Chain Complexes

Complexes of modules are a fundamental object in homological algebra. Conceptually, a complex is an infinite sequence of modules, indexed by integers, with maps between successive modules such that the composition of any two maps is zero.

• Creation of a complex from a list of A-modules
• Subcomplexes and quotient complexes
• Operations on complexes: Splice, shift, direct sum
• Exact extensions, zero extensions
• Dual of a complex
• Homology groups of a complex
• Boundary maps
• Construction of chain maps between complexes
• Composition of chain maps
• Image, kernel and cokernel of a chain map
• Predicates for chain maps: Surjection, injection, isomorphism
• Injective resolution (for modules over a basic algebra)
• Projective resolution (for modules over a basic algebra)
• Extending cohomology elements as chain maps
• Maps induced on homology by chain maps, long exact homology sequence