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ExteriorModules :: bassNumbers

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

Synopsis

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 :

For the programmer

The object bassNumbers is a method function.