betti(BettiTally) -- view and set the weight vector of a Betti diagram

Synopsis

Function: betti
Usage:

betti(t, Weights => w)
Inputs:
- t, a Betti tally,
Optional inputs:
- Weights => a list, default value null, with the same length as the multidegrees in t, used as the weight vector w
- Minimize => ..., default value false, minimal betti numbers of a non-minimal free resolution
Outputs:
- a Betti tally, with the same homological degrees, multidegrees, and ranks. If a weight vector w is provided, the total degree weights in the resulting Betti tally will be recomputed by taking the dot products of w with the multidegrees in the tally.

Description

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 : peek t

o4 = BettiTally{(0, {0, 0}, 0) => 1 }
                (1, {2, 2}, 4) => 2
                (1, {3, 3}, 6) => 2
                (2, {3, 7}, 10) => 2
                (2, {4, 4}, 8) => 1
                (2, {4, 5}, 9) => 4
                (2, {5, 4}, 9) => 4
                (2, {7, 3}, 10) => 2
                (3, {4, 7}, 11) => 4
                (3, {5, 5}, 10) => 6
                (3, {7, 4}, 11) => 4
                (4, {5, 7}, 12) => 2
                (4, {7, 5}, 12) => 2

The following three displays show the first degree, the second degree, and the total degree, respectively.

i5 : betti(t, Weights => {1,0})

            0 1  2  3 4
o5 = total: 1 4 13 14 4
         0: 1 .  .  . .
         1: . 2  2  4 2
         2: . 2  5  6 .
         3: . .  4  . 2
         4: . .  .  4 .
         5: . .  2  . .

o5 : BettiTally

i6 : betti(t, Weights => {0,1})

            0 1  2  3 4
o6 = total: 1 4 13 14 4
         0: 1 .  .  . .
         1: . 2  2  4 2
         2: . 2  5  6 .
         3: . .  4  . 2
         4: . .  .  4 .
         5: . .  2  . .

o6 : BettiTally

i7 : betti(t, Weights => {1,1})

            0 1  2  3 4
o7 = total: 1 4 13 14 4
         0: 1 .  .  . .
         1: . .  .  . .
         2: . .  .  . .
         3: . 2  .  . .
         4: . .  .  . .
         5: . 2  .  . .
         6: . .  1  . .
         7: . .  8  6 .
         8: . .  4  8 4

o7 : BettiTally

i8 : peek oo

o8 = BettiTally{(0, {0, 0}, 0) => 1 }
                (1, {2, 2}, 4) => 2
                (1, {3, 3}, 6) => 2
                (2, {3, 7}, 10) => 2
                (2, {4, 4}, 8) => 1
                (2, {4, 5}, 9) => 4
                (2, {5, 4}, 9) => 4
                (2, {7, 3}, 10) => 2
                (3, {4, 7}, 11) => 4
                (3, {5, 5}, 10) => 6
                (3, {7, 4}, 11) => 4
                (4, {5, 7}, 12) => 2
                (4, {7, 5}, 12) => 2

i9 : t' = multigraded t

         0         1                   2                   3                   4
o9 =  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

o9 : MultigradedBettiTally

i10 : betti(t', Weights => {1,0})

         0         1                 2         3         4
o10 = 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

o10 : MultigradedBettiTally

i11 : betti(t', Weights => {0,1})

         0         1                 2         3         4
o11 = 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

o11 : MultigradedBettiTally

i12 : betti(t', Weights => {1,1})

          0         1                   2                   3                   4
o12 =  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

o12 : MultigradedBettiTally

Ways to use this method: