# inducedMap(Module,Module,Matrix) -- compute the induced map

## Synopsis

• Function: inducedMap
• Usage:
inducedMap(M,N,f)
• Inputs:
• M,
• N,
• f, , a homomorphism P <-- Q
• Optional inputs:
• Degree => ..., default value null, specify the degree of a map
• Verify => ..., default value true, verify that a map is well-defined
• Outputs:
• , the homomorphism M <-- N induced by f.

## Description

The modules M and N must both be subquotient modules where M and P have the same ambient module, and N and Q have the same ambient module. If the optional argument Verify is given, check that the result defines a well defined homomorphism.

In this example, the module K2 is mapped via g into K1, and we construct the induced map from K2 to K1.

 i1 : R = ZZ/32003[x,y,z] o1 = R o1 : PolynomialRing i2 : g1 = matrix{{x,y,z}} o2 = | x y z | 1 3 o2 : Matrix R <--- R i3 : g2 = matrix{{x^2,y^2,z^2}} o3 = | x2 y2 z2 | 1 3 o3 : Matrix R <--- R i4 : K1 = ker g1 o4 = image {1} | -y 0 -z | {1} | x -z 0 | {1} | 0 y x | 3 o4 : R-module, submodule of R i5 : K2 = ker g2 o5 = image {2} | -y2 0 -z2 | {2} | x2 -z2 0 | {2} | 0 y2 x2 | 3 o5 : R-module, submodule of R i6 : f = map(ambient K1, ambient K2, {{x,0,0},{0,y,0},{0,0,z}}) o6 = {1} | x 0 0 | {1} | 0 y 0 | {1} | 0 0 z | 3 3 o6 : Matrix R <--- R i7 : h = inducedMap(K1,K2,f) o7 = {2} | xy 0 0 | {2} | 0 yz 0 | {2} | 0 0 xz | o7 : Matrix
If we omit the first argument, then it is understood to be the target of f, and if we omit the second argument, it is understood to be the source of f.
 i8 : h1 = inducedMap(target f,K2,f) o8 = {1} | -xy2 0 -xz2 | {1} | x2y -yz2 0 | {1} | 0 y2z x2z | o8 : Matrix i9 : h2 = inducedMap(,K2,f) o9 = {1} | -xy2 0 -xz2 | {1} | x2y -yz2 0 | {1} | 0 y2z x2z | o9 : Matrix i10 : h1 == h2 o10 = true
In this example, we cannot omit the second argument, since in that case the resulting object is not a homomorphism.