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D.5.7.4 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))
| 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:
See also:
dimH;
sheafCoh.
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