# bassNumbers -- compute the Bass numbers of a given graded module

## Synopsis

• Usage:
bassNumbers M
• Inputs:
• M, a graded module over an exterior algebra
• Outputs:
• , the Bass 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 and by $\mu_{i,j}(M) = \dim_K \mathrm{Ext}_E^i(K, M)_j$ the graded Bass numbers of M. Let $M^\ast$ be the right (left) $E$-module $\mathrm{Hom}_E(M,E).$ The duality between projective and injective resolutions implies the following relation between the graded Bass numbers of a module and the graded Betti numbers of its dual: $\beta_{i,j}(M) = \mu_{i,n-j}(M^\ast)$, for all $i, j.$

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=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 : bassNumbers M 0 1 2 3 4 5 o6 = total: 2 5 12 22 35 51 0: 2 1 1 1 1 1 1: . 4 11 21 34 50 o6 : BettiTally

## Ways to use bassNumbers :

• "bassNumbers(Module)"

## For the programmer

The object bassNumbers is .