Complexes : Index

• - ComplexMap -- perform arithmetic operations on complex maps
• arithmetic with complex maps -- perform arithmetic operations on complex maps
• augmentationMap -- map from a free resolution to a module regarded as a complex
• augmentationMap(Complex) -- map from a free resolution to a module regarded as a complex
• Base -- make a chain complex
• Basic invariants and properties -- information about accessing basic features
• betti(Complex) -- display of degrees in a complex
• Boundary -- a random map of chain complexes
• canonicalMap -- gets the natural map arising from various constructions
• canonicalMap(...,UseTarget=>...) -- gets the natural map arising from various constructions
• canonicalMap(Complex,Complex) -- gets the natural map arising from various constructions
• canonicalTruncation -- reducing the number of non-zero terms of a complex
• canonicalTruncation(Complex,InfiniteNumber,InfiniteNumber) -- reducing the number of non-zero terms of a complex
• canonicalTruncation(Complex,InfiniteNumber,ZZ) -- reducing the number of non-zero terms of a complex
• canonicalTruncation(Complex,Nothing,ZZ) -- reducing the number of non-zero terms of a complex
• canonicalTruncation(Complex,Sequence) -- reducing the number of non-zero terms of a complex
• canonicalTruncation(Complex,ZZ,InfiniteNumber) -- reducing the number of non-zero terms of a complex
• canonicalTruncation(Complex,ZZ,Nothing) -- reducing the number of non-zero terms of a complex
• canonicalTruncation(Complex,ZZ,ZZ) -- reducing the number of non-zero terms of a complex
• canonicalTruncation(ComplexMap,InfiniteNumber,InfiniteNumber) -- reducing the number of non-zero terms of a complex
• canonicalTruncation(ComplexMap,InfiniteNumber,ZZ) -- reducing the number of non-zero terms of a complex
• canonicalTruncation(ComplexMap,Nothing,ZZ) -- reducing the number of non-zero terms of a complex
• canonicalTruncation(ComplexMap,Sequence) -- reducing the number of non-zero terms of a complex
• canonicalTruncation(ComplexMap,ZZ,InfiniteNumber) -- reducing the number of non-zero terms of a complex
• canonicalTruncation(ComplexMap,ZZ,Nothing) -- 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 ** ComplexMap -- the map of complexes between tensor complexes
• Complex ** Matrix -- create the tensor product of a complex and a map of modules
• Complex ** Module -- tensor product of complexes
• Complex ** Ring -- tensor a complex along a ring map
• Complex ** RingMap -- tensor a complex along a ring map
• Complex ++ Complex -- direct sum of complexes
• Complex == Complex -- whether two complexes are equal
• Complex == ZZ -- 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 -- 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(...,Base=>...) -- make a chain complex
• 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(Ideal) -- make a chain complex of length zero
• complex(List) -- make a chain complex
• complex(Module) -- make a chain complex of length zero
• complex(Ring) -- make a chain complex of length zero
• Complexes -- development package for beta testing new version of chain complexes
• ComplexMap -- the class of all maps between chain complexes
• ComplexMap * ComplexMap -- composition of homomorphisms of complexes
• ComplexMap * Number -- perform arithmetic operations on complex maps
• ComplexMap * RingElement -- perform arithmetic operations on complex maps
• ComplexMap ** Complex -- the map of complexes between tensor complexes
• ComplexMap ** ComplexMap -- the map of complexes between tensor complexes
• ComplexMap ** Module -- the map of complexes between tensor complexes
• ComplexMap ** Ring -- tensor a map of complexes along a ring map
• ComplexMap ** RingMap -- tensor a map of complexes along a ring map
• ComplexMap + ComplexMap -- perform arithmetic operations on complex maps
• ComplexMap + Number -- perform arithmetic operations on complex maps
• ComplexMap + RingElement -- perform arithmetic operations on complex maps
• ComplexMap ++ ComplexMap -- direct sum of complex maps
• ComplexMap - ComplexMap -- perform arithmetic operations on complex maps
• ComplexMap - Number -- perform arithmetic operations on complex maps
• ComplexMap - RingElement -- perform arithmetic operations on complex maps
• ComplexMap // ComplexMap -- lift a map of chain complexes along a quasi-isomorphism
• ComplexMap == ComplexMap -- whether two complex maps are equal
• ComplexMap == ZZ -- whether two complex maps are equal
• ComplexMap ^ Array -- the composition with the canonical inclusion or projection map
• ComplexMap ^ ZZ -- the n-fold composition
• ComplexMap _ Array -- the composition with the canonical inclusion or projection map
• ComplexMap _ ZZ -- access individual matrices in a complex map
• ComplexMap | ComplexMap -- join or concatenate maps horizontally
• ComplexMap || ComplexMap -- join or concatenate maps vertically
• ComplexMap Array -- shift a complex or complex map
• 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(Complex) -- 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 -- makes the connecting maps in Ext
• connectingExtMap(...,Concentration=>...) -- makes the connecting maps in Ext
• connectingExtMap(Matrix,Matrix,Module) -- makes the connecting maps in Ext
• connectingExtMap(Module,Matrix,Matrix) -- makes the connecting maps in Ext
• connectingMap -- construct the connecting homomorphism on homology
• connectingMap(...,Concentration=>...) -- construct the connecting homomorphism on homology
• connectingMap(ComplexMap,ComplexMap) -- construct the connecting homomorphism on homology
• connectingTorMap -- makes the connecting maps in Tor
• connectingTorMap(...,Concentration=>...) -- makes the connecting maps in Tor
• connectingTorMap(Matrix,Matrix,Module) -- makes the connecting maps in Tor
• connectingTorMap(Module,Matrix,Matrix) -- makes the connecting maps in Tor
• Cycle -- a random map of chain complexes
• cylinder -- make the mapping cylinder of a morphism of chain complexes
• cylinder(ComplexMap) -- make the mapping cylinder of a morphism of chain complexes
• 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
• directSum(ComplexMap) -- direct sum of complex maps
• dual(Complex) -- make the dual of a complex
• dual(ComplexMap) -- the dual of a map of complexes
• extend(Complex,Complex,Matrix) -- extend a map of modules to a map of chain 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(...,DegreeLimit=>...) -- compute a free resolution of a module or ideal
• freeResolution(...,FastNonminimal=>...) -- compute a free resolution of a module or ideal
• freeResolution(...,HardDegreeLimit=>...) -- compute a free resolution of a module or ideal
• freeResolution(...,LengthLimit=>...) -- compute a free resolution of a module or ideal
• freeResolution(...,PairLimit=>...) -- compute a free resolution of a module or ideal
• freeResolution(...,SortStrategy=>...) -- compute a free resolution of a module or ideal
• freeResolution(...,StopBeforeComputation=>...) -- compute a free resolution of a module or ideal
• freeResolution(...,Strategy=>...) -- compute a free resolution of a module or ideal
• freeResolution(...,SyzygyLimit=>...) -- compute a free resolution of a module or ideal
• freeResolution(Ideal) -- compute a free resolution of a module or ideal
• freeResolution(Matrix) -- compute the induced map between free resolutions
• freeResolution(Module) -- compute a free resolution of a module or ideal
• freeResolution(MonomialIdeal) -- compute a free resolution of a module or ideal
• 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
• HH^ZZ ComplexMap -- induced map on homology or cohomology
• HH_ZZ Complex -- homology or cohomology module of a complex
• HH_ZZ ComplexMap -- induced map on homology or cohomology
• Hom(Complex,Complex) -- the complex of homomorphisms between two complexes
• Hom(Complex,ComplexMap) -- the map of complexes between Hom complexes
• Hom(Complex,Matrix) -- the map of complexes between Hom complexes
• Hom(Complex,Module) -- the complex of homomorphisms between two complexes
• Hom(Complex,Ring) -- the complex of homomorphisms between two complexes
• Hom(ComplexMap,Complex) -- the map of complexes between Hom complexes
• Hom(ComplexMap,ComplexMap) -- the map of complexes between Hom complexes
• Hom(ComplexMap,Matrix) -- the map of complexes between Hom complexes
• Hom(ComplexMap,Module) -- the map of complexes between Hom complexes
• Hom(ComplexMap,Ring) -- the map of complexes between Hom complexes
• Hom(Matrix,Complex) -- the map of complexes between Hom complexes
• Hom(Matrix,ComplexMap) -- the map of complexes between Hom complexes
• Hom(Module,Complex) -- the complex of homomorphisms between two complexes
• Hom(Module,ComplexMap) -- the map of complexes between Hom complexes
• Hom(Ring,Complex) -- the complex of homomorphisms between two complexes
• Hom(Ring,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
• homotopyMap -- lift a map of chain complexes along a quasi-isomorphism
• homotopyMap(ComplexMap) -- lift a map of chain complexes along a quasi-isomorphism
• horseshoeResolution -- make the horseshoe resolution
• horseshoeResolution(...,LengthLimit=>...) -- make the horseshoe resolution
• horseshoeResolution(Complex) -- make the horseshoe resolution
• horseshoeResolution(Matrix,Matrix) -- 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
• InternalDegree -- a random map of chain complexes
• isCommutative(ComplexMap) -- whether a complex map commutes with the differentials
• isComplexMorphism -- whether a complex map is a morphism of complexes
• isComplexMorphism(ComplexMap) -- whether a complex map is a morphism of complexes
• isExact -- whether a complex is exact
• isExact(Complex) -- whether a complex is exact
• isExact(Complex,InfiniteNumber,InfiniteNumber) -- whether a complex is exact
• isExact(Complex,InfiniteNumber,Number) -- whether a complex is exact
• isExact(Complex,Number,InfiniteNumber) -- whether a complex is exact
• isExact(Complex,Number,Number) -- whether a complex is exact
• isFree -- whether a complex consists of free modules
• 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 -- whether a map of complexes is null-homotopic
• isNullHomotopic(ComplexMap) -- whether a map of complexes is null-homotopic
• isNullHomotopyOf -- whether the first map of chain complexes is a null homotopy for the second
• isNullHomotopyOf(ComplexMap,ComplexMap) -- whether the first map of chain complexes is a null homotopy for the second
• isQuasiIsomorphism -- whether a map of complexes is a quasi-isomorphism
• isQuasiIsomorphism(...,Concentration=>...) -- whether a map of complexes is a quasi-isomorphism
• isQuasiIsomorphism(ComplexMap) -- whether a map of complexes is a quasi-isomorphism
• isShortExactSequence -- whether a chain complex is a short exact sequence
• 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 -- makes the Koszul complex
• koszulComplex(List) -- makes the Koszul complex
• koszulComplex(List,Concentration=>...) -- makes the Koszul complex
• koszulComplex(Matrix) -- makes the Koszul complex
• koszulComplex(Matrix,Concentration=>...) -- makes the Koszul complex
• length(Complex) -- length of a complex
• liftMapAlongQuasiIsomorphism -- lift a map of chain complexes along a quasi-isomorphism
• liftMapAlongQuasiIsomorphism(ComplexMap,ComplexMap) -- lift a map of chain complexes along a quasi-isomorphism
• longExactSequence -- make the long exact sequence in homology
• longExactSequence(ComplexMap,ComplexMap) -- make the long exact sequence in homology
• longExactSequence(ComplexMap,ComplexMap,Concentration=>...) -- 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
• Matrix ** Complex -- create the tensor product of a complex and a map of modules
• max(Complex) -- indices on which a complex may be non-zero
• min(Complex) -- indices on which a complex may be non-zero
• minimalPresentation(Complex) -- minimal presentation of all terms in a complex
• minimalPresentation(ComplexMap) -- minimal presentation of all terms in a complex
• minimize -- a quasi-isomorphic complex whose terms have minimal rank
• minimize(Complex) -- a quasi-isomorphic complex whose terms have minimal rank
• minimizingMap -- a quasi-isomorphic complex whose terms have minimal rank
• Module ** Complex -- tensor product of complexes
• Module ** ComplexMap -- the map of complexes between tensor complexes
• naiveTruncation -- drops all terms of a complex outside a given interval
• naiveTruncation(Complex,InfiniteNumber,InfiniteNumber) -- drops all terms of a complex outside a given interval
• naiveTruncation(Complex,InfiniteNumber,ZZ) -- drops all terms of a complex outside a given interval
• naiveTruncation(Complex,Nothing,ZZ) -- drops all terms of a complex outside a given interval
• naiveTruncation(Complex,Sequence) -- drops all terms of a complex outside a given interval
• naiveTruncation(Complex,ZZ,InfiniteNumber) -- drops all terms of a complex outside a given interval
• naiveTruncation(Complex,ZZ,Nothing) -- drops all terms of a complex outside a given interval
• naiveTruncation(Complex,ZZ,ZZ) -- drops all terms of a complex outside a given interval
• naiveTruncation(ComplexMap,InfiniteNumber,InfiniteNumber) -- drops all terms in the source of a complex outside a given interval
• naiveTruncation(ComplexMap,InfiniteNumber,ZZ) -- drops all terms in the source of a complex outside a given interval
• naiveTruncation(ComplexMap,Nothing,ZZ) -- drops all terms in the source of a complex outside a given interval
• naiveTruncation(ComplexMap,Sequence) -- drops all terms in the source of a complex outside a given interval
• naiveTruncation(ComplexMap,Sequence,Sequence) -- drops all terms in the source of a complex outside a given interval
• naiveTruncation(ComplexMap,ZZ,InfiniteNumber) -- drops all terms in the source of a complex outside a given interval
• naiveTruncation(ComplexMap,ZZ,Nothing) -- drops all terms in the source 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 -- a map which is a candidate for being a null homotopy
• nullHomotopy(ComplexMap) -- a map which is a candidate for being a null homotopy
• Number * ComplexMap -- perform arithmetic operations on complex maps
• Number + ComplexMap -- perform arithmetic operations on complex maps
• Number - ComplexMap -- perform arithmetic operations on complex maps
• part(List,Complex) -- extract a graded component of a complex
• part(List,ComplexMap) -- extract a graded component of a map of complexes
• part(ZZ,Complex) -- extract a graded component of a complex
• part(ZZ,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
• prune(Complex) -- minimal presentation of all terms in a complex
• prune(ComplexMap) -- minimal presentation of all terms in a complex
• quotient(ComplexMap,ComplexMap) -- lift a map of chain complexes along a quasi-isomorphism
• randomComplexMap -- a random map of chain complexes
• randomComplexMap(...,Boundary=>...) -- a random map of chain complexes
• randomComplexMap(...,Cycle=>...) -- a random map of chain complexes
• randomComplexMap(...,Degree=>...) -- a random map of chain complexes
• randomComplexMap(...,InternalDegree=>...) -- a random map of chain complexes
• 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
• resolutionMap(...,DegreeLimit=>...) -- map from a free resolution to the given complex
• resolutionMap(...,FastNonminimal=>...) -- map from a free resolution to the given complex
• resolutionMap(...,HardDegreeLimit=>...) -- map from a free resolution to the given complex
• resolutionMap(...,LengthLimit=>...) -- map from a free resolution to the given complex
• resolutionMap(...,PairLimit=>...) -- map from a free resolution to the given complex
• resolutionMap(...,SortStrategy=>...) -- map from a free resolution to the given complex
• resolutionMap(...,StopBeforeComputation=>...) -- map from a free resolution to the given complex
• resolutionMap(...,Strategy=>...) -- map from a free resolution to the given complex
• resolutionMap(...,SyzygyLimit=>...) -- map from a free resolution to the given complex
• resolutionMap(Complex) -- map from a free resolution to the given complex
• Ring ** Complex -- tensor a complex along a ring map
• Ring ** ComplexMap -- tensor a map of complexes along a ring map
• ring(Complex) -- access the ring of a complex or a complex map
• ring(ComplexMap) -- access the ring of a complex or a complex map
• RingElement * ComplexMap -- perform arithmetic operations on complex maps
• RingElement + ComplexMap -- perform arithmetic operations on complex maps
• RingElement - ComplexMap -- perform arithmetic operations on complex maps
• 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
• sum(Complex) -- make the direct sum of all terms
• sum(ComplexMap) -- make the direct sum of all terms
• Symbol ^ Complex -- get the maps between the terms in a complex
• target(ComplexMap) -- get the target of a map of chain complexes
• tensor(Complex,Complex) -- tensor product of complexes
• tensor(Complex,RingMap) -- tensor a complex along a ring map
• tensor(ComplexMap,ComplexMap) -- the map of complexes between tensor complexes
• tensor(ComplexMap,RingMap) -- tensor a map of complexes along a ring map
• tensor(RingMap,Complex) -- tensor a complex along a ring map
• tensor(RingMap,ComplexMap) -- tensor a map of complexes along a ring map
• tensorAssociativity(Complex,Complex,Complex) -- make the canonical isomorphism arising from associativity
• tensorCommutativity -- make the canonical isomorphism arising from commutativity
• 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
• Tor_ZZ(Module,Matrix) -- make the induced map on Tor modules
• torSymmetry -- makes the canonical isomorphism realizing the symmetry of Tor
• 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
• truncate(ZZ,Complex) -- truncation of a complex at a specified degree or set of degrees
• truncate(ZZ,ComplexMap) -- truncation of a complex map at a specified degree or set of degrees
• UseTarget -- gets the natural map arising from various constructions
• Working with Ext -- information about functorial properties
• Working with Tor -- information about functorial properties
• yonedaExtension -- creates a chain complex representing an extension of modules
• yonedaExtension' -- identifies the element of Ext corresponding to an extension
• yonedaExtension'(Complex) -- identifies the element of Ext corresponding to an extension
• yonedaExtension(Matrix) -- creates a chain complex representing an extension of modules
• yonedaMap -- creates a chain complex map representing an extension of modules
• yonedaMap' -- identifies the element of Ext corresponding to a map of free resolutions
• yonedaMap'(...,LengthLimit=>...) -- identifies the element of Ext corresponding to a map of free resolutions
• yonedaMap'(ComplexMap) -- identifies the element of Ext corresponding to a map of free resolutions
• yonedaMap(...,LengthLimit=>...) -- creates a chain complex map representing an extension of modules
• yonedaMap(Matrix) -- creates a chain complex map representing an extension of modules
• yonedaProduct -- make the product of two elements in Ext modules
• yonedaProduct(Matrix,Matrix) -- make the product of two elements in Ext modules
• yonedaProduct(Module,Module) -- make the product map between Ext modules
• ZZ == Complex -- whether two complexes are equal
• ZZ == ComplexMap -- whether two complex maps are equal