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      1      2      3

o4 : Complex

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

Authors

Version

This documentation describes version 0.1 of AInfinity.

Citation

If you have used this package in your research, please cite it as follows:

@misc{AInfinitySource,
  title = {{AInfinity: AInfinity structures on free resolutions. Version~0.1}},
  author = {David Eisenbud and Mike Stillman},
  howpublished = {A \emph{Macaulay2} package available at
    \url{https://github.com/Macaulay2/M2/tree/master/M2/Macaulay2/packages}}
}

Exports

Functions and commands
- aInfinity -- aInfinity algebra and module structures on free resolutions
- burkeDifferential -- see burkeResolution -- compute a resolution from A-infinity structures
- 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
- hasMinimalMult -- Determines if the A-infinity multiplication is minimal
- isGolodAInf -- Determines if the ring is Golod or not
- 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
- burkeDifferential(HashTable,HashTable,ZZ) -- see burkeResolution -- compute a resolution from A-infinity structures
- 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
- hasMinimalMult(Ideal) -- see hasMinimalMult -- Determines if the A-infinity multiplication is minimal
- hasMinimalMult(Ideal,ZZ) -- see hasMinimalMult -- Determines if the A-infinity multiplication is minimal
- hasMinimalMult(Ring) -- see hasMinimalMult -- Determines if the A-infinity multiplication is minimal
- hasMinimalMult(Ring,InfiniteNumber) -- see hasMinimalMult -- Determines if the A-infinity multiplication is minimal
- hasMinimalMult(Ring,ZZ) -- see hasMinimalMult -- Determines if the A-infinity multiplication is minimal
- isGolodAInf(Ring) -- see isGolodAInf -- Determines if the ring is Golod or not
- 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
- ShowRanks -- see picture -- displays information about the blocks of a map or maps between direct sum modules

For the programmer

The object AInfinity is a package, defined in AInfinity.m2.

The source of this document is in AInfinity.m2:1093:0.