# homomorphism -- get the homomorphism from element of Hom

## Synopsis

• Usage:
homomorphism f
• Inputs:
• f, or , corresponding to an element in $\mathrm{Hom}(M,N)$ or a map $R^1 \to \mathrm{Hom}(M,N)$
• Outputs:
• , the map $M \to N$, corresponding to the element $f \in \mathrm{Hom}(M,N)$

## Description

When H = Hom(M,N) is computed, information about computing the morphisms corresponding to its elements is stored in H.

 i1 : R = QQ[x,y,z, Degrees => {2,3,1}]/(y^2 - x^3) o1 = R o1 : QuotientRing i2 : H = Hom(ideal(x,y), R^1) o2 = image {-2} | x y | {-3} | y x2 | 2 o2 : R-module, submodule of R i3 : f = H_{1} o3 = {0} | 0 | {1} | 1 | 1 o3 : Matrix H <-- R i4 : g = homomorphism f o4 = | y x2 | o4 : Matrix

The source and target are what they should be.

 i5 : source g === module ideal(x,y) o5 = true i6 : target g === R^1 o6 = true

Except for a possible redistribution of degrees between the map and modules, we can undo the process with homomorphism'.

 i7 : f' = homomorphism' g o7 = {0} | 0 | {1} | 1 | 1 o7 : Matrix H <-- R i8 : f === f' o8 = false i9 : f - f' o9 = 0 1 o9 : Matrix H <-- R i10 : degree f, degree f' o10 = ({0}, {1}) o10 : Sequence i11 : degrees f, degrees f' o11 = ({{{0}, {1}}, {{1}}}, {{{0}, {1}}, {{0}}}) o11 : Sequence

After pruning a Hom module, one cannot use homomorphism directly. Instead, first apply the pruning map:

 i12 : H1 = prune H o12 = cokernel {0} | y x2 | {1} | -x -y | 2 o12 : R-module, quotient of R i13 : homomorphism(H1.cache.pruningMap * H1_{1}) o13 = | y x2 | o13 : Matrix