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Macaulay2Doc :: multigraded(BettiTally)

multigraded(BettiTally) -- 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 : 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
i5 : B = multigraded t

         0     1           2           3           4
o5 =  0: 1     .           .           .           .
      4: . 2a2b2           .           .           .
      6: . 2a3b3           .           .           .
      8: .     .        a4b4           .           .
      9: .     . 4a5b4+4a4b5           .           .
     10: .     . 2a7b3+2a3b7       6a5b5           .
     11: .     .           . 4a7b4+4a4b7           .
     12: .     .           .           . 2a7b5+2a5b7

o5 : MultigradedBettiTally
i6 : peek B

o6 = MultigradedBettiTally{(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
By changing the weights, we can reorder the columns of the diagram. The following three displays display the first degree, the second degree, and the total degree, respectively.
i7 : betti(B, Weights => {1,0})

        0     1          2     3     4
o7 = 0: 1     .          .     .     .
     2: . 2a2b2          .     .     .
     3: . 2a3b3      2a3b7     .     .
     4: .     . 4a4b5+a4b4 4a4b7     .
     5: .     .      4a5b4 6a5b5 2a5b7
     7: .     .      2a7b3 4a7b4 2a7b5

o7 : MultigradedBettiTally
i8 : betti(B, Weights => {0,1})

        0     1          2     3     4
o8 = 0: 1     .          .     .     .
     2: . 2a2b2          .     .     .
     3: . 2a3b3      2a7b3     .     .
     4: .     . 4a5b4+a4b4 4a7b4     .
     5: .     .      4a4b5 6a5b5 2a7b5
     7: .     .      2a3b7 4a4b7 2a5b7

o8 : MultigradedBettiTally
i9 : B' = betti(B, Weights => {1,1})

         0     1           2           3           4
o9 =  0: 1     .           .           .           .
      4: . 2a2b2           .           .           .
      6: . 2a3b3           .           .           .
      8: .     .        a4b4           .           .
      9: .     . 4a5b4+4a4b5           .           .
     10: .     . 2a7b3+2a3b7       6a5b5           .
     11: .     .           . 4a7b4+4a4b7           .
     12: .     .           .           . 2a7b5+2a5b7

o9 : MultigradedBettiTally

See also