ChainComplex

ChainComplex -- the class of all chain complexes.

If C is a chain complex, then C_i will produce the i-th module in the complex, and C.dd_i will produce the differential whose source is C_i.

A new chain complex can be made with C = new ChainComplex. This will automatically initialize C.dd, in which the differentials are stored. The modules can be installed with statements like C#i=M and the differentials can be installed with statements like C.dd#i=d.

See also ChainComplexMap for a discussion of maps between chain complexes. (The boundary map C.dd is regarded as one.)

Here are some functions for producing or manipulating chain complexes.

Module Array -- create a chain complex from a module

ChainComplex ++ ChainComplex -- direct sum

ChainComplex ** ChainComplex -- tensor product

ChainComplex ** GradedModule -- tensor product

GradedModule ** ChainComplex -- tensor product

dd -- obtain the differentials.

ChainComplex Array -- shift a chain complex

betti -- display degrees in a free resolution

chainComplex -- make a chain complex

complete -- complete the internal parts of a chain complex

dual -- dual complex

Hom(ChainComplex,Module) -- Hom

length -- length of a chain complex

max ChainComplex -- maximum index in a chain complex

min ChainComplex -- minimum index in a chain complex

poincare -- assemble degrees into polynomial

poincareN

new ChainComplex -- make a new chain complex from scratch

nullhomotopy -- produce a null homotopy

regularity -- compute the regularity

resolution -- make a projective resolution

status -- display the status of a resolution computation

syzygyScheme -- construct the syzygy scheme from some syzygies

tensorAssociativity -- associativity isomorphisms for tensor products

The default display for a chain complex shows the modules and the stage at which they appear.

i1 : R = ZZ/101[x,y,z]



o1 = R



o1 : PolynomialRing

i2 : C = resolution cokernel matrix {{x,y,z}}



      1      3      3      1

o2 = R  <-- R  <-- R  <-- R

                          

     0      1      2      3



o2 : ChainComplex

In order to see the matrices of the differentials, examine 'C.dd'.

i3 : C.dd



                    1

o3 = -1 : 0 <----- R  : 0

               0

     

          1                     3

     0 : R  <----------------- R  : 1

               {0} | x y z |

     

          3                        3

     1 : R  <-------------------- R  : 2

               {1} | -y -z 0  |

               {1} | x  0  -z |

               {1} | 0  x  y  |

     

          3                  1

     2 : R  <-------------- R  : 3

               {2} | z  |

               {2} | -y |

               {2} | x  |



o3 : ChainComplexMap