minimalBettiNumbers(Module) -- compute the minimal Betti numbers of a given graded module

Function: minimalBettiNumbers
Usage:

minimalBettiNumbers(M, LengthLimit => lim)
Inputs:
- M, a module, a graded module over an exterior algebra
Optional inputs:
- LengthLimit => an integer, default value infinity, only compute enough to determine the Betti table up to and including the column labelled lim
Outputs:
- a Betti tally, the Betti table of the module M computed using its minimal generators

Description

If M is a graded finitely generated module over an exterior algebra E, we denote by $\beta_{i,j}(M)=\dim_K\mathrm{Tor}_{i}^{E}(M,K)_j$ the graded Betti numbers of M.

Example:

i1 : E=QQ[e_1..e_4,SkewCommutative=>true]

o1 = E

o1 : PolynomialRing, 4 skew commutative variable(s)

i2 : F=E^{0,0}

      2
o2 = E

o2 : E-module, free

i3 : I_1=ideal(e_1*e_2,e_1*e_3,e_2*e_3)

o3 = ideal (e e , e e , e e )
             1 2   1 3   2 3

o3 : Ideal of E

i4 : I_2=ideal(e_1*e_2,e_1*e_3)

o4 = ideal (e e , e e )
             1 2   1 3

o4 : Ideal of E

i5 : M_1=createModule({I_1,I_2},F)

o5 = image | e_2e_3 e_1e_3 e_1e_2 0      0      |
           | 0      0      0      e_1e_3 e_1e_2 |

                             2
o5 : E-module, submodule of E

i6 : J=ideal(join(flatten entries gens I_1,{e_1*e_2*e_3}))

o6 = ideal (e e , e e , e e , e e e )
             1 2   1 3   2 3   1 2 3

o6 : Ideal of E

i7 : M_2=createModule({J,I_2},F)

o7 = image | e_2e_3 e_1e_3 e_1e_2 0      0      |
           | 0      0      0      e_1e_3 e_1e_2 |

                             2
o7 : E-module, submodule of E

i8 : M_1==M_2

o8 = true

i9 : betti M_1==betti M_2

o9 = true

i10 : minimalBettiNumbers(M_1, LengthLimit => 5) == minimalBettiNumbers(M_2, LengthLimit => 5)

o10 = true

Ways to use this method:

minimalBettiNumbers(Module) -- compute the minimal Betti numbers of a given graded module

The source of this document is in /__w/M2/M2/macaulay2-1.26.05+ds/M2/Macaulay2/packages/ExteriorModules.m2:911:0.