linearMF -- Constructs a Koszul d-fold factorization of a homogeneous element of degree d

Usage:

linearMF(RingElement)
Inputs:
- RingElement, a ring element, A ring element and a Symbol of RingElement, specifying the name of a distinguished root of unity if one needs to be adjoined.
Outputs:
- a ZZ/d graded factorization,

Description

This method constructs a Koszul d-fold factorization of a homogeneous element of degree $d$. The code first checks if the input element $f$ is homogeneous. If it is, it constructs a Koszul d-fold factorization obtained by taking the tensor product of the "trivial" factorizations of the constituent monomial terms (obtained by using the monomialMF function).

i1 : Q = QQ[x..z];

i2 : f = x*y^2*z^3 + x^3*y*z^2

      3   2      2 3
o2 = x y*z  + x*y z

o2 : Q

i3 : K = linearMF(f, t)

     /   Q[t]   \6     /   Q[t]   \6     /   Q[t]   \6     /   Q[t]   \6     /   Q[t]   \6     /   Q[t]   \6     /   Q[t]   \6
o3 = |----------|  <-- |----------|  <-- |----------|  <-- |----------|  <-- |----------|  <-- |----------|  <-- |----------|
     | 2        |      | 2        |      | 2        |      | 2        |      | 2        |      | 2        |      | 2        |
     \t  - t + 1/      \t  - t + 1/      \t  - t + 1/      \t  - t + 1/      \t  - t + 1/      \t  - t + 1/      \t  - t + 1/
                                                                                                                  
     0                 1                 2                 3                 4                 5                 0

o3 : ZZdFactorization

i4 : diffs = K.dd

         /   Q[t]   \6                                        /   Q[t]   \6
o4 = 5 : |----------|  <------------------------------------- |----------|  : 0
         | 2        |     {0, 2} | x z  0    0  0   0     |   | 2        |
         \t  - t + 1/     {0, 2} | 0 yt z    0  0   0     |   \t  - t + 1/
                          {0, 2} | 0 0  yt-y y  0   0     |
                          {0, 2} | 0 0  0    -z x   0     |
                          {0, 2} | 0 0  0    0  -zt x     |
                          {0, 2} | x 0  0    0  0   -zt+z |

         /   Q[t]   \6                                        /   Q[t]   \6
     0 : |----------|  <------------------------------------- |----------|  : 1
         | 2        |     {0, 2} | z z  0    0  0   0     |   | 2        |
         \t  - t + 1/     {0, 2} | 0 xt z    0  0   0     |   \t  - t + 1/
                          {0, 2} | 0 0  yt-y y  0   0     |
                          {0, 2} | 0 0  0    -y x   0     |
                          {0, 2} | 0 0  0    0  -zt x     |
                          {0, 2} | x 0  0    0  0   -zt+z |

         /   Q[t]   \6                                        /   Q[t]   \6
     1 : |----------|  <------------------------------------- |----------|  : 2
         | 2        |     {0, 2} | z z  0    0  0   0     |   | 2        |
         \t  - t + 1/     {0, 2} | 0 zt z    0  0   0     |   \t  - t + 1/
                          {0, 2} | 0 0  xt-x y  0   0     |
                          {0, 2} | 0 0  0    -y x   0     |
                          {0, 2} | 0 0  0    0  -yt x     |
                          {0, 2} | x 0  0    0  0   -zt+z |

         /   Q[t]   \6                                        /   Q[t]   \6
     2 : |----------|  <------------------------------------- |----------|  : 3
         | 2        |     {0, 2} | z z  0    0  0   0     |   | 2        |
         \t  - t + 1/     {0, 2} | 0 zt z    0  0   0     |   \t  - t + 1/
                          {0, 2} | 0 0  zt-z y  0   0     |
                          {0, 2} | 0 0  0    -x x   0     |
                          {0, 2} | 0 0  0    0  -yt x     |
                          {0, 2} | x 0  0    0  0   -yt+y |

         /   Q[t]   \6                                        /   Q[t]   \6
     3 : |----------|  <------------------------------------- |----------|  : 4
         | 2        |     {0, 2} | y z  0    0  0   0     |   | 2        |
         \t  - t + 1/     {0, 2} | 0 zt z    0  0   0     |   \t  - t + 1/
                          {0, 2} | 0 0  zt-z y  0   0     |
                          {0, 2} | 0 0  0    -z x   0     |
                          {0, 2} | 0 0  0    0  -xt x     |
                          {0, 2} | x 0  0    0  0   -yt+y |

         /   Q[t]   \6                                        /   Q[t]   \6
     4 : |----------|  <------------------------------------- |----------|  : 5
         | 2        |     {0, 2} | y z  0    0  0   0     |   | 2        |
         \t  - t + 1/     {0, 2} | 0 yt z    0  0   0     |   \t  - t + 1/
                          {0, 2} | 0 0  zt-z y  0   0     |
                          {0, 2} | 0 0  0    -z x   0     |
                          {0, 2} | 0 0  0    0  -zt x     |
                          {0, 2} | x 0  0    0  0   -xt+x |

o4 : ZZdFactorizationMap

i5 : diffs^6

         /   Q[t]   \6                                                                                         /   Q[t]   \6
o5 = 0 : |----------|  <-------------------------------------------------------------------------------------- |----------|  : 0
         | 2        |     {0, 2} | x3yz2+xy2z3 0           0           0           0           0           |   | 2        |
         \t  - t + 1/     {0, 2} | 0           x3yz2+xy2z3 0           0           0           0           |   \t  - t + 1/
                          {0, 2} | 0           0           x3yz2+xy2z3 0           0           0           |
                          {0, 2} | 0           0           0           x3yz2+xy2z3 0           0           |
                          {0, 2} | 0           0           0           0           x3yz2+xy2z3 0           |
                          {0, 2} | 0           0           0           0           0           x3yz2+xy2z3 |

         /   Q[t]   \6                                                                                         /   Q[t]   \6
     1 : |----------|  <-------------------------------------------------------------------------------------- |----------|  : 1
         | 2        |     {0, 2} | x3yz2+xy2z3 0           0           0           0           0           |   | 2        |
         \t  - t + 1/     {0, 2} | 0           x3yz2+xy2z3 0           0           0           0           |   \t  - t + 1/
                          {0, 2} | 0           0           x3yz2+xy2z3 0           0           0           |
                          {0, 2} | 0           0           0           x3yz2+xy2z3 0           0           |
                          {0, 2} | 0           0           0           0           x3yz2+xy2z3 0           |
                          {0, 2} | 0           0           0           0           0           x3yz2+xy2z3 |

         /   Q[t]   \6                                                                                         /   Q[t]   \6
     2 : |----------|  <-------------------------------------------------------------------------------------- |----------|  : 2
         | 2        |     {0, 2} | x3yz2+xy2z3 0           0           0           0           0           |   | 2        |
         \t  - t + 1/     {0, 2} | 0           x3yz2+xy2z3 0           0           0           0           |   \t  - t + 1/
                          {0, 2} | 0           0           x3yz2+xy2z3 0           0           0           |
                          {0, 2} | 0           0           0           x3yz2+xy2z3 0           0           |
                          {0, 2} | 0           0           0           0           x3yz2+xy2z3 0           |
                          {0, 2} | 0           0           0           0           0           x3yz2+xy2z3 |

         /   Q[t]   \6                                                                                         /   Q[t]   \6
     3 : |----------|  <-------------------------------------------------------------------------------------- |----------|  : 3
         | 2        |     {0, 2} | x3yz2+xy2z3 0           0           0           0           0           |   | 2        |
         \t  - t + 1/     {0, 2} | 0           x3yz2+xy2z3 0           0           0           0           |   \t  - t + 1/
                          {0, 2} | 0           0           x3yz2+xy2z3 0           0           0           |
                          {0, 2} | 0           0           0           x3yz2+xy2z3 0           0           |
                          {0, 2} | 0           0           0           0           x3yz2+xy2z3 0           |
                          {0, 2} | 0           0           0           0           0           x3yz2+xy2z3 |

         /   Q[t]   \6                                                                                         /   Q[t]   \6
     4 : |----------|  <-------------------------------------------------------------------------------------- |----------|  : 4
         | 2        |     {0, 2} | x3yz2+xy2z3 0           0           0           0           0           |   | 2        |
         \t  - t + 1/     {0, 2} | 0           x3yz2+xy2z3 0           0           0           0           |   \t  - t + 1/
                          {0, 2} | 0           0           x3yz2+xy2z3 0           0           0           |
                          {0, 2} | 0           0           0           x3yz2+xy2z3 0           0           |
                          {0, 2} | 0           0           0           0           x3yz2+xy2z3 0           |
                          {0, 2} | 0           0           0           0           0           x3yz2+xy2z3 |

         /   Q[t]   \6                                                                                         /   Q[t]   \6
     5 : |----------|  <-------------------------------------------------------------------------------------- |----------|  : 5
         | 2        |     {0, 2} | x3yz2+xy2z3 0           0           0           0           0           |   | 2        |
         \t  - t + 1/     {0, 2} | 0           x3yz2+xy2z3 0           0           0           0           |   \t  - t + 1/
                          {0, 2} | 0           0           x3yz2+xy2z3 0           0           0           |
                          {0, 2} | 0           0           0           x3yz2+xy2z3 0           0           |
                          {0, 2} | 0           0           0           0           x3yz2+xy2z3 0           |
                          {0, 2} | 0           0           0           0           0           x3yz2+xy2z3 |

o5 : ZZdFactorizationMap

These linear factorizations are interesting because they give an explicit construction of nontrivial Ulrich modules over hypersurface rings. These Ulrich modules are constructed by taking the cokernel of any of the matrices in the factorization, then tensoring with the hypersurface ring.

Ways to use linearMF:

linearMF(RingElement)
linearMF(RingElement,RingElement)
linearMF(RingElement,Symbol)

For the programmer

The object linearMF is a method function.

The source of this document is in MatrixFactorizations/MatrixFactorizationsDOC.m2:5493:0.

linearMF -- Constructs a Koszul d-fold factorization of a homogeneous element of degree d

Description

See also

Ways to use linearMF:

For the programmer