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 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
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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
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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
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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
|