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We have implemented functions to compute the dimensions of
cohomology groups of coherent sheaves on ordinary projective
space over an (exact) field. The sheaves are represented by
graded modules over the coordinate ring of the ambient
projective space. The sheaf may arise naturally as one
supported on a particular closed subscheme (eg, the
structure sheaf of a projective variety) but it is a matter
of indifference whether the sheaf is considered as lying on
the subscheme or the entire ambient space (equivalently,
whether the representing module is considered as a module over
the coordinate ring of the ambient or the quotient coordinate
ring of the subscheme) because the cohomology groups are naturally
isomorphic. We plan to add a fuller package of functionality for
coherent sheaves, but it is convenient to add the cohomology
function now as the algorithm we use works equally efficiently
(roughly speaking) when applied to any two graded modules that
represent the same coherent sheaf.
The algorithm we have implemented is that of Decker, Eisenbud,
Floystad and Schreyer which uses the Beilinson-Gelfand-Gelfand
(BGG) correspondence to reduce the computation of the
cohomology groups of the sheaf and its Serre twists to that
of various graded free modules in the projective resolution
of a module over a finite exterior (alternating) algebra.
SetVerbose("Cohom", n): Maximum: 1
M is a graded module over P = k[x0, .., xm] with k an exact field.
Let tilde(M) be the corresponding coherent sheaf on Proj(P) = Pmk.
The function returns the k-dimension of the cohomology group
Hr(Pkm, tilde(M)(n)) where tilde(M)(n) is the nth Serre twist
of tilde(M). n can be any integer and r a non-negative integer.
The algorithm used is based on the BGG correspondence. Details can be
found in [EFS03] or see [DE02] for a slightly more
computational description. Let A be the finite exterior algebra
with m + 1 generators, which is of dimension 2m + 1 over k. The
Tate resolution of tilde(M) is a doubly infinite exact
sequence of graded free A-modules. Each cohomology group of a twist
of tilde(M) is isomorphic as a k vector space to a particular graded
piece of a particular term in the Tate resolution. In fact, we never need
to explicitly compute the terms of the resolution of index ≥reg(M)
(the regularity of M) because they are pure graded of dimension given by
the Hilbert polynomial of M.
The algorithm computes two consecutive terms in the Tate resolution
at indices ≥reg(M), and the A-homomorphism between them, from
two corresponding graded pieces of M and the linear maps between them
coming from multiplication by the base variables. Then the resolution is
extended backwards as far as necessary by computing the A-projective
resolution of the kernel of this A-homomorphism. The projective
resolution is efficiently determined by non-commutative Gröbner basis
computations. This uses the new Magma machinery for exterior algebras
and their modules. The projective resolution information is cached so that
repeated calls to the function for the same module M will require either
no extra work or only an extension to the part of the resolution already
computed.
We consider a random surface X in a family of Enriques surfaces of
degree 9 in P4. It is defined by 15 degree 5 polynomials and we
work over F17 to keep the input of reasonable size (it is still
fairly large!).
The surface is non-singular with arithmetic genus (pa), geometric genus
(g) and irregularity q all zero. These are related generally for a
non-singular surface by g = pa + q and pa can be computed without
cohomology machinery (from Hilbert polynomials). But cohomology of the
structure sheaf of X and Serre duality is the easiest way to get g or q.
> R<x,y,z,t,u> := PolynomialRing(GF(17),5,"grevlex");
> I := ideal<R |[
> 2*x^3*z*t + 5*x^2*y*z*t + 14*x^2*z^2*t + x^3*z*u + 5*x^2*y*z*u +
> 10*x^2*z^2*u + 8*x*y*z^2*u + 15*y^2*z^2*u + 4*x*z^3*u + 2*y*z^3*u +
> 9*x^3*t*u + 14*x^2*y*t*u + 16*x^2*z*t*u + 10*x*y*z*t*u + 10*x*z^2*t*u +
> y*z^2*t*u + 13*x^3*u^2 + 14*x^2*y*u^2 + 11*x^2*z*u^2 + 15*x*y*z*u^2 +
> 10*y^2*z*u^2 + 8*x*z^2*u^2 + 8*y*z^2*u^2 + x^2*t*u^2 + 11*x*y*t*u^2 +
> 16*x*y*u^3 + 9*y^2*u^3 + 4*x*z*u^3 + 2*y*z*u^3 + 10*x*t*u^3 + y*t*u^3 +
> 8*x*u^4 + 8*y*u^4,
> 5*x^3*z*t + x^2*z^2*t + 5*x^3*z*u + 11*x^2*z^2*u + 15*x*y*z^2*u + 2*x*z^3*u
> + 14*x^3*t*u + 5*x^2*z*t*u + x*z^2*t*u + 14*x^3*u^2 + 10*x*y*z*u^2 +
> 8*x*z^2*u^2 + 15*x^2*t*u^2 + 11*x^2*u^3 + 9*x*y*u^3 + 2*x*z*u^3 +
> x*t*u^3 + 8*x*u^4,
> 14*x^3*z*t + x^2*y*z*t + 13*x^2*z^2*t + 7*x^2*z*t^2 + 3*x^3*z*u +
> 16*x^2*y*z*u + 4*x^2*z^2*u + 6*x^3*t*u + 16*x^2*y*t*u + 9*x^2*z*t*u +
> 9*x^2*t^2*u + 11*x^3*u^2 + x^2*y*u^2 + 14*x^2*z*u^2 + 2*x*z^2*u^2 +
> 11*y*z^2*u^2 + 6*z^3*u^2 + 4*x^2*t*u^2 + 4*x*z*t*u^2 + 14*y*z*t*u^2 +
> 6*z^2*t*u^2 + 15*x*t^2*u^2 + 10*z*t^2*u^2 + 3*x^2*u^3 + 11*x*z*u^3 +
> 16*y*z*u^3 + 4*z^2*u^3 + 16*x*t*u^3 + 3*y*t*u^3 + 14*z*t*u^3 + 6*t^2*u^3
> + 13*x*u^4 + 7*y*u^4 + 16*z*u^4 + 11*t*u^4 + 10*u^5,
> 15*x^3*z^2 + 12*x^2*y*z^2 + 3*x^2*z^3 + 12*x^3*z*u + 8*x^2*y*z*u +
> 11*x^2*z^2*u + x^3*u^2 + 14*x^2*y*u^2 + 3*x^2*z*u^2 + 11*x^2*u^3,
> 12*x^3*z^2 + 16*x^2*z^3 + 8*x^3*z*u + x^2*z^2*u + 14*x^3*u^2 + 16*x^2*z*u^2
> + x^2*u^3,
> 2*x^3*y*z + 5*x^2*y^2*z + 14*x^2*y*z^2 + 13*x^3*y*u + 12*x^2*y^2*u +
> 8*x^3*z*u + 4*x^2*y*z*u + 12*x^2*z^2*u + 3*x*y*z^2*u + 14*y^2*z^2*u +
> 15*x*z^3*u + 3*y*z^3*u + 15*x^2*y*t*u + 4*x^2*z*t*u + 15*x*y*z*t*u +
> 11*x*z^2*t*u + 10*y*z^2*t*u + 2*x^3*u^2 + 12*x^2*y*u^2 + 3*x^2*z*u^2 +
> 14*x*y*z*u^2 + 13*x*z^2*u^2 + 10*y*z^2*u^2 + x^2*t*u^2 + 15*x*y*t*u^2 +
> 16*y*z*t*u^2 + 12*x*y*u^3 + 3*y^2*u^3 + 15*x*z*u^3 + 4*y*z*u^3 +
> 11*x*t*u^3 + 6*y*t*u^3 + 13*x*u^4 + 14*y*u^4,
> 5*x^3*y*z + x^2*y*z^2 + 10*x^2*z^2*t + 4*x^4*u + 12*x^3*y*u + 5*x^3*z*u +
> 12*x^2*y*z*u + 14*x*y*z^2*u + 3*x*z^3*u + 15*x^3*t*u + 16*x^2*z*t*u +
> 10*x*z^2*t*u + 11*x^3*u^2 + 4*x^2*y*u^2 + 13*x^2*z*u^2 + 10*x*z^2*u^2 +
> 4*x^2*t*u^2 + 16*x*z*t*u^2 + 13*x^2*u^3 + 3*x*y*u^3 + 4*x*z*u^3 +
> 6*x*t*u^3 + 14*x*u^4,
> 10*x^2*z^3 + 8*x^2*z^2*u + 5*x^2*z*u^2 + 11*x^2*u^3,
> 16*x^3*z^2 + 12*x^2*y*z^2 + 7*x^2*z^3 + 9*x*y*z^3 + 2*y^2*z^3 + 13*x*z^4 +
> 15*y*z^4 + 13*x^3*z*t + 12*x^2*y*z*t + 7*x^2*z^2*t + 7*x*y*z^2*t +
> 7*x*z^3*t + 16*y*z^3*t + 4*x^3*z*u + 3*x^2*y*z*u + 6*x^2*z^2*u +
> 2*x*y*z^2*u + 7*y^2*z^2*u + 9*x*z^3*u + 9*y*z^3*u + 16*x^3*t*u +
> 3*x^2*y*t*u + 13*x^2*z*t*u + 6*x*y*z*t*u + x*y*z*u^2 + 8*y^2*z*u^2 +
> 13*x*z^2*u^2 + 15*y*z^2*u^2 + 6*x^2*t*u^2 + 7*x*z*t*u^2 + 16*y*z*t*u^2 +
> 9*x*z*u^3 + 9*y*z*u^3,
> 12*x^3*z^2 + 6*x^2*z^3 + 2*x*y*z^3 + 15*x*z^4 + 12*x^3*z*t + 11*x^2*z^2*t +
> 16*x*z^3*t + 3*x^3*z*u + 7*x*y*z^2*u + 9*x*z^3*u + 3*x^3*t*u +
> 3*x^2*z*t*u + 6*x^2*z*u^2 + 8*x*y*z*u^2 + 15*x*z^2*u^2 + 16*x^2*t*u^2 +
> 16*x*z*t*u^2 + 9*x*z*u^3,
> x^3*y*z + 5*x^2*y^2*z + 10*x^2*y*z^2 + 8*x*y^2*z^2 + 15*y^3*z^2 + 4*x*y*z^3
> + 2*y^2*z^3 + 13*x^3*y*t + 2*x^2*y^2*t + 9*x^3*z*t + 12*x^2*y*z*t +
> 10*x*y^2*z*t + 5*x^2*z^2*t + 7*x*y*z^2*t + 4*y^2*z^2*t + 2*x*z^3*t +
> 14*y*z^3*t + 2*x^2*y*t^2 + 13*x^2*z*t^2 + 2*x*y*z*t^2 + 6*x*z^2*t^2 +
> 7*y*z^2*t^2 + 13*x^3*y*u + 14*x^2*y^2*u + 11*x^2*y*z*u + 15*x*y^2*z*u +
> 10*y^3*z*u + 8*x*y*z^2*u + 8*y^2*z^2*u + 15*x^3*t*u + 6*x^2*y*t*u +
> 11*x*y^2*t*u + 14*x^2*z*t*u + 3*x*y*z*t*u + 4*x*z^2*t*u + 7*y*z^2*t*u +
> 16*x^2*t^2*u + 2*x*y*t^2*u + y*z*t^2*u + 16*x*y^2*u^2 + 9*y^3*u^2 +
> 4*x*y*z*u^2 + 2*y^2*z*u^2 + 15*x*y*t*u^2 + 15*y^2*t*u^2 + 2*x*z*t*u^2 +
> 13*y*z*t*u^2 + 6*x*t^2*u^2 + 11*y*t^2*u^2 + 8*x*y*u^3 + 8*y^2*u^3 +
> 4*x*t*u^3 + 3*y*t*u^3,
> 5*x^3*y*z + 11*x^2*y*z^2 + 15*x*y^2*z^2 + 2*x*y*z^3 + 13*x^4*t + 2*x^3*y*t +
> 7*x^3*z*t + 6*x^2*y*z*t + 16*x^2*z^2*t + 4*x*y*z^2*t + 14*x*z^3*t +
> 2*x^3*t^2 + 9*x^2*z*t^2 + 7*x*z^2*t^2 + 14*x^3*y*u + 5*x^3*z*u +
> 4*x^2*y*z*u + 10*x*y^2*z*u + 12*x^2*z^2*u + 8*x*y*z^2*u + 15*x*z^3*u +
> 6*y*z^3*u + 11*z^4*u + 16*x^3*t*u + 15*x^2*y*t*u + 13*x^2*z*t*u +
> 3*x*z^2*t*u + 3*y*z^2*t*u + 11*z^3*t*u + 13*x^2*t^2*u + 3*x*z*t^2*u +
> 7*z^2*t^2*u + 7*x^3*u^2 + 7*x^2*y*u^2 + 9*x*y^2*u^2 + x^2*z*u^2 +
> 2*x*y*z*u^2 + 6*x*z^2*u^2 + y*z^2*u^2 + 13*z^3*u^2 + 12*x^2*t*u^2 +
> 15*x*y*t*u^2 + 14*x*z*t*u^2 + 14*y*z*t*u^2 + 3*z^2*t*u^2 + 11*x*t^2*u^2
> + 11*z*t^2*u^2 + 9*x^2*u^3 + 8*x*y*u^3 + 4*x*z*u^3 + 10*y*z*u^3 +
> z^2*u^3 + 3*x*t*u^3 + 6*z*t*u^3 + 7*z*u^4,
> 4*x^4*z + 3*x^3*y*z + 10*x^3*z^2 + x^2*z^3 + 14*x*y*z^3 + 3*x*z^4 +
> 15*x^3*z*t + 8*x^2*z^2*t + 10*x*z^3*t + 14*x^3*y*u + 15*x^3*z*u +
> 11*x^2*z^2*u + 10*x*z^3*u + 16*x^2*z*t*u + 16*x*z^2*t*u + 6*x^3*u^2 +
> 14*x^2*z*u^2 + 3*x*y*z*u^2 + 4*x*z^2*u^2 + 6*x^2*t*u^2 + 6*x*z*t*u^2 +
> 15*x^2*u^3 + 14*x*z*u^3,
> 5*x^3*z^2 + 4*x^2*y*z^2 + 12*x^2*z^3 + 15*x*z^4 + 6*y*z^4 + 11*z^5 +
> 14*x^3*z*t + x^2*y*z*t + 7*x^2*z^2*t + 13*x*z^3*t + 3*y*z^3*t + 11*z^4*t
> + 12*x^2*z*t^2 + 2*x*z^2*t^2 + 7*z^3*t^2 + 7*x^3*z*u + 13*x^2*y*z*u +
> x^2*z^2*u + 6*x*z^3*u + y*z^3*u + 13*z^4*u + 6*x^3*t*u + 16*x^2*y*t*u +
> 9*x^2*z*t*u + x*z^2*t*u + 14*y*z^2*t*u + 3*z^3*t*u + 6*x^2*t^2*u +
> 11*z^2*t^2*u + 9*x^2*z*u^2 + 4*x*z^2*u^2 + 10*y*z^2*u^2 + z^3*u^2 +
> 15*x^2*t*u^2 + 6*z^2*t*u^2 + 7*z^2*u^3,
> 13*x^5 + 2*x^4*y + 7*x^4*z + 6*x^3*y*z + 16*x^3*z^2 + 4*x^2*y*z^2 +
> 14*x^2*z^3 + 2*x^4*t + 9*x^3*z*t + 7*x^2*z^2*t + 16*x^4*u + 15*x^3*y*u +
> x^3*z*u + 11*x^2*z^2*u + 3*x*y*z^2*u + 14*x*z^3*u + 13*x^3*t*u +
> 3*x^2*z*t*u + 7*x*z^2*t*u + 14*x^3*u^2 + 15*x^2*y*u^2 + 6*x^2*z*u^2 +
> 14*x*y*z*u^2 + 10*x*z^2*u^2 + 11*x^2*t*u^2 + 11*x*z*t*u^2 + 3*x*z*u^3]>;
> X := Scheme(Proj(R),I); // define the surface
> // The structure sheaf O_X of X is represented by graded module R/I
> O_X := GradedModule(I); // R/I
We first compute the dimension of H0(OX). This is fairly uninteresting
(it's just 1) but after this computation, the cached data will allow
the following cohomology calls to execute practically instantaneously.
> CohomologyDimension(O_X,0,0); // dim H^0(O_X)
1
> // get the geometric genus and irregularity
> time CohomologyDimension(O_X,2,0); // dim H^2(O_X) = g
0
Time: 0.000
> time CohomologyDimension(O_X,1,0); // dim H^1(O_X) = q
0
Time: 0.000
> // => p_a(X)=0. Verify this.
> ArithmeticGenus(X);
0
We now compute a module representative of the canonical sheaf KX
of X. We can just take Ext2R(M, R( - 5)). Then we check again that
H0(KX) = g is 0. Note that here the module representing KX is
maximal so that H0(KX(n)) is just the dimension of its nth
graded part for any n. However, in other cases (where X is again
not arithmetically Cohen-Macaulay) the Ext computation
for KX may not give the maximal representing module, so its 0th
graded piece might have dimension less than g.
> K_X := Ext(2,O_X,RModule(R,[5]));
> K_X;
Reduced Module R^6/<relations> with grading [1, 1, 1, 1, 1, 1]
Quotient Relations:
[12*z + 8*t + 7*u, 5*y + 7*z + 8*t + 6*u, 16*z + 4*u, 16*z + 7*u, 15*u,
7*u],
[14*u, 8*u, 13*x + 2*y + 4*t + 14*u, 4*y + 8*t + 2*u, 16*t + 6*u, 3*u],
[15*z + 9*t + 11*u, 2*y + 13*z + 8*t + 14*u, 7*t + 16*u, 8*x + 14*t + 6*
11*t + 15*u, 11*z + 14*t + 16*u],
[12*z + 15*t + 6*u, 5*y + 7*z + 4*t + 9*u, t + u, 2*t + 2*u, 7*x + 4*t +
12*z + t + 2*u],
[12*y + 10*z + 14*t + 14*u, 7*y + 15*z + 8*t, 4*u, 8*u, 16*u, 5*x + 15*z
+ 9*u],
[15*x, 15*x, 4*u, 8*u, 16*u, 0],
[14*z + 15*t + 9*u, 3*y + 11*z + 7*t + 13*u, 0, 11*z + 6*u, 2*z + 14*u,
10*t + 4*u],
[10*z + 14*t + 14*u, 4*x + 12*y + 15*z + 8*t, 0, 0, 0, 15*z + 16*t + 9*u
[5*z + 16*t + 4*u, 12*y + 10*z + 15*t + 5*u, 0, 2*z + 15*u, 6*u, 15*z +
7*u],
[13*z + t + 15*u, 4*y + 9*z + 16*t + 16*u, 0, 0, 0, 5*z + 11*t + 3*u]
> CohomologyDimension(K_X,0,0);
0
Finally, we verify some more cases of Serre duality which gives
dim Hr(OX(n)) = dim H2 - r(KX( - n)).
> [CohomologyDimension(K_X,0,i) eq CohomologyDimension(O_X,2,-i) : > i in [-1..5]];
[ true, true, true, true, true, true, true ]
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