Following Jesse Burke's paper "Higher Homotopies and Golod Rings", given a polynomial ring S and a factor ring R = S/I and an R-module X, we compute (finite) A-infinity algebra structure mR on an S-free resolution of R and the A-infinity mR-module structure on an S-free resolution of X, and use them to give a finite computation of the maps in an R-free resolution of X that we call the Burke resolution. Here is an example with the simplest Golod non-hypersurface in 3 variables
i1 : S = ZZ/101[a,b,c] o1 = S o1 : PolynomialRing |
i2 : R = S/(ideal(a)*ideal(a,b,c)) o2 = R o2 : QuotientRing |
i3 : mR = aInfinity R; |
i4 : res coker presentation R
1 3 3 1
o4 = S <-- S <-- S <-- S <-- 0
0 1 2 3 4
o4 : ChainComplex
|
i5 : mR#{2,2}
o5 = {3} | 0 -a 0 a 0 0 0 -c 0 |
{3} | 0 0 -a 0 0 0 a b 0 |
{3} | 0 0 0 0 0 -a 0 0 0 |
3 9
o5 : Matrix S <--- S
|
Given a module X over R, Jesse Burke constructed a possibly non-minimal R-free resolution of any length from the finite data mR and mX:
i6 : X = coker vars R
o6 = cokernel | a b c |
1
o6 : R-module, quotient of R
|
i7 : A = betti burkeResolution(X,8)
0 1 2 3 4 5 6 7 8
o7 = total: 1 3 6 13 28 60 129 277 595
0: 1 3 6 13 28 60 129 277 595
o7 : BettiTally
|
i8 : B = betti res(X, LengthLimit => 8)
0 1 2 3 4 5 6 7 8
o8 = total: 1 3 6 13 28 60 129 277 595
0: 1 3 6 13 28 60 129 277 595
o8 : BettiTally
|
i9 : A == B o9 = true |
This documentation describes version 0.1 of AInfinity.
The source code from which this documentation is derived is in the file AInfinity.m2.