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Packages » MatrixFactorizations :: map(ZZdFactorization,ZZdFactorization,ZZdFactorizationMap)
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map(ZZdFactorization,ZZdFactorization,ZZdFactorizationMap) -- make a new map of ZZ/d-graded factorizations from an existing one (for inducing maps)

Description

A map of factorizations $f : C' \rightarrow D'$ is a sequence of maps $f_i : C'_i \rightarrow D'_{d'+i}$. The new map $g : C \rightarrow D$ is the sequence of maps $g_i : C_i \rightarrow D_{d+i}$ induced by the matrix of $f_i$.

One use for this function is to get the new map of ZZ/d-graded factorizations obtained by shifting the source or target of an existing chain map. For example, one can regard the differential on a factorization can be regarded as a map of degree zero between shifted factorizations.

i1 : S = ZZ/101[a,b,c]

o1 = S

o1 : PolynomialRing
 
i2 : R = S/(a^2+b^2+c^2);
 
i3 : m = ideal vars R

o3 = ideal (a, b, c)

o3 : Ideal of R
 
i4 : C = tailMF m

      4      4      4
o4 = S  <-- S  <-- S
                    
     0      1      0

o4 : ZZdFactorization
 
i5 : D = ZZdfactorization(C, Base => 1)

      4      4      4
o5 = S  <-- S  <-- S
                    
     0      1      0

o5 : ZZdFactorization
 
i6 : f = map(D, C, dd^C, Degree => 0)

          4                          4
o6 = 0 : S  <---------------------- S  : 0
               {4} | 0  -b a -c |
               {4} | -a -c 0 b  |
               {4} | -c a  b 0  |
               {4} | b  0  c a  |

          4                          4
     1 : S  <---------------------- S  : 1
               {3} | 0  -a -c b |
               {3} | -b -c a  0 |
               {3} | a  0  b  c |
               {3} | -c b  0  a |

o6 : ZZdFactorizationMap
 
i7 : assert isWellDefined f
 
i8 : assert(degree f == 0)
 
i9 : assert isCommutative f
 
i10 : assert isFactorizationMorphism f
 
i11 : assert not isFactorizationMorphism dd^C

See also

Ways to use this method:

The source of this document is in MatrixFactorizations/MatrixFactorizationsDOC.m2:2677:0.