Chain complex in an abelian category

Depends on

• Abelian category

Used by

• Gaussian cancellation preserves homotopy type and abelian-category homology Corollary
• Morse homology recovers the Morse inequalities Corollary
• A chain complex with coefficients obtained by tensoring Definition
• A graded morphism of chain complexes Definition
• Augmented chain complexes over an object Definition
• Bounded, bounded below, and bounded above complexes Definition
• Chain map Definition
• Cochain complex in an abelian category Definition
• Cycle and boundary subobjects of a complex Definition
• Exactness of a complex at a degree and acyclic complexes Definition
• Finite based free complexes and contraction torsion Definition
• Homological double complex Definition
• Koszul Complex Of A Sequence With Coefficients Definition
• Minimal Free Resolution Over A Local Ring Definition
• Morse homology of a Morse--Smale pair Definition
• Simplicial objects, simplicial commutative rings and homotopy groups Definition
• Subcomplex Definition
• The BGG differential from signed Verma maps Definition
• The cotangent complex of a ring map Definition
• The Hom cochain complex of a chain complex Definition
• The shift of a chain complex Definition
• The standard induced resolution of the trivial module Definition
• The tensor product of a right and a left chain complex is totalized by direct sums with the Koszul differential Definition
• Zero complex and stalk complex Definition
• FALSE: any sequence of morphisms is a chain complex False statement
• A chain contraction makes the odd-to-even parity map invertible Lemma
• A lifted finite CW equivalence has a contractible group-ring mapping cone Lemma
• Arbitrary metric Morse--Smale end counts form finite Morse chain complexes Lemma
• Bicosimplicial total complexes agree with diagonal totalization Lemma
• Composition factors of the BGG kernel lie above the degree (BGG 10.6a) Lemma
• Long exact sequence of a triple in singular homology Lemma
• Normalized simplicial chains, prism homotopies and the abelian trivial-fibration criterion Lemma
• The boundary subobject factors through the cycle subobject Lemma
• The cycle-boundary short exact sequences for a free complex over a PID Lemma
• The differential descends to a quotient complex Lemma
• The low-degree filtration sequence of a first-quadrant bicomplex Lemma
• An additive functor applies degreewise to complexes and chain maps Proposition
• An exact sequence is a complex, and its exactness agrees with the earlier notion Proposition
• Products and coproducts of complexes are degreewise when they exist and preserve differentials Proposition
• The BGG differential squares to zero Proposition

…and 8 more results.

Dependency tree · two levels

3 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)