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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

- 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

- David Eisenbud <de@msri.org>
- Frank-Olaf Schreyer <schreyer@math.uni-sb.de>
- Mike Stillman <mike@math.cornell.edu>
- Courtney Gibbons <crgibbon@hamilton.edu>
- Branden Stone <bstone@adelphi.edu>

## Version

This documentation describes version 1.5 of BoijSoederberg, released April 01, 2015.

## 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/stable/M2/Macaulay2/packages}}
    }

## 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
  - 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
  - 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, defined in BoijSoederberg.m2.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/BoijSoederberg.m2:1543:0.