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# minimalBettiNumbers -- compute the minimal Betti numbers of a given graded ideal

- Usage:
    `minimalBettiNumbers(I, LengthLimit => lim)`
- Inputs:
  - I, a graded ideal of an exterior algebra
- Optional inputs:
  - LengthLimit => an integer, default value infinity, only compute enough to determine the Betti table up to and including the column labelled lim
- Outputs:
  - a Betti tally, the Betti table of the ideal I computed using its minimal generators

## Description

Example:

    i1 : E=QQ[e_1..e_4,SkewCommutative=>true]

    o1 = E

    o1 : PolynomialRing, 4 skew commutative variable(s)

    i2 : I=ideal(e_1*e_2,e_1*e_3,e_2*e_3)

    o2 = ideal (e e , e e , e e )
                 1 2   1 3   2 3

    o2 : Ideal of E

    i3 : J=ideal(join(flatten entries gens I,{e_1*e_2*e_3}))

    o3 = ideal (e e , e e , e e , e e e )
                 1 2   1 3   2 3   1 2 3

    o3 : Ideal of E

    i4 : I==J

    o4 = true

    i5 : betti I==betti J

    o5 = false

    i6 : minimalBettiNumbers(I, LengthLimit => 5) == minimalBettiNumbers(J, LengthLimit => 5)

    o6 = true

## Ways to use minimalBettiNumbers:

- `minimalBettiNumbers(Ideal)`

## For the programmer

The object minimalBettiNumbers is a method function with options.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/ExteriorIdeals.m2:615:0.