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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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