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D.5.18.5 sheafCohBGGregul
Procedure from library sheafcoh.lib (see sheafcoh_lib).
- Usage:
- sheafCohBGGregul(M,l,h); M module, l,h int, reg 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.
- 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;
def A=sheafCohBGGregul(M,-9,4,CM_regularity(M));
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
// 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);
def B=sheafCohBGGregul(M,-8,4,CM_regularity(M));
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
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See also:
dimH;
displayCohom;
sheafCoh.
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