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The homological delta-functor carried by homology of complexes

Definition

Fix an abelian category $A$ . For each integer $n$ , let $H_{n} : Ch (A) \to A$ be the homology functor of Homology is an additive functor. For every short exact sequence of complexes in $A$ , equip this family with the connecting morphisms $\partial_{n} : H_{n} (C) \to H_{n - 1} (A)$ of The connecting morphism in homology.

The resulting family $(H_{n}, \partial_{n})_{n \in Z}$ is the homological $δ$ -functor carried by homology of complexes. This page uses it only as this concrete example; the abstract theory is deferred to the later $δ$ -functor page.

Depends on

Used by

Homology as a homological delta functor Example
Homology of complexes satisfies the delta-functor naturality and exactness laws Proposition

Dependency tree · two levels

11 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)
Romyar Sharifi, Homological Algebra (standard reference, not scraped)