Singular Manual: sheafCohBGG

D.5.18.4 sheafCohBGG

Procedure from library sheafcoh.lib (see sheafcoh_lib).

Usage:

sheafCohBGG(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).

Example:

LIB "sheafcoh.lib";
// cohomology of structure sheaf on P^4:
//-------------------------------------------
ring r=0,x(1..5),dp;
module M=0;
intmat A=sheafCohBGG(M,-9,4);
A;
==> 70,35,15,5,1,0,0,0,0,0,-1,-1,-1,-1,
==> -1,0,0,0,0,0,0,0,0,0,0,-1,-1,-1,
==> -1,-1,0,0,0,0,0,0,0,0,0,0,-1,-1,
==> -1,-1,-1,0,0,0,0,0,0,0,0,0,0,-1,
==> -1,-1,-1,-1,0,0,0,0,0,1,5,15,35,70 
displayCohom(A,-9,4,nvars(r)-1);
==>       -9  -8  -7  -6  -5  -4  -3  -2  -1   0   1   2   3   4
==> ------------------------------------------------------------
==>   4:  70  35  15   5   1   -   -   -   -   -   *   *   *   *
==>   3:   *   -   -   -   -   -   -   -   -   -   -   *   *   *
==>   2:   *   *   -   -   -   -   -   -   -   -   -   -   *   *
==>   1:   *   *   *   -   -   -   -   -   -   -   -   -   -   *
==>   0:   *   *   *   *   -   -   -   -   -   1   5  15  35  70
==> ------------------------------------------------------------
==> chi:   *   *   *   *   1   0   0   0   0   1   *   *   *   *
// cohomology of cotangential bundle on P^3:
//-------------------------------------------
ring R=0,(x,y,z,u),dp;
resolution T1=mres(maxideal(1),0);
module M=T1[3];
intvec v=2,2,2,2,2,2;
attrib(M,"isHomog",v);
intmat B=sheafCohBGG(M,-8,4);
B;
==> 189,120,70,36,15,4,0,0,0,0,-1,-1,-1,
==> -1,0,0,0,0,0,0,0,0,0,0,-1,-1,
==> -1,-1,0,0,0,0,0,0,1,0,0,0,-1,
==> -1,-1,-1,0,0,0,0,0,0,0,6,20,45 
displayCohom(B,-8,4,nvars(R)-1);
==>        -8   -7   -6   -5   -4   -3   -2   -1    0    1    2    3    4
==> ---------------------------------------------------------------------
==>   3:  189  120   70   36   15    4    -    -    -    -    *    *    *
==>   2:    *    -    -    -    -    -    -    -    -    -    -    *    *
==>   1:    *    *    -    -    -    -    -    -    1    -    -    -    *
==>   0:    *    *    *    -    -    -    -    -    -    -    6   20   45
==> ---------------------------------------------------------------------
==> chi:    *    *    *  -36  -15   -4    0    0   -1    0    *    *    *