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TensorComplexes :: flattenedESTensor

flattenedESTensor -- make a flattened tensor from elementary symmetric functions

Synopsis

Description

Given a list L = {a, b1,…, bn} of positive integers with a= sumi bi, and a field (or ring of integers) kk, the script creates a ring S = kk[x1,…,xn] and a map

f: A →B1⊗…⊗Bn

of LabeledModules over S, where A is a free LabeledModule of rank a and Bi is a free LabeledModule of rank bi. The map f is constructed from symmetric functions, and corresponds to collection of linear forms on Pb1-1×…×ß Pbn-1 as used in the construction of pure resolutions in the paper “Betti numbers of graded modules and cohomology of vector bundles” of Eisenbud and Schreyer.

The format of F is the one required by tensorComplex1, namely f: A →B1⊗…⊗Bn, with a = rank A, bi = rank Bi.

There is an OPTION, MonSize => n (where n is 8,16, or 32). This sets the MonomialSize option when the base ring of flattenedESTensor is created.

i1 : kk = ZZ/101

o1 = kk

o1 : QuotientRing
i2 : f = flattenedESTensor({5,2,1,2},kk)

o2 = | x_0 x_1 x_2 0   0   |
     | 0   x_0 x_1 x_2 0   |
     | 0   x_0 x_1 x_2 0   |
     | 0   0   x_0 x_1 x_2 |

                            4                      5
o2 : Matrix (kk[x , x , x ])  <--- (kk[x , x , x ])
                 0   1   2              0   1   2
i3 : numgens ring f

o3 = 3
i4 : betti matrix f

            0 1
o4 = total: 4 5
        -1: . 5
         0: 4 .

o4 : BettiTally
i5 : S = ring f

o5 = S

o5 : PolynomialRing
i6 : g = tensorComplex1 f

o6 = | x_0^2 0     x_0x_1 0      x_1^2-x_0x_2 0            x_0x_2 0      x_1x_2 0      x_2^2        0            0      0      0      0      0      0      0     0     |
     | 0     x_0^2 x_0^2  x_0x_1 x_0x_1       x_1^2-x_0x_2 x_0x_1 x_0x_2 x_1^2  x_1x_2 x_1x_2       x_2^2        x_0x_2 0      x_1x_2 0      x_2^2  0      0     0     |
     | 0     0     0      x_0^2  x_0^2        x_0x_1       0      x_0x_1 x_0x_1 x_1^2  x_1^2-x_0x_2 x_1x_2       0      x_0x_2 x_0x_2 x_1x_2 x_1x_2 x_2^2  x_2^2 0     |
     | 0     0     0      0      0            x_0^2        0      0      0      x_0x_1 0            x_1^2-x_0x_2 0      0      0      x_0x_2 0      x_1x_2 0     x_2^2 |

             4       20
o6 : Matrix S  <--- S
i7 : betti res coker g

            0  1  2 3
o7 = total: 4 20 20 4
         0: 4  .  . .
         1: . 20 20 .
         2: .  .  . 4

o7 : BettiTally

See also

Ways to use flattenedESTensor :

  • flattenedESTensor(List,Ring) (missing documentation)