# homology(Matrix,Matrix) -- homology of a pair of maps

## Synopsis

• Function: homology
• Usage:
M = homology(f,g)
• Inputs:
• f,
• g,
• Outputs:
• M, , computes the homology module (kernel f)/(image g).

## Description

Here g and f should be composable maps with f*g equal to zero.

In the following example, we ensure that the source of f and the target of f are exactly the same, taking even the degrees into account, and we ensure that f is homogeneous.

 i1 : R = QQ[x]/x^5; i2 : f = map(R^1,R^1,{{x^3}}, Degree => 3) o2 = | x3 | 1 1 o2 : Matrix R <--- R i3 : M = homology(f,f) o3 = subquotient (| x2 |, | x3 |) 1 o3 : R-module, subquotient of R i4 : prune M o4 = cokernel {2} | x | 1 o4 : R-module, quotient of R