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naiveNorm(SimplicialModule,ZZ) -- compute the naive normalization of a simplicial object

Description

This function computes the naive normalization of a simplicial object S. The naive normalization is a complex built from the modules of the simplicial object, with a differential that is the alternating sum of the face maps. In general the naive normalization is homotopy equivalent to the normalization (see normalize), but is much bigger in general.

i1 : Q = ZZ/101[a..d]

o1 = Q

o1 : PolynomialRing
 
i2 : K = koszulComplex vars Q

      1      4      6      4      1
o2 = Q  <-- Q  <-- Q  <-- Q  <-- Q
                                  
     0      1      2      3      4

o2 : Complex
 
i3 : S = simplicialModule(K, Degeneracy => true)

      1      5      15      35      70
o3 = Q  <-- Q  <-- Q   <-- Q   <-- Q  <-- ...
                                    
     0      1      2       3       4

o3 : SimplicialModule
 
i4 : nK = naiveNorm(S)

      1      5      15      35      70
o4 = Q  <-- Q  <-- Q   <-- Q   <-- Q
                                    
     0      1      2       3       4

o4 : Complex
 
i5 : isWellDefined nK

o5 = true
 
i6 : prune HH nK

                                                 46
o6 = cokernel | d c b a | <-- 0 <-- 0 <-- 0 <-- Q
                                                 
     0                        1     2     3     4

o6 : Complex

Note that in the above, the naive normalization will always be an infinite complex, so there will always be extraneous homology at the tail end. Note in this case that the homology of the naive normalization is precisely the homology of K, as it should be (in fact, it is homotopy equivalent to K).

See also

Ways to use this method:

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/SimplicialModules/SimplicialModuleDOC.m2:3438:0.