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TensorComplexes :: pureResTC1

pureResTC1 -- computes the first map of a balanced tensor complex with pure resolution of a given type

Synopsis

Description

Given a degree sequence d∈ℤn+1 and a field k of arbtirary characteristic, this produces the first map of a balanced tensor complex with a pure resolution of type d, as constructed in Section 3 of the paper “Tensor Complexes: Multilinear free resolutions constructed from higher tensors by Berkesch-Erman-Kummini-Sam. The cokernel of the output is an indecomposable module of codimension n.

The code gives an error if d is not strictly increasing with d0=0.

i1 : d={0,2,4,5};
i2 : p=pureResTC1(d,ZZ/32003)

o2 = | -x_(0,1,0,0)x_(1,0,0,0)+x_(0,0,0,0)x_(1,1,0,0)                                               -x_(0,1,0,0)x_(2,0,0,0)+x_(0,0,0,0)x_(2,1,0,0)                                               -x_(1,1,0,0)x_(2,0,0,0)+x_(1,0,0,0)x_(2,1,0,0)                                               -x_(0,1,0,0)x_(3,0,0,0)+x_(0,0,0,0)x_(3,1,0,0)                                               -x_(1,1,0,0)x_(3,0,0,0)+x_(1,0,0,0)x_(3,1,0,0)                                               -x_(2,1,0,0)x_(3,0,0,0)+x_(2,0,0,0)x_(3,1,0,0)                                               -x_(0,1,0,0)x_(4,0,0,0)+x_(0,0,0,0)x_(4,1,0,0)                                               -x_(1,1,0,0)x_(4,0,0,0)+x_(1,0,0,0)x_(4,1,0,0)                                               -x_(2,1,0,0)x_(4,0,0,0)+x_(2,0,0,0)x_(4,1,0,0)                                               -x_(3,1,0,0)x_(4,0,0,0)+x_(3,0,0,0)x_(4,1,0,0)                                               |
     | -x_(0,1,1,0)x_(1,0,0,0)-x_(0,1,0,0)x_(1,0,1,0)+x_(0,0,1,0)x_(1,1,0,0)+x_(0,0,0,0)x_(1,1,1,0) -x_(0,1,1,0)x_(2,0,0,0)-x_(0,1,0,0)x_(2,0,1,0)+x_(0,0,1,0)x_(2,1,0,0)+x_(0,0,0,0)x_(2,1,1,0) -x_(1,1,1,0)x_(2,0,0,0)-x_(1,1,0,0)x_(2,0,1,0)+x_(1,0,1,0)x_(2,1,0,0)+x_(1,0,0,0)x_(2,1,1,0) -x_(0,1,1,0)x_(3,0,0,0)-x_(0,1,0,0)x_(3,0,1,0)+x_(0,0,1,0)x_(3,1,0,0)+x_(0,0,0,0)x_(3,1,1,0) -x_(1,1,1,0)x_(3,0,0,0)-x_(1,1,0,0)x_(3,0,1,0)+x_(1,0,1,0)x_(3,1,0,0)+x_(1,0,0,0)x_(3,1,1,0) -x_(2,1,1,0)x_(3,0,0,0)-x_(2,1,0,0)x_(3,0,1,0)+x_(2,0,1,0)x_(3,1,0,0)+x_(2,0,0,0)x_(3,1,1,0) -x_(0,1,1,0)x_(4,0,0,0)-x_(0,1,0,0)x_(4,0,1,0)+x_(0,0,1,0)x_(4,1,0,0)+x_(0,0,0,0)x_(4,1,1,0) -x_(1,1,1,0)x_(4,0,0,0)-x_(1,1,0,0)x_(4,0,1,0)+x_(1,0,1,0)x_(4,1,0,0)+x_(1,0,0,0)x_(4,1,1,0) -x_(2,1,1,0)x_(4,0,0,0)-x_(2,1,0,0)x_(4,0,1,0)+x_(2,0,1,0)x_(4,1,0,0)+x_(2,0,0,0)x_(4,1,1,0) -x_(3,1,1,0)x_(4,0,0,0)-x_(3,1,0,0)x_(4,0,1,0)+x_(3,0,1,0)x_(4,1,0,0)+x_(3,0,0,0)x_(4,1,1,0) |
     | -x_(0,1,1,0)x_(1,0,1,0)+x_(0,0,1,0)x_(1,1,1,0)                                               -x_(0,1,1,0)x_(2,0,1,0)+x_(0,0,1,0)x_(2,1,1,0)                                               -x_(1,1,1,0)x_(2,0,1,0)+x_(1,0,1,0)x_(2,1,1,0)                                               -x_(0,1,1,0)x_(3,0,1,0)+x_(0,0,1,0)x_(3,1,1,0)                                               -x_(1,1,1,0)x_(3,0,1,0)+x_(1,0,1,0)x_(3,1,1,0)                                               -x_(2,1,1,0)x_(3,0,1,0)+x_(2,0,1,0)x_(3,1,1,0)                                               -x_(0,1,1,0)x_(4,0,1,0)+x_(0,0,1,0)x_(4,1,1,0)                                               -x_(1,1,1,0)x_(4,0,1,0)+x_(1,0,1,0)x_(4,1,1,0)                                               -x_(2,1,1,0)x_(4,0,1,0)+x_(2,0,1,0)x_(4,1,1,0)                                               -x_(3,1,1,0)x_(4,0,1,0)+x_(3,0,1,0)x_(4,1,1,0)                                               |

               ZZ                                                                                                                                                                                                          3         ZZ                                                                                                                                                                                                          10
o2 : Matrix (-----[x       , x       , x       , x       , x       , x       , x       , x       , x       , x       , x       , x       , x       , x       , x       , x       , x       , x       , x       , x       ])  <--- (-----[x       , x       , x       , x       , x       , x       , x       , x       , x       , x       , x       , x       , x       , x       , x       , x       , x       , x       , x       , x       ])
             32003  0,0,0,0   0,0,1,0   0,1,0,0   0,1,1,0   1,0,0,0   1,0,1,0   1,1,0,0   1,1,1,0   2,0,0,0   2,0,1,0   2,1,0,0   2,1,1,0   3,0,0,0   3,0,1,0   3,1,0,0   3,1,1,0   4,0,0,0   4,0,1,0   4,1,0,0   4,1,1,0          32003  0,0,0,0   0,0,1,0   0,1,0,0   0,1,1,0   1,0,0,0   1,0,1,0   1,1,0,0   1,1,1,0   2,0,0,0   2,0,1,0   2,1,0,0   2,1,1,0   3,0,0,0   3,0,1,0   3,1,0,0   3,1,1,0   4,0,0,0   4,0,1,0   4,1,0,0   4,1,1,0
i3 : betti res coker p

            0  1  2 3
o3 = total: 3 10 15 8
         0: 3  .  . .
         1: . 10  . .
         2: .  . 15 8

o3 : BettiTally

See also

  • pureResTC -- constructs the balanced tensor complex of a given type

Ways to use pureResTC1 :

  • pureResTC1(List,Ring) (missing documentation)