ZZdFactorizationMap * ZZdFactorizationMap -- composition of homomorphisms of ZZ/d-graded factorizations

Operator: *
Usage:

f = h * g
Inputs:
- h, a map of ZZ/d-graded factorizations, if a ring element or integer, then we multiply the ring element by the appropriate identity map
- g, a map of ZZ/d-graded factorizations,
Outputs:
- f, a map of ZZ/d-graded factorizations, the composition of $g$ followed by $h$

Description

If $g_i : C_i \rightarrow D_{d+i}$, and $h_j : D_j \rightarrow E_{e+j}$, then the composition corresponds to $f_i := h_{d+i} * g_i : C_i \rightarrow E_{i+d+e}$. In particular, the degree of the composition $f$ is the sum of the degrees of $g$ and $h$.

i1 : R = ZZ/101[a..d]

o1 = R

o1 : PolynomialRing

i2 : C = koszulMF({a,b,c}, a^3 + b^3 + c^3)

      4      4      4
o2 = R  <-- R  <-- R
                    
     0      1      0

o2 : ZZdFactorization

i3 : 3 * dd^C

          4                            4
o3 = 1 : R  <------------------------ R  : 0
               | 3c2 0   3b  3a   |
               | 0   3c2 3a2 -3b2 |
               | 3b2 3a  -3c 0    |
               | 3a2 -3b 0   -3c  |

          4                             4
     0 : R  <------------------------- R  : 1
               | 3c  0   3b   3a   |
               | 0   3c  3a2  -3b2 |
               | 3b2 3a  -3c2 0    |
               | 3a2 -3b 0    -3c2 |

o3 : ZZdFactorizationMap

i4 : 0 * dd^C

o4 = 0

o4 : ZZdFactorizationMap

i5 : dd^C * dd^C

          4                                               4
o5 = 0 : R  <------------------------------------------- R  : 0
               | a3+b3+c3 0        0        0        |
               | 0        a3+b3+c3 0        0        |
               | 0        0        a3+b3+c3 0        |
               | 0        0        0        a3+b3+c3 |

          4                                               4
     1 : R  <------------------------------------------- R  : 1
               | a3+b3+c3 0        0        0        |
               | 0        a3+b3+c3 0        0        |
               | 0        0        a3+b3+c3 0        |
               | 0        0        0        a3+b3+c3 |

o5 : ZZdFactorizationMap

Ways to use this method:

ZZdFactorizationMap * ZZdFactorizationMap -- composition of homomorphisms of ZZ/d-graded factorizations

The source of this document is in MatrixFactorizations/MatrixFactorizationsDOC.m2:3082:0.

ZZdFactorizationMap * ZZdFactorizationMap -- composition of homomorphisms of ZZ/d-graded factorizations

Description

See also

Ways to use this method: