Complexes : Table of Contents

• Complexes -- development package for beta testing new version of chain complexes
• arithmetic with complex maps -- perform arithmetic operations on complex maps
• augmentationMap -- map from a free resolution to a module regarded as a complex
• Basic invariants and properties -- information about accessing basic features
• betti(Complex) -- display of degrees in a complex
• canonicalMap -- gets the natural map arising from various constructions
• canonicalTruncation(Complex,ZZ,ZZ) -- reducing the number of non-zero terms of a complex
• canonicalTruncation(ComplexMap,ZZ,ZZ) -- reducing the number of non-zero terms of a complex
• chainComplex(Complex) -- translate between data types for chain complexes
• chainComplex(ComplexMap) -- translate between data types for chain complexes
• coimage(ComplexMap) -- make the coimage of a map of complexes
• cokernel(ComplexMap) -- make the cokernel of a map of complexes
• Complex -- the class of all chain complexes
• complex -- make a chain complex
• Complex ** Complex -- tensor product of complexes
• Complex ** Matrix -- create the tensor product of a complex and a map of modules
• Complex == Complex -- whether two complexes are equal
• Complex _ Array -- the canonical inclusion or projection map of a direct sum
• Complex _ ZZ -- access individual object in a complex
• Complex Array -- shift a complex or complex map
• complex(ChainComplex) -- translate between data types for chain complexes
• complex(ChainComplexMap) -- translate between data types for chain complex maps
• complex(Complex) -- make a complex by reindexing the terms of the complex
• complex(ComplexMap) -- make a complex by specifying the differential
• complex(HashTable) -- make a chain complex
• complex(Module) -- make a chain complex of length zero
• ComplexMap -- the class of all maps between chain complexes
• ComplexMap * ComplexMap -- composition of homomorphisms of complexes
• ComplexMap ** ComplexMap -- the map of complexes between tensor complexes
• ComplexMap ++ ComplexMap -- direct sum of complex maps
• ComplexMap == ComplexMap -- whether two complex maps are equal
• ComplexMap ^ Array -- the composition with the canonical inclusion or projection map
• ComplexMap ^ ZZ -- the n-fold composition
• ComplexMap _ ZZ -- access individual matrices in a complex map
• ComplexMap | ComplexMap -- join or concatenate maps horizontally
• ComplexMap || ComplexMap -- join or concatenate maps vertically
• components(Complex) -- list the components of a direct sum
• components(ComplexMap) -- list the components of a direct sum
• Concentration -- optional argument used to specify the concentration
• concentration -- indices on which a complex may be non-zero
• concentration(ComplexMap) -- indices on which a complex map may be non-zero
• cone(ComplexMap) -- make the mapping cone of a morphism of chain complexes
• connectingExtMap(Module,Matrix,Matrix) -- makes the connecting maps in Ext
• connectingMap(ComplexMap,ComplexMap) -- construct the connecting homomorphism on homology
• connectingTorMap(Module,Matrix,Matrix) -- makes the connecting maps in Tor
• cylinder -- make the mapping cylinder of a morphism of chain complexes
• Default strategy for free resolutions of homogeneous modules -- algorithm for computing free resolutions exploiting the Schreyer frame
• degree(ComplexMap) -- get the degree of a map of chain complexes
• differential of a chain complex -- get the maps between the terms in a complex
• directSum(Complex) -- direct sum of complexes
• dual(Complex) -- make the dual of a complex
• dual(ComplexMap) -- the dual of a map of complexes
• extend(Complex,Complex,Matrix,Sequence) -- extend a map of modules to a map of chain complexes
• freeResolution -- compute a free resolution of a module or ideal
• freeResolution(..., Strategy => Engine) -- algorithm for computing a free resolution
• freeResolution(...,LengthLimit=>...) -- optional arguments for freeResolution
• freeResolution(Matrix) -- compute the induced map between free resolutions
• gradedModule(Complex) -- a new complex in which the differential is zero
• HH Complex -- homology of a complex
• HH ComplexMap -- induced map on homology or cohomology
• HH_ZZ Complex -- homology or cohomology module of a complex
• Hom(Complex,Complex) -- the complex of homomorphisms between two complexes
• Hom(ComplexMap,ComplexMap) -- the map of complexes between Hom complexes
• homomorphism'(ComplexMap) -- get the element of Hom from a map of complexes
• homomorphism(ComplexMap) -- get the homomorphism from an element of Hom
• homomorphism(ZZ,Matrix,Complex) -- get the homomorphism from an element of Hom
• horseshoeResolution(Complex) -- make the horseshoe resolution
• id _ Complex -- the identity map of a chain complex
• image(ComplexMap) -- make the image of a map of complexes
• inducedMap(Complex,Complex) -- make the map of complexes induced at each term by the identity map
• isCommutative(ComplexMap) -- whether a complex map commutes with the differentials
• isComplexMorphism(ComplexMap) -- whether a complex map is a morphism of complexes
• isExact -- whether a complex is exact
• isFree(Complex) -- whether a complex consists of free modules
• isHomogeneous(Complex) -- whether a complex is homogeneous
• isHomogeneous(ComplexMap) -- whether a map of complexes is homogeneous
• isNullHomotopic(ComplexMap) -- whether a map of complexes is null-homotopic
• isNullHomotopyOf(ComplexMap,ComplexMap) -- whether the first map of chain complexes is a null homotopy for the second
• isQuasiIsomorphism(ComplexMap) -- whether a map of complexes is a quasi-isomorphism
• isShortExactSequence(Complex) -- whether a chain complex is a short exact sequence
• isShortExactSequence(ComplexMap,ComplexMap) -- whether a pair of complex maps forms a short exact sequence
• isShortExactSequence(Matrix,Matrix) -- whether a pair of matrices forms a short exact sequence
• isWellDefined(Complex) -- whether a complex is well-defined
• isWellDefined(ComplexMap) -- whether a map of chain complexes is well-defined
• kernel(ComplexMap) -- make the kernel of a map of complexes
• koszulComplex(Matrix) -- makes the Koszul complex
• length(Complex) -- length of a complex
• liftMapAlongQuasiIsomorphism(ComplexMap,ComplexMap) -- lift a map of chain complexes along a quasi-isomorphism
• longExactSequence(ComplexMap,ComplexMap) -- make the long exact sequence in homology
• Making chain complexes -- information about the basic constructors
• Making maps between chain complexes -- information about the basic constructors
• map(Complex,Complex,ComplexMap) -- make a new map of chain complexes from an existing one
• map(Complex,Complex,Function) -- make a map of chain complexes
• map(Complex,Complex,HashTable) -- make a map of chain complexes
• map(Complex,Complex,List) -- make a map of chain complexes
• map(Complex,Complex,ZZ) -- make the zero map or identity between chain complexes
• minimalPresentation(Complex) -- minimal presentation of all terms in a complex
• minimize(Complex) -- a quasi-isomorphic complex whose terms have minimal rank
• naiveTruncation(Complex,ZZ,ZZ) -- drops all terms of a complex outside a given interval
• naiveTruncation(ComplexMap,ZZ,ZZ) -- drops all terms in the source of a complex outside a given interval
• nullHomotopy(ComplexMap) -- a map which is a candidate for being a null homotopy
• part(List,Complex) -- extract a graded component of a complex
• part(List,ComplexMap) -- extract a graded component of a map of complexes
• poincare(Complex) -- assemble degrees of a chain complex into a polynomial
• poincareN(Complex) -- assemble degrees of a chain complex into a polynomial
• randomComplexMap(Complex,Complex) -- a random map of chain complexes
• regularity(Complex) -- compute the Castelnuovo-Mumford regularity
• resolution(Complex) -- minimal free resolution of a complex
• resolutionMap -- map from a free resolution to the given complex
• ring(Complex) -- access the ring of a complex or a complex map
• RingMap ** Complex -- tensor a complex along a ring map
• RingMap ** ComplexMap -- tensor a map of complexes along a ring map
• RingMap Complex -- apply a ring map
• RingMap ComplexMap -- apply a ring map to a map of complexes
• source(ComplexMap) -- get the source of a map of chain complexes
• Strategies for free resolutions -- overview of the different algorithms for computing free resolutions
• Strategy for free resolutions of homogeneous modules aided by Hilbert functions -- algorithm for computing free resolutions step by step aided by Hilbert functions
• Strategy for free resolutions of homogeneous modules via successive syzygies -- algorithm for computing free resolutions step by step
• Strategy for free resolutions over a field -- algorithm for computing free resolutions over a field
• Strategy for free resolutions over the integers -- algorithm for computing free resolutions of ZZ-modules
• Strategy for free resolutions via homogenization -- algorithm for computing free resolutions by first homogenizing
• Strategy for free resolutions via Schreyer-Lascala -- algorithm for computing free resolutions exploiting the Schreyer frame
• Strategy for free resolutions via syzygies -- algorithm for computing free resolutions step by step using syzygies
• Strategy for nonminimal free resolutions -- algorithm for computing nonminimal free resolutions
• sum(Complex) -- make the direct sum of all terms
• target(ComplexMap) -- get the target of a map of chain complexes
• tensorAssociativity(Complex,Complex,Complex) -- make the canonical isomorphism arising from associativity
• tensorCommutativity(Complex,Complex) -- make the canonical isomorphism arising from commutativity
• tensorCommutativity(Module,Module) -- make the canonical isomorphism arising from commutativity
• Tor_ZZ(Matrix,Module) -- make the induced map on Tor modules
• torSymmetry(ZZ,Module,Module) -- makes the canonical isomorphism realizing the symmetry of Tor
• Towards computing in the derived category
• truncate(List,Complex) -- truncation of a complex at a specified degree or set of degrees
• truncate(List,ComplexMap) -- truncation of a complex map at a specified degree or set of degrees
• Working with Ext -- information about functorial properties
• Working with Tor -- information about functorial properties
• yonedaExtension'(Complex) -- identifies the element of Ext corresponding to an extension
• yonedaExtension(Matrix) -- creates a chain complex representing an extension of modules
• yonedaMap'(ComplexMap) -- identifies the element of Ext corresponding to a map of free resolutions
• yonedaMap(Matrix) -- creates a chain complex map representing an extension of modules
• yonedaProduct(Matrix,Matrix) -- make the product of two elements in Ext modules
• yonedaProduct(Module,Module) -- make the product map between Ext modules