golodBetti -- list the ranks of the free modules in the resolution of a Golod module

Synopsis

• Usage:
B = golodBetti(M,b)
• Inputs:
• M, , R-module
• b, an integer, homological degree to which to carry the computation
• Outputs:
• B, , This would be betti table of the free res of M over R, if M were a Golod module over R

Description

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(B), where B is the truncated resolution F_1 <- F_2... and T(B) 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 : R = S/I o4 = R o4 : QuotientRing i5 : F = burkeResolution (coker vars R, 6) 1 3 9 27 81 243 729 o5 = R <-- R <-- R <-- R <-- R <-- R <-- R 0 1 2 3 4 5 6 o5 : Complex i6 : golodBetti(coker vars R,6) 0 1 2 3 4 5 6 o6 = 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 o6 : BettiTally i7 : betti res (coker vars R, LengthLimit => 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 F 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