kernel(Matrix) -- kernel of a map of modules

Function: kernel
Usage:

kernel f

kernel a
Inputs:
- f, a matrix or a ring element, a map of modules $M \to N$, or a ring element interpreted as a $1 \times 1$ matrix
Optional inputs:
- DegreeLimit => ..., default value {}, stop after finding enough elements of a subring
- Strategy => ..., default value {}, strategy for Groebner Basis computations used in kernel computations
- SubringLimit => ..., default value infinity, stop after finding enough elements of a subring
Outputs:
- a module, the kernel of the map, as a submodule of $M$

Description

The kernel is the submodule of $M$ consisting of all elements that map to zero.

Over polynomial rings this is computed using a Groebner basis computation.

i1 : R = ZZ/32003[a,b]/(ideal(a,b))^3

o1 = R

o1 : QuotientRing

i2 : M = R^1/(ideal a^2)

o2 = cokernel | a2 |

                            1
o2 : R-module, quotient of R

i3 : mat = matrix{{a^2,b^2},{b,a}}

o3 = | a2 b2 |
     | b  a  |

             2      2
o3 : Matrix R  <-- R

i4 : ker mat

o4 = image {2} | a  b2 ab -ab 0  0  |
           {2} | -b 0  0  b2  ab a2 |

                             2
o4 : R-module, submodule of R

i5 : presentation ker mat

o5 = {3} | b a2 0 0 0 0 0 0 0 0 0 0 |
     {4} | 0 0  b a 0 0 0 0 0 0 0 0 |
     {4} | 0 0  0 0 b a 0 0 0 0 0 0 |
     {4} | 1 0  0 0 0 0 b a 0 0 0 0 |
     {4} | 0 0  0 0 0 0 0 0 b a 0 0 |
     {4} | 0 0  0 0 0 0 0 0 0 0 b a |

             6      12
o5 : Matrix R  <-- R

i6 : syz mat

o6 = {2} | a  b2 ab -ab 0  0  |
     {2} | -b 0  0  b2  ab a2 |

             2      6
o6 : Matrix R  <-- R

i7 : f = map(M++M, M++M, mat)

o7 = | 0 b2 |
     | b a  |

o7 : Matrix

i8 : ker f

o8 = subquotient (| -a 0 b2 ab -a2 a2 |, | a2 0  |)
                  | b  a 0  0  0   0  |  | 0  a2 |

                               2
o8 : R-module, subquotient of R

Ways to use this method:

kernel(Matrix) -- kernel of a map of modules
kernel(RingElement)

The source of this document is in /build/reproducible-path/macaulay2-1.26.06+ds/M2/Macaulay2/packages/Macaulay2Doc/functions/kernel-doc.m2:54:0.

kernel(Matrix) -- kernel of a map of modules

Description

See also

Ways to use this method: