# inducedMap(Module,Module) -- compute the map induced by the identity

## Synopsis

• Function: inducedMap
• Usage:
inducedMap(M,N)
• Inputs:
• M,
• N,
• 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 the identity.

## Description

The modules M and N must both be subquotient modules of the same ambient free module F. If M = M1/M2 and N = N1/N2, where M1, M2, N1, N2 are all submodules of F, then return the map induced by F --> F. If the optional argument Verify is given, check that the result defines a well defined homomorphism.

In this example, we make the inclusion map between two submodules of R^3. M is defined by two elements and N is generated by one element in M

 i1 : R = ZZ/32003[x,y,z]; i2 : P = R^3; i3 : M = image(x*P_{1}+y*P_{2} | z*P_{0}) o3 = image | 0 z | | x 0 | | y 0 | 3 o3 : R-module, submodule of R i4 : N = image(x^4*P_{1} + x^3*y*P_{2} + x*y*z*P_{0}) o4 = image | xyz | | x4 | | x3y | 3 o4 : R-module, submodule of R i5 : h = inducedMap(M,N) o5 = | x3 | | xy | o5 : Matrix i6 : source h == N o6 = true i7 : target h == M o7 = true i8 : ambient M == ambient N o8 = true