Cohomology Of Coherent Sheaves

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.

CohomologyDimension(M, r, n): ModMPolGrd, RngIntElt, RngIntElt -> RngIntElt

verbose: Cohom                    Default: Verbose : 1

\(M\) is a graded module over \(P = k[x_0,..,x_m]\) with \(k\) an exact field. Let \(\tilde{M}\) be the corresponding coherent sheaf on \(Proj(P) = {\mathbb{P}}^m_k\). The function returns the \(k\)-dimension of the cohomology group \(H^r({\mathbb{P}}_k^m,\tilde{M}(n))\) where \(\tilde{M}(n)\) is the \(n\)th 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 [Eisenbud et al., 2003] or see [Decker and Eisenbud, 2002] for a slightly more computational description. Let \(A\) be the finite exterior algebra with \(m+1\) generators, which is of dimension \(2^{m+1}\) 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 \(\ge reg(M)\) (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 \(\ge reg(M)\), 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.

Example: cohom (ex-f9ee7a)

We consider a random surface \(X\) in a family of Enriques surfaces of degree 9 in \({\mathbb{P}}^4\). It is defined by 15 degree 5 polynomials and we work over \({\mathbb{F}}_{17}\) to keep the input of reasonable size (it is still fairly large!).

The surface is non-singular with arithmetic genus (\(p_a\)), geometric genus (\(g\)) and irregularity \(q\) all zero. These are related generally for a non-singular surface by \(g = p_a + q\) and \(p_a\) 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

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We first compute the dimension of \(H^0(O_X)\). 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

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We now compute a module representative of the canonical sheaf \(K_X\) of \(X\). We can just take \(Ext^2_R(M,R(-5))\). Then we check again that \(H^0(K_X) = g\) is 0. Note that here the module representing \(K_X\) is maximal so that \(H^0(K_X(n))\) is just the dimension of its \(n\)th graded part for any \(n\). However, in other cases (where \(X\) is again not arithmetically Cohen-Macaulay) the Ext computation for \(K_X\) 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

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Finally, we verify some more cases of Serre duality which gives

\[\hbox{dim } H^r(O_X(n)) = \hbox{dim } H^{2-r}(K_X(-n)).\]

> [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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