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# 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

## See also

- Making ZZdFactorizations -- Information about the basic constructors

## 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 /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/MatrixFactorizations/MatrixFactorizationsDOC.m2:5783:0.