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# differentialModule -- converts a square zero matrix into a differential module

- Usage:
    `differentialModule(f)`
- Inputs:
  - f, a matrix, representing a module map with the same source and target
- Outputs:
  - a differential module,

## Description

Given a module map $f: M \rightarrow M$ of degree $a$ this creates a degree $a$ differential module from $f$ represented as a 3-term chain complex in homological degrees $-1, 0$, and $1$. If $a \neq 0$, then since the source and target of $f$ are required to be equal, we must specify the degree of the differential to be $a$ in order for the differential to be homogeneous.

    i1 : R = QQ[x]

    o1 = R

    o1 : PolynomialRing

    i2 : phi = map(R^1/(x^2),R^1/(x^2), x, Degree=>1)

    o2 = | x |

    o2 : Matrix

    i3 : differentialModule(phi)

    o3 = cokernel | x2 | <-- cokernel | x2 | <-- cokernel | x2 |

         -1                  0                   1

    o3 : DifferentialModule

## See also

- differentialModule(Complex) -- converts a complex into a differential module
- differential(DifferentialModule) -- returns the differential of a differential module

## Ways to use differentialModule:

- `differentialModule(Matrix)`
- differentialModule(Complex) -- converts a complex into a differential module

## For the programmer

The object differentialModule is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/MultigradedBGG.m2:608:0.