# module homomorphisms

A homomorphism f : M --> N is represented as a matrix from the generators of M to the generators of N.
 i1 : R = QQ[x,y]/(y^2-x^3); i2 : M = module ideal(x,y) o2 = image | x y | 1 o2 : R-module, submodule of R
One homomorphism F : M --> R is x |--> y, y |--> x^2 (this is multiplication by the fraction y/x). We write this in the following way.
 i3 : F = map(R^1,M,matrix{{y,x^2}}) o3 = | y x2 | o3 : Matrix
Notice that as is usual in Macaulay2, the target comes before the source.

Macaulay2 doesn't display the source and target, unless they are both free modules. Use target and source to get them. The matrix routine recovers the matrix of free modules between the generators of the source and target.

 i4 : source F o4 = image | x y | 1 o4 : R-module, submodule of R i5 : target F == R^1 o5 = true i6 : matrix F o6 = | y x2 | 1 2 o6 : Matrix R <--- R
Macaulay2 also does not check that the homomorphism is well defined (i.e. the relations of the source map into the relations of the target). Use isWellDefined to check. This generally requires a Gröbner basis computation (which is performed automatically, if it is required and has not already been done).
 i7 : isWellDefined F o7 = true i8 : isIsomorphism F o8 = false
The image of F lies in the submodule M of R^1. Suppose we wish to define this new map G : M --> M. How does one do this?

To obtain the map M --> M, we use Matrix // Matrix. In order to do this, we need the inclusion map of M into R^1.

 i9 : inc = inducedMap(R^1, M) o9 = | x y | o9 : Matrix
Now we use // to lift F : M --> R^1 along inc : M --> R^1, to obtain a map G : M --> M, such that inc * G == F.
 i10 : G = F // inc o10 = {1} | 0 x | {1} | 1 0 | o10 : Matrix i11 : target G == M and source G == M o11 = true i12 : inc * G == F o12 = true
Let's make sure that this map G is well defined.
 i13 : isWellDefined G o13 = true i14 : isIsomorphism G o14 = false i15 : prune coker G o15 = cokernel | y x | 1 o15 : R-module, quotient of R i16 : kernel G == 0 o16 = true