Triplets -- Betti diagrams and hypercohomology tables associated to triplets of degree sequences

Description

Triplets is a package to calculate

1) Betti diagrams of triplets of pure free squarefree complexes, as introduced in math.AC/1207.2071 "Triplets of pure free squarefree complexes"

2) hypercohomology tables associated to homology triplets, as given in math.AC/1212.3675 "Zipping Tate resolutions and exterior coalgebras"

by Gunnar Floeystad.

Degree sequences

strands -- strand span of degree sequence
strandsL -- strand span as a subset of [0,n]
conj -- conjugate of degree sequence

Degree triplets and homology triplets

triplet -- make a triplet
rotForw -- forward cyclic permutation
rotBack -- backward cyclic permutation
toHomology -- from degree triplet to homology triplet
toDegree -- from homology triplet to degree triplet
dualHomTriplet -- the dual homology triplet
type -- number of variables

Checking triplets

isDegreeTriplet -- checks if it is a degree triplet
isHomologyTriplet -- checks if it is a homology triplet

Betti diagrams

Betti1 -- Betti numbers of first pure complex
Betti3 -- Betti numbers of the three pure complexes
BettiDiagram1 -- Betti diagram of first pure complex
BettiDiagram3 -- Betti diagrams of the three pure complexes

Polynomials

binPol -- product of two binomial polynomials
hilbCoeff -- coefficients of Hilbert polynomial
hilbPol -- Hilbert polynomial
chiPol -- Hilbert polynomial of cohomology sheaves

Cohomology tables

cohMatrix -- cohomology table in matrix form
cohTable -- cohomology table

We create a Triplet using the triplet function:

i1 : T = triplet({1,2,3}, {0,2}, {0,2,3})

o1 = {{1, 2, 3}, {0, 2}, {0, 2, 3}}

o1 : Triplet

i2 : isDegreeTriplet T

o2 = true

We can rotate this degree triplet forwards or backwards:

i3 : rotForw T

o3 = {{0, 2}, {0, 2, 3}, {1, 2, 3}}

o3 : Triplet

i4 : rotBack T

o4 = {{0, 2, 3}, {1, 2, 3}, {0, 2}}

o4 : Triplet

We can compute the Betti numbers and Betti diagrams associated to the degree sequences of this triplet:

i5 : Betti3 T
{1, 2, 3}   ===>   {3, 6, 2}
{0, 2}   ===>   {1, 3}
{0, 2, 3}   ===>   {2, 3, 1}

i6 : BettiDiagram3 T
       0 1 2            0 1            0 1 2
total: 3 6 2     total: 1 3     total: 2 3 1
    1: 3 6 2         0: 1 .         0: 2 . .
                     1: . 3         1: . 3 1

We convert it to a homology triplet:

i7 : Th = toHomology T

o7 = {{1, 2, 3}, {1, 3}, {0, 2, 3}}

o7 : Triplet

i8 : isHomologyTriplet Th

o8 = true

We compute the hypercohomology table of a complex of coherent sheaves associated to this homology triplet:

i9 : cohTable (-7, 5,Th)

         -7 -6 -5 -4 -3 -2 -1 0 1  2  3  4  5
o9 =  2: 77 50 30 16  7  2  . . .  .  .  .  .
      1:  2  2  2  2  2  2  2 1 .  .  .  .  .
      0:  .  .  .  .  .  .  . . 1  2  3  4  5
     -1:  .  .  .  .  .  .  . 1 4 10 20 35 56

o9 : CohomologyTally

The dual homology triplet and its hypercohomology table:

i10 : Thd = dualHomTriplet Th

o10 = {{0, 1, 2}, {0, 2, 3}, {1, 3}}

o10 : Triplet

i11 : cohTable (-7,5,Thd)

          -7 -6 -5 -4 -3 -2 -1 0 1  2  3  4  5
o11 =  2: 56 35 20 10  4  1  . . .  .  .  .  .
       1:  5  4  3  2  1  .  . . .  .  .  .  .
       0:  .  .  .  .  .  1  2 2 2  2  2  2  2
      -1:  .  .  .  .  .  .  . 2 7 16 30 50 77

o11 : CohomologyTally

Author

Gunnar Floystad <[email protected]>

Version

This documentation describes version 0.1 of Triplets.

Citation

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

@misc{TripletsSource,
  title = {{Triplets: A \emph{Macaulay2} package. Version~0.1}},
  author = {Gunnar Floystad},
  howpublished = {A \emph{Macaulay2} package available at
    \url{https://github.com/Macaulay2/M2/tree/master/M2/Macaulay2/packages}}
}

Exports

Types
- Triplet -- triplet
Functions and commands
- Betti1 -- see Betti1(Triplet) -- Betti numbers of first pure complex
- Betti3 -- see Betti3(Triplet) -- Betti numbers of the three pure complexes
- BettiDiagram1 -- see BettiDiagram1(Triplet) -- Betti diagram of first pure complex
- BettiDiagram3 -- see BettiDiagram3(Triplet) -- Betti diagrams of the three pure complexes
- binPol -- see binPol(RingElement,ZZ,ZZ) -- product of two binomial polynomials
- chiPol -- see chiPol(RingElement,ZZ,List,List) -- Hilbert polynomial of cohomology sheaves
- cohMatrix -- see cohMatrix(ZZ,ZZ,Triplet) -- cohomology table in matrix form
- cohTable -- see cohTable(ZZ,ZZ,Triplet) -- cohomology table
- conj -- see conj(List,ZZ) -- conjugate of degree sequence
- dualHomTriplet -- see dualHomTriplet(Triplet) -- the dual homology triplet
- hilbCoeff -- see hilbCoeff(Triplet) -- coefficients of Hilbert polynomial
- hilbPol -- see hilbPol(RingElement,ZZ,List,List) -- Hilbert polynomial
- isDegreeTriplet -- see isDegreeTriplet(Triplet) -- checks if it is a degree triplet
- isHomologyTriplet -- see isHomologyTriplet(Triplet) -- checks if it is a homology triplet
- rotBack -- see rotBack(Triplet) -- backward cyclic permutation
- rotForw -- see rotForw(Triplet) -- forward cyclic permutation
- strands -- see strands(List) -- strand span of degree sequence
- strandsL -- see strandsL(ZZ,List) -- strand span as a subset of [0,n]
- toDegree -- see toDegree(Triplet) -- from homology triplet to degree triplet
- toHomology -- see toHomology(Triplet) -- from degree triplet to homology triplet
- triplet -- see triplet(List,List,List) -- make a triplet
- type -- see type(Triplet) -- number of variables
Methods
- Betti1(Triplet) -- Betti numbers of first pure complex
- Betti3(Triplet) -- Betti numbers of the three pure complexes
- BettiDiagram1(Triplet) -- Betti diagram of first pure complex
- BettiDiagram3(Triplet) -- Betti diagrams of the three pure complexes
- binPol(RingElement,ZZ,ZZ) -- product of two binomial polynomials
- chiPol(RingElement,ZZ,List,List) -- Hilbert polynomial of cohomology sheaves
- cohMatrix(ZZ,ZZ,Triplet) -- cohomology table in matrix form
- cohTable(ZZ,ZZ,Triplet) -- cohomology table
- conj(List,ZZ) -- conjugate of degree sequence
- dualHomTriplet(Triplet) -- the dual homology triplet
- hilbCoeff(Triplet) -- coefficients of Hilbert polynomial
- hilbPol(RingElement,ZZ,List,List) -- Hilbert polynomial
- isDegreeTriplet(Triplet) -- checks if it is a degree triplet
- isHomologyTriplet(Triplet) -- checks if it is a homology triplet
- rotBack(Triplet) -- backward cyclic permutation
- rotForw(Triplet) -- forward cyclic permutation
- strands(List) -- strand span of degree sequence
- strandsL(ZZ,List) -- strand span as a subset of [0,n]
- toDegree(Triplet) -- from homology triplet to degree triplet
- toHomology(Triplet) -- from degree triplet to homology triplet
- triplet(List,List,List) -- make a triplet
- type(Triplet) -- number of variables

For the programmer

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

The source of this document is in Triplets.m2:700:0.