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The following section describes some basic functionality for
homomorphisms between sheaves defined on the same scheme. A sheaf homomorphism
is represented by a module homomorphism between representing modules (defining, maximal or global section modules) for the two sheaves. Strictly, only (degree 0) homomorphisms that preserve the module gradings should be allowed but,
for flexibility, we allow "homogeneous" homomorphisms that uniformly shift the
grading by d and should be thought of as sheaf homomorphisms from the domain sheaf
to the dth Serre twist of the codomain sheaf. We then say that the homomorphism
is of degree d as for module homomorphisms.
There are some basic constructors and accessor functions, a function to "expand" a
chain of homomorphisms to a single homomorphism and image, kernel and cokernel
functions. The type of a sheaf homomorphism is ShfHom. Note that this is
NOT a Map subtype, so that a sheaf homomorphism doesn't automatically
inherit all of the usual map properties.
A simple basic constructor. S and T are sheaves on the same scheme X and
h is a module homomorphism between M0 and N0, where M0 is one of
the defining, maximal or global section modules of S and N0 is one of these
modules for T. h must be a homogeneous module homomorphism as returned by
IsHomogeneous and if d is its degree then the homomorphism returned is
really one from S to T(d) in the category of OX-sheaves.
For the construction of sheaf homomorphisms see also SheafHoms.
The domain of f.
The codomain of f.
The degree of f as defined in the introduction to this section.
The underlying homogeneous graded module homomorphism of f.
Returns the kernel of f and its inclusion homomorphism into the domain of f.
Returns the image, I, of f and two sheaf homomorphisms g and h.
If f has degree d and S, T are its domain and codomain, then
I is a subsheaf of T(d). g is the restriction of f from S to
I and h is the inclusion of I in T(d), so that g also has
degree d and h has degree 0.
Returns the cokernel of f and the quotient homomorphism from the codomain
to it.
Strictly speaking, if f has degree d and S, T are its domain and codomain,
here we are thinking of f as a homomorphism from S(d) <- T rather than
S <- T(d).
If hms = [h1, ..., hn] is a sequence of sheaf homomorphisms, then this returns
the combined homomorphism h1 * h2 * ... hn. The domain of h2 must be the
codomain of h1 etc. and the stronger condition that the underlying module
homomorphisms must be composable also holds. So the domain of
ModuleHomomorphism(h2) must be the codomain of ModuleHomomorphism(h1) etc.
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