Predicates

This subsection describes tests for several important properties of coherent sheaves. It contains an isomorphism test that, combined with DivisorToSheaf, can be used as a test for linear equivalence of Cartier divisors.

IsLocallyFree(S) : ShfCoh -> BoolElt, RngIntElt

    UseFitting: BoolElt                 Default: true

Given a sheaf S on ordinary projective scheme X, this function returns true if and only if S is a locally free sheaf on X of constant rank and, if so, also returns its rank.

Our original implementation was very fast but unfortunately incorrect! We have modified it to the algorithm described below which uses an "{étale stratification" of X.

The more straightforward method uses Fitting ideals of the module M (or M_max) of S. If the possible rank is d (as determined from Hilbert polynomials), it is required to check that the saturation of the dth Fitting ideal is the full ring and that the (d - 1)th lies in the saturated ideal of X. This is now the default method, but can be extremely slow and use a large amount of memory. Our alternative method is much slower than it was but we still find that it can be much faster than the Fitting ideal method for a low dimensional X in a high dimensional ambient and a sheaf S whose (maximal) module has a presentation with a reasonably large minimal number of both generators and relations.

To use the alternative method, the user can set the UseFitting parameter to false. For this method, it is assumed that X is equidimensional (all of its primary components have the same dimension), (locally) Cohen-Macaulay and connected. Magma does not check these conditions. The alternative method is described below.

The equidimensional and locally Cohen-Macaulay assumptions imply that X is faithfully flat over P₀ = Proj(R₀) for a Noether normalisation R₀ of the coordinate ring of X. Standard flatness properties mean that S being locally free over X implies that it is locally free as a sheaf over P₀, which is just a full projective space. Serre's criterion (see [Ser55]) states that the latter is true if and only if all intermediate cohomology rings Hⁱ(P₀, S(q)) vanish for q ll 0. If M_max is the maximal graded module of S, this translates to all intermediate Ext(M_max, R₀) R₀-modules being finite length, which in turn translates to the dual complex to the minimal free resolution of M_max as an R₀-module, having finite-length homology groups at all intermediate places. Rather than actually computing homology modules, we can further translate this condition into a number of equality tests for Hilbert polynomials of cokernels of the maps between free modules in the dual complex. This operation seems to be fairly fast and efficient in practice.

The above gives a necessary condition for local freeness over X (and gives the rank) but it is only sufficient over the Zariski open subset of X over which X -> P₀ is unramified. We have adapted it to be applied inductively over a chain of closed subschemes of X, which is what we refer to as the {étale stratification of X. At each level, we have a subscheme Y of X and a possibly empty collection of polynomials {F_i} which are non-zero divisors on Y and such that a chosen Noether normalisation of Y is unramified (equivalently, {étale) outside of the subschemes Y_i defined by F_i adjoined to the equations of Y. These subschemes lie at the next level down. The conditions imply that all of the Y_i are equidimensional and Cohen-Macaulay. For Y_i of positive dimension, we apply the above test to the restriction of S to Y_i, checking that the rank is the same as the rank determined for X at the top level if the test is passed and also that F_i is not a zero-divisor on the module of the restricted sheaf. For Y_i of dimension 0, we perform a more direct test.

The overall algorithm generally takes much longer than just applying the main test to X. The {étale stratification can take some time to compute when X lies in higher dimensional ambients. However, the main problem is that the degrees of the Y_i increase as we go down the chain (the subscheme defined by F_i has the degree of Y_i multiplied by the degree of F_i as its degree) and in practice (mainly working with surfaces X), the most time is taken in the bottom level checks on high degree zero-dimensional subschemes. Once computed, the {étale stratification is stored with X, so does not need to be recomputed for tests on other sheaves. However, it relies on finding Noether normalisations at each level that are generically unramified, i.e. separable. Currently we do not check the condition and it can fail in small positive characteristic leading to a crash or wrong results.

IsIsomorphic(S, T) : ShfCoh, ShfCoh -> BoolElt, ShfHom

IsIsomorphicWithTwist(S, T) : ShfCoh, ShfCoh -> BoolElt, RngIntElt, ShfHom

For S and T coherent sheaves on the same base scheme X, this function returns true if and only if S is isomorphic to T or (for the second intrinsic) to a Serre twist T(d) of T. In either case, an isomorphism is returned, if one exists and for the second intrinsic, the twist d is also returned as the second return value (so the isomorphism is between S and T(d)).

For the implementation, we first do a quick Hilbert polynomial check and then a Betti number check for the maximal modules M_max and N_max of S and T. This gives necessary conditions for an isomorphism and the possible d in the "with twist" case. Then we look for an isomorphism in the finite dimensional space of homomorphisms between M_max and N_max. We could have chosen to work with the homomorphisms between the truncated modules with gradings greater than or equal to N, for some N greater than or equal to the regularity of any defining modules for the two sheaves, but these have much larger presentations in general so the computation of homomorphisms is slower.

To look for isomorphisms, we look at the "zero degree" subblocks of the matrices giving a basis to the space of all homomorphisms. This reduces the problem to determining whether there is an invertible matrix in a space of n x n matrices over the base field. This is known to be a difficult problem in general and currently our implementation is rather weak at this point. We hope to improve it for future releases.

IsArithmeticallyCohenMacaulay(S) : ShfCoh -> BoolElt

Given a sheaf S on an ordinary projective scheme X, this function returns true if and only if the maximal graded module M_max of S is a Cohen-Macaulay module over the coordinate ring of X.

A scheme X is called arithmetically Cohen-Macaulay if and only if its coordinate ring is a Cohen-Macaulay ring. This is then true if and only if its coordinate ring is equal to the maximal module of OX and the intrinsic returns true for OX.

This is a fairly straightforward computation once M_max has been determined. If we already know the structure of M_max as a module over a Noether normalisation of the coordinate ring of X, it is an immediate freeness check. Otherwise it is a straightforward depth calculation from a minimal free resolution of M_max as a graded module over the coordinate ring of the ambient of X.