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BettiTally -- the class of all Betti tallies

Description

A Betti tally is a special type of Tally that is printed as a display of graded Betti numbers. The class was created so the function betti could return something that both prints nicely and from which information can be extracted. The keys are triples (i,d,h) encoding:

i, the column labels, representing the homological degree;

d, a list of integers giving a multidegree; and

h, the row labels, representing the dot product of a weight covector and d.

Only i and h are used in printing, and the weight covector can be modified by specifying the betti(...,Weights=>...) option to betti(BettiTally).

i1 : t = new BettiTally from { (0,{0},0) => 1, (1,{1},1) => 2, (2,{3},3) => 3, (2,{4},4) => 4 }

            0 1 2
o1 = total: 1 2 7
         0: 1 2 .
         1: . . 3
         2: . . 4

o1 : BettiTally
 
i2 : betti(t, Weights => {2})

            0 1 2
o2 = total: 1 2 7
         0: 1 . .
         1: . 2 .
         2: . . .
         3: . . .
         4: . . 3
         5: . . .
         6: . . 4

o2 : BettiTally
 
i3 : peek oo

o3 = BettiTally{(0, {0}, 0) => 1}
                (1, {1}, 2) => 2
                (2, {3}, 6) => 3
                (2, {4}, 8) => 4

For convenience, the operations of direct sum (++), tensor product (**), codim, degree, dual, pdim, poincare, regularity, and degree shifting (numbers in brackets or parentheses), have been implemented for Betti tallies. These operations mimic the corresponding operations on chain complexes.

i4 : t(5)

            0 1 2
o4 = total: 1 2 7
        -5: 1 2 .
        -4: . . 3
        -3: . . 4

o4 : BettiTally
 
i5 : t[-5]

            5 6 7
o5 = total: 1 2 7
        -5: 1 2 .
        -4: . . 3
        -3: . . 4

o5 : BettiTally
 
i6 : dual oo

            -7 -6 -5
o6 = total:  7  2  1
         3:  4  .  .
         4:  3  .  .
         5:  .  2  1

o6 : BettiTally
 
i7 : t ++ oo

            -7 -6 -5 -4 -3 -2 -1 0 1 2
o7 = total:  7  2  1  .  .  .  . 1 2 7
         0:  .  .  .  .  .  .  . 1 2 .
         1:  .  .  .  .  .  .  . . . 3
         2:  .  .  .  .  .  .  . . . 4
         3:  4  .  .  .  .  .  . . . .
         4:  3  .  .  .  .  .  . . . .
         5:  .  2  1  .  .  .  . . . .

o7 : BettiTally
 
i8 : t ** t

            0 1  2  3  4
o8 = total: 1 4 18 28 49
         0: 1 4  4  .  .
         1: . .  6 12  .
         2: . .  8 16  9
         3: . .  .  . 24
         4: . .  .  . 16

o8 : BettiTally
 
i9 : pdim t

o9 = 2
 
i10 : codim t

o10 = 0
 
i11 : degree t

o11 = 6
 
i12 : poincare t

                 3     4
o12 = 1 - 2T + 3T  + 4T

o12 : ZZ[T]
 
i13 : regularity t

o13 = 2

If the Betti tally represents the Betti numbers of a resolution of a module $M$ on a polynomial ring $R = K[x_0,...,x_n]$, then while the data does not uniquely determine $M$, it suffices to compute the Hilbert polynomial and Hilbert series of $M$.

i14 : n = 3

o14 = 3
 
i15 : hilbertSeries(n, t)

                 3     4
      1 - 2T + 3T  + 4T
o15 = ------------------
                  3
           (1 - T)

o15 : Expression of class Divide
 
i16 : hilbertPolynomial(n, t)

o16 = 33*P  - 23*P  + 6*P
          0       1      2

o16 : ProjectiveHilbertPolynomial

A Betti tally can be multiplied by an integer or by a rational number, and the values can be lifted to integers, when possible.

i17 : (1/2) * t

               0 1   2
o17 = total: 1/2 1 7/2
          0: 1/2 1   .
          1:   . . 3/2
          2:   . .   2

o17 : BettiTally
 
i18 : 2 * oo

             0 1 2
o18 = total: 1 2 7
          0: 1 2 .
          1: . . 3
          2: . . 4

o18 : BettiTally
 
i19 : lift(oo,ZZ)

             0 1 2
o19 = total: 1 2 7
          0: 1 2 .
          1: . . 3
          2: . . 4

o19 : BettiTally

Given a ring $R$, a chain complex with zero maps over $R$ that has a prescribed Betti table can be constructed. Negative entries are ignored and rational entries produce an error. Multigraded rings work only if the Betti tally contains degrees of the correct degree length.

i20 : R = QQ[x,y]

o20 = R

o20 : PolynomialRing
 
i21 : C = R^t

       1      2      7
o21 = R  <-- R  <-- R
                     
      0      1      2

o21 : ChainComplex
 
i22 : betti C

             0 1 2
o22 = total: 1 2 7
          0: 1 2 .
          1: . . 3
          2: . . 4

o22 : BettiTally
 
i23 : C.dd

           1         2
o23 = 0 : R  <----- R  : 1
                0

           2         7
      1 : R  <----- R  : 2
                0

o23 : ChainComplexMap

Contributors

Hans-Christian von Bothmer donated the last feature.

Types of Betti tally :

Functions and methods returning a Betti tally :

Methods that use a Betti tally :

For the programmer

The object BettiTally is a type, with ancestor classes VirtualTally < HashTable < Thing.