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BoijSoederberg :: BoijSoederberg

BoijSoederberg -- Betti diagram routines

Description

BoijSoederberg is a package designed to help with the investigation of the Boij-Soederberg conjectures and theorems. For the definitions and conjectures, see math.AC/0611081, "Graded Betti numbers of Cohen-Macaulay modules and the Multiplicity conjecture", by Mats Boij, Jonas Soederberg.

Manipulation of Betti diagrams

Pure Betti diagrams

  • pureBetti -- list of smallest integral Betti numbers corresponding to a degree sequence
  • makePureBetti -- list of Betti numbers corresponding to a degree sequence
  • pureBettiDiagram -- pure Betti diagram given a list of degrees
  • makePureBettiDiagram -- makes a pure Betti diagram given a list of degrees
  • isPure -- is a Betti diagram pure?

Cohomology tables

Decomposition into pure diagrams

  • decompose(BettiTally) -- write a Betti diagram as a positive combination of pure integral diagrams
  • decomposeBetti -- write a Betti diagram as a positive combination of pure integral diagrams
  • decomposeDegrees -- Find the degree sequences of pure diagrams occuring in a Boij-Soederberg decomposition of B
  • eliminateBetti -- elimination table for a Betti diagram
  • isMassEliminate -- determines whether the Boij-Soederberg decomposition algorithm eliminates multiple Betti numbers at the same time

Three constructions for pure resolutions. These routines provide the zero-th betti number given a degree sequence.

  • pureTwoInvariant -- first betti number of specific exact complex
  • pureWeyman -- first betti number of specific exact complex
  • pureCharFree -- first betti number of specific exact complex
  • pureAll -- Vector of first betti number of our three specific exact complexes

Constructions often leading to pure resolutions

  • randomModule -- module with random relations in prescribed degrees
  • randomSocleModule -- random finite length module with prescribed number of socle elements in single degree

Facet equation and the dot product between Betti diagrams and cohomology tables

  • facetEquation -- The upper facet equation corresponding to (L,i)
  • dotProduct -- entry by entry dot product of two Betti diagrams
  • supportFunctional (missing documentation)
  • BettiTally * CohomologyTally (missing documentation)

Authors

Version

This documentation describes version 1.5 of BoijSoederberg.

Source code

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

Exports

  • Types
  • Functions and commands
    • bott -- cohomology of Schur functors of tautological bundle on P^n
    • decomposeBetti -- write a Betti diagram as a positive combination of pure integral diagrams
    • decomposeDegrees -- Find the degree sequences of pure diagrams occuring in a Boij-Soederberg decomposition of B
    • dotProduct -- entry by entry dot product of two Betti diagrams
    • eliminateBetti -- elimination table for a Betti diagram
    • facetEquation, see facetEquation(List,ZZ,ZZ,ZZ) -- The upper facet equation corresponding to (L,i)
    • highestDegrees, see highestDegrees(BettiTally) -- list of highest degree shifts
    • isMassEliminate -- determines whether the Boij-Soederberg decomposition algorithm eliminates multiple Betti numbers at the same time
    • isPure, see isPure(BettiTally) -- is a Betti diagram pure?
    • lowestDegrees, see lowestDegrees(BettiTally) -- list of lowest degree shifts
    • makeCI -- Make the Betti diagram of a complete intersection ideal
    • makePureBetti, see makePureBetti(List) -- list of Betti numbers corresponding to a degree sequence
    • makePureBettiDiagram -- makes a pure Betti diagram given a list of degrees
    • mat2betti, see mat2betti(Matrix,ZZ) -- matrix to Betti diagram
    • mat2cohom (missing documentation)
    • pureAll -- Vector of first betti number of our three specific exact complexes
    • pureBetti, see pureBetti(List) -- list of smallest integral Betti numbers corresponding to a degree sequence
    • pureBettiDiagram, see pureBettiDiagram(List) -- pure Betti diagram given a list of degrees
    • pureCharFree -- first betti number of specific exact complex
    • pureCohomologyTable, see pureCohomologyTable(List,ZZ,ZZ) -- pure cohomology table given zeros of Hilbert polynomial
    • pureTwoInvariant -- first betti number of specific exact complex
    • pureWeyman -- first betti number of specific exact complex
    • randomModule, see randomModule(List,ZZ) -- module with random relations in prescribed degrees
    • randomSocleModule, see randomSocleModule(List,ZZ) -- random finite length module with prescribed number of socle elements in single degree
    • supportFunctional (missing documentation)
  • Methods
    • BettiTally * CohomologyTally (missing documentation)
    • CohomologyTally * BettiTally (missing documentation)
    • CohomologyTally ++ CohomologyTally (missing documentation)
    • CohomologyTally == CohomologyTally (missing documentation)
    • CohomologyTally ZZ (missing documentation)
    • net(CohomologyTally) (missing documentation)
    • ZZ * CohomologyTally (missing documentation)
  • Symbols