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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

Cohomology tables

Decomposition into pure diagrams

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

Constructions often leading to pure resolutions

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

Authors

Version

This documentation describes version 1.5 of BoijSoederberg.

Citation

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

@misc{BoijSoederbergSource,
  title = {{BoijSoederberg: Betti diagram operations useful for investigating the Boij-Soederberg conjectures. Version~1.5}},
  author = {David Eisenbud and Frank-Olaf Schreyer and Mike Stillman and Courtney Gibbons and Branden Stone},
  howpublished = {A \emph{Macaulay2} package available at
    \url{https://github.com/Macaulay2/M2/tree/master/M2/Macaulay2/packages}}
}

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

For the programmer

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

The source of this document is in BoijSoederberg.m2:1543:0.