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flattenedESTensor -- make a flattened tensor from elementary symmetric functions

Synopsis

Description

Given a list $L = \{a, b_1,\dots, b_n\}$ of positive integers with $ a= sum_i b_i, $ and a field (or ring of integers) kk, the script creates a ring $S = kk[x_1,\dots,x_n]$ and a map $$ f: A \to B_1\otimes\cdots \otimes B_n $$ of LabeledModules over $S$, where $A$ is a free LabeledModule of rank $a$ and $B_i$ is a free LabeledModule of rank $b_i$. The map $f$ is constructed from symmetric functions, and corresponds to collection of linear forms on $P^{b_1-1}\times\cdots\timesß P^{b_n-1}$ 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 $f: A \to B_1\otimes \cdots \otimes B_n$, with $a = rank A, b_i = rank B_i$.

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 ])  <--- (kk[x ..x ])
                 0   2              0   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 :

For the programmer

The object flattenedESTensor is a method function with options.