A basic algebra is a finite dimensional algebra A over a field, all of whose simple modules have dimension one. In the literature such an algebra is known as a "split" basic algebra. The type in Magma is optimized for the purposes of doing homological calculations.
Creation from a sequence of projective modules and a path tree for each module
Creation of the basic algebra corresponding to the group algebra of a p-group over GF(p).
Arithmetic
Extension and restriction of the coefficient ring
Tensor product
Opposite algebra
Construction of modules over basic algebras
Submodules, quotient modules, radicals and socles
Projective covers and injective hulls
Algebra as a right regular module over itself
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