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resolutionCohomological -- constructs a free resolution in cohomological Mackey functors of specified length

Description

This method returns a free resolution of a cohomological Mackey functor by free cohomological Mackey functors. The resolution is implemented as a list of Mackey functor homomorphisms.

i1 : N = cokernel(matrix({{3}}));
 
i2 : M = makeZeroOnUnderlyingMackeyFunctor (3,N);
 
i3 : resolutionCohomological(M,4)

o3 = {                  fix : | 1 |       ,         fix : | 3 |           ,
      cokernel | 3 | <---------------   1     1  <-------------------   1  
            ^ |                       ZZ    ZZ                        ZZ   
            | |                       ^ |   ^ |                       ^ |  
            | |                       | |   | |                       | |  
            | v                       | v   | v                       | v  
             0       <---------------   1     1  <-------------------   3  
            ^ |         und : 0       ZZ    ZZ      und : | 1 1 1 |   ZZ   
            └-┘                       ^ |   ^ |                       ^ |  
                                      └-┘   └-┘                       └-┘  
                                                                           
                                                                           
     ------------------------------------------------------------------------
             fix : 0                  ,         fix : | 3 1 |           ,    
       1  <----------------------   1     1  <---------------------   2      
     ZZ                           ZZ    ZZ                          ZZ      2
     ^ |                          ^ |   ^ |                         ^ |   ZZ 
     | |                          | |   | |                         | |   ^ |
     | v                          | v   | v                         | v   | |
       3  <----------------------   3     3  <---------------------   4   | v
     ZZ      und : | -1 0  1  |   ZZ    ZZ      und : | 1 1 1 1 |   ZZ      4
     ^ |           | 1  -1 0  |   ^ |   ^ |           | 1 1 1 1 |   ^ |   ZZ 
     └-┘           | 0  1  -1 |   └-┘   └-┘           | 1 1 1 1 |   └-┘   ^ |
                                                                          └-┘
                                                                             
     ------------------------------------------------------------------------
        fix : | -1 |             }
              | 3  |
     <----------------------   1
                             ZZ
                             ^ | 
                             | | 
                             | v 
     <----------------------   3
        und : | -1 0  0  |   ZZ
              | 0  -1 0  |   ^ | 
              | 0  0  -1 |   └-┘ 
              | 1  1  1  |

o3 : List

See also

Ways to use resolutionCohomological:

  • resolutionCohomological(CpMackeyFunctor,ZZ)

For the programmer

The object resolutionCohomological is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/CpMackeyFunctors/Documentation/ResolutionsDoc.m2:67:0.