Complexes -- development package for beta testing new version of chain complexes

Description

This package develops new data types and routines for homological algebra. Eventually, it will replace the current facilities for homological algebra. We are making this available in order to get feedback from users before making this change. Please email the authors with any and all comments or suggestions.

The overarching goal is to make all of the homological algebra routines functorial. For instance, we have canonical maps associated to kernels, cokernels, images, coimages, cones, and cylinders.

Authors

Version

This documentation describes version 0.999995 of Complexes.

Citation

If you have used this package in your research, please cite it as follows:

@misc{ComplexesSource,
  title = {{Complexes: beta testing new version of chain complexes. Version~0.999995}},
  author = {Gregory G. Smith and Mike Stillman},
  howpublished = {A \emph{Macaulay2} package available at
    \url{https://github.com/Macaulay2/M2/tree/master/M2/Macaulay2/packages}}
}

Exports

Types
- Complex -- the class of all chain complexes
- ComplexMap -- the class of all maps between chain complexes
Functions and commands
- augmentationMap -- map from a free resolution to a module regarded as a complex
- canonicalMap -- gets the natural map arising from various constructions
- canonicalTruncation -- see canonicalTruncation(Complex,ZZ,ZZ) -- reducing the number of non-zero terms of a complex
- complex -- see complex(List) -- make a chain complex
- concentration -- indices on which a complex may be non-zero
- connectingExtMap -- see connectingExtMap(Module,Matrix,Matrix) -- makes the connecting maps in Ext
- connectingMap -- see connectingMap(ComplexMap,ComplexMap) -- construct the connecting homomorphism on homology
- connectingTorMap -- see connectingTorMap(Module,Matrix,Matrix) -- makes the connecting maps in Tor
- constantStrand (missing documentation)
- cylinder -- make the mapping cylinder of a morphism of chain complexes
- epicResolutionMap (missing documentation)
- freeResolution -- compute a free resolution of a module or ideal
- horseshoeResolution -- see horseshoeResolution(Complex) -- make the horseshoe resolution
- isComplexMorphism -- see isComplexMorphism(ComplexMap) -- whether a complex map is a morphism of complexes
- isFree -- see isFree(Complex) -- whether a complex consists of free modules
- isNullHomotopic -- see isNullHomotopic(ComplexMap) -- whether a map of complexes is null-homotopic
- isNullHomotopyOf -- see isNullHomotopyOf(ComplexMap,ComplexMap) -- whether the first map of chain complexes is a null homotopy for the second
- isQuasiIsomorphism -- see isQuasiIsomorphism(ComplexMap) -- whether a map of complexes is a quasi-isomorphism
- isShortExactSequence -- see isShortExactSequence(Complex) -- whether a chain complex is a short exact sequence
- koszulComplex -- see koszulComplex(Matrix) -- makes the Koszul complex
- homotopyMap -- see liftMapAlongQuasiIsomorphism(ComplexMap,ComplexMap) -- lift a map of chain complexes along a quasi-isomorphism
- liftMapAlongQuasiIsomorphism -- see liftMapAlongQuasiIsomorphism(ComplexMap,ComplexMap) -- lift a map of chain complexes along a quasi-isomorphism
- longExactSequence -- see longExactSequence(ComplexMap,ComplexMap) -- make the long exact sequence in homology
- naiveTruncation -- see naiveTruncation(Complex,ZZ,ZZ) -- drops all terms of a complex outside a given interval
- nullHomotopy -- see nullHomotopy(ComplexMap) -- a map which is a candidate for being a null homotopy
- randomComplexMap -- see randomComplexMap(Complex,Complex) -- a random map of chain complexes
- resolutionMap -- map from a free resolution to the given complex
- tensorCommutativity -- see tensorCommutativity(Module,Module) -- make the canonical isomorphism arising from commutativity
- torSymmetry -- see torSymmetry(ZZ,Module,Module) -- makes the canonical isomorphism realizing the symmetry of Tor
- yonedaExtension' -- see yonedaExtension'(Complex) -- identifies the element of Ext corresponding to an extension
- yonedaExtension -- see yonedaExtension(Matrix) -- creates a chain complex representing an extension of modules
- yonedaMap' -- see yonedaMap'(ComplexMap) -- identifies the element of Ext corresponding to a map of free resolutions
- yonedaMap -- see yonedaMap(Matrix) -- creates a chain complex map representing an extension of modules
- yonedaProduct -- see yonedaProduct(Matrix,Matrix) -- make the product of two elements in Ext modules
Methods
- - ComplexMap -- see arithmetic with complex maps -- perform arithmetic operations on complex maps
- ComplexMap * Number -- see arithmetic with complex maps -- perform arithmetic operations on complex maps
- ComplexMap * RingElement -- see arithmetic with complex maps -- perform arithmetic operations on complex maps
- ComplexMap + ComplexMap -- see arithmetic with complex maps -- perform arithmetic operations on complex maps
- ComplexMap + Number -- see arithmetic with complex maps -- perform arithmetic operations on complex maps
- ComplexMap + RingElement -- see arithmetic with complex maps -- perform arithmetic operations on complex maps
- ComplexMap - ComplexMap -- see arithmetic with complex maps -- perform arithmetic operations on complex maps
- ComplexMap - Number -- see arithmetic with complex maps -- perform arithmetic operations on complex maps
- ComplexMap - RingElement -- see arithmetic with complex maps -- perform arithmetic operations on complex maps
- Number * ComplexMap -- see arithmetic with complex maps -- perform arithmetic operations on complex maps
- Number + ComplexMap -- see arithmetic with complex maps -- perform arithmetic operations on complex maps
- Number - ComplexMap -- see arithmetic with complex maps -- perform arithmetic operations on complex maps
- RingElement * ComplexMap -- see arithmetic with complex maps -- perform arithmetic operations on complex maps
- RingElement + ComplexMap -- see arithmetic with complex maps -- perform arithmetic operations on complex maps
- RingElement - ComplexMap -- see arithmetic with complex maps -- perform arithmetic operations on complex maps
- augmentationMap(Complex) -- see augmentationMap -- map from a free resolution to a module regarded as a complex
- basis(List,Complex) (missing documentation)
- basis(List,ComplexMap) (missing documentation)
- basis(ZZ,Complex) (missing documentation)
- basis(ZZ,ComplexMap) (missing documentation)
- betti(Complex) -- display of degrees in a complex
- canonicalMap(Complex,Complex) -- see canonicalMap -- gets the natural map arising from various constructions
- canonicalTruncation(Complex,InfiniteNumber,InfiniteNumber) -- see canonicalTruncation(Complex,ZZ,ZZ) -- reducing the number of non-zero terms of a complex
- canonicalTruncation(Complex,InfiniteNumber,ZZ) -- see canonicalTruncation(Complex,ZZ,ZZ) -- reducing the number of non-zero terms of a complex
- canonicalTruncation(Complex,Nothing,ZZ) -- see canonicalTruncation(Complex,ZZ,ZZ) -- reducing the number of non-zero terms of a complex
- canonicalTruncation(Complex,Sequence) -- see canonicalTruncation(Complex,ZZ,ZZ) -- reducing the number of non-zero terms of a complex
- canonicalTruncation(Complex,ZZ,InfiniteNumber) -- see canonicalTruncation(Complex,ZZ,ZZ) -- reducing the number of non-zero terms of a complex
- canonicalTruncation(Complex,ZZ,Nothing) -- see canonicalTruncation(Complex,ZZ,ZZ) -- 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) -- see canonicalTruncation(ComplexMap,ZZ,ZZ) -- reducing the number of non-zero terms of a complex
- canonicalTruncation(ComplexMap,InfiniteNumber,ZZ) -- see canonicalTruncation(ComplexMap,ZZ,ZZ) -- reducing the number of non-zero terms of a complex
- canonicalTruncation(ComplexMap,Nothing,ZZ) -- see canonicalTruncation(ComplexMap,ZZ,ZZ) -- reducing the number of non-zero terms of a complex
- canonicalTruncation(ComplexMap,Sequence) -- see canonicalTruncation(ComplexMap,ZZ,ZZ) -- reducing the number of non-zero terms of a complex
- canonicalTruncation(ComplexMap,ZZ,InfiniteNumber) -- see canonicalTruncation(ComplexMap,ZZ,ZZ) -- reducing the number of non-zero terms of a complex
- canonicalTruncation(ComplexMap,ZZ,Nothing) -- see canonicalTruncation(ComplexMap,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
- coimage(ComplexMap) -- make the coimage of a map of complexes
- cokernel(ComplexMap) -- make the cokernel of a map of complexes
- complete(Complex) (missing documentation)
- complete(ComplexMap) (missing documentation)
- Complex ** Complex -- tensor product of complexes
- Complex ** Module -- see Complex ** Complex -- tensor product of complexes
- Module ** Complex -- see Complex ** Complex -- tensor product of complexes
- tensor(Complex,Complex) -- see Complex ** Complex -- tensor product of complexes
- Complex ** Matrix -- create the tensor product of a complex and a map of modules
- Matrix ** Complex -- see Complex ** Matrix -- create the tensor product of a complex and a map of modules
- Complex == Complex -- whether two complexes are equal
- Complex == ZZ -- see Complex == Complex -- whether two complexes are equal
- ZZ == Complex -- see Complex == Complex -- whether two complexes are equal
- Complex ^ Array -- see Complex _ Array -- the canonical inclusion or projection map of a direct sum
- Complex _ Array -- the canonical inclusion or projection map of a direct sum
- Complex ^ ZZ -- see Complex _ ZZ -- access individual object in a complex
- Complex _ ZZ -- access individual object in a complex
- Complex Array -- shift a complex or complex map
- ComplexMap Array -- see Complex Array -- shift a complex or complex map
- 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(List) -- make a chain complex
- complex(Matrix) -- see complex(List) -- make a chain complex
- complex(Ideal) -- see complex(Module) -- make a chain complex of length zero
- complex(Module) -- make a chain complex of length zero
- complex(Ring) -- see complex(Module) -- make a chain complex of length zero
- ComplexMap * ComplexMap -- composition of homomorphisms of complexes
- Complex ** ComplexMap -- see ComplexMap ** ComplexMap -- the map of complexes between tensor complexes
- ComplexMap ** Complex -- see ComplexMap ** ComplexMap -- the map of complexes between tensor complexes
- ComplexMap ** ComplexMap -- the map of complexes between tensor complexes
- ComplexMap ** Module -- see ComplexMap ** ComplexMap -- the map of complexes between tensor complexes
- Module ** ComplexMap -- see ComplexMap ** ComplexMap -- the map of complexes between tensor complexes
- tensor(ComplexMap,ComplexMap) -- see ComplexMap ** ComplexMap -- the map of complexes between tensor complexes
- ComplexMap ++ ComplexMap -- direct sum of complex maps
- directSum(ComplexMap) -- see ComplexMap ++ ComplexMap -- direct sum of complex maps
- ComplexMap == ComplexMap -- whether two complex maps are equal
- ComplexMap == ZZ -- see ComplexMap == ComplexMap -- whether two complex maps are equal
- ZZ == ComplexMap -- see ComplexMap == ComplexMap -- whether two complex maps are equal
- ComplexMap ^ Array -- the composition with the canonical inclusion or projection map
- ComplexMap _ Array -- see 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(Complex) -- see concentration -- indices on which a complex may be non-zero
- max(Complex) -- see concentration -- indices on which a complex may be non-zero
- min(Complex) -- see 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(Matrix,Matrix,Module) -- see connectingExtMap(Module,Matrix,Matrix) -- makes the connecting maps in Ext
- connectingExtMap(Module,Matrix,Matrix) -- makes the connecting maps in Ext
- connectingMap(ComplexMap,ComplexMap) -- construct the connecting homomorphism on homology
- connectingTorMap(Matrix,Matrix,Module) -- see connectingTorMap(Module,Matrix,Matrix) -- makes the connecting maps in Tor
- connectingTorMap(Module,Matrix,Matrix) -- makes the connecting maps in Tor
- constantStrand(Complex,List) (missing documentation)
- constantStrand(Complex,ZZ) (missing documentation)
- cylinder(ComplexMap) -- see cylinder -- make the mapping cylinder of a morphism of chain complexes
- degree(ComplexMap) -- get the degree of a map of chain complexes
- Symbol ^ Complex -- see differential of a chain complex -- get the maps between the terms in a complex
- Complex ++ Complex -- see directSum(Complex) -- direct sum of complexes
- directSum(Complex) -- direct sum of complexes
- dual(Complex) -- make the dual of a complex
- dual(ComplexMap) -- the dual of a map of complexes
- transpose(ComplexMap) -- see dual(ComplexMap) -- the dual of a map of complexes
- epicResolutionMap(Complex) (missing documentation)
- extend(Complex,Complex,Matrix) -- see extend(Complex,Complex,Matrix,Sequence) -- 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
- formation(Complex) (missing documentation)
- freeResolution(Ideal) -- see freeResolution -- compute a free resolution of a module or ideal
- freeResolution(Module) -- see freeResolution -- compute a free resolution of a module or ideal
- freeResolution(MonomialIdeal) -- see freeResolution -- compute a free resolution of a module or ideal
- freeResolution(Complex) -- minimal free resolution of a complex
- 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 ComplexMap -- see HH ComplexMap -- induced map on homology or cohomology
- HH_ZZ ComplexMap -- see HH ComplexMap -- induced map on homology or cohomology
- HH^ZZ Complex -- see HH_ZZ Complex -- homology or cohomology module of a complex
- HH_ZZ Complex -- homology or cohomology module of a complex
- Hom(Complex,Complex) -- the complex of homomorphisms between two complexes
- Hom(Complex,Module) -- see Hom(Complex,Complex) -- the complex of homomorphisms between two complexes
- Hom(Complex,Ring) -- see Hom(Complex,Complex) -- the complex of homomorphisms between two complexes
- Hom(Module,Complex) -- see Hom(Complex,Complex) -- the complex of homomorphisms between two complexes
- Hom(Ring,Complex) -- see Hom(Complex,Complex) -- the complex of homomorphisms between two complexes
- Hom(Complex,ComplexMap) -- see Hom(ComplexMap,ComplexMap) -- the map of complexes between Hom complexes
- Hom(Complex,Matrix) -- see Hom(ComplexMap,ComplexMap) -- the map of complexes between Hom complexes
- Hom(ComplexMap,Complex) -- see Hom(ComplexMap,ComplexMap) -- the map of complexes between Hom complexes
- Hom(ComplexMap,ComplexMap) -- the map of complexes between Hom complexes
- Hom(ComplexMap,Matrix) -- see Hom(ComplexMap,ComplexMap) -- the map of complexes between Hom complexes
- Hom(ComplexMap,Module) -- see Hom(ComplexMap,ComplexMap) -- the map of complexes between Hom complexes
- Hom(ComplexMap,Ring) -- see Hom(ComplexMap,ComplexMap) -- the map of complexes between Hom complexes
- Hom(Matrix,Complex) -- see Hom(ComplexMap,ComplexMap) -- the map of complexes between Hom complexes
- Hom(Matrix,ComplexMap) -- see Hom(ComplexMap,ComplexMap) -- the map of complexes between Hom complexes
- Hom(Module,ComplexMap) -- see Hom(ComplexMap,ComplexMap) -- the map of complexes between Hom complexes
- Hom(Ring,ComplexMap) -- see 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
- horseshoeResolution(Matrix,Matrix) -- see horseshoeResolution(Complex) -- make the horseshoe resolution
- 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(Complex) -- see isExact(Complex,Number,Number) -- whether a complex is exact
- isExact(Complex,InfiniteNumber,InfiniteNumber) -- see isExact(Complex,Number,Number) -- whether a complex is exact
- isExact(Complex,InfiniteNumber,Number) -- see isExact(Complex,Number,Number) -- whether a complex is exact
- isExact(Complex,Number,InfiniteNumber) -- see isExact(Complex,Number,Number) -- whether a complex is exact
- isExact(Complex,Number,Number) -- 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(List) -- see koszulComplex(Matrix) -- makes the Koszul complex
- koszulComplex(Matrix) -- makes the Koszul complex
- length(Complex) -- length of a complex
- ComplexMap // ComplexMap -- see liftMapAlongQuasiIsomorphism(ComplexMap,ComplexMap) -- lift a map of chain complexes along a quasi-isomorphism
- homotopyMap(ComplexMap) -- see liftMapAlongQuasiIsomorphism(ComplexMap,ComplexMap) -- lift a map of chain complexes along a quasi-isomorphism
- liftMapAlongQuasiIsomorphism(ComplexMap,ComplexMap) -- lift a map of chain complexes along a quasi-isomorphism
- quotient(ComplexMap,ComplexMap) -- see liftMapAlongQuasiIsomorphism(ComplexMap,ComplexMap) -- lift a map of chain complexes along a quasi-isomorphism
- longExactSequence(ComplexMap,ComplexMap) -- make the long exact sequence in homology
- 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
- mathML(Complex) (missing documentation)
- mathML(ComplexMap) (missing documentation)
- minimalPresentation(Complex) -- minimal presentation of all terms in a complex
- minimalPresentation(ComplexMap) -- see minimalPresentation(Complex) -- minimal presentation of all terms in a complex
- prune(Complex) -- see minimalPresentation(Complex) -- minimal presentation of all terms in a complex
- prune(ComplexMap) -- see minimalPresentation(Complex) -- minimal presentation of all terms in a complex
- minimize(Complex) -- a quasi-isomorphic complex whose terms have minimal rank
- naiveTruncation(Complex,InfiniteNumber,InfiniteNumber) -- see naiveTruncation(Complex,ZZ,ZZ) -- drops all terms of a complex outside a given interval
- naiveTruncation(Complex,InfiniteNumber,ZZ) -- see naiveTruncation(Complex,ZZ,ZZ) -- drops all terms of a complex outside a given interval
- naiveTruncation(Complex,Nothing,ZZ) -- see naiveTruncation(Complex,ZZ,ZZ) -- drops all terms of a complex outside a given interval
- naiveTruncation(Complex,Sequence) -- see naiveTruncation(Complex,ZZ,ZZ) -- drops all terms of a complex outside a given interval
- naiveTruncation(Complex,ZZ,InfiniteNumber) -- see naiveTruncation(Complex,ZZ,ZZ) -- drops all terms of a complex outside a given interval
- naiveTruncation(Complex,ZZ,Nothing) -- see naiveTruncation(Complex,ZZ,ZZ) -- 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) -- see naiveTruncation(ComplexMap,ZZ,ZZ) -- drops all terms in the source of a complex outside a given interval
- naiveTruncation(ComplexMap,InfiniteNumber,ZZ) -- see naiveTruncation(ComplexMap,ZZ,ZZ) -- drops all terms in the source of a complex outside a given interval
- naiveTruncation(ComplexMap,Nothing,ZZ) -- see naiveTruncation(ComplexMap,ZZ,ZZ) -- drops all terms in the source of a complex outside a given interval
- naiveTruncation(ComplexMap,Sequence) -- see naiveTruncation(ComplexMap,ZZ,ZZ) -- drops all terms in the source of a complex outside a given interval
- naiveTruncation(ComplexMap,Sequence,Sequence) -- see naiveTruncation(ComplexMap,ZZ,ZZ) -- drops all terms in the source of a complex outside a given interval
- naiveTruncation(ComplexMap,ZZ,InfiniteNumber) -- see naiveTruncation(ComplexMap,ZZ,ZZ) -- drops all terms in the source of a complex outside a given interval
- naiveTruncation(ComplexMap,ZZ,Nothing) -- see naiveTruncation(ComplexMap,ZZ,ZZ) -- 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(ComplexMap) -- a map which is a candidate for being a null homotopy
- nullhomotopy(ComplexMap) (missing documentation)
- part(List,Complex) -- extract a graded component of a complex
- part(ZZ,Complex) -- see part(List,Complex) -- extract a graded component of a complex
- part(List,ComplexMap) -- extract a graded component of a map of complexes
- part(ZZ,ComplexMap) -- see 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
- rank(Complex) (missing documentation)
- regularity(Complex) -- compute the Castelnuovo-Mumford regularity
- resolutionMap(Complex) -- see resolutionMap -- map from a free resolution to the given complex
- ring(Complex) -- access the ring of a complex or a complex map
- ring(ComplexMap) -- see ring(Complex) -- access the ring of a complex or a complex map
- Complex ** Ring -- see RingMap ** Complex -- tensor a complex along a ring map
- Complex ** RingMap -- see RingMap ** Complex -- tensor a complex along a ring map
- Ring ** Complex -- see RingMap ** Complex -- tensor a complex along a ring map
- RingMap ** Complex -- tensor a complex along a ring map
- tensor(Complex,RingMap) -- see RingMap ** Complex -- tensor a complex along a ring map
- tensor(RingMap,Complex) -- see RingMap ** Complex -- tensor a complex along a ring map
- ComplexMap ** Ring -- see RingMap ** ComplexMap -- tensor a map of complexes along a ring map
- ComplexMap ** RingMap -- see RingMap ** ComplexMap -- tensor a map of complexes along a ring map
- Ring ** ComplexMap -- see RingMap ** ComplexMap -- tensor a map of complexes along a ring map
- RingMap ** ComplexMap -- tensor a map of complexes along a ring map
- tensor(ComplexMap,RingMap) -- see RingMap ** ComplexMap -- tensor a map of complexes along a ring map
- tensor(RingMap,ComplexMap) -- see 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
- status(Complex) (missing documentation)
- sum(Complex) -- make the direct sum of all terms
- sum(ComplexMap) -- see 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
- torSymmetry(ZZ,Module,Module) -- makes the canonical isomorphism realizing the symmetry of Tor
- truncate(List,Complex) -- truncation of a complex at a specified degree or set of degrees
- truncate(ZZ,Complex) -- see 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,ComplexMap) -- see truncate(List,ComplexMap) -- truncation of a complex map at a specified degree or set of degrees
- 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
Symbols
- UseTarget -- see canonicalMap -- gets the natural map arising from various constructions
- Concentration -- optional argument used to specify the concentration
- FreeToExact (missing documentation)
- minimizingMap -- see minimize(Complex) -- a quasi-isomorphic complex whose terms have minimal rank
- Boundary -- see randomComplexMap(Complex,Complex) -- a random map of chain complexes
- Cycle -- see randomComplexMap(Complex,Complex) -- a random map of chain complexes
- InternalDegree -- see randomComplexMap(Complex,Complex) -- a random map of chain complexes
- OverField -- see Strategy for free resolutions over a field -- algorithm for computing free resolutions over a field
- OverZZ -- see Strategy for free resolutions over the integers -- algorithm for computing free resolutions of ZZ-modules
- Homogenization -- see Strategy for free resolutions via homogenization -- algorithm for computing free resolutions by first homogenizing
- Nonminimal -- see Strategy for nonminimal free resolutions -- algorithm for computing nonminimal free resolutions

For the programmer

The object Complexes is a package, defined in Complexes.m2, with auxiliary files in Complexes/.

The source of this document is in Complexes/ChainComplexDoc.m2:38:0.