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# kernel(ChainComplexMap) -- kernel of a chain complex map

- Function: kernel
- Usage:
    `kernel f`
- Inputs:
  - f, a chain complex map, a map of chain complexes C --> D
- Optional inputs:
  - DegreeLimit (missing documentation) => ..., default value {},
  - Strategy (missing documentation) => ..., default value {},
  - SubringLimit => ..., default value infinity,
- Outputs:
  - a chain complex, the kernel of f

## Description

If f is a graded module map, then the result will be a graded module.

    i1 : R = QQ[a..d]

    o1 = R

    o1 : PolynomialRing

    i2 : I = ideal(a^3,b^3,c^3)

                 3   3   3
    o2 = ideal (a , b , c )

    o2 : Ideal of R

    i3 : C = res coker gens I

          1      3      3      1
    o3 = R  <-- R  <-- R  <-- R  <-- 0

         0      1      2      3      4

    o3 : ChainComplex

    i4 : D = res coker gens (I + ideal(a*b*c))

          1      4      6      3
    o4 = R  <-- R  <-- R  <-- R  <-- 0

         0      1      2      3      4

    o4 : ChainComplex

    i5 : F = extend(D,C,map(D_0,C_0,1))

              1             1
    o5 = 0 : R  <--------- R  : 0
                   | 1 |

              4                     3
         1 : R  <----------------- R  : 1
                   {3} | 1 0 0 |
                   {3} | 0 1 0 |
                   {3} | 0 0 0 |
                   {3} | 0 0 1 |

              6                     3
         2 : R  <----------------- R  : 2
                   {5} | 0 0 0 |
                   {5} | 0 0 0 |
                   {5} | 0 0 0 |
                   {6} | 1 0 0 |
                   {6} | 0 1 0 |
                   {6} | 0 0 1 |

              3                   1
         3 : R  <--------------- R  : 3
                   {7} | c2  |
                   {7} | -b2 |
                   {7} | a2  |

         4 : 0 <----- 0 : 4
                  0

    o5 : ChainComplexMap

## See also

- syz -- the syzygy matrix
- genericSkewMatrix -- make a generic skew symmetric matrix of variables

## Ways to use this method:

- kernel(ChainComplexMap) -- kernel of a chain complex map
- `kernel(GradedModuleMap)`

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/OldChainComplexes/docs/kernel-doc.m2:25:0.