extend(ChainComplex,ChainComplex,Matrix) -- extend a module map to a chain map, if possible

Synopsis

Function: extend
Usage:

extend(D,C,f0)
Inputs:
- D, a chain complex
- C, a chain complex
- f0, a matrix, a map from C_0 to D_0
Optional inputs:
- Verify => a Boolean value, default value true, whether to check that the map extends
Outputs:
- a chain complex map, a chain complex map f: D <--- C of degree 0 extending f0 in the sense that f_0==f0

Description

i1 : R = ZZ/101[a..c]

o1 = R

o1 : PolynomialRing

i2 : I = image vars R

o2 = image | a b c |

                             1
o2 : R-module, submodule of R

i3 : J = image symmetricPower (2,vars R)

o3 = image | a2 ab ac b2 bc c2 |

                             1
o3 : R-module, submodule of R

i4 : g = extend( resolution (R^1/I), resolution (R^1/J), id_(R^1))

          1             1
o4 = 0 : R  <--------- R  : 0
               | 1 |

          3                           6
     1 : R  <----------------------- R  : 1
               {1} | a b 0 0 0 0 |
               {1} | 0 0 b 0 0 0 |
               {1} | 0 0 0 a b c |

          3                               8
     2 : R  <--------------------------- R  : 2
               {2} | 0 b 0 0 0 0 0 0 |
               {2} | 0 0 a b 0 0 0 0 |
               {2} | 0 0 0 0 0 b 0 0 |

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

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

o4 : ChainComplexMap

i5 : g_1

o5 = {1} | a b 0 0 0 0 |
     {1} | 0 0 b 0 0 0 |
     {1} | 0 0 0 a b c |

             3      6
o5 : Matrix R  <-- R

i6 : g_2

o6 = {2} | 0 b 0 0 0 0 0 0 |
     {2} | 0 0 a b 0 0 0 0 |
     {2} | 0 0 0 0 0 b 0 0 |

             3      8
o6 : Matrix R  <-- R

Ways to use this method: