Module ^** ZZ -- tensor power

Synopsis

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

                            3
o3 : R-module, quotient of R

i4 : M^**2

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

                            9
o4 : R-module, quotient of R

Ways to use this method:

Module ^** ZZ -- tensor power