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free resolutions of modules

The function resolution (also called res), can be used to produce a free resolution of a module.
i1 : R = ZZ/101[x,y];
i2 : m = ideal vars R

o2 = ideal (x, y)

o2 : Ideal of R
i3 : M = m/m^3

o3 = subquotient (| x y |, | x3 x2y xy2 y3 |)

                               1
o3 : R-module, subquotient of R
i4 : C = resolution M

      2      5      3
o4 = R  <-- R  <-- R  <-- 0
                           
     0      1      2      3

o4 : ChainComplex
The default display for a chain complex shows the modules and the number of the stage at which they appear. See the documentation of resolution for details on the options that can be used to control the computation.

The same function, applied to a map f, will produce a map from a free resolution of the source of f to a free resolution of the target of f.

i5 : h = resolution inducedMap(M, m^2/m^4)

          2                     3
o5 = 0 : R  <----------------- R  : 0
               {1} | x y 0 |
               {1} | 0 0 y |

          5                             7
     1 : R  <------------------------- R  : 1
               {2} | 0 y 0 0 0 0 0 |
               {3} | 0 0 x y 0 0 0 |
               {3} | 0 0 0 0 0 0 0 |
               {3} | 0 0 0 0 x y 0 |
               {3} | 0 0 0 0 0 0 y |

          3                       4
     2 : R  <------------------- R  : 2
               {4} | 0 y 0 0 |
               {4} | 0 x 0 0 |
               {4} | 0 0 0 y |

     3 : 0 <----- 0 : 3
              0

o5 : ChainComplexMap