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HH_ZZ ZZdFactorization -- homology or cohomology module of a ZZ/d-graded factorization

Description

The homology of a 2-fold factorization $H$ is defined by ker dd^C/image dd^C. The differential of the homology complex is the zero map. If the differentials of the factorization do not compose to 0, then the homology is not defined. Similarly, in order to make sense of the homology of a d-fold factorization for d > 2, one should specify both the spot to take homology in, and also how many compositions of the differentials to use (since there are multiple ways to compose the differentials for a longer factorization)

An easy way to construct complexes from factorizations is to compute the endomorphisms of the factorization. In general, if $F$ is a factorization of some polynomial $f$ and $G$ is a factorization of some polynomial $g$, then $\operatorname{Hom} (F,G)$ is a factorization of $g-f$.

i1 : Q = ZZ/11[x,y];
 
i2 : F = randomLinearMF(2,Q)

      4      4      4
o2 = Q  <-- Q  <-- Q
                    
     0      1      0

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

      32      32      32
o3 = Q   <-- Q   <-- Q
                      
     0       1       0

o3 : ZZdFactorization
 
i4 : isZZdComplex E

o4 = true

Thus $E$ is a $2$-periodic complex. This means that the homology is well-defined, and moreover the indices of homology are also taken modulo 2:

i5 : prune HH_0 E

o5 = cokernel | y x 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
              | 0 0 y x 0 0 0 0 0 0 0 0 0 0 0 0 |
              | 0 0 0 0 y x 0 0 0 0 0 0 0 0 0 0 |
              | 0 0 0 0 0 0 y x 0 0 0 0 0 0 0 0 |
              | 0 0 0 0 0 0 0 0 y x 0 0 0 0 0 0 |
              | 0 0 0 0 0 0 0 0 0 0 y x 0 0 0 0 |
              | 0 0 0 0 0 0 0 0 0 0 0 0 y x 0 0 |
              | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 y x |

                            8
o5 : Q-module, quotient of Q
 
i6 : prune HH_1 E

o6 = cokernel | y x 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
              | 0 0 y x 0 0 0 0 0 0 0 0 0 0 0 0 |
              | 0 0 0 0 y x 0 0 0 0 0 0 0 0 0 0 |
              | 0 0 0 0 0 0 y x 0 0 0 0 0 0 0 0 |
              | 0 0 0 0 0 0 0 0 y x 0 0 0 0 0 0 |
              | 0 0 0 0 0 0 0 0 0 0 y x 0 0 0 0 |
              | 0 0 0 0 0 0 0 0 0 0 0 0 y x 0 0 |
              | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 y x |

                            8
o6 : Q-module, quotient of Q
 
i7 : prune HH_4 E

o7 = cokernel | y x 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
              | 0 0 y x 0 0 0 0 0 0 0 0 0 0 0 0 |
              | 0 0 0 0 y x 0 0 0 0 0 0 0 0 0 0 |
              | 0 0 0 0 0 0 y x 0 0 0 0 0 0 0 0 |
              | 0 0 0 0 0 0 0 0 y x 0 0 0 0 0 0 |
              | 0 0 0 0 0 0 0 0 0 0 y x 0 0 0 0 |
              | 0 0 0 0 0 0 0 0 0 0 0 0 y x 0 0 |
              | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 y x |

                            8
o7 : Q-module, quotient of Q
 
i8 : R = Q/(potential F);
 
i9 : netList for i to 3 list prune Ext^i (coker vars R, coker vars R)

     +-------------------------+
o9 = |cokernel | y x |         |
     +-------------------------+
     |cokernel {-1} | y x 0 0 ||
     |         {-1} | 0 0 y x ||
     +-------------------------+
     |cokernel {-2} | y x 0 0 ||
     |         {-2} | 0 0 y x ||
     +-------------------------+
     |cokernel {-3} | y x 0 0 ||
     |         {-3} | 0 0 y x ||
     +-------------------------+

In the above case, the homology of the endomorphism factorization is computing the stable Ext of the residue field over the hypersurface $R/(f)$, ie, the Ext module obtained by taking sufficiently high Ext values.

See also

Ways to use this method:

  • HH_ZZ ZZdFactorization -- homology or cohomology module of a ZZ/d-graded factorization
  • HH_ZZ(ZZ,ZZdFactorization)

The source of this document is in MatrixFactorizations/MatrixFactorizationsDOC.m2:1118:0.