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

mat2betti -- matrix to Betti diagram
matrix(BettiTally,ZZ,ZZ)
lowestDegrees -- list of lowest degree shifts
highestDegrees -- list of highest degree shifts
BettiTally -- the class of all Betti tallies
makeCI -- Make the Betti diagram of a complete intersection ideal

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

CohomologyTally -- cohomology table
pureCohomologyTable -- pure cohomology table given zeros of Hilbert polynomial
bott -- cohomology of Schur functors of tautological bundle on P^n

Decomposition into pure diagrams

decompose(BettiTally)
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
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
- BettiEliminationTally -- Betti elimination table
- CohomologyTally -- cohomology table
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
- 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
- decompose(BettiTally) -- write a Betti diagram as a positive combination of pure integral diagrams
- 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
- 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
- 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
- EliminationSequence -- option for eliminateBetti
- HerzogKuhl -- An argument for the option TableEntries
- LeastIntegerEntries -- An argument for the option TableEntries
- RealizationModules -- An argument for the option TableEntries
- TableEntries -- Set the convention for what kind of pure Betti diagrams to use in a decomposition.

For the programmer

The object BoijSoederberg is a package.