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# Hom -- module of homomorphisms

## Synopsis

• Usage:
Hom(M, N)
• Inputs:
• Optional inputs:
• DegreeLimit => ..., default value null, an integer or a list, if set, stop after homomorphisms in this degree have been computed
• MinimalGenerators => , default value true, whether to trim the resulting module
• Strategy => , default value null
• Outputs:
• , isomorphic to $\mathrm{Hom}_R(M,N), where$M$and$N$are both$R$-modules ## Description If$M$or$N$is an ideal or ring, it is regarded as a module in the evident way.  i1 : R = QQ[x,y]/(y^2 - x^3); i2 : M = image matrix{{x, y}} o2 = image | x y | 1 o2 : R-module, submodule of R i3 : H = Hom(M, M, MinimalGenerators => true) o3 = image {-1} | x y | {-1} | y x2 | 2 o3 : R-module, submodule of R To recover the modules used to create a Hom-module, use the function formation.  i4 : formation H o4 = Hom (image | x y |, image | x y |, DegreeLimit => null) o4 : Expression of class FunctionApplication Specific homomorphisms may be obtained using homomorphism, as follows.  i5 : f0 = homomorphism H_{0} o5 = {1} | 1 0 | {1} | 0 1 | o5 : Matrix M <-- M i6 : f1 = homomorphism H_{1} o6 = {1} | 0 x | {1} | 1 0 | o6 : Matrix M <-- M In the example above, f0 is the identity map, and f1 maps$x$to$y$and$y$to$x^2\$.

 i7 : M_0, M_1 o7 = (| x |, | y |) o7 : Sequence i8 : f0 M_0, f0 M_1 o8 = (| x |, | y |) o8 : Sequence i9 : f1 M_0, f1 M_1 o9 = (| y |, | x2 |) o9 : Sequence

## Contributors

Devlin Mallory implemented the strategy which accepts a degree limit.