AInfinity -- A-infinity algebra and module structures on free resolutions

Description

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

• aInfinity -- aInfinity algebra and module structures on free resolutions

Version

This documentation describes version 0.1 of AInfinity.

Source code

The source code from which this documentation is derived is in the file AInfinity.m2.

Exports

• Functions and commands
• aInfinity -- aInfinity algebra and module structures on free resolutions
• burkeResolution -- compute a resolution from A-infinity structures
• displayBlocks -- prints a matrix showing the source and target decomposition
• extractBlocks -- displays components of a map in a labeled complex
• golodBetti -- list the ranks of the free modules in the resolution of a Golod module
• picture -- displays information about the blocks of a map or maps between direct sum modules
• Methods
• "aInfinity(HashTable,Module)" -- see aInfinity -- aInfinity algebra and module structures on free resolutions
• "aInfinity(Module)" -- see aInfinity -- aInfinity algebra and module structures on free resolutions
• "aInfinity(Ring)" -- see aInfinity -- aInfinity algebra and module structures on free resolutions
• "burkeResolution(Module,ZZ)" -- see burkeResolution -- compute a resolution from A-infinity structures
• "displayBlocks(Matrix)" -- see displayBlocks -- prints a matrix showing the source and target decomposition
• "extractBlocks(Matrix,List)" -- see extractBlocks -- displays components of a map in a labeled complex
• "extractBlocks(Matrix,List,List)" -- see extractBlocks -- displays components of a map in a labeled complex
• "golodBetti(Module,ZZ)" -- see golodBetti -- list the ranks of the free modules in the resolution of a Golod module
• "picture(ChainComplex)" -- see picture -- displays information about the blocks of a map or maps between direct sum modules
• "picture(Complex)" -- see picture -- displays information about the blocks of a map or maps between direct sum modules
• "picture(Matrix)" -- see picture -- displays information about the blocks of a map or maps between direct sum modules
• "picture(Module)" -- see picture -- displays information about the blocks of a map or maps between direct sum modules
• Symbols
• Check -- Option for burkeResolution

For the programmer

The object AInfinity is .