next | previous | forward | backward | up | top | index | toc | Macaulay2 website
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

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.

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
    • bott(List,ZZ) -- cohomology of Schur functor of tautological bundle on P^n
    • bott(List,ZZ,ZZ) -- cohomology table of Schur functor of tautolgical bundle on P^n
    • "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
    • 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
    • "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
  • Symbols

For the programmer

The object BoijSoederberg is a package.