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BettiBounds :: BettiBounds

BettiBounds -- Bounds for the Betti numbers of iterated stellar subdivisions of the boundary of a simplex.


Using unprojection theory we give in [6] bounds for the Betti numbers of the Stanley-Reisner ring of a stellar subdivision of a Gorenstein* simplicial complex. Applying this result we obtain a bound for the total Betti numbers of iterated stellar subdivisions of the boundary complex of a simplex. This bound depends only on the number of stellars and we construct examples which prove that it is sharp. The purpose of this package is to give an implementation of this construction. Using Kustin-Miller addition of Betti tables, we provide an efficient way to compute the graded Betti tables of the examples. This works far beyond the range in which the resolution or even ideal can be computed directly.

This package requires the package SimplicialComplexes.m2 version 1.2 or higher, so install this first.


For the Kustin-Miller complex see:

[1] A. Kustin and M. Miller, Constructing big Gorenstein ideals from small ones, J. Algebra 85 (1983), 303-322.

[2] S. Papadakis, Kustin-Miller unprojection with complexes, J. Algebraic Geometry 13 (2004) 249-268,

[3] J. Boehm, S. Papadakis: Implementing the Kustin-Miller complex construction, , J. Softw. Algebra Geom. 4 (2012), 6-11,

For the stellar subdivisions and unprojection see:

[5] J. Boehm, S. Papadakis: Stellar subdivisions and Stanley-Reisner rings of Gorenstein complexes, to appear in Australas. J. Combin.,

For the bounds on the Betti numbers see

[6] J. Boehm, S. Papadakis: Bounds on the Betti numbers of successive stellar subdivisions of a simplex,


This documentation describes version 1.0 of BettiBounds.

Source code

The source code from which this documentation is derived is in the file BettiBounds.m2.


  • Functions and commands
    • bettiBounds -- Compute the Betti bounds
    • champion -- Compute simplicial complexes with maximal total Betti numbers
    • gradedBettiChampion -- Compute Betti tables of the examples with maximal total Betti numbers
    • stellarSubdivision -- Compute the stellar subdivision of a simplicial complex.
  • Other things