Hom(Module,ChainComplex) -- the Hom functor

Synopsis

Function: Hom
Usage:

Hom(M, C)

Hom(C, M)
Inputs:
- M, a module,
- C, a chain complex,
Optional inputs:
- DegreeLimit => ..., default value null, an integer or a list, see Hom(...,DegreeLimit=>...)
- MinimalGenerators => a Boolean value, default value true, see Hom(...,MinimalGenerators=>...)
- Strategy => a thing, default value null, see Hom(...,Strategy=>...)
Outputs:
- a chain complex, the chain complex whose $i$-th spot is $\mathrm{Hom}(M, C_i)$, in the first case, or $\mathrm{Hom}(C_(-i), M)$ in the second case.

Description

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

i2 : C = res coker vars R

      1      4      6      4      1
o2 = R  <-- R  <-- R  <-- R  <-- R  <-- 0
                                         
     0      1      2      3      4      5

o2 : ChainComplex

i3 : M = R^1/(a,b)

o3 = cokernel | a b |

                            1
o3 : R-module, quotient of R

i4 : C' = Hom(C,M)

o4 = 0  <-- cokernel {-4} | b a | <-- cokernel {-3} | b a 0 0 0 0 0 0 | <-- cokernel {-2} | b a 0 0 0 0 0 0 0 0 0 0 | <-- cokernel {-1} | b a 0 0 0 0 0 0 | <-- cokernel | b a |
                                               {-3} | 0 0 b a 0 0 0 0 |              {-2} | 0 0 b a 0 0 0 0 0 0 0 0 |              {-1} | 0 0 b a 0 0 0 0 |      
     -5     -4                                 {-3} | 0 0 0 0 b a 0 0 |              {-2} | 0 0 0 0 b a 0 0 0 0 0 0 |              {-1} | 0 0 0 0 b a 0 0 |     0
                                               {-3} | 0 0 0 0 0 0 b a |              {-2} | 0 0 0 0 0 0 b a 0 0 0 0 |              {-1} | 0 0 0 0 0 0 b a |
                                                                                     {-2} | 0 0 0 0 0 0 0 0 b a 0 0 |      
                                      -3                                             {-2} | 0 0 0 0 0 0 0 0 0 0 b a |     -1
                                                                             
                                                                            -2

o4 : ChainComplex

i5 : C'.dd_-1

o5 = {-2} | 0 0 0 0  |
     {-2} | c 0 0 0  |
     {-2} | 0 c 0 0  |
     {-2} | d 0 0 0  |
     {-2} | 0 d 0 0  |
     {-2} | 0 0 d -c |

o5 : Matrix

i6 : C'.dd^2 == 0

o6 = true

induced map on Hom

Usage:

Hom(f,M)

Hom(M,f)
Inputs:
- f, a chain complex map,
- M, a module,
- DegreeLimit => ...
- MinimalGenerators => ...
- Strategy => ...
Outputs:
- a chain complex map, the induced map on Hom

Caveat

See Hom(Complex,Complex) for Hom of two chain complexes.

Ways to use this method: