# betti(...,Minimize=>...) -- minimal betti numbers of a non-minimal free resolution

## Synopsis

• Usage:
betti(C, Minimize => true)
• Inputs:
• C, , computed using FastNonminimal (and therefore a non-minimal free resolution of an ideal or module in a polynomial ring or skew commuting polynomial ring, over a finite prime field)
• Outputs:
• ,

## Description

Given a chain complex computed using res(I, FastNonminimal => true) (FastNonminimal), returns the minimal graded Betti numbers of this complex.

 i1 : I = Grassmannian(1,6, CoefficientRing => ZZ/101); ZZ o1 : Ideal of ---[p ..p , p , p , p , p , p , p , p , p , p , p , p , p , p , p , p , p , p , p , p ] 101 0,1 0,2 1,2 0,3 1,3 2,3 0,4 1,4 2,4 3,4 0,5 1,5 2,5 3,5 4,5 0,6 1,6 2,6 3,6 4,6 5,6 i2 : S = ring I o2 = S o2 : PolynomialRing i3 : elapsedTime C = res(I, FastNonminimal => true) -- 2.45327 seconds elapsed 1 35 241 841 1781 2464 2294 1432 576 135 14 o3 = S <-- S <-- S <-- S <-- S <-- S <-- S <-- S <-- S <-- S <-- S <-- 0 0 1 2 3 4 5 6 7 8 9 10 11 o3 : ChainComplex

For a non-minimal resolution, betti gives the actual ranks of the complex. If the option Minimize => true is given, the minimal Betti numbers are returned.

 i4 : betti C 0 1 2 3 4 5 6 7 8 9 10 o4 = total: 1 35 241 841 1781 2464 2294 1432 576 135 14 0: 1 . . . . . . . . . . 1: . 35 140 290 402 402 293 152 53 11 1 2: . . 101 514 1174 1577 1365 780 287 62 6 3: . . . 37 204 479 621 480 221 56 6 4: . . . . 1 6 15 20 15 6 1 o4 : BettiTally i5 : betti(C, Minimize => true) 0 1 2 3 4 5 6 7 8 9 10 o5 = total: 1 35 140 385 819 1080 819 385 140 35 1 0: 1 . . . . . . . . . . 1: . 35 140 189 84 . . . . . . 2: . . . 196 735 1080 735 196 . . . 3: . . . . . . 84 189 140 35 . 4: . . . . . . . . . . 1 o5 : BettiTally

This command is useful if the non-minimal free resolution has already been computed. However, to get the minimal betti numbers of an ideal or module, it is recommended to use the function minimalBetti as that avoids much computation and allows the use of length and degree limits.

## Further information

• Default value: false
• Function: betti -- display or modify a Betti diagram
• Option key: Minimize -- an optional argument

## Caveat

Released in M2 1.9, still experimental. Only works over finite prime field. If the complex is the resolution of a non-homogeneous or multi-homogeneous object, then this function will result in an error.