D.15.22.4 sheafCohBGGsres

Usage:

sheafCohBGGsres(M,l,h); M module, l,h int

Assume:

M is graded, and it comes assigned with an admissible degree vector as an attribute, h>=l, and the basering has n+1 variables.

Return:

intmat, cohomology of twists of the coherent sheaf F on P^n associated to coker(M). The range of twists is determined by l, h.

Display:

The intmat is displayed in a diagram of the following form: with displayCohom(A,l,h,nvars(r)-1);

l l+1 h ---------------------------------------------------------- n: h^n(F(l)) h^n(F(l+1)) ...... h^n(F(h)) ............................................... 1: h^1(F(l)) h^1(F(l+1)) ...... h^1(F(h)) 0: h^0(F(l)) h^0(F(l+1)) ...... h^0(F(h)) ---------------------------------------------------------- chi: chi(F(l)) chi(F(l+1)) ...... chi(F(h))

A '-' in the diagram refers to a zero entry; a '*' refers to a negative entry (= dimension not yet determined). refers to a not computed dimension.

Note:

This procedure is based on the Bernstein-Gel'fand-Gel'fand correspondence and on Tate resolution ( see [Eisenbud, Floystad, Schreyer: Sheaf cohomology and free resolutions over exterior algebras, Trans AMS 355 (2003)] ).
sheafCohBGG(M,l,h) does not compute all values in the above table. To determine all values of h^i(F(d)), d=l..h,
use sheafCohBGG(M,l-n,h+n).

LIB "sresext.lib";
// cohomology of structure sheaf on P^4:
//-------------------------------------------
ring r=0,x(1..3),dp;
module M=0;
intmat A=sheafCohBGGsres(M,-4,1);
A;
==> 3,1,0,0,0,-1,
==> -1,0,0,0,0,0,
==> -1,-1,0,0,1,3 
displayCohom(A,-4,1,nvars(r)-1);
==>       -4  -3  -2  -1   0   1
==> ----------------------------
==>   2:   3   1   -   -   -   *
==>   1:   *   -   -   -   -   -
==>   0:   *   *   -   -   1   3
==> ----------------------------
==> chi:   *   *   0   0   1   *