# 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.

## 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?

## 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)

## 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