences for a free complex over a PID Lemma
• The homological universal-coefficient edge map is well defined Lemma
• The low-degree filtration sequence of a first-quadrant bicomplex Lemma
• The preconnecting arrow annihilates boundaries Lemma
• The strict simplicial algebra adjunctions underlying the cotangent construction Lemma
• Degree-zero Hochschild homology is bimodule coinvariants Proposition
• Homology of a shift is shifted homology Proposition
• The handle chain complex computes singular homology Proposition
• A chain map induces a well-defined map on homology Theorem
• A chain map is a quasi-isomorphism exactly when its cone is acyclic Theorem
• A complex is exact at n exactly when its nth homology is zero Theorem
• An exact functor commutes with homology Theorem
• Chain-homotopic maps induce the same map on homology Theorem

…and 5 more results.

Dependency tree · two levels

8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• Charles A. Weibel, Chapter 1 of An Introduction to Homological Algebra (standard reference, not scraped)
• The Stacks Project, Section 12.13: Complexes (standard reference, not scraped)