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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)).
If M and N are reduced (a common case), then they equal their
respective presentation modules, so there is no difference between
the above two cases (the ambient matrix and the presentation matrix
are identical). So the only difference between (a) and (b) occurs when
at least one of M and N is a non-ambient (proper) submodule of an
embedded module.
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.
Given a module homomorphism f:M -> N, return the domain M.
Given a module homomorphism f:M -> N, return the codomain N.
Matrix(f) : ModMPolHom -> ModMatRngElt
Given a module homomorphism f:M -> N, return the presentation
matrix AP 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.
Matrix(f) : ModMPolHom -> ModMatRngElt
Given a module homomorphism f:M -> N, return the ambient
matrix AA 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).
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.
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.
Given a module homomorphism f:M -> N, return the image of f
as a submodule of N (which will be reduced iff N is).
Given a module homomorphism f:M -> N, return the kernel of f
as a submodule of M (which will be reduced iff M is).
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).
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.
Given a module homomorphism f:M -> N, return whether f is
injective (whether the kernel of f is the zero module).
Given a module homomorphism f:M -> N, return whether f is
surjective (whether the image of f equals N).
Given a module homomorphism f:M -> N, return whether f is
bijective (injective and surjective).
IsHomogeneous(f) : ModMPolHom -> BoolElt
Given a module homomorphism f:M -> N, where M and N are
graded modules, return whether f is graded of some degree d;
that is, whether for every element v∈M, f(v)=0 or
Degree(f(v)) equals Degree(v) + d.
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 graded, then the `maximum' concept is unnecessary, since the degree
will be consistent for all elements of M (see the previous function).
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