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differentialModule(Complex) -- converts a complex into a differential module

Description

Given a complex of modules in homological degrees $-1, 0$, and $1$, with the differentials being identical, this method produces the corresponding DifferentialModule.

i1 : S = QQ[x,y]

o1 = S

o1 : PolynomialRing
 
i2 : del = map(S^{-1,0,0,1},S^{-1,0,0,1},matrix{{0,y,x,-1},{0,0,0,x},{0,0,0,-y},{0,0,0,0}}, Degree=>2)

o2 = {1}  | 0 y x -1 |
     {0}  | 0 0 0 x  |
     {0}  | 0 0 0 -y |
     {-1} | 0 0 0 0  |

             4      4
o2 : Matrix S  <-- S
 
i3 : C = complex{-del, -del}[1]

      4      4      4
o3 = S  <-- S  <-- S
                    
     -1     0      1

o3 : Complex
 
i4 : D = differentialModule C

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

o4 : DifferentialModule
 
i5 : D.dd

           4                         4
o5 = -1 : S  <--------------------- S  : 0
                {1}  | 0 y x -1 |
                {0}  | 0 0 0 x  |
                {0}  | 0 0 0 -y |
                {-1} | 0 0 0 0  |

          4                         4
     0 : S  <--------------------- S  : 1
               {1}  | 0 y x -1 |
               {0}  | 0 0 0 x  |
               {0}  | 0 0 0 -y |
               {-1} | 0 0 0 0  |

o5 : ComplexMap

See also

Ways to use this method:

The source of this document is in MultigradedBGG.m2:624:0.