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

## Synopsis

• Function: minimalBettiNumbers
• Usage:
minimalBettiNumbers M
• Inputs:
• M, , a graded module over an exterior algebra
• Outputs:
• , 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 variables 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==minimalBettiNumbers M_2 o10 = true