• « Previous
• Up
• Contents
• Index
• Search
• Next »

The Homomorphism Type

Magma has a special type for a homomorphism between two R-modules. The type of such a homomorphism is ModMPolHom. In general, functions such as Morphism return a homomorphism of type ModMPolHom, while the boundary maps of complexes are also of type ModMPolHom (see the function FreeResolution).

A homomorphism f: M -> N is represented by a matrix A. There are two ways in which A can be defined:

(a)

A is an ambient matrix: in this case, A gives the explicit map on the ambient modules of M and N. Thus A is m x n, where m=Degree(M), n=Degree(N).

(b)

A is a presentation matrix: in this case, A gives the explicit map on the presentation modules of M and N. Thus A is m x n, where m=Degree(Presentation(M)), n=Degree(Presentation(N)).

When M and N are graded - that is, generated by elements homogeneous with respect to the ambient column weightings and with a relation module that is also generated by homogeneous elements - all homomorphisms as non-graded modules are still allowed. However there are functions to test if a given homomorphism preserves the gradings on the domain and codomain up to a constant degree shift. See IsHomogeneous and Degree below.

Homomorphism(M, N, A): ModMPol, ModMPol, Mtrx -> ModMPolHom

    Presentation: BoolElt               Default: true

Given R-modules M and N and an m x n matrix A over R, construct the homomorphism f:M -> N (with type ModMPolHom) defined by A.

By default, A is assumed to be a presentation matrix (see the comments above), in which case m and n must equal the degrees of the presentation modules of M and N, respectively. Alternatively, setting the parameter Presentation to {false} specifies that A is an ambient matrix; in this case, m and n must equal the degrees of M and N, respectively.

Domain(f) : ModMPolHom -> ModMPol

Given a module homomorphism f:M -> N, return the domain M.

Codomain(f) : ModMPolHom -> ModMPol

Given a module homomorphism f:M -> N, return the codomain N.

PresentationMatrix(f) : ModMPolHom -> ModMatRngElt

Matrix(f) : ModMPolHom -> ModMatRngElt

Given a module homomorphism f:M -> N, return the presentation matrix A_P of f as an m x n matrix corresponding to the presentation modules of M and N, respectively. This presentation matrix is always well-defined and computed, even if f is constructed via an ambient matrix.

AmbientMatrix(f) : ModMPolHom -> ModMatRngElt

Matrix(f) : ModMPolHom -> ModMatRngElt

Given a module homomorphism f:M -> N, return the ambient matrix A_A of f as an m x n corresponding to the ambient modules of M and N, respectively. If M and N are reduced (as commonly happens), this will be the same as the presentation matrix above. But if M and N are not reduced and f is constructed via a presentation matrix, then an error may result (since it may be impossible to give a matrix over the base ring R which gives the mapping for the ambient modules).

f(v) : ModMPolHom, RngMPolElt -> RngMPolElt

v * f : ModMPolHom, RngMPolElt -> RngMPolElt

Given a module homomorphism f:M -> N and an element v of M, return the image of v under f, as an element of N.

f[i] : ModMPolHom, RngIntElt -> RngMPolElt

Given a module homomorphism f:M -> N and an integer i, return the element of N corresponding to the i-th row of the ambient matrix of f.

Image(f) : ModMPolHom -> ModMPol

Given a module homomorphism f:M -> N, return the image of f as a submodule of N (which will be reduced iff N is).

Kernel(f) : ModMPolHom -> ModMPol

Given a module homomorphism f:M -> N, return the kernel of f as a submodule of M (which will be reduced iff M is).

Cokernel(f) : ModMPolHom -> ModMPol

Given a module homomorphism f:M -> N, return the cokernel of f as a quotient module of N (which will be reduced iff N is).

IsZero(f) : ModMPolHom -> BoolElt

Given a module homomorphism f:M -> N, return whether f is the zero map. Note that f may be the zero map even if the presentation or ambient matrices of f are non-zero.

IsInjective(f) : ModMPolHom -> BoolElt

Given a module homomorphism f:M -> N, return whether f is injective (whether the kernel of f is the zero module).

IsSurjective(f) : ModMPolHom -> BoolElt

Given a module homomorphism f:M -> N, return whether f is surjective (whether the image of f equals N).

IsBijective(f) : ModMPolHom -> BoolElt

Given a module homomorphism f:M -> N, return whether f is bijective (injective and surjective).

IsGraded(f) : ModMPolHom -> BoolElt

IsHomogeneous(f) : ModMPolHom -> BoolElt

Given a module homomorphism f:M -> N, where M and N are graded modules, return whether f is homogeneous of some degree d; that is, whether for every pure degree element v∈M, f(v)=0 or Degree(f(v)) equals Degree(v) + d.

Degree(f) : ModMPolHom -> RngIntElt

Given a module homomorphism f:M -> N, return the degree of f, which is the maximum d such that an element of M of degree e is mapped via f to zero or an element of degree e + d. If f is homogeneous, then the `maximum' concept is unnecessary, since the degree will be consistent for all elements of M (see the previous function).

> R<x,y> := PolynomialRing(RationalField(), 2, "grevlex");
> F := EModule(R, 3);
> // get a submodule M1 generated by a single non-zero element of F
> M1 := sub<F|[x^2,y^2,x*y]>;
> // and a second submodule M2 containing M1
> M2 := sub<F|[x,0,y],[0,y,0]>;
> incl_hm :=  Homomorphism(M1,M2,IdentityMatrix(R,3) :
>               Presentation := false);
> incl_hm;
Module homomorphism (3 by 3)
Ambient matrix:
[1 0 0]
[0 1 0]
[0 0 1]

>  PresentationMatrix(incl_hm);
[y x]
> // check homogeneity of incl_hm
> IsHomogeneous(incl_hm);
true
> Degree(incl_hm);
0

• « Previous
• Up
• Contents
• Index
• Search
• Next »