The long exact sequence in homology

Depends on

• The connecting morphism in homology
• A chain map induces a well-defined map on homology
• Exactness at the homology of the left complex
• Exactness at the homology of the middle complex
• Exactness at the homology of the right complex
• Exactness at the target of the connecting map

Used by

• A short exact sequence with acyclic middle complex identifies neighbouring homology Corollary
• Homology of a chain-split direct-sum sequence Corollary
• The long exact homology sequence is natural Corollary
• Two-out-of-three for acyclicity in a short exact sequence of complexes Corollary
• FALSE: the homology functor is exact on short exact sequences of complexes False statement
• A collar constructs the relative orientation class and its boundary class Lemma
• bounded finite length complex euler identities Lemma
• Cap product and the Mayer–Vietoris duality ladder Lemma
• Cellular reduction for a highly connected pair Lemma
• Cubical pinch is additive on relative homology Lemma
• Derived tensor product of abelian sheaves Lemma
• koszul euler characteristic first element reduction Lemma
• Long exact sequence of a triple in singular homology Lemma
• Reduction of an exact free complex by a nonzerodivisor stays exact above degree one Lemma
• Relative homology Mayer–Vietoris for closed supports Lemma
• shifted adic koszul filtration euler comparison Lemma
• The cycle-boundary tensor sequence has the Kunneth kernel and cokernel Lemma
• The double cover branched over a slice disk is a rational homology ball Lemma
• The Khovanov-Seidel grid resolutions of the vertex modules Lemma
• The Kunneth Tor map Lemma
• A short exact sequence of complexes gives six-term exact sequences when homology is concentrated in two degrees Proposition
• An exact functor carries the long exact homology sequence to the corresponding long exact sequence Proposition
• Homology of complexes satisfies the delta-functor naturality and exactness laws Proposition
• The connecting morphism vanishes for a chain-split short exact sequence Proposition
• The Tor balance isomorphism is natural and coherent under a change of resolutions Proposition
• A filtered complex produces an exact couple Theorem
• A left module is flat exactly when Tor one against every right module vanishes Theorem
• Canonical truncations fit a distinguished triangle Theorem
• Excision and Mayer–Vietoris with local coefficients Theorem
• Finite-generation cohomological UCT gives integral cone comparison Theorem
• Functoriality and coefficient long exact sequences for Hochschild homology Theorem
• Integral finite generation of universal real and oriented Thom homology Theorem
• Long exact sequence of a pair Theorem
• Mayer–Vietoris sequence in singular homology Theorem
• Simplicial and singular homology agree Theorem
• The acyclic-resolution theorem for left derived functors Theorem
• The cone long exact sequence Theorem
• The Kunneth theorem for free complexes over a PID Theorem
• The left and right projective constructions of Tor are naturally isomorphic Theorem
• The long exact sequence in cohomology Theorem

…and 3 more results.

Dependency tree · two levels

13 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)