Singular Manual: sheafCohBGGsres

  | #### D.15.24.4 sheafCohBGGsres Procedure from library `sresext.lib` (see sresext_lib). Usage: sheafCohBGGsres(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 "sresext.lib";
    // cohomology of structure sheaf on P^4:
    //-------------------------------------------
    ring r=0,x(1..3),dp;
    module M=0;
    intmat A=sheafCohBGGsres(M,-4,1);
    A;
    ==> 3,1,0,0,0,-1,
    ==> -1,0,0,0,0,0,
    ==> -1,-1,0,0,1,3
    displayCohom(A,-4,1,nvars(r)-1);
    ==>       -4  -3  -2  -1   0   1
    ==> ----------------------------
    ==>   2:   3   1   -   -   -   *
    ==>   1:   *   -   -   -   -   -
    ==>   0:   *   *   -   -   1   3
    ==> ----------------------------
    ==> chi:   *   *   0   0   1   *
See also: dimH; displayCohom; sheafCoh.

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