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

Synopsis

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

See also

Ways to use multigraded :

For the programmer

The object multigraded is a method function.