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Complexes :: betti(Complex)

betti(Complex) -- display of degrees in a complex

Synopsis

Description

Column $j$ of the top row of the diagram gives the rank of the $j$-th component $C_j$ of the complex $C$. The entry in column $j$ in the row labelled $i$ is the number of basis elements of (weighted) degree $i+j$ in $C_j$. When the complex is the free resolution of a module the entries are the total and the graded Betti numbers of the module.

As a first example, we consider the ideal in 18 variables which cuts out the variety of commuting 3 by 3 matrices.

i1 : S = ZZ/101[vars(0..17)]

o1 = S

o1 : PolynomialRing
i2 : m1 = genericMatrix(S,a,3,3)

o2 = | a d g |
     | b e h |
     | c f i |

             3       3
o2 : Matrix S  <--- S
i3 : m2 = genericMatrix(S,j,3,3)

o3 = | j m p |
     | k n q |
     | l o r |

             3       3
o3 : Matrix S  <--- S
i4 : J = ideal(m1*m2-m2*m1)

o4 = ideal (d*k + g*l - b*m - c*p, b*j - a*k + e*k + h*l - b*n - c*q, c*j +
     ------------------------------------------------------------------------
     f*k - a*l + i*l - b*o - c*r, - d*j + a*m - e*m + d*n + g*o - f*p, - d*k
     ------------------------------------------------------------------------
     + b*m + h*o - f*q, - d*l + c*m + f*n - e*o + i*o - f*r, - g*j - h*m +
     ------------------------------------------------------------------------
     a*p - i*p + d*q + g*r, - g*k - h*n + b*p + e*q - i*q + h*r, - g*l - h*o
     ------------------------------------------------------------------------
     + c*p + f*q)

o4 : Ideal of S
i5 : C0 = freeResolution J

      1      8      33      60      61      32      5
o5 = S  <-- S  <-- S   <-- S   <-- S   <-- S   <-- S
                                                    
     0      1      2       3       4       5       6

o5 : Complex
i6 : betti C0

            0 1  2  3  4  5 6
o6 = total: 1 8 33 60 61 32 5
         0: 1 .  .  .  .  . .
         1: . 8  2  .  .  . .
         2: . . 31 32  3  . .
         3: . .  . 28 58 32 4
         4: . .  .  .  .  . 1

o6 : BettiTally

From the display, we see that $J$ has 8 minimal generators, all in degree 2, and that there are 2 linear syzygies on these generators, and 31 quadratic syzygies. Since this complex is the free resolution of $S/J$, the projective dimension is 6, the index of the last column, and the regularity of $S/J$ is 4, the index of the last row in the diagram.

i7 : length C0

o7 = 6
i8 : pdim betti C0

o8 = 6
i9 : regularity betti C0

o9 = 4

The betti display still makes sense if the complex is not a free resolution.

i10 : betti dual C0

             -6 -5 -4 -3 -2 -1 0
o10 = total:  5 32 61 60 33  8 1
         -4:  1  .  .  .  .  . .
         -3:  4 32 58 28  .  . .
         -2:  .  .  3 32 31  . .
         -1:  .  .  .  .  2  8 .
          0:  .  .  .  .  .  . 1

o10 : BettiTally
i11 : C1 = Hom(C0, image matrix{{a,b}});
i12 : betti C1

             -6 -5  -4  -3 -2 -1 0
o12 = total: 10 64 122 120 66 16 2
         -3:  2  .   .   .  .  . .
         -2:  8 64 116  56  .  . .
         -1:  .  .   6  64 62  . .
          0:  .  .   .   .  4 16 .
          1:  .  .   .   .  .  . 2

o12 : BettiTally
i13 : C1_-6

o13 = image {-9}  | 0 0 b a 0 0 0 0 0 0 |
            {-9}  | 0 0 0 0 b a 0 0 0 0 |
            {-9}  | 0 0 0 0 0 0 b a 0 0 |
            {-9}  | 0 0 0 0 0 0 0 0 b a |
            {-10} | b a 0 0 0 0 0 0 0 0 |

                              5
o13 : S-module, submodule of S

This module has 10 generators, 2 in degree $-9=(-6)+(-3)$, and 8 in degree $-8=(-6)+(-2)$.

In the multi-graded case, the heft vector is used, by default, as the weight vector for weighting the components of the degree vectors of basis elements.

The following example is a nonstandard $\mathbb{Z}$-graded polynomial ring.

i14 : R = ZZ/101[a,b,c,Degrees=>{-1,-2,-3}];
i15 : heft R

o15 = {-1}

o15 : List
i16 : C2 = freeResolution coker vars R

       1      3      3      1
o16 = R  <-- R  <-- R  <-- R
                            
      0      1      2      3

o16 : Complex
i17 : betti C2

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

o17 : BettiTally
i18 : betti(C2, Weights => {1})

             0 1 2 3
o18 = total: 1 3 3 1
         -9: . . . 1
         -8: . . . .
         -7: . . 1 .
         -6: . . 1 .
         -5: . . 1 .
         -4: . 1 . .
         -3: . 1 . .
         -2: . 1 . .
         -1: . . . .
          0: 1 . . .

o18 : BettiTally

The following example is the Cox ring of the second Hirzebruch surface, and the complex is the free resolution of the irrelevant ideal.

i19 : T = QQ[a,b,c,d,Degrees=>{{1,0},{-2,1},{1,0},{0,1}}];
i20 : B = intersect(ideal(a,c),ideal(b,d))

o20 = ideal (b*c, a*b, c*d, a*d)

o20 : Ideal of T
i21 : C3 = freeResolution B

       1      4      4      1
o21 = T  <-- T  <-- T  <-- T
                            
      0      1      2      3

o21 : Complex
i22 : dd^C3

           1                       4
o22 = 0 : T  <------------------- T  : 1
                | ab bc ad cd |

           4                               4
      1 : T  <--------------------------- T  : 2
                {-1, 1} | -c -d 0  0  |
                {-1, 1} | a  0  0  -d |
                {1, 1}  | 0  b  -c 0  |
                {1, 1}  | 0  0  a  b  |

           4                      1
      2 : T  <------------------ T  : 3
                {0, 1}  | d  |
                {-1, 2} | -c |
                {2, 1}  | -b |
                {-1, 2} | a  |

o22 : ComplexMap
i23 : heft T

o23 = {1, 3}

o23 : List
i24 : betti C3

             0 1 2 3
o24 = total: 1 4 4 1
          0: 1 . . .
          1: . 2 1 .
          2: . . . .
          3: . 2 3 1

o24 : BettiTally
i25 : betti(C3, Weights => {1,0})

             0 1 2 3
o25 = total: 1 4 4 1
         -3: . . 2 1
         -2: . 2 1 .
         -1: . . . .
          0: 1 2 1 .

o25 : BettiTally
i26 : betti(C3, Weights => {0,1})

             0 1 2 3
o26 = total: 1 4 4 1
         -1: . . 2 1
          0: 1 4 2 .

o26 : BettiTally
i27 : degrees C3_1

o27 = {{-1, 1}, {-1, 1}, {1, 1}, {1, 1}}

o27 : List

See also