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ExactCouples :: toChainComplex

toChainComplex -- converts a module for R[d]/d^2 to a chain complex

Synopsis

Usage:

toChainComplex M
Inputs:
- M, a module, for a ring of the form R[d]/d^2
Outputs:
- C, a chain complex, consisting of the the degrees of M that are multiples of (degree d), together with a differential reflecting the action of d.

Description

Suppose d has degree v. The output chain complex C has C_0 = M_{0*v}, and since the differential in a chain complex has degree -1, it has generally

$C_i = M_{-iv}$.

i1 : R = ZZ[d,Degrees=>{2}]/d^2;

i2 : M = cokernel map(R^(-{{0},{1},{2},{3}}),,{{4,0,d,0},{0,6,0,d},{0,0,8,0},{0,0,0,10}})

o2 = cokernel {0} | 4 0 d 0  |
              {1} | 0 6 0 d  |
              {2} | 0 0 8 0  |
              {3} | 0 0 0 10 |

                            4
o2 : R-module, quotient of R

i3 : isHomogeneous M

o3 = true

i4 : prune toChainComplex M

o4 = cokernel | 8 | <-- cokernel | 32 | <-- cokernel | 4 |
                                             
     -2                 -1                  0

o4 : ChainComplex

i5 : apply(10,d->prune evaluateInDegree({d},M))

o5 = {cokernel | 4 |, cokernel | 6 |, cokernel | 32 |, cokernel | 60 |,
     ------------------------------------------------------------------------
     cokernel | 8 |, cokernel | 10 |, 0, 0, 0, 0}

o5 : List

Caveat

M must be homogeneous

See also

chainModule -- writes a chain complex of R-modules as an R[d]/d^2-module

Ways to use `toChainComplex`:

toChainComplex(Module)

For the programmer

The object toChainComplex is a method function.