ComplexMap ** ZZdFactorizationMap -- the map of ZZ/d-graded factorizations between tensor factorizations, obtained by first folding the relevant complex map

Operator: **
Usage:

h = f ** g

h = tensor(f, g)
Inputs:
- f, a map of complexes, or a map of ZZ/d-graded factorizations
- g, a map of ZZ/d-graded factorizations, or a map of complexes
Outputs:
- h, a map of ZZ/d-graded factorizations,

Description

The maps $f : C \to D$ and $g : E \to F$ of ZZ/d-graded factorizations induce the map $h = f \otimes g : C \otimes E \to D \otimes F$ defined by $c \otimes e \mapsto f(c) \otimes g(e)$.

i1 : S = ZZ/101[a..c]

o1 = S

o1 : PolynomialRing

i2 : C = freeResolution coker vars S

      1      3      3      1
o2 = S  <-- S  <-- S  <-- S
                           
     0      1      2      3

o2 : Complex

i3 : D = (freeResolution coker matrix{{a^2,a*b,b^3}})[-1]

      1      3      2
o3 = S  <-- S  <-- S
                    
     1      2      3

o3 : Complex

i4 : f = randomComplexMap(D,C)

                   1
o4 = 0 : 0 <----- S  : 0
              0

          1                                                3
     1 : S  <-------------------------------------------- S  : 1
               | 24a-36b-30c -29a+19b+19c -10a-29b-8c |

          3                           3
     2 : S  <----------------------- S  : 2
               {2} | -22 -24 -16 |
               {2} | -29 -38 39  |
               {3} | 0   0   0   |

          2                  1
     3 : S  <-------------- S  : 3
               {3} | 21 |
               {4} | 0  |

o4 : ComplexMap

i5 : m = ideal vars (S/(a^3+b^3))

o5 = ideal (a, b, c)

                 S
o5 : Ideal of -------
               3    3
              a  + b

i6 : g = randomFactorizationMap(tailMF m, tailMF (m^2), Cycle => true)

          4                                                                                                                                                                                9
o6 = 0 : S  <---------------------------------------------------------------------------------------------------------------------------------------------------------------------------- S  : 0
               {3} | -47a-18b+50c 47a-48b-41c 6a+44b+42c 13a2+47ab+49b2 7a2-47ab+17b2+47ac-41bc -28a2+41ab-45b2 -45a2-50ab-28b2 -11a2+2ab+47b2-50ac+47bc 11a2-15ab-34b2-16ac+7bc+47c2 |
               {4} | 40           19          11         -16a-2b-38c    -47a+34b+23c            39a-47b+19c     -43a+48b+15c    -22a+35b-16c             39a-23b+2c                   |
               {4} | 47           -2          15         -11a+38b       -30a-25b                5a-19b          26a-15b         -7a-37b                  -28a+33b+48c                 |
               {4} | -34          -35         23         -47a-37b+11c   40a+47b-23c             -48a-37b-5c     2a-4b-26c       19a+22b-40c              11a-39b-33c                  |

          4                                                                                                                                   9
     1 : S  <------------------------------------------------------------------------------------------------------------------------------- S  : 1
               {5} | -47a-25b-38c -28a-15b    47a-37b+15c  6a+33b-16c   -13a-19b     -7a+47b-19c  45a-38b      11a-2b-38c   -11a+15b+23c |
               {5} | -40a-47b-31c -39a+4b-47c -19a-22b-11c -11a+39b-12c -16a+37b+50c -47a+34b-46c -43a-37b-41c -22a+35b+11c 39a-23b+38c  |
               {5} | -47a+17b+11c -5a-28b     2a+47b-26c   -15a-34b-40c -11a-45b     -30a-18b+5c  26a-49b      -7a-48b+11c  -28a+44b-23c |
               {6} | 34           48          35           -23          -47          40           2            19           11           |

o6 : ZZdFactorizationMap

i7 : isWellDefined (g**f)

o7 = true

Ways to use this method:

Complex ** ZZdFactorizationMap
ComplexMap ** ZZdFactorizationMap -- the map of ZZ/d-graded factorizations between tensor factorizations, obtained by first folding the relevant complex map
ZZdFactorizationMap ** Complex
ZZdFactorizationMap ** ComplexMap

The source of this document is in MatrixFactorizations/MatrixFactorizationsDOC.m2:3904:0.

ComplexMap ** ZZdFactorizationMap -- the map of ZZ/d-graded factorizations between tensor factorizations, obtained by first folding the relevant complex map

Description

See also

Ways to use this method: