Making chain complexes -- information about the basic constructors

Basic constructors

• complex(HashTable) -- make a chain complex
• complex(List) -- make a chain complex
• complex(Module) -- make a chain complex of length zero
• complex(Complex) -- make a complex by reindexing the terms of the complex
• Complex Array -- shift a complex or complex map
• isWellDefined(Complex) -- whether a complex is well-defined

Important computations creating new complexes

• freeResolution(Module) -- compute a free resolution of a module or ideal
• resolution(Complex) -- minimal free resolution of a complex
• HH Complex -- homology of a complex

More advanced constructors

• Complex ++ Complex -- direct sum of complexes
• Complex ** Complex -- tensor product of complexes
• Hom(Complex,Complex) -- the complex of homomorphisms between two complexes
• dual(Complex) -- make the dual of a complex
• RingMap Complex -- apply a ring map
• RingMap ** Complex -- tensor a complex along a ring map
• koszulComplex(Matrix) -- makes the Koszul complex
• naiveTruncation(Complex,ZZ,ZZ) -- drops all terms of a complex outside a given interval
• canonicalTruncation(Complex,ZZ,ZZ) -- reducing the number of non-zero terms of a complex
• minimalPresentation(Complex) -- minimal presentation of all terms in a complex
• minimize(Complex) -- a quasi-isomorphic complex whose terms have minimal rank
• gradedModule(Complex) -- a new complex in which the differential is zero
• part(List,Complex) -- extract a graded component of a complex
• truncate(List,Complex) -- truncation of a complex at a specified degree or set of degrees
• yonedaExtension(Matrix) -- creates a chain complex representing an extension of modules

Extracting complexes from complex maps

• source(ComplexMap) -- get the source of a map of chain complexes
• target(ComplexMap) -- get the target of a map of chain complexes
• kernel(ComplexMap) -- make the kernel of a map of complexes
• cokernel(ComplexMap) -- make the cokernel of a map of complexes
• image(ComplexMap) -- make the image of a map of complexes
• coimage(ComplexMap) -- make the coimage of a map of complexes
• cone(ComplexMap) -- make the mapping cone of a morphism of chain complexes
• cylinder(ComplexMap) -- make the mapping cylinder of a morphism of chain complexes