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Triplets :: Triplets

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

Checking triplets

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

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

Version

This documentation describes version 0.1 of Triplets.

Source code

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

Exports