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isZZdComplex -- Checks if the differentials of a factorization compose to 0

Description

This method evaluates whether all d-fold compositions of the differentials are equal to 0. A simple way to obtain a ZZ/d-graded complex is to form the endomorphisms of a factorization:

i1 : Q = ZZ/101[x_1..x_3];
 
i2 : F = randomTailMF(x_1^3+x_2^3 + x_3^3 , 2, 5, 2)

      9      9      9
o2 = Q  <-- Q  <-- Q
                    
     0      1      0

o2 : ZZdFactorization
 
i3 : E = Hom(F,F)

      162      162      162
o3 = Q    <-- Q    <-- Q
                        
     0        1        0

o3 : ZZdFactorization
 
i4 : isZZdComplex E

o4 = true
 
i5 : F' = linearMF(x_1^3 + x_2^3 , t)

     /   Q[t]   \3     /   Q[t]   \3     /   Q[t]   \3     /   Q[t]   \3
o5 = |----------|  <-- |----------|  <-- |----------|  <-- |----------|
     | 2        |      | 2        |      | 2        |      | 2        |
     \t  + t + 1/      \t  + t + 1/      \t  + t + 1/      \t  + t + 1/
                                                            
     0                 1                 2                 0

o5 : ZZdFactorization
 
i6 : E' = Hom(F', F')

     /   Q[t]   \27     /   Q[t]   \27     /   Q[t]   \27     /   Q[t]   \27
o6 = |----------|   <-- |----------|   <-- |----------|   <-- |----------|
     | 2        |       | 2        |       | 2        |       | 2        |
     \t  + t + 1/       \t  + t + 1/       \t  + t + 1/       \t  + t + 1/
                                                               
     0                  1                  2                  0

o6 : ZZdFactorization
 
i7 : isZZdComplex E'

o7 = true

Folding a complex is also another simple way to obtain a ZZ/d-graded complex:

i8 : K = koszulComplex vars Q

      1      3      3      1
o8 = Q  <-- Q  <-- Q  <-- Q
                           
     0      1      2      3

o8 : Complex
 
i9 : fK = Fold(K, 2)

      4      4      4
o9 = Q  <-- Q  <-- Q
                    
     0      1      0

o9 : ZZdFactorization
 
i10 : isZZdComplex fK

o10 = true
 
i11 : Fold(K**K,4)

       16      12      16      20      16
o11 = Q   <-- Q   <-- Q   <-- Q   <-- Q
                                       
      0       1       2       3       0

o11 : ZZdFactorization
 
i12 : isZZdComplex oo

o12 = true

See also

Ways to use isZZdComplex:

  • isZZdComplex(ZZdFactorization)

For the programmer

The object isZZdComplex is a method function.

The source of this document is in MatrixFactorizations/MatrixFactorizationsDOC.m2:5703:0.