Accessor Functions

The following functions provide an interface to conveniently extract the basic data from a coherent sheaf.

Module(S) : ShfCoh -> ModMPol
Returns the graded module that was used to define sheaf S.
Scheme(S) : ShfCoh -> Sch
Returns the ordinary projective scheme X on which the sheaf S is defined.
FullModule(S) : ShfCoh -> ModMPol
Computes and returns the maximal module Mmax giving sheaf S. The nth graded piece of Mmax is equal to the global sections of the Serre twist S(n) as a finite dimensional vector space over k, the base field of the scheme X of S. Thus Mmax isomorphic to direct-sum n ∈Z H0(X, S(n)) as in [Har77]. Here, it is implicitly assumed that the exact support of S on X has no irreducible components of dimension 0 and that there are no embedded associated prime places of dimension 0. More concretely, if M is a defining module for S with a possible non-zero finite torsion module for the redundant maximal ideal having been divided out, then no (homogeneous) associated prime of M has dimension 1. This assumption means that the terms in the above direct sum are 0 for n ll 0 or equivalently that Mmax is a finitely-generated module.

As mentioned in the introduction, a further assumption, which isn't checked, for the computation of Mmax is that S is equidimensional. That is, M has no embedded associated primes (except possibly the redundant ideal) and the irreducible components of its exact support have the same non-zero dimension. It may be possible to avoid this assumption with more complex (and computationally heavy) code that works with an equidimensional decomposition of the defining module, but it suffices for many cases of interest (e.g., torsion-free sheaves on a variety or equidimensional scheme).

The method used is basically the computation of the double dual of the defining module over an appropriate polynomial algebra A. A possible approach is to take A as the exact "supporting" algebra k[x0, ..., xn]/I where the polynomial ring is the coordinate ring of the ambient of X and I is the exact annihilator of M. This would involve stronger assumptions on the support of S and the computation of the dualising module for this A. We choose instead to work with A as a Noether normalisation of the above A, which means that A is a simple polynomial ring and is its own dualising module (up to a shift in grading). Then M is re-expressed as a module over this A, Mmax is computed as a module over A and finally is recovered as a module over k[x0, ..., xn] by keeping track of the multiplication maps by the xi variables which don't occur in A.

The module Mmax is stored so that it is only computed once.

GlobalSectionSubmodule(S) : ShfCoh -> ModMPol
Given a sheaf S, this function returns the submodule of the maximal module Mmax generated in degrees ≥0, that is direct-sum n ≥0 H0(X, S(n)).
SaturateSheaf(~S) : ShfCoh ->
Procedure to compute and store (but not return) the maximal module Mmax of the sheaf S.

Example Sheaf_access_exs (H120E2)

The classic example of a natural module which is unsaturated (non-maximal) defining a sheaf is the coordinate ring R of a non-projectively normal non-singular projective variety X. The ring R defines the structure sheaf as usual, but not maximally. By definition, R isn't integrally closed. Its integral closure R1 is an extension ring, inheriting its natural grading and agreeing with R in all but finitely many graded parts. In fact, R1 considered as an R-module is precisely the maximal graded module of the structure sheaf!

Such a situation can very commonly arise when a non-singular variety is projected down isomorphically into a subspace of its ambient projective space. The projected down image X is then not even linearly-normal: the degree one graded part of its coordinate ring is missing coordinates that were eliminated in the projection. These must reoccur in the graded R-module that is computed as the maximal module of the structure sheaf.

In the following example, X is taken as the non-singular projection into P3 of a degree 4 rational normal curve (which naturally lives in P4). We can see the difference between the maximal module of the structure sheaf and the coordinate ring using Hilbert series. In fact, they just differ by dimension 1 in the 1-graded part, corresponding to that missing coordinate! 

> P3<x,y,z,t> := ProjectiveSpace(Rationals(),3);
> X := Scheme(P3,[
> y^3 - y*z^2 - 2*y^2*t - 2*x*z*t - 3*y*z*t + z^2*t - y*t^2 + 2*z*t^2 + 2*t^3,
> x^2*z + x*z^2 + y*z^2 + 3*x*z*t + 2*y*z*t - z^2*t + y*t^2 - 2*z*t^2 - 2*t^3,
> y^2*z - y*z^2 + y^2*t - x*z*t - 4*y*z*t + z^2*t - 3*y*t^2 + 2*z*t^2 + 2*t^3,
> x*y - x*z - x*t + y*t]);
> OX := StructureSheaf(X);
> M1 := Module(OX);
> M2 := FullModule(OX);
> h1 := HilbertSeries(M1); h1;
(-t^3 + 2*t^2 + 2*t + 1)/(t^2 - 2*t + 1)
> h2 := HilbertSeries(M2); h2;
(3*t + 1)/(t^2 - 2*t + 1)
> h2-h1;
t
V2.28, 13 July 2023