ChainComplex

ChainComplex -- the class of all chain complexes.

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.

++ -- direct sum

ChainComplex ** Module -- tensor product

C_i -- select a module

C.dd_i -- select a differential

C[i] -- shift a chain complex

HH_i C -- homology of a chain complex

HH^i C -- cohomology of a chain complex

Hom(C,M) -- Hom 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

length -- length of a chain complex

poincare -- assemble degrees into polynomial

poincareN -- assemble degrees into polynomial

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

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                 3
     o3 = 1: R  <--| x y z |-- R
          
              3                    3
          2: R  <--| -y -z 0  |-- R
                   | x  0  -z |
                   | 0  x  y  |
          
              3              1
          3: R  <--| z  |-- R
                   | -y |
                   | x  |
     
     o3 : ChainComplexMap