trivialMF -- Constructs the trivial matrix factorization of an element f

Usage:

trivialMF(d, i, M, f)
Inputs:
- d, an integer, The period of the desired factorization
- i, an integer, The homological degree that the module M will appear in
- M, a module, A module
- RingElement, a ring element, An element in the ring of M
Outputs:
- a ZZ/d graded factorization,

Description

Constructs the trivial matrix factorization associated to M, which is the factorization: $$ M \xrightarrow{1} M \xrightarrow{1} \cdots \xrightarrow{1} M \xrightarrow{f} M.$$ The user can specify using the index $i$ which differential should be multiplication by the element $f$. If only one integer is specified, it is assumed that multiplication by $f$ is the first differential in the factorization. If no integers are specified, then the output is the length 2 factorization with first differential given by multiplication by $f$.

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

o1 = Q

o1 : PolynomialRing

i2 : f = a^3 + b^3 + c^3

      3    3    3
o2 = a  + b  + c

o2 : Q

i3 : T = trivialMF(Q^3, f)

      3      3      3
o3 = Q  <-- Q  <-- Q
                    
     0      1      0

o3 : ZZdFactorization

i4 : T.dd

          3                 3
o4 = 1 : Q  <------------- Q  : 0
               | 1 0 0 |
               | 0 1 0 |
               | 0 0 1 |

          3                                      3
     0 : Q  <---------------------------------- Q  : 1
               | a3+b3+c3 0        0        |
               | 0        a3+b3+c3 0        |
               | 0        0        a3+b3+c3 |

o4 : ZZdFactorizationMap

i5 : isdFactorization T

             3    3    3
o5 = (true, a  + b  + c )

o5 : Sequence

i6 : isNullHomotopic id_T

o6 = true

i7 : nullHomotopy id_T

          3         3
o7 = 1 : Q  <----- Q  : 0
               0

          3                 3
     0 : Q  <------------- Q  : 1
               | 1 0 0 |
               | 0 1 0 |
               | 0 0 1 |

o7 : ZZdFactorizationMap

i8 : T1 = trivialMF(4, 2, Q^4, f)

      4      4      4      4      4
o8 = Q  <-- Q  <-- Q  <-- Q  <-- Q
                                  
     0      1      2      3      0

o8 : ZZdFactorization

i9 : T1.dd

          4                   4
o9 = 3 : Q  <--------------- Q  : 0
               | 1 0 0 0 |
               | 0 1 0 0 |
               | 0 0 1 0 |
               | 0 0 0 1 |

          4                   4
     0 : Q  <--------------- Q  : 1
               | 1 0 0 0 |
               | 0 1 0 0 |
               | 0 0 1 0 |
               | 0 0 0 1 |

          4                                               4
     1 : Q  <------------------------------------------- Q  : 2
               | 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
     2 : Q  <--------------- Q  : 3
               | 1 0 0 0 |
               | 0 1 0 0 |
               | 0 0 1 0 |
               | 0 0 0 1 |

o9 : ZZdFactorizationMap

Ways to use trivialMF:

trivialMF(Module,RingElement)
trivialMF(ZZ,Module,RingElement)
trivialMF(ZZ,ZZ,Module,RingElement)
trivialMF(ZZ,ZZ,Module,ZZ)

For the programmer

The object trivialMF is a method function.

The source of this document is in MatrixFactorizations/MatrixFactorizationsDOC.m2:5783:0.

trivialMF -- Constructs the trivial matrix factorization of an element f

Description

See also

Ways to use trivialMF:

For the programmer