chainComplex(Complex) -- translate between data types for chain complexes

Synopsis

Function: chainComplex
Usage:

C = chainComplex D
Inputs:
- D, a complex,
Outputs:
- C, a chain complex,

Description

Both ChainComplex and Complex are Macaulay2 types that implement chain complexes of modules over rings. The plan is to replace ChainComplex with this new type. Before this happens, this function allows interoperability between these types.

The first example is the minimal free resolution of the twisted cubic curve.

i1 : R = ZZ/32003[a..d];

i2 : I = monomialCurveIdeal(R, {1,2,3})

             2                    2
o2 = ideal (c  - b*d, b*c - a*d, b  - a*c)

o2 : Ideal of R

i3 : M = R^1/I

o3 = cokernel | c2-bd bc-ad b2-ac |

                            1
o3 : R-module, quotient of R

i4 : C = resolution M

      1      3      2
o4 = R  <-- R  <-- R  <-- 0
                           
     0      1      2      3

o4 : ChainComplex

i5 : D = freeResolution M

      1      3      2
o5 = R  <-- R  <-- R
                    
     0      1      2

o5 : Complex

i6 : C1 = chainComplex D

      1      3      2
o6 = R  <-- R  <-- R
                    
     0      1      2

o6 : ChainComplex

i7 : assert(C == C1)

The tensor products make the same choice of signs.

i8 : D2 = D ** D

      1      6      13      12      4
o8 = R  <-- R  <-- R   <-- R   <-- R
                                    
     0      1      2       3       4

o8 : Complex

i9 : C2 = chainComplex D2

      1      6      13      12      4
o9 = R  <-- R  <-- R   <-- R   <-- R
                                    
     0      1      2       3       4

o9 : ChainComplex

i10 : assert(C2 == C1 ** C1)

Caveat

This is a temporary method to allow comparisons among the data types, and will be removed once the older data structure is replaced

Ways to use this method: