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# HH DifferentialModule -- computes the homology of a differential module

- Function: homology
- Usage:
    `HH D`
- Inputs:
  - D, a differential module,
- Outputs:
  - a module,

## Description

This computes the homology of a differential module. More specifically: since we interpret differential modules as 3-term complexes, this returns the zeroth homology module.

    i1 : R = QQ[x,y]

    o1 = R

    o1 : PolynomialRing

    i2 : M = R^1/ideal(x^2,y^2)

    o2 = cokernel | x2 y2 |

                                1
    o2 : R-module, quotient of R

    i3 : phi = map(M,M,x*y)

    o3 = | xy |

    o3 : Matrix M <-- M

    i4 : D = differentialModule phi

    o4 = M  <-- M <-- M

         -1     0     1

    o4 : DifferentialModule

    i5 : HH D

    o5 = subquotient (| y -x |, | xy x2 y2 |)

                                   1
    o5 : R-module, subquotient of R

## See also

- image(DifferentialModule) -- computes the image of the differential of a differential module
- kernel(DifferentialModule) -- computes the kernel of the differential in a differential module.

## Ways to use this method:

- HH DifferentialModule -- computes the homology of a differential module

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/MultigradedBGG.m2:510:0.