Let S be a standard graded polynomial ring. A module M over R = S/I is Golod if the resolution H of M has maximal betti numbers given the betti numbers of the S-free resolutions F of R and K of M. This resolution, H, has underlying graded module H = R**K**T(F'), where F' is the truncated resolution F_1 <- F_2... and T(F') is the tensor algebra.
Since the component modules of H are given, the computation only requires the computation of the minimal S-free resolution of M, and then is purely numeric; the differentials in the R-free resolution of M are not computed.
In case M = coker vars R, the result is the Betti table of the Golod-Shamash-Eagon resolution of the residue field.
We say that M is a Golod module (over R) if the ranks of the free modules in a minimal R-free resolution of M are equal to the numbers produced by golodBetti. Theorems of Levin and Lescot assert that if R has a Golod module, then R is a Golod ring; and that if R is Golod, then the d-th syzygy of any R-module M is Golod for all d greater than or equal to the projective dimension of M as an S-module (more generally, the co-depth of M) (Avramov, 6 lectures, 5.3.2).
i1 : S = ZZ/101[a,b,c] o1 = S o1 : PolynomialRing |
i2 : I = (ideal(a,b,c^2))^2
2 2 2 2 4
o2 = ideal (a , a*b, a*c , b , b*c , c )
o2 : Ideal of S
|
i3 : F = res(S^1/I)
1 6 8 3
o3 = S <-- S <-- S <-- S <-- 0
0 1 2 3 4
o3 : ChainComplex
|
i4 : K = res coker vars S
1 3 3 1
o4 = S <-- S <-- S <-- S <-- 0
0 1 2 3 4
o4 : ChainComplex
|
i5 : R = S/I o5 = R o5 : QuotientRing |
i6 : E = eagon(R,6); |
i7 : golodBetti(F,K,6)
0 1 2 3 4 5 6
o7 = total: 1 3 9 27 81 243 729
0: 1 3 6 12 24 48 96
1: . . 2 10 32 88 224
2: . . 1 5 20 72 232
3: . . . . 4 28 128
4: . . . . 1 7 42
5: . . . . . . 6
6: . . . . . . 1
o7 : BettiTally
|
i8 : betti res (coker vars R, LengthLimit => 6)
0 1 2 3 4 5 6
o8 = total: 1 3 9 27 81 243 729
0: 1 3 6 12 24 48 96
1: . . 2 10 32 88 224
2: . . 1 5 20 72 232
3: . . . . 4 28 128
4: . . . . 1 7 42
5: . . . . . . 6
6: . . . . . . 1
o8 : BettiTally
|
i9 : betti eagonResolution E
0 1 2 3 4 5 6
o9 = total: 1 3 9 27 81 243 729
0: 1 3 6 12 24 48 96
1: . . 2 10 32 88 224
2: . . 1 5 20 72 232
3: . . . . 4 28 128
4: . . . . 1 7 42
5: . . . . . . 6
6: . . . . . . 1
o9 : BettiTally
|
The object golodBetti is a method function.