Module ^** ZZ -- tensor power

Operator: ^**
Usage:

M^**i
Inputs:
- M, a module
- i, an integer
Outputs:
- a module, the i-th tensor power of M

Description

The second symmetric power of the canonical module of the rational quartic:

i1 : R = QQ[a..d];

i2 : I = monomialCurveIdeal(R,{1,3,4})

                        3      2     2    2    3    2
o2 = ideal (b*c - a*d, c  - b*d , a*c  - b d, b  - a c)

o2 : Ideal of R

i3 : M = Ext^1(I,R^{-4})

o3 = subquotient (| c a  0 b 0  0 |, | b2 0  -a -c |)
                  | d 0  a 0 b  0 |  | ac 0  -b -d |
                  | 0 -d c 0 0  b |  | c2 -b d  0  |
                  | 0 0  0 d -c a |  | bd -a c  0  |

                               4
o3 : R-module, subquotient of R

i4 : M^**2

o4 = cokernel {2} | 1 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  1 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  |
              {2} | 0 1 0 c a -d 0 -b 0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  1 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  |
              {2} | 0 0 0 d b 0  c 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  1 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  |
              {2} | 0 0 0 0 0 c  b a  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  1 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  |
              {2} | 0 1 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  1 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  |
              {2} | 0 0 1 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  1 0 0 0 0 0  0 0  |
              {2} | 0 0 0 0 0 0  0 0  1 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 1 0 c a -d 0 -b 0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  |
              {2} | 0 0 0 0 0 0  0 0  0 1 0 c a -d 0 -b 0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 1 0 c a -d 0 -b 0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  |
              {2} | 0 0 0 0 0 0  0 0  0 0 0 d b 0  c 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 1 0 c a -d 0 -b 0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  |
              {2} | 0 0 0 0 0 0  0 0  0 0 0 0 0 c  b a  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 1 0 c a -d 0 -b 0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  |
              {2} | 0 0 0 0 0 0  0 0  0 1 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 1 0 c a -d 0 -b 0 0 0 0 0 0  0 0  |
              {2} | 0 0 0 0 0 0  0 0  0 0 1 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 1 0 c a -d 0 -b |
              {2} | 0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  1 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 d b 0  c 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  |
              {2} | 0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 1 0 c a -d 0 -b 0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 d b 0  c 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  |
              {2} | 0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 d b 0  c 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 d b 0  c 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  |
              {2} | 0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 c  b a  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 d b 0  c 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  |
              {2} | 0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 1 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 d b 0  c 0  0 0 0 0 0 0  0 0  |
              {2} | 0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 1 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 d b 0  c 0  |
              {2} | 0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  1 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 c  b a  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  |
              {2} | 0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 1 0 c a -d 0 -b 0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 c  b a  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  |
              {2} | 0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 d b 0  c 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 c  b a  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  |
              {2} | 0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 c  b a  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 c  b a  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  |
              {2} | 0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 1 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 c  b a  0 0 0 0 0 0  0 0  |
              {2} | 0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 1 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 c  b a  |
              {2} | 0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  1 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 1 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  |
              {2} | 0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 1 0 c a -d 0 -b 0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 1 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  |
              {2} | 0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 d b 0  c 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 1 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  |
              {2} | 0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 c  b a  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 1 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  |
              {2} | 0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 1 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 1 0 0 0 0  0 0  0 0 0 0 0 0  0 0  |
              {2} | 0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 1 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 1 0 0 0 0  0 0  |
              {2} | 0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  1 0 0 0 0 0  0 0  0 0 1 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  |
              {2} | 0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 1 0 c a -d 0 -b 0 0 0 0 0 0  0 0  0 0 1 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  |
              {2} | 0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 d b 0  c 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 1 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  |
              {2} | 0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 c  b a  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 1 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  |
              {2} | 0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 1 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 1 0 0 0  0 0  0 0 0 0 0 0  0 0  |
              {2} | 0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 1 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 0 0 0 0  0 0  0 0 1 0 0 0  0 0  |

                            36
o4 : R-module, quotient of R

Ways to use this method:

Module ^** ZZ -- tensor power

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/Macaulay2Doc/functions/tensor-doc.m2:444:0.