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reverseFactors -- The isomorphism from F**G to G**F when F,G are complexes

Synopsis

Description

maps F_{n-i}**G_i \to G_i**F_{n-i} changing the basis order and putting in sign (-1)^{i*(n-i)}. In reverseFactors(M,N,p,q) the integers p and q specify the homological degrees of M and N respectively.

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

o1 = S

o1 : PolynomialRing
 
i2 : F = chainComplex{map(S^1,S^{-1},a)}

      1      1
o2 = S  <-- S
             
     0      1

o2 : ChainComplex
 
i3 : G = chainComplex{map(S^1,S^{-1},b)}[3]

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

o3 : ChainComplex
 
i4 : phi = reverseFactors(F,G)

           1             1
o4 = -3 : S  <--------- S  : -3
                | 1 |

           2                    2
     -2 : S  <---------------- S  : -2
                {1} | 0 -1 |
                {1} | 1 0  |

           1                 1
     -1 : S  <------------- S  : -1
                {2} | 1 |

o4 : ChainComplexMap
 
i5 : G**F

      1      2      1
o5 = S  <-- S  <-- S
                    
     -3     -2     -1

o5 : ChainComplex
 
i6 : F**G

      1      2      1
o6 = S  <-- S  <-- S
                    
     -3     -2     -1

o6 : ChainComplex
 
i7 : apply(1+length(F**G), i->(
                 (phi_i)*((F**G).dd_(i+1)) ==  ((G**F).dd_(i+1))*phi_(i+1)
                 ))

o7 = {true, true, true}

o7 : List
 
i8 : apply(length (F**G), i -> (rank phi_i) == rank ((F**G)_i))

o8 = {true, true}

o8 : List

Ways to use reverseFactors :

For the programmer

The object reverseFactors is a method function.