# multigraded -- convert a Betti tally into a multigraded Betti tally

## Synopsis

• Usage:
multigraded t
• Inputs:
• t, ,
• Outputs:
• , different from the input only in the printed diagram

## Description

A multigraded Betti tally is a special type of BettiTally that both prints nicely and from which multigraded Betti numbers could be easily extracted.

 i1 : R = ZZ/101[a..d, Degrees => {2:{1,0},2:{0,1}}]; i2 : I = ideal random(R^1, R^{2:{-2,-2},2:{-3,-3}}); o2 : Ideal of R i3 : t = betti res I 0 1 2 3 4 o3 = total: 1 4 13 14 4 0: 1 . . . . 1: . . . . . 2: . . . . . 3: . 2 . . . 4: . . . . . 5: . 2 . . . 6: . . 1 . . 7: . . 8 6 . 8: . . 4 8 4 o3 : BettiTally i4 : B = multigraded t 0 1 2 3 4 o4 = 0: 1 . . . . 4: . 2*a^2*b^2 . . . 6: . 2*a^3*b^3 . . . 8: . . a^4*b^4 . . 9: . . 4*a^5*b^4+4*a^4*b^5 . . 10: . . 2*a^7*b^3+2*a^3*b^7 6*a^5*b^5 . 11: . . . 4*a^7*b^4+4*a^4*b^7 . 12: . . . . 2*a^7*b^5+2*a^5*b^7 o4 : MultigradedBettiTally

By changing the weights, we can reorder the columns of the diagram. The following three displays show the first degree, the second degree, and the total degree, respectively.

 i5 : betti(B, Weights => {1,0}) 0 1 2 3 4 o5 = 0: 1 . . . . 2: . 2*a^2*b^2 . . . 3: . 2*a^3*b^3 2*a^3*b^7 . . 4: . . 4*a^4*b^5+a^4*b^4 4*a^4*b^7 . 5: . . 4*a^5*b^4 6*a^5*b^5 2*a^5*b^7 7: . . 2*a^7*b^3 4*a^7*b^4 2*a^7*b^5 o5 : MultigradedBettiTally i6 : betti(B, Weights => {0,1}) 0 1 2 3 4 o6 = 0: 1 . . . . 2: . 2*a^2*b^2 . . . 3: . 2*a^3*b^3 2*a^7*b^3 . . 4: . . 4*a^5*b^4+a^4*b^4 4*a^7*b^4 . 5: . . 4*a^4*b^5 6*a^5*b^5 2*a^7*b^5 7: . . 2*a^3*b^7 4*a^4*b^7 2*a^5*b^7 o6 : MultigradedBettiTally i7 : betti(B, Weights => {1,1}) 0 1 2 3 4 o7 = 0: 1 . . . . 4: . 2*a^2*b^2 . . . 6: . 2*a^3*b^3 . . . 8: . . a^4*b^4 . . 9: . . 4*a^5*b^4+4*a^4*b^5 . . 10: . . 2*a^7*b^3+2*a^3*b^7 6*a^5*b^5 . 11: . . . 4*a^7*b^4+4*a^4*b^7 . 12: . . . . 2*a^7*b^5+2*a^5*b^7 o7 : MultigradedBettiTally