resolutionPoset -- generates a poset from a resolution

Synopsis

Usage:

P = resolutionPoset C

P = resolutionPoset I
Inputs:
- C, a chain complex,
- I, a monomial ideal,
- I, an ideal,
Outputs:
- P, an instance of the type Poset,

Description

Given a resolution $C$, a poset can be defined by the non-zero entries of the matrices of each component of the resolution.

i1 : R = QQ[x,y,z];

i2 : C = res ideal(y*z,x*z,x^2*y)

      1      3      2
o2 = R  <-- R  <-- R  <-- 0
                           
     0      1      2      3

o2 : ChainComplex

i3 : resolutionPoset C

o3 = Relation Matrix: | 1 1 1 1 1 1 |
                      | 0 1 0 0 1 0 |
                      | 0 0 1 0 1 1 |
                      | 0 0 0 1 0 1 |
                      | 0 0 0 0 1 0 |
                      | 0 0 0 0 0 1 |

o3 : Poset

i4 : (resolutionPoset C).GroundSet

o4 = {{0, 0}, {1, 0}, {1, 1}, {1, 2}, {2, 0}, {2, 1}}

o4 : List

Moreover, the resolution-poset of a MonomialIdeal can be labeled as the lcm of the generators involved at each level. As the lcm needn't be unique at each step, we simply append it to the base labeling, as above.

i5 : P = resolutionPoset monomialIdeal(y*z,x*z,x^2*y)

o5 = P

o5 : Poset

i6 : P.GroundSet

                              2                                      2
o6 = {{0, 0, {0, 0}}, {1, 0, x y}, {1, 1, x*z}, {1, 2, y*z}, {2, 0, x y*z},
     ------------------------------------------------------------------------
     {2, 1, x*y*z}}

o6 : List

Ways to use resolutionPoset :

resolutionPoset(ChainComplex)
resolutionPoset(Ideal)
resolutionPoset(MonomialIdeal)

For the programmer

The object resolutionPoset is a method function.