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D.5.10.4 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:
| | 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))
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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;
def A=sheafCohBGG(M,-9,4);
==> -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);
def B=sheafCohBGG(M,-8,4);
==> -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 * * *
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See also:
dimH;
sheafCoh.
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