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euler(ZZdFactorization) -- Calculates the Euler characteristic of a ZZdFactorization

Description

Calculates the Euler characteristic of a ZZdFactorization. To be well-defined, the input should be a complex. This method calculates the Euler characteristic by computing $$\ell_R (H_0 (C)) - \ell_R (H_1 (C))$$ where $C$ is some ZZ/2-graded complex. An easy way to construct such complexes is to take the endomorphisms of a matrix factorization arising from an isolated singularity.

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

o1 = S

o1 : PolynomialRing
 
i2 : C = koszulMF(a^3+b^3+c^3)

      4      4      4
o2 = S  <-- S  <-- S
                    
     0      1      0

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

      32      32      32
o3 = S   <-- S   <-- S
                      
     0       1       0

o3 : ZZdFactorization
 
i4 : prune HH E

o4 = cokernel | c b a 0 0 0 0 0 0 0 0 0 | <-- cokernel | c b a 0 0 0 0 0 0 0 0 0 | <-- cokernel | c b a 0 0 0 0 0 0 0 0 0 |
              | 0 0 0 c b a 0 0 0 0 0 0 |              | 0 0 0 c b a 0 0 0 0 0 0 |              | 0 0 0 c b a 0 0 0 0 0 0 |
              | 0 0 0 0 0 0 c b a 0 0 0 |              | 0 0 0 0 0 0 c b a 0 0 0 |              | 0 0 0 0 0 0 c b a 0 0 0 |
              | 0 0 0 0 0 0 0 0 0 c b a |              | 0 0 0 0 0 0 0 0 0 c b a |              | 0 0 0 0 0 0 0 0 0 c b a |
                                                                                        
     0                                        1                                        0

o4 : ZZdFactorization
 
i5 : euler E

o5 = 0
 
i6 : Q = ZZ/101[a,b,c,d]

o6 = Q

o6 : PolynomialRing
 
i7 : D = koszulMF(a*b - c*d)

      2      2      2
o7 = Q  <-- Q  <-- Q
                    
     0      1      0

o7 : ZZdFactorization
 
i8 : F = Hom(D,D)

      8      8      8
o8 = Q  <-- Q  <-- Q
                    
     0      1      0

o8 : ZZdFactorization
 
i9 : euler(F)

o9 = 1

Caveat

In order to be well-defined, the input should be a complex (ie, the differential should square to 0).

See also

Ways to use this method:

The source of this document is in MatrixFactorizations/MatrixFactorizationsDOC.m2:5602:0.